Finite Group Databasebeta

← #7652150

Subgroups — #7652150

Order 512.

5588 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#42874C218yesmaximal, complemented
2#42835C218yesmaximal, complemented
3#42023C218yesmaximal, complemented
4#40125C218yesmaximal, complemented
5#39659C218yesmaximal, complemented
6#38487C218yesmaximal, complemented
7#35660C212yesmaximal, complemented
8#35592C212yesmaximal, complemented
9#35557C212yesmaximal, complemented
10#35273C212yesmaximal, complemented
11#35153C211yesmaximal, characteristic, complemented
12C2×#2567C211yesmaximal, characteristic, complemented
13#34744C212yesmaximal, complemented
14#34268C212yesmaximal, complemented
15#34257C211yesmaximal, characteristic, complemented
16C22×2−1+4C2211yescharacteristic
17C2×#537C2212yes—
18C2×#537C2211yescharacteristic
19C2×#536C2212yes—
20C2×#532C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×C4×#41C2212yes—
24C22×C4×Q8C2211yescharacteristic
25#2857C2214yes—
26#2856C2214yes—
27#2854C2214yes—
28#2832C2218yes—
29#2830C2218yes—
30#2808C2218yes—
31#2805C2218yes—
32#2803C22116yes—
33#2788C22116yes—
34#2788C22116yes—
35#2784C22116yes—
36#2727C2218yes—
37#2715C22116yes—
38#2686C22116yes—
39#2682C2218yes—
40#2665C22116yes—
41#2662C22116yes—
42#2649C22116yes—
43#2620C2214yes—
44#2620C2214yes—
45#2618C2214yes—
46#2591C2218yes—
47#2587C22116yes—
48#2586C2218yes—
49#2567C2218yes—
50#2567C2218yes—
51#2559C22116yes—
52#2555C2218yes—
53#2538C2218yes—
54#2537C2218yes—
55#2512C2218yes—
56#2511C2218yes—
57#2497C2218yes—
58#2496C2218yes—
59#2492C2218yes—
60#2492C2218yes—
61#2491C2218yes—
62#2491C2218yes—
63#2481C22116yes—
64#2448C2214yes—
65#2447C2214yes—
66#2446C2214yes—
67#2445C2214yes—
68#2444C2218yes—
69#2443C2218yes—
70#2436C2218yes—
71#2436C2214yes—
72#2434C2214yes—
73#2432C2214yes—
74#2431C2214yes—
75#2430C22116yes—
76#2429C2218yes—
77#2426C2218yes—
78#2416C22116yes—
79#2409C2218yes—
80#2407C2218yes—
81#2395C2218yes—
82C2×#395C2214yes—
83#2367C2218yes—
84#2365C2218yes—
85#2359C22116yes—
86#2341C2218yes—
87Q8×C22⋊C4C2218yes—
88#2335C22116yes—
89#2323C2218yes—
90#2316C2211yescharacteristic
91#2311C2212yes—
92#2311C2212yes—
93#2311C2214yes—
94#2311C2212yes—
95C4×C22⋊Q8C2218yes—
96C4×#123C2218yes—
97#2297C2212yes—
98#2297C2212yes—
99#2297C2212yes—
100#2297C2214yes—
101#2296C2212yes—
102C2×#391C2212yes—
103C2×#389C2214yes—
104C2×#384C2212yes—
105#2274C2212yes—
106C2×C4×C4⋊C4C2211yescharacteristic
107C2×2−1+4—28no—
108C2×2−1+4—44no—
109C2×2−1+4—44no—
110C22×#41C2312yes—
111C22×#41C2312yes—
112C22×#41C2314yes—
113C22×#41C2314yes—
114C23×Q8C2311yescharacteristic
115C23×Q8C2314yes—
116C23×Q8C2312yes—
117#537—44no—
118#537—48no—
119#536—216no—
120#532—48no—
121C2×C4⋊Q8—44no—
122C2×C4⋊Q8C2314yes—
123C2×C4⋊Q8C2318yes—
124C2×C4⋊Q8C2312yes—
125C2×#127C2312yes—
126C2×#126C23116yes—
127C2×#124C2314yes—
128C2×#124C2312yes—
129C2×#124C2312yes—
130C2×#124C23116yes—
131C2×#123C2318yes—
132C2×#123C2318yes—
133C2×C22⋊Q8C2318yes—
134C2×C22⋊Q8C2314yes—
135C2×C22⋊Q8C2314yes—
136C2×C22⋊Q8C2314yes—
137C2×#121C2318yes—
138C2×#121C2314yes—
139C2×#121C2314yes—
140C2×#121C2314yes—
141C2×#121C2318yes—
142C2×#121C2314yes—
143C2×#121C23116yes—
144#519—44no—
145C4×#41—48no—
146C2×C4×Q8—44no—
147C2×C4×Q8C2318yes—
148C2×C4×Q8C2318yes—
149C2×C4×Q8C2314yes—
150C2×C4×D4C2318yes—
151C2×C4×D4C2314yes—
152C2×C4×D4C2318yes—
153C2×#117C2312yes—
154C2×#117C2312yes—
155C2×#117C2314yes—
156C2×#117C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2312yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2312yes—
162C22×C4⋊C4C2318yes—
163C22×C42C2311yescharacteristic
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2314yes—
167#400C23116yes—
168#400C23116yes—
169#400C2318yes—
170#400C23116yes—
171#399C2318yes—
172#398C2318yes—
173#398C2318yes—
174#397C2318yes—
175#397C23116yes—
176#397C2318yes—
177#397C2318yes—
178#396C23116yes—
179#396C23116yes—
180#396C23116yes—
181#396C23116yes—
182#395C2318yes—
183#395C2318yes—
184#395C23116yes—
185#395C23116yes—
186#394C23116yes—
187#394C2318yes—
188#394C23116yes—
189#394C23116yes—
190#394C23116yes—
191#394C2318yes—
192#394C2318yes—
193#393C23116yes—
194#393C23116yes—
195#393C23116yes—
196#392C23116yes—
197#391C2318yes—
198#391C2314yes—
199#391C2314yes—
200#391C2314yes—
201#391C2314yes—
202#391C2318yes—
203#391C2314yes—
204#391C23116yes—
205#391C2314yes—
206#391C2318yes—
207#391C2314yes—
208#391C2314yes—
209#390C2318yes—
210#390C2314yes—
211#390C2314yes—
212#390C2314yes—
213#390C23116yes—
214#390C2318yes—
215#390C2314yes—
216#390C2318yes—
217#390C2314yes—
218#390C2314yes—
219#389C23116yes—
220#389C2318yes—
221#389C2318yes—
222#389C2318yes—
223#389C2314yes—
224#389C23116yes—
225#389C23116yes—
226#389C2318yes—
227#389C2314yes—
228#388C23116yes—
229#388C23116yes—
230#388C2318yes—
231#388C23116yes—
232#388C23116yes—
233#388C23116yes—
234#388C23116yes—
235#388C2318yes—
236#387C23116yes—
237#387C2318yes—
238#387C23116yes—
239#387C23116yes—
240#387C2318yes—
241#387C23116yes—
242#386C2318yes—
243#386C2314yes—
244#386C2318yes—
245#386C2318yes—
246#386C23116yes—
247#386C2314yes—
248#386C2314yes—
249#386C2314yes—
250#386C23116yes—
251#386C2318yes—
252#386C2318yes—
253#385C23116yes—
254#385C23116yes—
255#385C2318yes—
256#385C23116yes—
257#385C2318yes—
258#384C2314yes—
259#384C2314yes—
260#384C2318yes—
261#381C2318yes—
262#380C2318yes—
263#380C23116yes—
264#380C2318yes—
265#380C2318yes—
266#380C2314yes—
267#380C23116yes—
268C4×C4⋊C4C2314yes—
269C4×C4⋊C4C2314yes—
270C4×C4⋊C4C2318yes—
271C4×C4⋊C4C23116yes—
272C4×C4⋊C4C2318yes—
273C4×C4⋊C4C2318yes—
274C4×C22⋊C4C23116yes—
275C4×C22⋊C4C23116yes—
276C4×C22⋊C4C2318yes—
277C4×C22⋊C4C2318yes—
278C4×C22⋊C4C2314yes—
279C4×C22⋊C4C2318yes—
280C4×C22⋊C4C2314yes—
281C2×#95C2318yes—
282C2×#95C2312yes—
2832−1+4—464no—
284C2×#41—216no—
285C2×#41—28no—
286C2×#41—216no—
287C2×#41—28no—
288C2×#41—28no—
289C2×#41—28no—
290C2×#41—28no—
291C2×#41—28no—
292C2×#41—48no—
293C2×#41—48no—
294C2×#41—48no—
295C2×#41—48no—
296C22×Q8—216no—
297C22×Q8—24no—
298C22×Q8—24no—
299C22×Q8—44no—
300C22×Q8—48no—
301C22×Q8—48no—
302C22×Q8—84no—
303C22×Q8#4118yes—
304C22×Q8C2412yes—
305C22×Q8C2412yes—
306C22×Q8C2414yes—
307C22×Q8C2414yes—
308C22×Q8C2418yes—
309C22×D4C2414yes—
310C22×D4C2412yes—
311C22×D4C2414yes—
312C22×D4C2412yes—
313C22×D4C2418yes—
314C22×D4C2414yes—
315C22×D4C2414yes—
316C22×D4C2418yes—
317C23×C4C2411yescharacteristic
318C23×C4C2414yes—
319C23×C4C2414yes—
320C23×C4C2414yes—
321C23×C4C2412yes—
322C23×C4C2412yes—
323C23×C4C2414yes—
324C23×C4C2411yescharacteristic
325C23×C4C2418yes—
326C23×C4C2418yes—
327C23×C4C2411yescharacteristic
328C4⋊Q8—216no—
329C4⋊Q8—416no—
330C4⋊Q8—44no—
331#127—44no—
332#126—816no—
333#124—216no—
334#124—44no—
335#124—44no—
336#124—816no—
337#123—232no—
338#123—232no—
339C22⋊Q8—232no—
340C22⋊Q8—48no—
341C22⋊Q8—48no—
342C22⋊Q8—48no—
343#121—48no—
344#121—48no—
345#121—48no—
346#121—416no—
347#121—48no—
348#121—416no—
349#121—816no—
350C4×Q8—416no—
351C4×Q8—48no—
352C4×Q8—416no—
353C4×D4—48no—
354C4×D4—416no—
355C4×D4—416no—
356#117—28no—
357#117—28no—
358#117—48no—
359#117—48no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—28no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—88no—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C24116yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2418yes—
376C2×C4⋊C4C24116yes—
377C2×C4⋊C4C24116yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C2418yes—
382C2×C4⋊C4C2418yes—
383C2×C4⋊C4C2414yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2418yes—
386C2×C4⋊C4C2412yes—
387C2×C22⋊C4C2414yes—
388C2×C22⋊C4C2414yes—
389C2×C22⋊C4C24116yes—
390C2×C22⋊C4C2418yes—
391C2×C22⋊C4C2418yes—
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2418yes—
394C2×C22⋊C4C2418yes—
395C2×C22⋊C4C2418yes—
396C2×C22⋊C4C2418yes—
397C2×C22⋊C4C2418yes—
398C2×C22⋊C4C2418yes—
399C2×C22⋊C4C24116yes—
400C2×C22⋊C4C24116yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C2418yes—
403C2×C22⋊C4C24116yes—
404C2×C22⋊C4C2418yes—
405C2×C42—24no—
406C2×C42—28no—
407C2×C42—48no—
408C2×C42—44no—
409C2×C42C2414yes—
410C2×C42C2412yes—
411C2×C42C2414yes—
412C2×C42C2414yes—
413C2×C42C2414yes—
414C2×C42C2412yes—
415C2×C42C2418yes—
416C2×C42C2418yes—
417C2×C42C2418yes—
418C2×C42C24116yes—
419#95C24116yes—
420#95C2414yes—
421#95C2412yes—
422#95C2412yes—
423#95C24116yes—
424#95C2418yes—
425#95C2418yes—
426#95C24116yes—
427#95C24116yes—
428#95C2418yes—
429#95C2418yes—
430#95C2418yes—
431#95C2418yes—
432#95C2418yes—
433#95C24116yes—
434#95C2418yes—
435#95C24116yes—
436#95C2418yes—
437C24C2514yes—
438C24C2514yes—
439C24C2511yescharacteristic
440#41—464no—
441#41—464no—
442#41—432no—
443#41—432no—
444C2×Q8—232no—
445C2×Q8—24no—
446C2×Q8—28no—
447C2×Q8—24no—
448C2×Q8—28no—
449C2×Q8—28no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—416no—
457C2×Q8—48no—
458C2×Q8—464no—
459C2×Q8—816no—
460C2×Q8C2×#41116yes—
461C2×D4—28no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—28no—
466C2×D4—28no—
467C2×D4—24no—
468C2×D4—28no—
469C2×D4—24no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—216no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—48no—
478C2×D4—48no—
479C2×D4—44no—
480C2×D4—48no—
481C2×D4—48no—
482C2×D4—416no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—28no—
489C22×C4—24no—
490C22×C4—28no—
491C22×C4—28no—
492C22×C4—216no—
493C22×C4—28no—
494C22×C4—28no—
495C22×C4—28no—
496C22×C4—28no—
497C22×C4—28no—
498C22×C4—28no—
499C22×C4—216no—
500C22×C4—44no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—42no—
504C22×C4—48no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4C2×#4118yes—
509C22×C4C2×#4114yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C4H(5,𝔽2)14yes—
512C22×C4C2514yes—
513C22×C4C2518yes—
514C22×C4C2512yes—
515C22×C4C2518yes—
516C22×C4C2514yes—
517C22×C4C2518yes—
518C22×C4C2514yes—
519C22×C4C2514yes—
520C22×C4C2514yes—
521C22×C4C2514yes—
522C22×C4C2512yes—
523C22×C4C2512yes—
524C4⋊C4—28no—
525C4⋊C4—232no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—216no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—416no—
536C4⋊C4—416no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C42−1+418yescomplemented
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—232no—
548C22⋊C4—216no—
549C22⋊C4—216no—
550C22⋊C4—216no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—416no—
558C22⋊C4—416no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—416no—
562C22⋊C4—48no—
563C42—28no—
564C42—24no—
565C42—24no—
566C42—28no—
567C42—28no—
568C42—28no—
569C42—48no—
570C42—48no—
571C42—48no—
572C42—416no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—28no—
582C23—28no—
583C23—28no—
584C23—28no—
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)12yes—
587C23C2×H(5,𝔽2)12yes—
588C23C2611yescharacteristic
589Q8—232no—
590Q8—416no—
591Q8—48no—
592Q8—48no—
593Q8—816no—
594D4—416no—
595D4—416no—
596D4—416no—
597D4—48no—
598D4—48no—
599D4—816no—
600D4—164no—
601D4—168no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—216no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—216no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—28no—
631C2×C4—416no—
632C2×C4—432no—
633C2×C4—44no—
634C2×C4—416no—
635C2×C4—416no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4#538116yes—
640C2×C4C22×#4118yes—
641C2×C4C22×#4114yes—
642C2×C4C2×H(5,𝔽2)14yes—
643C2×C4C2×H(5,𝔽2)14yes—
644C2×C4C2×2−1+414yes—
645C2×C4C2×2−1+418yes—
646C22—24no—
647C22—22no—
648C22—22no—
649C22—24no—
650C22—48no—
651C22—48no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22C2×#53814yes—
661C22C23×#4111yescharacteristic
662C22C22×H(5,𝔽2)11yescharacteristic
663C22C22×H(5,𝔽2)11yescharacteristic
664C22C22×2−1+412yes—
665C22H(7,𝔽2)12yes—
666C4—24no—
667C4—28no—
668C4—28no—
669C4—48no—
670C4—42no—
671C4—44no—
672C4—48no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—48no—
677C4C2×#53818yes—
678C2—24no—
679C2—84no—
680C2—84no—
681C2C22×#53811yescharacteristic
682C2#6283311yescharacteristic
683C2#6287611yescharacteristic
684C2#6294412yes—
685C2#6299912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #42874C2semidirect
2. #42835C2semidirect
3. #42023C2semidirect
4. #40125C2semidirect
5. #39659C2semidirect
6. #38487C2semidirect
7. #35660C2semidirect
8. #35592C2semidirect
9. #35557C2semidirect
10. #35273C2semidirect
11. #35153C2semidirect
12. C2×#2567C2semidirect
13. #34744C2semidirect
14. #34268C2semidirect
15. #34257C2semidirect
544. C4⋊C42−1+4semidirect

Maximal subgroups

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

subgroupindex
#428742
#428352
#420232
#401252
#396592
#384872
#356602
#355922
#355572
#352732
#351532
C2×#25672
#347442
#342682
#342572