Finite Group Databasebeta

← #8095312

Subgroups — #8095312

Order 512.

5384 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53022C218yesmaximal, complemented
2#53008C218yesmaximal, complemented
3#52913C218yesmaximal, complemented
4#45669C218yesmaximal, complemented
5#45569C218yesmaximal, complemented
6#45503C218yesmaximal, complemented
7#35672C211yesmaximal, characteristic, complemented
8#35667C212yesmaximal, complemented
9#35634C211yesmaximal, characteristic
10#35630C212yesmaximal, complemented
11#35316C212yesmaximal, complemented
12#35308C212yesmaximal, complemented
13#35242C212yesmaximal, complemented
14#34742C212yesmaximal, complemented
15C2×#2434C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C2×#537C2211yescharacteristic
18C2×#536C2212yes—
19C2×#536C2212yes—
20C2×#536C2212yes—
21C22×C4⋊Q8C2212yes—
22C2×#525C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×C4×#41C2212yes—
26#2857C2218yescomplemented
27#2857C2214yescomplemented
28#2850C2214yescomplemented
29#2849C2218yes—
30#2843C2218yescomplemented
31#2841C22116yescomplemented
32#2838C22116yescomplemented
33#2838C22116yescomplemented
34#2834C2218yes—
35#2830C2218yescomplemented
36#2826C2218yes—
37#2807C22116yes—
38#2807C2218yes—
39#2792C22116yescomplemented
40#2792C2218yescomplemented
41#2784C2218yescomplemented
42#2778C22116yes—
43#2757C22116yes—
44#2757C22116yes—
45#2748C22116yes—
46#2742C22116yescomplemented
47#2737C22116yescomplemented
48#2693C2218yes—
49#2621C2214yescomplemented
50#2621C2218yescomplemented
51#2620C2214yescomplemented
52#2614C2218yes—
53#2547C2218yescomplemented
54#2546C2214yescomplemented
55#2546C2218yescomplemented
56#2546C2214yescomplemented
57#2545C2214yescomplemented
58#2545C2214yescomplemented
59#2542C2218yes—
60#2540C2218yes—
61#2539C2214yes—
62#2514C2218yescomplemented
63#2514C2218yescomplemented
64#2511C2218yescomplemented
65#2511C2218yescomplemented
66#2501C2214yes—
67#2493C22116yes—
68#2486C22116yes—
69#2484C2214yes—
70#2482C2214yes—
71#2436C2218yes—
72#2434C2218yes—
73#2434C2218yes—
74#2434C2218yes—
75#2429C2218yescomplemented
76#2429C2218yescomplemented
77#2429C2218yescomplemented
78#2427C2218yescomplemented
79#2425C22116yes—
80#2423C2218yes—
81#2423C2214yes—
82#2422C2214yes—
83#2421C2214yes—
84#2421C2218yes—
85#2420C2214yes—
86#2419C2218yes—
87#2418C2218yes—
88#2413C22116yes—
89#2409C2218yes—
90#2409C2218yes—
91#2395C2218yes—
92C2×#395C2214yes—
93#2376C22116yes—
94#2317C2212yes—
95#2310C2212yes—
96#2310C2212yes—
97#2310C2211yescharacteristic
98C4×#123C2218yes—
99#2297C2212yes—
100#2296C2212yes—
101C2×#391C2212yes—
102C2×#385C2214yes—
103#2285C2212yes—
104#2280C2212yes—
105#2274C2211yescharacteristic
106C2×C4×C22⋊C4C2211yescharacteristic
107C22×#41C2312yes—
108C23×Q8—44no—
109C23×Q8—48no—
110C23×Q8C2312yes—
111C23×Q8C2311yescharacteristic
112C23×Q8C2312yes—
113C23×Q8C2318yes—
114C24×C4C2311yescharacteristic
115C24×C4C2312yes—
116#537—44no—
117#536—48no—
118#536—48no—
119#536—48no—
120C2×C4⋊Q8—48no—
121C2×C4⋊Q8C2314yes—
122C2×C4⋊Q8C2314yes—
123C2×C4⋊Q8C2314yes—
124C2×C4⋊Q8C2318yes—
125C2×C4⋊Q8C2314yes—
126C2×C4⋊Q8C2314yes—
127C2×#124C2314yes—
128C2×#124C2314yes—
129C2×#124C2314yes—
130#525—48no—
131C2×#123C2318yes—
132C2×#123C2318yes—
133C2×#123C2318yes—
134C2×#123C2314yes—
135C2×#123C2318yes—
136C2×C22⋊Q8—44no—
137C2×C22⋊Q8C2314yes—
138C2×C22⋊Q8C2314yes—
139C2×C22⋊Q8C2318yes—
140C2×C22⋊Q8C2318yes—
141C2×C22⋊Q8C2314yes—
142C2×C22⋊Q8C2318yes—
143C2×C22⋊Q8C2314yes—
144C2×#121C2314yes—
145C2×#121C2314yes—
146C2×#121C2314yes—
147C2×#120C2314yes—
148#519—44no—
149C4×#41—48no—
150C2×C4×Q8C2314yes—
151C2×C4×Q8C2314yes—
152C2×C4×Q8C2314yes—
153C2×C4×D4C2318yes—
154C2×C4×D4C2318yes—
155C2×C4×D4C2314yes—
156C2×#117C2312yes—
157C2×#117C2312yes—
158C2×#117C2312yes—
159C2×#117C2314yes—
160C2×#117C2314yes—
161C22×C4⋊C4C2311yescharacteristic
162C22×C4⋊C4C2312yes—
163C22×C4⋊C4C2318yes—
164C22×C22⋊C4C2312yes—
165C22×C22⋊C4C2312yes—
166C22×C42C2312yes—
167C22×C42C2314yes—
168#401C23116yes—
169#400C23116yes—
170#400C2318yes—
171#400C2318yes—
172#400C23116yes—
173#400C23116yes—
174#399C2318yes—
175#399C2318yes—
176#399C23116yes—
177#399C2318yes—
178#398C2318yes—
179#398C23116yes—
180#398C23116yes—
181#398C23116yes—
182#398C2318yes—
183#398C23116yes—
184#398C23116yes—
185#397C23116yes—
186#397C2318yes—
187#397C23116yes—
188#397C2318yes—
189#397C2318yes—
190#397C23116yes—
191#397C2318yes—
192#396C23116yes—
193#396C23116yes—
194#396C23116yes—
195#395C2318yes—
196#395C23116yes—
197#395C23116yes—
198#395C23116yes—
199#395C2318yes—
200#394C2318yes—
201#394C23116yes—
202#394C2318yes—
203#393C23116yes—
204#393C23116yes—
205#392C2318yes—
206#391C2318yes—
207#391C2318yes—
208#391C2314yes—
209#391C2318yes—
210#391C2314yes—
211#391C2318yes—
212#391C2314yes—
213#391C2312yes—
214#391C2314yes—
215#391C2312yes—
216#391C2318yes—
217#390C2318yes—
218#390C2312yes—
219#390C2312yes—
220#390C23116yes—
221#389C2318yes—
222#389C23116yes—
223#389C23116yes—
224#389C23116yes—
225#389C23116yes—
226#389C23116yes—
227#389C2314yes—
228#389C2314yes—
229#388C23116yes—
230#388C23116yes—
231#388C2318yes—
232#388C23116yes—
233#388C23116yes—
234#388C2318yes—
235#387C23116yes—
236#387C2318yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C2318yes—
241#386C23116yes—
242#386C23116yes—
243#386C23116yes—
244#386C2318yes—
245#386C2318yes—
246#385C2318yes—
247#385C2318yes—
248#385C23116yes—
249#385C23116yes—
250#385C2318yes—
251#385C2318yes—
252#385C2318yes—
253#385C23116yes—
254#384C2314yes—
255#384C2318yes—
256#384C2314yes—
257#383C23116yes—
258#382C2314yes—
259#382C2314yes—
260#381C2318yes—
261#381C2318yes—
262#381C2314yes—
263#381C2318yes—
264#381C2314yes—
265#380C23116yes—
266#380C23116yes—
267#380C23116yes—
268#380C2314yes—
269#380C2314yes—
270#380C2318yes—
271#379C2314yes—
272C4×C4⋊C4C2314yes—
273C4×C4⋊C4C2318yes—
274C4×C4⋊C4C2318yes—
275C4×C4⋊C4C2314yes—
276C4×C22⋊C4C2314yes—
277C4×C22⋊C4C2318yes—
278C4×C22⋊C4C2318yes—
279C4×C22⋊C4C2314yes—
280C4×C22⋊C4C2314yes—
281C2×#95C2312yes—
282C2×#95C2318yes—
283C25C2411yescharacteristic
284C2×#41—28no—
285C2×#41—28no—
286C2×#41—28no—
287C22×Q8—24no—
288C22×Q8—28no—
289C22×Q8—28no—
290C22×Q8—28no—
291C22×Q8—28no—
292C22×Q8—44no—
293C22×Q8—48no—
294C22×Q8—416no—
295C22×Q8—416no—
296C22×Q8—416no—
297C22×Q8—464no—
298C22×Q8C2414yes—
299C22×Q8C2412yes—
300C22×Q8C2414yes—
301C22×Q8C2414yes—
302C22×Q8C2418yes—
303C22×D4C2414yes—
304C22×D4C2412yes—
305C22×D4C2418yes—
306C22×D4C2414yes—
307C23×C4—24no—
308C23×C4—28no—
309C23×C4—48no—
310C23×C4—44no—
311C23×C4C2412yes—
312C23×C4C2411yescharacteristic
313C23×C4C2414yes—
314C23×C4C2414yes—
315C23×C4C2414yes—
316C23×C4C2414yes—
317C23×C4C2412yes—
318C23×C4C2418yes—
319C23×C4C2418yes—
320C23×C4C2411yescharacteristic
321C4⋊Q8—48no—
322C4⋊Q8—48no—
323C4⋊Q8—416no—
324C4⋊Q8—48no—
325C4⋊Q8—48no—
326C4⋊Q8—48no—
327#124—48no—
328#124—48no—
329#124—48no—
330#123—232no—
331#123—48no—
332#123—416no—
333#123—416no—
334#123—416no—
335C22⋊Q8—416no—
336C22⋊Q8—48no—
337C22⋊Q8—416no—
338C22⋊Q8—48no—
339C22⋊Q8—48no—
340C22⋊Q8—416no—
341C22⋊Q8—48no—
342#121—48no—
343#121—48no—
344#121—48no—
345#120—48no—
346C4×Q8—48no—
347C4×Q8—48no—
348C4×Q8—48no—
349C4×D4—48no—
350C4×D4—416no—
351C4×D4—416no—
352#117—28no—
353#117—28no—
354#117—28no—
355#117—48no—
356#117—48no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—416no—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2418yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C24116yes—
369C2×C4⋊C4C24116yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2414yes—
375C2×C4⋊C4C24116yes—
376C2×C4⋊C4C24116yes—
377C2×C4⋊C4C2418yes—
378C2×C4⋊C4C2418yes—
379C2×C4⋊C4C2418yes—
380C2×C4⋊C4C2418yes—
381C2×C4⋊C4C24116yes—
382C2×C22⋊C4—28no—
383C2×C22⋊C4—44no—
384C2×C22⋊C4C2414yes—
385C2×C22⋊C4C2412yes—
386C2×C22⋊C4C2414yes—
387C2×C22⋊C4C2414yes—
388C2×C22⋊C4C2414yes—
389C2×C22⋊C4C2418yes—
390C2×C22⋊C4C2418yes—
391C2×C22⋊C4C2418yes—
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2418yes—
394C2×C22⋊C4C2418yes—
395C2×C22⋊C4C2418yes—
396C2×C22⋊C4C24116yes—
397C2×C22⋊C4C2418yes—
398C2×C22⋊C4C24116yes—
399C2×C22⋊C4C2418yes—
400C2×C22⋊C4C2418yes—
401C2×C22⋊C4C2418yes—
402C2×C22⋊C4C2412yes—
403C2×C42—28no—
404C2×C42—48no—
405C2×C42C2414yes—
406C2×C42C2412yes—
407C2×C42C2414yes—
408C2×C42C2414yes—
409C2×C42C2414yes—
410C2×C42C2412yes—
411C2×C42C2418yes—
412C2×C42C2418yes—
413C2×C42C2418yes—
414#95C2414yes—
415#95C2412yes—
416#95C2412yes—
417#95C24116yes—
418#95C24116yes—
419#95C2418yes—
420#95C2418yes—
421#95C2418yes—
422#95C2418yes—
423#95C2418yes—
424#95C2418yes—
425#95C2418yes—
426#95C2418yes—
427#95C2418yes—
428#95C2418yes—
429#95C2418yes—
430#95C24116yes—
431#95C24116yes—
432#95C24116yes—
433#95C24116yes—
434#95C24116yes—
435C24—22no—
436C24—24no—
437C24—24no—
438C24—42no—
439C24C2514yes—
440C24C2512yes—
441C24C2511yescharacteristic
442#41—432no—
443C2×Q8—28no—
444C2×Q8—24no—
445C2×Q8—28no—
446C2×Q8—24no—
447C2×Q8—28no—
448C2×Q8—28no—
449C2×Q8—28no—
450C2×Q8—28no—
451C2×Q8—28no—
452C2×Q8—24no—
453C2×Q8—28no—
454C2×Q8—28no—
455C2×Q8—416no—
456C2×Q8—48no—
457C2×Q8—4128no—
458C2×Q8—432no—
459C2×Q8—416no—
460C2×Q8—432no—
461C2×D4—28no—
462C2×D4—28no—
463C2×D4—28no—
464C2×D4—216no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—48no—
469C2×D4—48no—
470C2×D4—44no—
471C2×D4—48no—
472C22×C4—28no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—24no—
481C22×C4—28no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—28no—
486C22×C4—28no—
487C22×C4—28no—
488C22×C4—216no—
489C22×C4—48no—
490C22×C4—416no—
491C22×C4—44no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—44no—
495C22×C4—432no—
496C22×C4—416no—
497C22×C4H(5,𝔽2)18yes—
498C22×C4H(5,𝔽2)14yes—
499C22×C4C2514yes—
500C22×C4C2518yes—
501C22×C4C2514yes—
502C22×C4C2518yes—
503C22×C4C2514yes—
504C22×C4C2512yes—
505C22×C4C2518yes—
506C22×C4C2514yes—
507C22×C4C2514yes—
508C22×C4C2514yes—
509C22×C4C2514yes—
510C22×C4C2512yes—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—216no—
517C4⋊C4—216no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—416no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—416no—
532C4⋊C4—416no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—24no—
537C22⋊C4—216no—
538C22⋊C4—28no—
539C22⋊C4—216no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—416no—
550C22⋊C4—416no—
551C22⋊C42−1+418yes—
552C42—24no—
553C42—24no—
554C42—28no—
555C42—28no—
556C42—28no—
557C42—28no—
558C42—48no—
559C42—48no—
560C42—48no—
561C23—22no—
562C23—22no—
563C23—28no—
564C23—22no—
565C23—24no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—28no—
572C23—28no—
573C23—416no—
574C23—44no—
575C23—44no—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)14yes—
578C23C2×2−1+414yes—
579C23C2611yescharacteristic
580Q8—432no—
581Q8—416no—
582Q8—416no—
583Q8—416no—
584Q8—48no—
585D4—416no—
586D4—48no—
587D4—816no—
588D4—164no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—416no—
617C2×C4—432no—
618C2×C4—44no—
619C2×C4—416no—
620C2×C4—48no—
621C2×C4—416no—
622C2×C4—416no—
623C2×C4—816no—
624C2×C4—84no—
625C2×C4#560116yes—
626C2×C4C2×H(5,𝔽2)14yes—
627C2×C4C2×H(5,𝔽2)14yes—
628C2×C4C2×H(5,𝔽2)14yes—
629C2×C4C2×2−1+418yes—
630C22—22no—
631C22—22no—
632C22—28no—
633C22—24no—
634C22—28no—
635C22—416no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22C2×#56014yes—
645C22C22×H(5,𝔽2)11yescharacteristic
646C22C22×H(5,𝔽2)12yes—
647C22C22×2−1+412yes—
648C222−1+612yes—
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4—48no—
656C4—42no—
657C4—48no—
658C4—44no—
659C4—44no—
660C4C2×#56018yes—
661C2—24no—
662C2—44no—
663C2—84no—
664C2C22×#56011yescharacteristic
665C2#6283311yescharacteristic
666C2#6288711yescharacteristic
667C2#6294214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radical#35634C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #53022C2semidirect
2. #53008C2semidirect
3. #52913C2semidirect
4. #45669C2semidirect
5. #45569C2semidirect
6. #45503C2semidirect
7. #35672C2semidirect
8. #35667C2semidirect
10. #35630C2semidirect
11. #35316C2semidirect
12. #35308C2semidirect
13. #35242C2semidirect
14. #34742C2semidirect
15. C2×#2434C2semidirect
26. #2857C22semidirect
27. #2857C22semidirect
28. #2850C22semidirect
30. #2843C22semidirect
31. #2841C22semidirect
32. #2838C22semidirect
33. #2838C22semidirect
35. #2830C22semidirect
39. #2792C22semidirect
40. #2792C22semidirect
41. #2784C22semidirect
46. #2742C22semidirect
47. #2737C22semidirect
49. #2621C22semidirect
50. #2621C22semidirect
51. #2620C22semidirect
53. #2547C22semidirect
54. #2546C22semidirect
55. #2546C22semidirect
56. #2546C22semidirect
57. #2545C22semidirect
58. #2545C22semidirect
62. #2514C22semidirect
63. #2514C22semidirect
64. #2511C22semidirect
65. #2511C22semidirect
75. #2429C22semidirect
76. #2429C22semidirect
77. #2429C22semidirect
78. #2427C22semidirect

Maximal subgroups

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

subgroupindex
#530222
#530082
#529132
#456692
#455692
#455032
#356722
#356672
#356342
#356302
#353162
#353082
#352422
#347422
C2×#24342