Finite Group Databasebeta

← #72320993

Subgroups — #72320993

Order 1024.

759 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#483617C211yesmaximal, characteristic
2#483466C211yesmaximal, characteristic, complemented
3#483463C211yesmaximal, characteristic, complemented
4#225773C211yesmaximal, characteristic
5#225720C211yesmaximal, characteristic
6#140503C211yesmaximal, characteristic, complemented
7#140497C211yesmaximal, characteristic, complemented
8#136464C211yesmaximal, characteristic, complemented
9#136464C211yesmaximal, characteristic, complemented
10#136463C211yesmaximal, characteristic, complemented
11#136462C211yesmaximal, characteristic, complemented
12#135801C211yesmaximal, characteristic, complemented
13#135074C211yesmaximal, characteristic, complemented
14#135073C211yesmaximal, characteristic, complemented
15#129183C211yesmaximal, characteristic, complemented
16#33485C2211yescharacteristic
17#33484C2211yescharacteristic
18#22688—21no—
19#18061—21no—
20C8×#135C2211yescharacteristic
21#17835C2211yescharacteristic
22#17834C2211yescharacteristic
23#17826C2211yescharacteristic
24#17826C2211yescharacteristic
25C4×#365C2211yescharacteristic
26#12063C411yescharacteristic, complemented
27#12063C411yescharacteristic, complemented
28#12060C411yescharacteristic, complemented
29#12060C411yescharacteristic, complemented
30#12058C411yescharacteristic, complemented
31#12058C411yescharacteristic, complemented
32SD16⋊C16C411yescharacteristic, complemented
33SD16⋊C16C411yescharacteristic, complemented
34#12030—21no—
35#12030—21no—
36#12030C2211yescharacteristic
37#12030C2211yescharacteristic
38#12029—21no—
39#12029C2211yescharacteristic
40#12029C2211yescharacteristic
41#12028—21no—
42#12020—21no—
43#12020C2211yescharacteristic
44#12020C2211yescharacteristic
45#12020C2211yescharacteristic
46#12020C2211yescharacteristic
47#12019—21no—
48#12013C2211yescharacteristic
49#12000—21no—
50#12000C2211yescharacteristic
51C2×#1368—21no—
52#11615C2211yescharacteristic
53#11615C2211yescharacteristic
54#11402C2211yescharacteristic
55#9636—21no—
56#9628—21no—
57#8904—21no—
58#8904—21no—
59#8901—21no—
60#8900—21no—
61#8824C2211yescharacteristic
62#8822C2211yescharacteristic
63#8670C2211yescharacteristic
64#8670C2211yescharacteristic
65#8666C2211yescharacteristic
66#8666C2211yescharacteristic
67#8293C2211yescharacteristic
68C8×#106C2211yescharacteristic
69#7302C2211yescharacteristic
70#7302C2211yescharacteristic
71#7300C2211yescharacteristic
72#7300C2211yescharacteristic
73#7174C2211yescharacteristic
74#7173C2211yescharacteristic
75#2991—21no—
76#2990—21no—
77C8×#41C2311yescharacteristic
78C8×#41C2311yescharacteristic
79C2×#443—41no—
80C4×#135D411yescharacteristic
81#2906—21no—
82#2906—21no—
83#2183C2×C411yescharacteristic
84#2183C2×C411yescharacteristic
85Q8×C16C2×C411yescharacteristic
86Q8×C16C2×C411yescharacteristic
87#2169—21no—
88#2169—21no—
89#2169C2×C411yescharacteristic
90#2169C2×C411yescharacteristic
91#2168—21no—
92#2168—21no—
93D4×C16C2×C411yescharacteristic
94D4×C16C2×C411yescharacteristic
95#2163—21no—
96#2163—21no—
97#2163C2311yescharacteristic
98#2163C2311yescharacteristic
99C2×#365—21no—
100C4×M32—21no—
101C4×M32C2311yescharacteristic
102C4×M32C2311yescharacteristic
103C2×#346—21no—
104#1945—41no—
105#1866—41no—
106#1866—41no—
107#1848—21no—
108#1837—21no—
109#1806—21no—
110#1806—21no—
111C4×#108C2311yescharacteristic
112C4×#106C2311yescharacteristic
113C8×C4⋊C4—21no—
114#1592—41no—
115#1584—41no—
116#1581—41no—
117SD16⋊C8—41no—
118C8×Q16C2×C411yescharacteristic
119C8×Q16Q811yescharacteristic
120C8×SD16C2×C411yescharacteristic
121C8×SD16C2×C411yescharacteristic
122C8×SD16Q811yescharacteristic
123C8×SD16Q811yescharacteristic
124C8×D8C2×C411yescharacteristic
125C8×D8Q811yescharacteristic
126#1574—21no—
127#1574—21no—
128#1574—41no—
129#1573—41no—
130#1570—21no—
131#1570—21no—
132#1570—41no—
133#1570—41no—
134#1568—21no—
135#1567—21no—
136#1566—21no—
137#1566—21no—
138#1566—21no—
139#1566—21no—
140#1566C2311yescharacteristic
141#1566C2311yescharacteristic
142#1566C2311yescharacteristic
143#1566C2311yescharacteristic
144#1564—21no—
145#1564D411yescharacteristic
146#1564C2311yescharacteristic
147C2×#334—21no—
148C2×#334—21no—
149C2×#334—21no—
150C2×#334C2311yescharacteristic
151#1489—21no—
152#1489—21no—
153#1476D411yescharacteristic
154#1476D411yescharacteristic
155#1451—21no—
156C2×C82C2311yescharacteristic
157#1383—21no—
158#1370C2×C411yescharacteristic
159#1370C2×C411yescharacteristic
160#1368—21no—
161#1368—21no—
162#1368—21no—
163#1368—21no—
164#1368—21no—
165#1368—21no—
166#1368—21no—
167#1368—21no—
168#1368—21no—
169#1368—21no—
170#1368—21no—
171#1368—21no—
172#1368—41no—
173#1368—41no—
174#1368D411yescharacteristic
175#1368D411yescharacteristic
176#1368D411yescharacteristic
177#1368D411yescharacteristic
178#1367C2×C411yescharacteristic
179#1367C2×C411yescharacteristic
180Q8⋊C16C2×C411yescharacteristic
181Q8⋊C16C2×C411yescharacteristic
182Q8⋊C16C2×C411yescharacteristic
183Q8⋊C16C2×C411yescharacteristic
184D4⋊C16C2×C411yescharacteristic
185D4⋊C16C2×C411yescharacteristic
186D4⋊C16C2×C411yescharacteristic
187D4⋊C16C2×C411yescharacteristic
188#1312C2×C411yescharacteristic
189#1312C2×C411yescharacteristic
190#1312C2×C411yescharacteristic
191#1312C2×C411yescharacteristic
192#1303—21no—
193#1303—21no—
194#1299—21no—
195#1299—21no—
196#1279—21no—
197#1278—21no—
198C2×#135SD1611yescharacteristic
199C2×#131—41no—
200C2×#131—41no—
201C4×#41C2×D411yescharacteristic
202C4×#41C2×D411yescharacteristic
203C2×M32—21no—
204C2×M32—21no—
205C2×M32—21no—
206C2×M32#4111yescharacteristic
207C2×M32#4111yescharacteristic
208C22×C16—21no—
209C22×C16—21no—
210#446—41no—
211C8×Q8—41no—
212C8×Q8C22×C411yescharacteristic
213C8×Q8C22×C411yescharacteristic
214C8×Q8C2×Q811yescharacteristic
215C8×Q8C2×Q811yescharacteristic
216#443—84no—
217C4×Q16C4⋊C411yescharacteristic
218C4×Q16SD1611yescharacteristic
219C4×SD16C4⋊C411yescharacteristic
220C4×SD16C4⋊C411yescharacteristic
221C4×SD16SD1611yescharacteristic
222C4×SD16SD1611yescharacteristic
223C4×D8C4⋊C411yescharacteristic
224C4×D8SD1611yescharacteristic
225#436—41no—
226#436—81no—
227#435—81no—
228C8×D4—21no—
229C8×D4—21no—
230C8×D4—21no—
231C8×D4—21no—
232C8×D4—41no—
233C8×D4C22×C411yescharacteristic
234C8×D4C22×C411yescharacteristic
235C8×D4C2×Q811yescharacteristic
236C8×D4C2×Q811yescharacteristic
237#432—41no—
238#431—21no—
239#431—21no—
240#431—21no—
241#431—41no—
242#431C2×D411yescharacteristic
243#431C2×D411yescharacteristic
244C2×#108—21no—
245C2×#108—41no—
246C2×#108—41no—
247#427SD1611yescharacteristic
248C2×#106—21no—
249#424—41no—
250#424—41no—
251#424—81no—
252#424—81no—
253#423—21no—
254#423—21no—
255#423—21no—
256#423—21no—
257#423—41no—
258#423—41no—
259#423—41no—
260C2×C4⋊C8—21no—
261C2×C4⋊C8—41no—
262C2×C4⋊C8—41no—
263C2×C4⋊C8—41no—
264C2×#104—41no—
265C2×#104—41no—
266#416SD1611yescharacteristic
267#416SD1611yescharacteristic
268C4×M16—21no—
269C2×C4×C8—21no—
270C2×C4×C8—21no—
271C2×C4×C8—41no—
272C2×C4×C8C2×D411yescharacteristic
273C2×C4×C8C2×D411yescharacteristic
274C2×C4×C8C2411yescharacteristic
275C4×C4⋊C4—21no—
276#365—42no—
277C4⋊C16—21no—
278C4⋊C16—21no—
279C4⋊C16—21no—
280C4⋊C16—21no—
281C4⋊C16C22×C411yescharacteristic
282C4⋊C16C22×C411yescharacteristic
283C4⋊C16C22×C411yescharacteristic
284C4⋊C16C22×C411yescharacteristic
285C22⋊C16—21no—
286C22⋊C16—21no—
287C22⋊C16—21no—
288C22⋊C16—21no—
289C22⋊C16—21no—
290C22⋊C16—21no—
291#346—21no—
292#346—21no—
293#346—21no—
294#346—21no—
295#346—21no—
296#346C2×D411yescharacteristic
297#346C2×D411yescharacteristic
298C4×C16—21no—
299C4×C16—21no—
300C4×C16C22×C411yescharacteristic
301C4×C16C22×C411yescharacteristic
302C4×C16C22×C411yescharacteristic
303C4×C16C22×C411yescharacteristic
304#341—21no—
305#341—81no—
306#340—21no—
307#339—41no—
308#339—41no—
309#336—41no—
310#335—21no—
311#335—41no—
312#335—41no—
313#335—41no—
314#335—41no—
315#335C4⋊C411yescharacteristic
316#335C4⋊C411yescharacteristic
317#334—21no—
318#334—21no—
319#334—21no—
320#334—21no—
321#334—21no—
322#334—21no—
323#334—21no—
324#334—21no—
325#334—21no—
326#334—41no—
327#334—41no—
328#334—41no—
329#334—41no—
330#334—41no—
331#334—41no—
332#334—81no—
333#334C4⋊C411yescharacteristic
334#334C4⋊C411yescharacteristic
335#334C2×D411yescharacteristic
336#334C2×D411yescharacteristic
337#334C2×D411yescharacteristic
338#334C2×D411yescharacteristic
339Q8⋊C8—41no—
340Q8⋊C8—41no—
341Q8⋊C8C4⋊C411yescharacteristic
342Q8⋊C8C4⋊C411yescharacteristic
343Q8⋊C8C4⋊C411yescharacteristic
344Q8⋊C8C4⋊C411yescharacteristic
345D4⋊C8—41no—
346D4⋊C8—41no—
347D4⋊C8C4⋊C411yescharacteristic
348D4⋊C8C4⋊C411yescharacteristic
349D4⋊C8C4⋊C411yescharacteristic
350D4⋊C8C4⋊C411yescharacteristic
351#322—41no—
352#322—41no—
353C82C22×C411yescharacteristic
354C82C22×C411yescharacteristic
355C82C2×Q811yescharacteristic
356C82C2×Q811yescharacteristic
357C2×#41C2×SD1611yescharacteristic
358C2×#41C2×SD1611yescharacteristic
359#135—24no—
360C2×Q16#10611yescharacteristic
361C2×SD16#10611yescharacteristic
362C2×SD16#10611yescharacteristic
363C2×D8#10611yescharacteristic
364#131—82no—
365#131—82no—
366#131—161no—
367#131—161no—
368C2×M16—41no—
369C2×M16—41no—
370C2×M16—41no—
371C2×M16—41no—
372C2×M16—41no—
373C2×M16—81no—
374C2×M16—81no—
375C2×M16—81no—
376C22×C8—21no—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—41no—
382C22×C8—81no—
383C22×C8—81no—
384C22×C8—81no—
385C22×C8C2×SD1611yescharacteristic
386C22×C8C2×SD1611yescharacteristic
387C22×C8C2×#4111yescharacteristic
388C22×C82−1+411yescharacteristic
389C4×Q8C2×C4⋊C411yescharacteristic
390C4×Q8C2×C4⋊C411yescharacteristic
391C4×Q8C2×SD1611yescharacteristic
392C4×Q8C2×SD1611yescharacteristic
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4C2×C4⋊C411yescharacteristic
398C4×D4C2×C4⋊C411yescharacteristic
399C4×D4C2×SD1611yescharacteristic
400C4×D4C2×SD1611yescharacteristic
401#117—21no—
402#117—21no—
403#117C2×SD1611yescharacteristic
404#117C2×SD1611yescharacteristic
405C2×C4⋊C4—41no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—41no—
409C2×C42C22×D411yescharacteristic
410M32—41no—
411M32—41no—
412M32—42no—
413M32—81no—
414M32—81no—
415C2×C16—21no—
416C2×C16—21no—
417C2×C16—21no—
418C2×C16—21no—
419C2×C16—21no—
420C2×C16—21no—
421C2×C16—42no—
422C2×C16—81no—
423C2×C16C4×D411yescharacteristic
424C2×C16C4×D411yescharacteristic
425C2×C16C4×D411yescharacteristic
426C2×C16C4×D411yescharacteristic
427#108—41no—
428#108—41no—
429#108—82no—
430#108—161no—
431#107#10611yescharacteristic
432#107#10611yescharacteristic
433#106—22no—
434#106—41no—
435#106#10611yescharacteristic
436#106#10611yescharacteristic
437C4⋊C8—21no—
438C4⋊C8—21no—
439C4⋊C8—21no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442C4⋊C8—21no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—41no—
446C4⋊C8—41no—
447C4⋊C8—41no—
448C4⋊C8—41no—
449C4⋊C8—41no—
450C4⋊C8—41no—
451C4⋊C8—41no—
452C4⋊C8—41no—
453C4⋊C8—41no—
454C4⋊C8—81no—
455C4⋊C8—81no—
456C4⋊C8—81no—
457C4⋊C8—81no—
458C4⋊C8—81no—
459C4⋊C8—81no—
460C4⋊C8C2×C4⋊C411yescharacteristic
461C4⋊C8C2×C4⋊C411yescharacteristic
462C4⋊C8C2×C4⋊C411yescharacteristic
463C4⋊C8C2×C4⋊C411yescharacteristic
464#104—82no—
465#104—82no—
466#103#10611yescharacteristic
467#103#10611yescharacteristic
468#103#10611yescharacteristic
469#103#10611yescharacteristic
470#102#10611yescharacteristic
471#102#10611yescharacteristic
472#102#10611yescharacteristic
473#102#10611yescharacteristic
474C22⋊C8—21no—
475C22⋊C8—21no—
476C22⋊C8—21no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479C22⋊C8—21no—
480C22⋊C8—41no—
481C22⋊C8—81no—
482C22⋊C8—81no—
483#97—41no—
484#97—81no—
485C4×C8—21no—
486C4×C8—21no—
487C4×C8—21no—
488C4×C8—21no—
489C4×C8—21no—
490C4×C8—21no—
491C4×C8—21no—
492C4×C8—41no—
493C4×C8—41no—
494C4×C8—81no—
495C4×C8—82no—
496C4×C8C2×C4⋊C411yescharacteristic
497C4×C8C2×C4⋊C411yescharacteristic
498C4×C8C2×C4⋊C411yescharacteristic
499C4×C8C2×C4⋊C411yescharacteristic
500C4×C8C2×SD1611yescharacteristic
501C4×C8C2×SD1611yescharacteristic
502C4×C8C23×C411yescharacteristic
503C4×C8C22×D411yescharacteristic
504C4×C8C22×Q811yescharacteristic
505C4×C8H(5,𝔽2)11yescharacteristic
506#95—41no—
507#41—22no—
508#41—22no—
509#41—41no—
510#41—41no—
511C2×Q8C2×#10611yescharacteristic
512C2×Q8C2×#10611yescharacteristic
513C2×D4—21no—
514C2×D4—21no—
515C2×D4C2×#10611yescharacteristic
516C2×D4C2×#10611yescharacteristic
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—81no—
524C22×C4#54711yescharacteristic
525C22×C4C22×SD1611yescharacteristic
526C22×C4#57711yescharacteristic
527Q16—22no—
528SD16—22no—
529SD16#36514yescomplemented
530D8—22no—
531M16—81no—
532M16—81no—
533M16—81no—
534M16—81no—
535M16—81no—
536M16—81no—
537M16—162no—
538M16—162no—
539M16—161no—
540M16—161no—
541M16—321no—
542C2×C8—22no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—41no—
552C2×C8—41no—
553C2×C8—41no—
554C2×C8—41no—
555C2×C8—41no—
556C2×C8—41no—
557C2×C8—41no—
558C2×C8—41no—
559C2×C8—41no—
560C2×C8—41no—
561C2×C8—81no—
562C2×C8—81no—
563C2×C8—81no—
564C2×C8—81no—
565C2×C8—81no—
566C2×C8—81no—
567C2×C8—81no—
568C2×C8—81no—
569C2×C8—81no—
570C2×C8—81no—
571C2×C8—162no—
572C2×C8—162no—
573C2×C8—321no—
574C2×C8C2×#10611yescharacteristic
575C2×C8C2×#10611yescharacteristic
576C2×C8C2×#10611yescharacteristic
577C2×C8C2×#10611yescharacteristic
578C2×C8C2×C4×D411yescharacteristic
579C2×C8#52011yescharacteristic
580C2×C8D4×Q811yescharacteristic
581C2×C8#55411yescharacteristic
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—42no—
587C4⋊C4—81no—
588C4⋊C4C2×#10611yescharacteristic
589C4⋊C4C2×#10611yescharacteristic
590C4⋊C4C2×#10611yescharacteristic
591C4⋊C4C2×#10611yescharacteristic
592C22⋊C4—21no—
593C22⋊C4—21no—
594C22⋊C4—21no—
595C22⋊C4—21no—
596C22⋊C4—21no—
597C22⋊C4—21no—
598C42—21no—
599C42—21no—
600C42—21no—
601C42—21no—
602C42—21no—
603C42—21no—
604C42—81no—
605C42—81no—
606C42C22×C4⋊C411yescharacteristic
607C42D4211yescharacteristic
608C42C22×SD1611yescharacteristic
609C42#57711yescharacteristic
610C16—21no—
611C16—21no—
612C16—21no—
613C16—21no—
614C16—41no—
615C16—41no—
616C16—41no—
617C16—41no—
618C16—41no—
619C16—41no—
620C23—21no—
621C23—21no—
622C23#333511yescharacteristic
623Q8—21no—
624Q8—21no—
625Q8—21no—
626Q8C2×#36512yes—
627D4—21no—
628D4—21no—
629D4—21no—
630D4—42no—
631D4—42no—
632D4C2×#36512yes—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—82no—
653C2×C4—161no—
654C2×C4—161no—
655C2×C4D4×C4⋊C411yescharacteristic
656C2×C4C22×#10611yescharacteristic
657C2×C4#291211yescharacteristic
658C2×C4D4×SD1611yescharacteristic
659C2×C4D4×SD1611yescharacteristic
660C2×C4#333511yescharacteristic
661C8—21no—
662C8—21no—
663C8—21no—
664C8—21no—
665C8—21no—
666C8—21no—
667C8—41no—
668C8—41no—
669C8—41no—
670C8—41no—
671C8—41no—
672C8—41no—
673C8—81no—
674C8—81no—
675C8—81no—
676C8—81no—
677C8—161no—
678C8—161no—
679C8—161no—
680C8—161no—
681C8—161no—
682C8C2×#36512yes—
683C8#295511yescharacteristic
684C8#295611yescharacteristic
685C22—21no—
686C22—22no—
687C22—21no—
688C22—21no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22D4×#10611yescharacteristic
695C4—21no—
696C4—22no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—162no—
706C4#2268711yescharacteristic
707C4#2269511yescharacteristic
708C4C22×#36511yescharacteristic
709C4#3340411yescharacteristic
710C2—41no—
711C2—42no—
712C2—81no—
713C2#22571411yescharacteristic
714C2D4×#36511yescharacteristic
715C2#49783911yescharacteristic
subgroupG/H
center, socleC22D4×#106
commutatorC4×C8C23×C4
Frattini, cosocleC2×C4×C8C24
prime radicalC2×#135SD16
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#135 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #483466C2semidirect
3. #483463C2semidirect
6. #140503C2semidirect
7. #140497C2semidirect
8. #136464C2semidirect
9. #136464C2semidirect
10. #136463C2semidirect
11. #136462C2semidirect
12. #135801C2semidirect
13. #135074C2semidirect
14. #135073C2semidirect
15. #129183C2semidirect
26. #12063C4semidirect
27. #12063C4semidirect
28. #12060C4semidirect
29. #12060C4semidirect
30. #12058C4semidirect
31. #12058C4semidirect
32. SD16⋊C16C4semidirect
33. SD16⋊C16C4semidirect
529. SD16#365semidirect

Maximal subgroups

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

subgroupindex
#4836172
#4834662
#4834632
#2257732
#2257202
#1405032
#1404972
#1364642
#1364632
#1364622
#1358012
#1350742
#1350732
#1291832