Finite Group Databasebeta

← #72225393

Subgroups — #72225393

Order 4096.

1878 conjugacy classes of subgroups.

The subgroup lattice has 788 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72225387C212yesmaximal, complemented
2#72225385C212yesmaximal, complemented
3#72225383C212yesmaximal, complemented
4#72225381—22no—
5#72225381—22no—
6#72225380—22no—
7#72225379—22no—
8#72225378C211yesmaximal, characteristic, complemented
9#72225376C211yesmaximal, characteristic, complemented
10#72225373C2212yes—
11#72225372C211yesmaximal, characteristic
12#72225369C418yescomplemented
13#72225366C211yesmaximal, characteristic, complemented
14#72225363C2212yes—
15#72225363C2212yes—
16#72225362C2211yescharacteristic
17#72225360C211yesmaximal, characteristic
18#72225356C414yescomplemented
19#72225355C414yescomplemented
20#72225353C2212yes—
21#72225353C2212yes—
22#72225352C2211yescharacteristic
23#72225315C211yesmaximal, characteristic
24#72225314C411yescharacteristic
25#72225313C2211yescharacteristic
26#72225312C412yes—
27#72225311C2212yes—
28#72225310C411yescharacteristic
29#72225308C2211yescharacteristic
30#72225305C411yescharacteristic
31#72225304C412yes—
32#72223679C411yescharacteristic
33#72216537C212yesmaximal, complemented
34#72216535C2212yes—
35#72216534C2212yes—
36C2×#93752C2212yes—
37C2×#93751C2212yes—
38#72216243C2212yes—
39#72216231C211yesmaximal, characteristic, complemented
40#72216226C2211yescharacteristic
41#72216226C2211yescharacteristic
42C8×#1398C2211yescharacteristic
43C8×#1398C2211yescharacteristic
44#72216220C2211yescharacteristic
45#72216211C2212yes—
46C16×M64C2211yescharacteristic
47#72216182C2211yescharacteristic
48#512778C812yes—
49#512773C2×C411yescharacteristic
50#512751—22no—
51#512751—22no—
52#512548—42no—
53#493242C2×C412yes—
54#493217—22no—
55C2×#12047C2×C412yes—
56C2×#12046C2312yes—
57#151999C812yes—
58#151998C2×C411yescharacteristic
59#151989—22no—
60#151989—22no—
61#151981—22no—
62#151973C812yes—
63#151971C812yes—
64#151968C2×C412yes—
65#151904—42no—
66#151904—42no—
67#151895—42no—
68C4×#1398C2×C411yescharacteristic
69C4×#1398C2×C411yescharacteristic
70C4×#1398D411yescharacteristic
71C4×#1398D411yescharacteristic
72#151086C818yes—
73#151082C2×C414yes—
74#151081C2×C414yes—
75#151073—24no—
76#151073C2×C418yescomplemented
77#151072—24no—
78#151072C2×C418yescomplemented
79#151040C2×C411yescharacteristic
80#151035C2×C411yescharacteristic
81C2×#7459C2312yes—
82C2×#7459C2312yes—
83#150069C2×C412yes—
84#150067C2×C412yes—
85#150051—22no—
86#150051—22no—
87#149495C2×C411yescharacteristic
88#149495C2×C411yescharacteristic
89#149495D411yescharacteristic
90#149495D411yescharacteristic
91C8×M64C2311yescharacteristic
92C8×M64C2311yescharacteristic
93C8×M64C2311yescharacteristic
94C8×M64C2311yescharacteristic
95#129143C2×C412yes—
96#129142C2312yes—
97#129101C2×C412yes—
98#128674—22no—
99#128673—22no—
100C4×#1335—22no—
101C8×#346—21no—
102C8×#346C2311yescharacteristic
103C2×#7052—21no—
104C2×#7052C2311yescharacteristic
105C2×C162C2311yescharacteristic
106#94054C2×C412yes—
107#94053C2×C412yes—
108#94041—22no—
109#94041—22no—
110#94029C2×C412yes—
111#94019—22no—
112#93885C2×C414yescomplemented
113#93882C2×C414yescomplemented
114#93752C2×C418yescomplemented
115#93751C2×C414yescomplemented
116#93751C2×C414yescomplemented
117#93666—22no—
118#93666C2×C414yescomplemented
119#93665—22no—
120#93664C2×C412yescomplemented
121C16×C32C2×C412yescomplemented
122#33981C2×C811yescharacteristic
123#33974—42no—
124#33974—42no—
125C2×#2179C2×C812yes—
126C2×#2178C22×C412yes—
127#33451C2×C812yes—
128C2×#2170—22no—
129C2×#2170—22no—
130#33326—42no—
131C4×#443—22no—
132#13658—24no—
133#13657—24no—
134#13656C2×C814yes—
135#13655C4214yes—
136#13648—44no—
137#13648—44no—
138#13647—28no—
139#13647—28no—
140#13638C2×C812yes—
141#13636C2×C811yescharacteristic
142C2×#1422—22no—
143C2×#1422—22no—
144#13625C2×C812yes—
145#13622C2×C812yes—
146#13620C2×C812yes—
147#13558—42no—
148#13558—42no—
149#13551—42no—
150#13551—42no—
151#13551—84no—
152C2×#1398—22no—
153C2×#1398—22no—
154C2×#1398—42no—
155C2×#1398—84no—
156C2×#1398C2×C812yes—
157C2×#1398C2×C812yes—
158C2×#1398D812yes—
159C2×#1398D812yes—
160C4×M64C22×C411yescharacteristic
161C4×M64C22×C411yescharacteristic
162C4×M64C22×C411yescharacteristic
163C4×M64C22×C411yescharacteristic
164C4×M64C2×D411yescharacteristic
165C4×M64C2×D411yescharacteristic
166C4×M64C2×D411yescharacteristic
167C4×M64C2×D411yescharacteristic
168#12676C2×C812yes—
169#12671C2×C812yes—
170#12670C22×C412yes—
171#12664—22no—
172#12664—22no—
173C4×#364—22no—
174#12642—22no—
175#12641—22no—
176#12640—22no—
177#12615—42no—
178#12615—42no—
179#12604—42no—
180C4×#346—22no—
181C4×#346C22×C411yescharacteristic
182C4×#346C2×D411yescharacteristic
183#12047—24no—
184#12047C2×C818yes—
185#12046—22no—
186#12046—22no—
187#12046C22×C414yes—
188#12046C22×C414yes—
189C2×#1373C22×C412yes—
190C2×#1373C22×C412yes—
191C2×#1335—22no—
192C2×#1335—22no—
193C2×#1311—22no—
194C2×#1311C22×C411yescharacteristic
195C2×#1311C2×D411yescharacteristic
196C2×C8×C16—21no—
197C2×C8×C16—21no—
198C2×C8×C16#4111yescharacteristic
199C2×C8×C16C2411yescharacteristic
200C4×C82#4111yescharacteristic
201#7480C2×C812yes—
202#7479C2×C812yes—
203#7476C2×C811yescharacteristic
204#7475C2×C811yescharacteristic
205#7469—42no—
206#7469—42no—
207#7468—42no—
208#7464C2×C814yes—
209#7462C4214yes—
210#7459—22no—
211#7459—22no—
212#7459C22×C414yes—
213#7459C22×C414yes—
214#7388C2×C818yes—
215#7387C4218yes—
216#7381—28no—
217#7381—28no—
218#7330—22no—
219#7330—22no—
220#7330—22no—
221#7330C2×C814yes—
222#7329—21no—
223#7329C22×C412yes—
224#7329C22×C412yes—
225#7329#4112yes—
226C8×C32—21no—
227C8×C32C22×C412yes—
228C8×C32C22×C412yes—
229C8×C32#4112yes—
230#7179—22no—
231#7179—22no—
232#7160—22no—
233#7052—24no—
234#7052—21no—
235#7052C22×C412yes—
236C162—21no—
237C162C22×C411yescharacteristic
238C162C22×C411yescharacteristic
239C2×#508C22×C812yes—
240C2×#504—42no—
241C2×#504—42no—
242C2×#443—22no—
243C2×#443—22no—
244C4×#135—22no—
245C4×#131—42no—
246#2264M3214yes—
247#2263M3214yes—
248#2262C4×C814yes—
249#2258—48no—
250#2258—48no—
251#2258—84no—
252#2258—84no—
253C2×M64—21no—
254C2×M64—21no—
255C2×M64—21no—
256C2×M64—21no—
257C2×M64—42no—
258C2×M64—42no—
259C2×M64—84no—
260C2×M64—84no—
261C2×M64C22×C812yes—
262C2×M64C22×C811yescharacteristic
263C2×M64C22×C811yescharacteristic
264C2×M64C2×D811yescharacteristic
265C2×M64C2×D812yes—
266C2×M64C2×D811yescharacteristic
267#2179—24no—
268#2179—42no—
269#2178—22no—
270#2178—22no—
271#2178C2×C4214yes—
272#2178C22×C814yes—
273#2177—22no—
274#2176—22no—
275#2175—42no—
276#2170—24no—
277#2170—24no—
278#2170—28no—
279#2170—42no—
280#2170—42no—
281#2170—44no—
282C2×#368C22×C812yes—
283C2×#368C22×C812yes—
284#2158C22×C812yes—
285C2×#364—22no—
286C2×#364—22no—
287C2×#359—22no—
288C2×#359—22no—
289#2138—22no—
290C2×#350—42no—
291C2×#350—42no—
292#2108—42no—
293#2108—42no—
294#2108—82no—
295#2108—82no—
296C4×M32—42no—
297C4×M32—82no—
298C2×#346—44no—
299C2×#346—82no—
300C2×#346C22×C812yes—
301C2×#346#13112yes—
302C2×#346C2×D812yes—
303C2×#346#13512yes—
304C2×C4×C16—22no—
305C2×C4×C16—22no—
306C2×C4×C16C4×D411yescharacteristic
307C2×C4×C16C4×D411yescharacteristic
308C2×C4×C16C23×C411yescharacteristic
309C2×C4×C16C22×D411yescharacteristic
310C4×#108—22no—
311#1772—22no—
312#1772—22no—
313#1764—42no—
314#1764—42no—
315C4×#104—42no—
316C42×C8—21no—
317C42×C8C4×D411yescharacteristic
318C2×C82—21no—
319C2×C82C4×D411yescharacteristic
320C2×C82C2×#4111yescharacteristic
321#1426—24no—
322#1425C4×C814yes—
323#1422—24no—
324#1422—24no—
325#1422—42no—
326#1422—42no—
327#1420—22no—
328#1419—22no—
329#1418—24no—
330#1417C4×C818yes—
331#1401—48no—
332#1401—48no—
333#1398—24no—
334#1398—24no—
335#1398—42no—
336#1398—42no—
337#1398—44no—
338#1398—44no—
339#1398—82no—
340#1398—82no—
341#1398—84no—
342#1398—84no—
343#1397—21no—
344#1397—21no—
345#1397—21no—
346#1397—21no—
347#1397C22×C812yes—
348#1397C22×C812yes—
349#1397#13512yes—
350#1397#13512yes—
351C4×C32—21no—
352C4×C32—21no—
353C4×C32C2×C4212yes—
354C4×C32C2×C4212yes—
355C4×C32C4×D412yes—
356C4×C32C4×D412yes—
357C4×C32C4×D412yes—
358C4×C32C4×D412yes—
359#1382—22no—
360#1382—22no—
361#1381—22no—
362#1379—42no—
363#1379—42no—
364#1376—42no—
365#1373—22no—
366#1373—22no—
367#1373C2×C4214yes—
368#1373C22×C814yes—
369#1335—24no—
370#1335—24no—
371#1335—28no—
372#1311—21no—
373#1311—21no—
374#1311—44no—
375#1311C22×C812yes—
376#1311#13112yes—
377C8×C16—22no—
378C8×C16—22no—
379C8×C16—22no—
380C8×C16—21no—
381C8×C16—21no—
382C8×C16—22no—
383C8×C16C23×C411yescharacteristic
384C8×C16C23×C411yescharacteristic
385C8×C16C2×#4111yescharacteristic
386C8×C16C2×#4111yescharacteristic
387C2×#135—22no—
388C2×#131—42no—
389C2×#131—42no—
390C4×#41—42no—
391#508—22no—
392#508—22no—
393#508C2×C4×C814yes—
394#508C2×M3214yes—
395C2×Q32C2×M3212yes—
396C2×SD32—22no—
397C2×D16C2×M3212yes—
398#504—48no—
399#504—44no—
400#504—44no—
401#504—84no—
402#504—82no—
403#504—82no—
404C2×M32—42no—
405C2×M32—42no—
406C2×M32—82no—
407C2×M32—82no—
408C22×C16—24no—
409C22×C16—42no—
410C22×C16—82no—
411C22×C16—82no—
412C22×C16C8×D412yes—
413C22×C16C4×D812yes—
414C22×C16C23×C811yescharacteristic
415C22×C16C2×#13111yescharacteristic
416C22×C16C22×D811yescharacteristic
417C22×C16C2×#13511yescharacteristic
418#443—24no—
419#443—24no—
420#443—44no—
421#443—42no—
422#443—42no—
423#443—82no—
424C4×Q16—42no—
425C4×SD16—82no—
426C4×D8—42no—
427C2×#108—22no—
428C2×#108—22no—
429#427—22no—
430C2×#104—42no—
431C2×#104—42no—
432#416—42no—
433#405—42no—
434#405—42no—
435#405—82no—
436#405—82no—
437C4×M16—42no—
438C4×M16—82no—
439C2×C4×C8—21no—
440C2×C4×C8—21no—
441C2×C4×C8—42no—
442C2×C4×C8—82no—
443C2×C4×C8C8×D412yes—
444C2×C4×C8C4×D812yes—
445C2×C4×C8#44311yescharacteristic
446C2×C4×C8C2×C4×D411yescharacteristic
447C43#44311yescharacteristic
448Q64#34614yescomplemented
449SD64#34618yescomplemented
450D32#34614yescomplemented
451M64—24no—
452M64—24no—
453M64—22no—
454M64—22no—
455M64—22no—
456M64—22no—
457M64—22no—
458M64—22no—
459M64—42no—
460M64—42no—
461M64—44no—
462M64—44no—
463M64—88no—
464M64—82no—
465M64—82no—
466M64—88no—
467M64C2×M3214yes—
468M64#50914yes—
469C2×C32—22no—
470C2×C32—22no—
471C2×C32—22no—
472C2×C32—22no—
473C2×C32—22no—
474C2×C32—88no—
475C2×C32—82no—
476C2×C32—82no—
477C2×C32—88no—
478C2×C32C2×C4×C812yes—
479C2×C32C2×C4×C812yes—
480C2×C32C4×D812yes—
481C2×C32C4×D814yes—
482C2×C32C4×D812yes—
483#368—22no—
484#368—22no—
485#368—22no—
486#368C2×C4×C814yes—
487#367—22no—
488#366—22no—
489#364—24no—
490#364—24no—
491#364—42no—
492#364—42no—
493#359—24no—
494#359—24no—
495#359—44no—
496#358—42no—
497#357—42no—
498#350—48no—
499#350—44no—
500#350—44no—
501#346—22no—
502#346—22no—
503#346—22no—
504#346—22no—
505#346—42no—
506#346—42no—
507#346—82no—
508#346—82no—
509#346—88no—
510#346—82no—
511C4×C16—22no—
512C4×C16—22no—
513C4×C16—24no—
514C4×C16—24no—
515C4×C16—21no—
516C4×C16—21no—
517C4×C16—42no—
518C4×C16—82no—
519C4×C16—82no—
520C4×C16—82no—
521C4×C16C8×D412yes—
522C4×C16#44312yes—
523C4×C16C22×C4211yescharacteristic
524C4×C16C2×C4×D411yescharacteristic
525C4×C16C2×C4×D411yescharacteristic
526C4×C16C4×#4111yescharacteristic
527C4×C16C23×C811yescharacteristic
528C4×C16C2×#13111yescharacteristic
529C4×C16C2×#13511yescharacteristic
530C4×C16C2×#13511yescharacteristic
531#341—42no—
532#339—42no—
533#339—42no—
534C82—21no—
535C82—21no—
536C82—42no—
537C82C8×D412yes—
538C82C4×#4111yescharacteristic
539C82C4×#4111yescharacteristic
540C2×#41—42no—
541#135—24no—
542#135—24no—
543#135—42no—
544#135—42no—
545C2×Q16—22no—
546C2×SD16—42no—
547C2×D8—22no—
548#131—44no—
549#131—44no—
550#131—84no—
551#131—82no—
552#131—82no—
553#131—162no—
554C2×M16—42no—
555C2×M16—42no—
556C2×M16—82no—
557C2×M16—82no—
558C22×C8—22no—
559C22×C8—41no—
560C22×C8—82no—
561C22×C8—82no—
562C22×C8C8×D812yes—
563C22×C8C2×C8×D411yescharacteristic
564C22×C8C2×C4×D811yescharacteristic
565C4×Q8—82no—
566C4×D4—82no—
567C4×D4—162no—
568#117—42no—
569#117—82no—
570C2×C42—41no—
571C2×C42—82no—
572C2×C42C8×D812yes—
573C2×C42C2×#44311yescharacteristic
574Q32—22no—
575Q32C2×#34614yes—
576SD32—24no—
577SD32—24no—
578D16—22no—
579D16C2×#34614yes—
580M32—44no—
581M32—44no—
582M32—88no—
583M32—82no—
584M32—82no—
585M32—84no—
586M32—84no—
587C2×C16—24no—
588C2×C16—22no—
589C2×C16—22no—
590C2×C16—22no—
591C2×C16—22no—
592C2×C16—24no—
593C2×C16—22no—
594C2×C16—22no—
595C2×C16—22no—
596C2×C16—48no—
597C2×C16—44no—
598C2×C16—88no—
599C2×C16—84no—
600C2×C16—82no—
601C2×C16—82no—
602C2×C16—84no—
603C2×C16—84no—
604C2×C16C8×D814yes—
605C2×C16C22×C4×C811yescharacteristic
606C2×C16C4×#13111yescharacteristic
607C2×C16C2×C4×D811yescharacteristic
608C2×C16C2×C4×D811yescharacteristic
609C2×C16C4×#13511yescharacteristic
610C2×C16C4×#13511yescharacteristic
611C2×C16C8×#4111yescharacteristic
612C2×C16C8×#4111yescharacteristic
613C2×C16C22×M3212yes—
614C2×C16#340712yes—
615C2×C16C2×#50912yes—
616C2×C16#341412yes—
617#108—24no—
618#108—42no—
619#108—42no—
620#108—42no—
621#107—42no—
622#106—42no—
623#104—44no—
624#104—44no—
625#104—84no—
626#103—82no—
627#102—82no—
628#97—82no—
629#97—82no—
630#97—82no—
631C4×C8—22no—
632C4×C8—22no—
633C4×C8—22no—
634C4×C8—22no—
635C4×C8—22no—
636C4×C8—21no—
637C4×C8—21no—
638C4×C8—22no—
639C4×C8—41no—
640C4×C8—41no—
641C4×C8—82no—
642C4×C8—82no—
643C4×C8—84no—
644C4×C8—82no—
645C4×C8C42×D411yescharacteristic
646C4×C8C2×C8×D411yescharacteristic
647C4×C8C4×#13511yescharacteristic
648C4×C8C2×#44311yescharacteristic
649C32—24no—
650C32—24no—
651C32—22no—
652C32—22no—
653C32—24no—
654C32—42no—
655C32—42no—
656C32—42no—
657C32—84no—
658C32—84no—
659C32C2×#34614yes—
660C32#217714yes—
661#41—44no—
662#41—44no—
663#41—82no—
664#41—82no—
665C2×Q8—42no—
666C2×D4—42no—
667C2×D4—82no—
668C22×C4—21no—
669C22×C4—22no—
670C22×C4—82no—
671C22×C4—162no—
672C22×C4C2×C8×D811yescharacteristic
673Q16—24no—
674Q16—42no—
675SD16—44no—
676SD16—44no—
677D8—24no—
678D8—42no—
679M16—44no—
680M16—82no—
681M16—84no—
682M16—84no—
683M16—82no—
684M16—82no—
685M16—164no—
686C2×C8—22no—
687C2×C8—22no—
688C2×C8—22no—
689C2×C8—22no—
690C2×C8—22no—
691C2×C8—44no—
692C2×C8—42no—
693C2×C8—82no—
694C2×C8—82no—
695C2×C8—84no—
696C2×C8—84no—
697C2×C8—82no—
698C2×C8—164no—
699C2×C8C2×C8×D811yescharacteristic
700C2×C8C8×#13511yescharacteristic
701C2×C8C4×C8×D411yescharacteristic
702C2×C8C42×D811yescharacteristic
703C2×C8D4×M3212yes—
704C2×C8#3344312yes—
705C2×C8C4×#50912yes—
706C2×C8#3346312yes—
707C4⋊C4—82no—
708C4⋊C4—162no—
709C22⋊C4—82no—
710C22⋊C4—162no—
711C42—21no—
712C42—22no—
713C42—22no—
714C42—82no—
715C42—82no—
716C42—162no—
717C42C8×#13511yescharacteristic
718C42C4×#44311yescharacteristic
719C16—28no—
720C16—21no—
721C16—21no—
722C16—24no—
723C16—21no—
724C16—21no—
725C16—44no—
726C16—42no—
727C16—84no—
728C16—84no—
729C16—84no—
730C16C22×#34611yescharacteristic
731C16#3332611yescharacteristic
732C16C2×#217711yescharacteristic
733C16#3345111yescharacteristic
734C16C4×#50912yes—
735C16#3348412yes—
736C23—82no—
737C23#48351211yescharacteristic
738Q8—44no—
739Q8—82no—
740D4—44no—
741D4—84no—
742D4—84no—
743D4—82no—
744C2×C4—21no—
745C2×C4—21no—
746C2×C4—42no—
747C2×C4—42no—
748C2×C4—82no—
749C2×C4—84no—
750C2×C4—84no—
751C2×C4—82no—
752C2×C4—162no—
753C2×C4—642no—
754C2×C4C4×C8×D811yescharacteristic
755C2×C4C8×#50912yes—
756C2×C4#48351211yescharacteristic
757C2×C4D8×M3212yes—
758C8—24no—
759C8—42no—
760C8—42no—
761C8—82no—
762C8—84no—
763C8—162no—
764C8C8×#50912yes—
765C8#48354412yes—
766C8D4×#34611yescharacteristic
767C8#49321711yescharacteristic
768C8C4×#217711yescharacteristic
769C8#49324211yescharacteristic
770C22—41no—
771C22—84no—
772C22—322no—
773C22#7221621111yescharacteristic
774C22#7222539112yes—
775C4—22no—
776C4—41no—
777C4—82no—
778C4—84no—
779C4—322no—
780C4C8×#217711yescharacteristic
781C4D8×#34611yescharacteristic
782C4#7222539112yes—
783C2—21no—
784C2—22no—
785C2—164no—
786C2#7222539211yescharacteristic
subgroupG/H
centerC8C4×#2177
commutatorC2×C16C22×C4×C8
Frattini, cosocleC2×C8×C16C24
socleC2#72225392
prime radical#33981C2×C8
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×M64 ◁ C4×#1398 ◁ C8×#1398 ◁ G
LECSG ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #72225387C2semidirect
2. #72225385C2semidirect
3. #72225383C2semidirect
8. #72225378C2semidirect
9. #72225376C2semidirect
12. #72225369C4semidirect
13. #72225366C2semidirect
18. #72225356C4semidirect
19. #72225355C4semidirect
33. #72216537C2semidirect
39. #72216231C2semidirect
76. #151073C2×C4semidirect
78. #151072C2×C4semidirect
112. #93885C2×C4semidirect
113. #93882C2×C4semidirect
114. #93752C2×C4semidirect
115. #93751C2×C4semidirect
116. #93751C2×C4semidirect
118. #93666C2×C4semidirect
120. #93664C2×C4semidirect
121. C16×C32C2×C4semidirect
448. Q64#346semidirect
449. SD64#346semidirect
450. D32#346semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#722253872
#722253852
#722253832
#722253782
#722253762
#722253722
#722253662
#722253602
#722253152
#722165372
#722162312