Finite Group Databasebeta

← #356979

Subgroups — #356979

Order 512.

1465 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2#21218C211yesmaximal, characteristic
3#21212C211yesmaximal, characteristic
4#20441C211yesmaximal, characteristic, complemented
5#20431C211yesmaximal, characteristic, complemented
6C2×#1757C212yesmaximal, complemented
7#10961C212yesmaximal, complemented
8#10961C212yesmaximal, complemented
9#10848C211yesmaximal, characteristic, complemented
10#10764C212yesmaximal, complemented
11C4×#344C211yesmaximal, characteristic, complemented
12C22×#131C2212yes—
13C2×#532C2211yescharacteristic
14#2935—21no—
15#2935—21no—
16#2935—22no—
17#2935—21no—
18#2935—21no—
19#2935—22no—
20C2×#436C412yescomplemented
21C2×#435C414yescomplemented
22C2×C8×D4C412yescomplemented
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#413C2212yes—
26C2×#413C2212yes—
27C2×#410C2212yes—
28C2×#410C2212yes—
29C2×#408C2212yes—
30C2×#405C2212yes—
31C2×#405C2212yes—
32C2×C4×M16C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2285C2211yescharacteristic
35C4×#117C2211yescharacteristic
36#1967C414yescomplemented
37#1966C412yescomplemented
38#1965C412yescomplemented
39#1964C414yes—
40#1964C414yes—
41#1771—22no—
42#1770—21no—
43C8×C4⋊C4—21no—
44#1768—22no—
45#1767—21no—
46C4×C4⋊C8—21no—
47#1757—28no—
48C2×#344C2211yescharacteristic
49C2×#344C2212yes—
50C2×#344C2211yescharacteristic
51C2×#344C2211yescharacteristic
52C2×#344C2212yes—
53C2×#344C2211yescharacteristic
54#1729C2212yes—
55#1729C2212yes—
56C22×#41C2312yes—
57C2×#131—24no—
58C2×#131—24no—
59C2×#131—22no—
60C2×#131—22no—
61C2×#131—22no—
62C2×#131—22no—
63C2×#131—42no—
64C2×#131—42no—
65C22×M16C2×C414yes—
66C22×M16C2311yescharacteristic
67C22×M16C2312yes—
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8C2×C414yes—
72C23×C8C2312yes—
73#532—21no—
74#532—21no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532—21no—
80#532—21no—
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#127C2×C411yescharacteristic
83C2×#124C2×C412yes—
84C2×#121C2×C412yes—
85C2×#121C2×C412yes—
86C2×C4×D4C2×C412yes—
87C2×C4×D4C2×C412yes—
88C2×#117C2312yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#436—22no—
93#436—22no—
94#436—24no—
95#435—24no—
96#435—24no—
97#435—24no—
98#435—24no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—24no—
102#423—21no—
103#423—21no—
104#423C2×C412yes—
105#423C2×C412yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2×C411yescharacteristic
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8C2×C411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×C4⋊C8D412yes—
114C2×C4⋊C8D411yescharacteristic
115C2×C4⋊C8D411yescharacteristic
116C2×C4⋊C8D412yes—
117#413—24no—
118#413—24no—
119#413—42no—
120#413—42no—
121C2×#101C2×C414yes—
122C2×#101C2×C414yes—
123C2×#100C2×C414yes—
124C2×#100C2×C414yes—
125#410—24no—
126#410—24no—
127#410—42no—
128#410—42no—
129C2×#99C2×C418yes—
130C2×#99C2×C418yes—
131#408—22no—
132#408—22no—
133#408—22no—
134#408—22no—
135C2×C22⋊C8C2×C414yes—
136C2×C22⋊C8C2×C414yes—
137#405—24no—
138#405—24no—
139#405—42no—
140#405—42no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16—21no—
146C4×M16—21no—
147C4×M16C2×C412yes—
148C4×M16C2×C412yes—
149C2×#97—21no—
150C2×#97—22no—
151C2×#97—22no—
152C2×#97C2×C412yes—
153C2×C4×C8—21no—
154C2×C4×C8—22no—
155C2×C4×C8—22no—
156C2×C4×C8C2×C412yes—
157#384—21no—
158#382—21no—
159#380C2×C412yes—
160#380C2×C412yes—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C22⋊C4C2×C412yes—
166C4×C22⋊C4C2×C412yes—
167#376—21no—
168C43—21no—
169#344—24no—
170#344—22no—
171#344—22no—
172#344—22no—
173#344—42no—
174#344D414yescomplemented
175#336—22no—
176#336—22no—
177#336—22no—
178#336—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—22no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C22×Q8C22×C412yes—
192C22×D4C4212yes—
193C22×D4C4212yes—
194C22×D4C22×C414yes—
195C22×D4C22×C412yes—
196C23×C4C22×C414yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C2411yescharacteristic
200#131—44no—
201#131—44no—
202#131—44no—
203#131—44no—
204#131—44no—
205#131—44no—
206#131—44no—
207#131—44no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—22no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—24no—
224C2×M16—24no—
225C2×M16—28no—
226C2×M16—22no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—22no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—44no—
235C2×M16C22×C412yes—
236C2×M16C22×C414yes—
237C2×M16C22×C412yes—
238C2×M16C2×D412yes—
239C2×M16C2×D412yes—
240C2×M16#4112yes—
241C2×M16#4112yes—
242C22×C8—22no—
243C22×C8—21no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—21no—
248C22×C8—22no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8C22×C412yes—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C2×D412yes—
258C22×C8C2×D412yes—
259C22×C8C2×D412yes—
260C22×C8C2×D412yes—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265#127—21no—
266#127—21no—
267#127—21no—
268#127—21no—
269#124—22no—
270#124—22no—
271#124—22no—
272#124—22no—
273#121—22no—
274#121—22no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289#117—21no—
290#117—21no—
291#117—22no—
292#117—21no—
293#117—21no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—21no—
298#117—21no—
299#117C2×D412yes—
300#117#4112yes—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4C4212yes—
307C2×C4⋊C4C4212yes—
308C2×C22⋊C4C4214yes—
309C2×C22⋊C4C4214yes—
310C2×C22⋊C4C22×C412yes—
311C2×C22⋊C4C22×C414yes—
312C2×C22⋊C4C22×C412yes—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42#4111yescharacteristic
328C2×C42#4111yescharacteristic
329C2×C42#4111yescharacteristic
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8C22⋊C411yescharacteristic
335C4⋊C8C22⋊C411yescharacteristic
336C4⋊C8C22⋊C412yes—
337C4⋊C8C22⋊C412yes—
338C4⋊C8C22⋊C412yes—
339C4⋊C8C22⋊C412yes—
340C4⋊C8C22⋊C412yes—
341C4⋊C8C22⋊C411yescharacteristic
342C4⋊C8C22⋊C411yescharacteristic
343C4⋊C8C22⋊C412yes—
344#101—28no—
345#101—44no—
346#100—28no—
347#100—44no—
348#99—216no—
349#99—48no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354#97—22no—
355#97—22no—
356#97—22no—
357#97—22no—
358#97—44no—
359#97—44no—
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—44no—
365C4×C8—44no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24C2×C4214yes—
371C24C23×C411yescharacteristic
372#41—42no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41—42no—
385#41—42no—
386#41—42no—
387#41—42no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—44no—
411C2×D4—82no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—81no—
423C22×C4C2×C4214yes—
424C22×C4C2×C4211yescharacteristic
425C22×C4C2×C4211yescharacteristic
426C22×C4C2×C4211yescharacteristic
427C22×C4C2×C4211yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C23×C411yescharacteristic
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C42−1+411yescharacteristic
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—24no—
458M16—22no—
459M16—22no—
460M16—24no—
461M16—48no—
462M16—48no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—42no—
468M16—41no—
469M16—41no—
470M16—42no—
471M16C4×D414yes—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C8—48no—
496C2×C8—42no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—42no—
500C2×C8C2×C22⋊C412yes—
501C2×C8C2×C22⋊C412yes—
502C2×C8C2×C22⋊C412yes—
503C2×C8C2×C22⋊C412yes—
504C2×C8C4×D414yes—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C22⋊C4—22no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—24no—
526C22⋊C4C4×D414yes—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—42no—
531C42—41no—
532C42—41no—
533C42—42no—
534C42—82no—
535C42C2×C22⋊C411yescharacteristic
536C42C2×C22⋊C411yescharacteristic
537C42C2×C22⋊C411yescharacteristic
538C42C2×C22⋊C411yescharacteristic
539C42#11711yescharacteristic
540C42#11711yescharacteristic
541C42#11711yescharacteristic
542C42#11711yescharacteristic
543C42C4×D411yescharacteristic
544C42C4×D411yescharacteristic
545C42C4×D411yescharacteristic
546C42C4×D411yescharacteristic
547C42C4×D411yescharacteristic
548C42C4×D411yescharacteristic
549C42C4×D411yescharacteristic
550C42C4×D411yescharacteristic
551C23—24no—
552C23—24no—
553C23—24no—
554C23—41no—
555C23—41no—
556C23—44no—
557C23—44no—
558C23C22×C4211yescharacteristic
559C23C2×C4×D411yescharacteristic
560C23C2×C4×D411yescharacteristic
561C23C2×C4×D411yescharacteristic
562C23C4×#4111yescharacteristic
563C23#51811yescharacteristic
564C23#52011yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—44no—
573D4—44no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—44no—
579D4—82no—
580D4—82no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4C4×C22⋊C411yescharacteristic
610C2×C4C4×C22⋊C411yescharacteristic
611C2×C4C4×C22⋊C411yescharacteristic
612C2×C4C4×C22⋊C411yescharacteristic
613C2×C4C4×C22⋊C411yescharacteristic
614C2×C4C4×C22⋊C411yescharacteristic
615C2×C4C4×C22⋊C411yescharacteristic
616C2×C4C4×C22⋊C411yescharacteristic
617C2×C4C22×C22⋊C411yescharacteristic
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C4×#4111yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4#51811yescharacteristic
626C2×C4#51811yescharacteristic
627C2×C4#52011yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4D4211yescharacteristic
630C2×C4D4211yescharacteristic
631C2×C4#54611yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54711yescharacteristic
634C2×C4#54811yescharacteristic
635C2×C4#54811yescharacteristic
636C2×C4#55011yescharacteristic
637C2×C4#55011yescharacteristic
638C2×C4#55011yescharacteristic
639C2×C4#55311yescharacteristic
640C2×C4#55311yescharacteristic
641C2×C4#55511yescharacteristic
642C2×C4#55611yescharacteristic
643C2×C4#55911yescharacteristic
644C2×C4#55911yescharacteristic
645C8—22no—
646C8—22no—
647C8—22no—
648C8—22no—
649C8—24no—
650C8—24no—
651C8—42no—
652C8—42no—
653C8—44no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—84no—
668C22—84no—
669C22C2×C4×C22⋊C411yescharacteristic
670C22C42×D411yescharacteristic
671C22#227511yescharacteristic
672C22#236111yescharacteristic
673C22#236311yescharacteristic
674C22#236611yescharacteristic
675C22#236711yescharacteristic
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—21no—
683C4—44no—
684C4—44no—
685C4—84no—
686C4—82no—
687C4#175712yescomplemented
688C4D4×C22⋊C411yescharacteristic
689C4#234111yescharacteristic
690C4#235411yescharacteristic
691C4#235711yescharacteristic
692C4#236111yescharacteristic
693C4#236311yescharacteristic
694C4#236611yescharacteristic
695C4#236711yescharacteristic
696C2—21no—
697C2—21no—
698C2—21no—
699C2—21no—
700C2—21no—
701C2—21no—
702C2—84no—
703C2#1382011yescharacteristic
704C2C2×#175711yescharacteristic
705C2#2040111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20441C2semidirect
5. #20431C2semidirect
6. C2×#1757C2semidirect
7. #10961C2semidirect
8. #10961C2semidirect
9. #10848C2semidirect
10. #10764C2semidirect
11. C4×#344C2semidirect
20. C2×#436C4semidirect
21. C2×#435C4semidirect
22. C2×C8×D4C4semidirect
36. #1967C4semidirect
37. #1966C4semidirect
38. #1965C4semidirect
174. #344D4semidirect
687. C4#1757semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
#212182
#212122
#204412
#204312
C2×#17572
#109612
#108482
#107642
C4×#3442