Finite Group Databasebeta

← #7712140

Subgroups — #7712140

Order 512.

4708 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59679C214yesmaximal, complemented
2#59445C214yesmaximal, complemented
3#46159C218yesmaximal, complemented
4#39816C218yesmaximal, complemented
5#39513C212yesmaximal, complemented
6#39506C212yesmaximal, complemented
7#39474C218yesmaximal, complemented
8#39385C212yesmaximal, complemented
9#39379C212yesmaximal, complemented
10#37640C212yesmaximal
11#37393C214yesmaximal, complemented
12#36645C211yesmaximal, characteristic, complemented
13#36640C211yesmaximal, characteristic, complemented
14#36477C214yesmaximal, complemented
15#36454C214yesmaximal, complemented
16#35309C214yesmaximal, complemented
17#34374C212yesmaximal, complemented
18C4×#532C211yesmaximal, characteristic
19C2×#541C2214yes—
20C2×#533C2212yes—
21C2×#532C2211yescharacteristic
22C2×C4×#41C2212yes—
23#2868C2211yescharacteristic
24#2868C2211yescharacteristic
25#2864C2214yes—
26#2862C2212yes—
27#2859C2211yescharacteristic
28#2859C2212yes—
29#2859C2211yescharacteristic
30#2859C2214yes—
31#2857C2212yes—
32#2857C2214yes—
33#2857C2212yes—
34#2855C2218yes—
35#2851C2212yes—
36#2851C2214yes—
37#2851C2212yes—
38#2820C2218yes—
39#2800C22116yes—
40#2793C22116yes—
41#2783C22116yes—
42#2781C22116yes—
43#2779C22116yes—
44#2774C22116yes—
45#2725C22116yes—
46#2721C22116yes—
47#2629C2214yes—
48#2624C2218yes—
49#2621C2214yes—
50#2620C2218yes—
51#2616C2218yes—
52#2608C22116yes—
53#2587C22116yes—
54#2575C22116yes—
55#2570C2218yes—
56#2567C2214yes—
57#2566C2218yes—
58#2552C2218yes—
59#2548C2214yes—
60#2547C2214yes—
61#2545C2218yes—
62#2542C2214yes—
63#2538C2214yes—
64#2538C2214yes—
65#2517C22116yes—
66#2513C2218yes—
67#2512C2218yes—
68#2512C2218yes—
69#2511C2218yes—
70#2508C2218yes—
71#2506C2218yes—
72#2497C2218yes—
73#2489C2218yes—
74#2468C22116yes—
75#2466C22116yes—
76#2462C22116yes—
77#2460C22116yes—
78#2447C2218yes—
79#2446C2218yes—
80#2429C2218yes—
81#2421C2218yes—
82#2382C2218yes—
83#2370C2218yes—
84#2367C2218yes—
85#2365C2214yes—
86#2365C2214yes—
87#2357C2218yes—
88#2355C2214yes—
89#2355C2218yes—
90#2355C2214yes—
91Q8×C4⋊C4C2214yes—
92Q8×C4⋊C4C2214yes—
93#2323C2218yes—
94#2323C2218yes—
95#2314C2218yes—
96#2314C2214yes—
97C4×C4⋊Q8C2214yes—
98C4×C4⋊Q8C2212yes—
99C4×C4⋊Q8C2212yes—
100C4×C4⋊Q8C2212yes—
101C4×#127C2212yes—
102C4×#125C2214yes—
103C4×#125C2212yes—
104C4×#125C2212yes—
105C4×#124C2214yes—
106C4×C22⋊Q8C2218yes—
107C4×#121C2214yes—
108C4×#121C2214yes—
109#2297C2212yes—
110#2296C2214yes—
111#2293C2212yes—
112#2293C2212yes—
113#2285C2212yes—
114#2285C2214yes—
115#2285C2211yescharacteristic
116C42×D4C2212yes—
117C42×D4C2212yes—
118C2×C43C2211yescharacteristic
119C22×#41C2312yes—
120#541—416no—
121#533—216no—
122#532—28no—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2312yes—
126C2×C4⋊Q8C2312yes—
127C2×C4⋊Q8C2312yes—
128C2×C4⋊Q8C2314yes—
129C2×#127C2311yescharacteristic
130C2×#126C23116yes—
131C2×#125C2314yes—
132C2×#125C2312yes—
133C2×#125C2314yes—
134C2×#125C2312yes—
135C2×#125C23116yes—
136C2×#124C2312yes—
137C2×#124C2318yes—
138C2×C22⋊Q8C2314yes—
139C2×C22⋊Q8C2314yes—
140C2×C22⋊Q8C2318yes—
141C2×C22⋊Q8C2318yes—
142C2×#121C2312yes—
143C2×#121C2312yes—
144C4×#41—48no—
145C2×C4×Q8C2314yes—
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2314yes—
148C2×C4×Q8C2318yes—
149C2×C4×Q8C2318yes—
150C2×C4×D4C2314yes—
151C2×C4×D4C2312yes—
152C2×C4×D4C2312yes—
153C2×C4×D4C2314yes—
154C2×C4×D4C2314yes—
155C2×C4×D4C2318yes—
156C2×#117C2314yes—
157C2×#117C2314yes—
158C2×#117C2312yes—
159C2×#117C2314yes—
160C22×C42C2311yescharacteristic
161C22×C42C2312yes—
162C22×C42C2311yescharacteristic
163C22×C42C2314yes—
164C22×C42C2312yes—
165C22×C42C2311yescharacteristic
166#400C23116yes—
167#400C23116yes—
168#400C23116yes—
169#399C23116yes—
170#398C23116yes—
171#398C23116yes—
172#398C23116yes—
173#397C23116yes—
174#397C23116yes—
175#397C23116yes—
176#397C23116yes—
177#396C23116yes—
178#396C23116yes—
179#396C23116yes—
180#396C23116yes—
181#396C23116yes—
182#395C23116yes—
183#394C23116yes—
184#394C23116yes—
185#394C23116yes—
186#391C2314yes—
187#391C2314yes—
188#391C2314yes—
189#391C2314yes—
190#391C2314yes—
191#391C2318yes—
192#391C2318yes—
193#391C2318yes—
194#390C2314yes—
195#390C2314yes—
196#390C2314yes—
197#390C2318yes—
198#390C2318yes—
199#390C2318yes—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2314yes—
204#389C2318yes—
205#389C2318yes—
206#389C2318yes—
207#389C2318yes—
208#389C2314yes—
209#389C2314yes—
210#389C2318yes—
211#389C2318yes—
212#389C23116yes—
213#389C2318yes—
214#389C2318yes—
215#389C23116yes—
216#389C2314yes—
217#389C2314yes—
218#389C2314yes—
219#389C2314yes—
220#389C23116yes—
221#388C23116yes—
222#388C23116yes—
223#388C23116yes—
224#388C23116yes—
225#388C23116yes—
226#388C23116yes—
227#388C23116yes—
228#388C2318yes—
229#388C2318yes—
230#387C2318yes—
231#387C2318yes—
232#387C2318yes—
233#387C2318yes—
234#387C23116yes—
235#387C23116yes—
236#387C23116yes—
237#387C23116yes—
238#387C23116yes—
239#387C23116yes—
240#387C23116yes—
241#387C23116yes—
242#387C23116yes—
243#387C23116yes—
244#386C2318yes—
245#385C23116yes—
246#385C23116yes—
247#385C2318yes—
248#385C2318yes—
249#384C2312yes—
250#384C2312yes—
251#384C2312yes—
252#384C2314yes—
253#384C2318yes—
254#384C2318yes—
255#384C2312yes—
256#384C2312yes—
257#384C2314yes—
258#384C2314yes—
259#382C2312yes—
260#382C2314yes—
261#382C2312yes—
262#382C2314yes—
263#382C2318yes—
264#382C2314yes—
265#382C2314yes—
266#382C23116yes—
267#382C2312yes—
268#381C23116yes—
269#380C2312yes—
270#380C2314yes—
271#380C2318yes—
272#380C2312yes—
273#380C2314yes—
274#380C2318yes—
275#380C2318yes—
276#380C2318yes—
277C4×C4⋊C4C2314yes—
278C4×C4⋊C4C2318yes—
279C4×C4⋊C4C2314yes—
280C4×C4⋊C4C2318yes—
281C4×C4⋊C4C2318yes—
282C4×C4⋊C4C2318yes—
283C4×C4⋊C4C2314yes—
284C4×C4⋊C4C2314yes—
285C4×C4⋊C4C2318yes—
286C4×C4⋊C4C2318yes—
287C4×C4⋊C4C2318yes—
288C4×C4⋊C4C23116yes—
289C4×C4⋊C4C2318yes—
290C4×C4⋊C4C2312yes—
291C4×C4⋊C4C2312yes—
292C4×C22⋊C4C2318yes—
293C4×C22⋊C4C2318yes—
294C4×C22⋊C4C23116yes—
295C4×C22⋊C4C2314yes—
296C4×C22⋊C4C2314yes—
297C43C2312yes—
298C43C2311yescharacteristic
299C43C2311yescharacteristic
300C43C2311yescharacteristic
301C43C2311yescharacteristic
302C43C2312yes—
303C2×#41—28no—
304C2×#41—44no—
305C2×#41—44no—
306C22×Q8C2414yes—
307C22×Q8C2414yes—
308C22×Q8C2412yes—
309C22×D4C2412yes—
310C22×D4C2412yes—
311C22×D4C2414yes—
312C22×D4C2412yes—
313C23×C4C2414yes—
314C23×C4C2412yes—
315C23×C4C2411yescharacteristic
316C23×C4C2411yescharacteristic
317C23×C4C2411yescharacteristic
318C23×C4C2418yes—
319C23×C4C2412yes—
320C4⋊Q8—24no—
321C4⋊Q8—28no—
322C4⋊Q8—28no—
323C4⋊Q8—28no—
324C4⋊Q8—48no—
325C4⋊Q8—44no—
326#127—24no—
327#126—432no—
328#125—216no—
329#125—28no—
330#125—216no—
331#125—44no—
332#125—432no—
333#124—28no—
334#124—416no—
335C22⋊Q8—216no—
336C22⋊Q8—216no—
337C22⋊Q8—416no—
338C22⋊Q8—416no—
339#121—28no—
340#121—28no—
341C4×Q8—216no—
342C4×Q8—216no—
343C4×Q8—48no—
344C4×Q8—416no—
345C4×Q8—416no—
346C4×D4—28no—
347C4×D4—28no—
348C4×D4—216no—
349C4×D4—48no—
350C4×D4—48no—
351C4×D4—416no—
352#117—216no—
353#117—216no—
354#117—48no—
355#117—44no—
356C2×C4⋊C4C2418yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2418yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2414yes—
373C2×C4⋊C4C2418yes—
374C2×C4⋊C4C2418yes—
375C2×C4⋊C4C2414yes—
376C2×C4⋊C4C2414yes—
377C2×C4⋊C4C2414yes—
378C2×C4⋊C4C24116yes—
379C2×C4⋊C4C24116yes—
380C2×C4⋊C4C24116yes—
381C2×C4⋊C4C24116yes—
382C2×C4⋊C4C2418yes—
383C2×C4⋊C4C2418yes—
384C2×C4⋊C4C2418yes—
385C2×C4⋊C4C2418yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C2418yes—
388C2×C22⋊C4C2414yes—
389C2×C22⋊C4C2414yes—
390C2×C22⋊C4C2418yes—
391C2×C22⋊C4C2414yes—
392C2×C22⋊C4C2418yes—
393C2×C22⋊C4C2418yes—
394C2×C22⋊C4C24116yes—
395C2×C22⋊C4C24116yes—
396C2×C22⋊C4C24116yes—
397C2×C22⋊C4C24116yes—
398C2×C42—24no—
399C2×C42—28no—
400C2×C42—48no—
401C2×C42—44no—
402C2×C42—42no—
403C2×C42#4118yes—
404C2×C42C2414yes—
405C2×C42C2418yes—
406C2×C42C2418yes—
407C2×C42C2412yes—
408C2×C42C2412yes—
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2412yes—
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416C2×C42C2418yes—
417C2×C42C24116yes—
418C2×C42C24116yes—
419C2×C42C2412yes—
420C2×C42C2412yes—
421C2×C42C2412yes—
422C2×C42C2411yescharacteristic
423C2×C42C2411yescharacteristic
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427C2×C42C2412yes—
428#95C2418yes—
429#95C2414yes—
430#95C2414yes—
431#95C2418yes—
432#95C2418yes—
433#95C2418yes—
434#95C2418yes—
435#95C2418yes—
436#95C2418yes—
437#95C2418yes—
438#95C2418yes—
439#95C24116yes—
440#95C24116yes—
441#95C24116yes—
442#95C24116yes—
443#95C24116yes—
444#95C2418yes—
445#95C2418yes—
446#95C2418yes—
447#95C24116yes—
448C24C2514yes—
449C24C2511yescharacteristic
450#41—432no—
451C2×Q8—28no—
452C2×Q8—28no—
453C2×Q8—24no—
454C2×Q8—28no—
455C2×Q8—28no—
456C2×Q8—28no—
457C2×Q8—28no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×D4—28no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—28no—
465C2×D4—28no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—44no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—24no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—28no—
480C22×C4—216no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2512yes—
490C22×C4C2518yes—
491C22×C4C2518yes—
492C22×C4C2518yes—
493C22×C4C2514yes—
494C22×C4C2518yes—
495C22×C4C2518yes—
496C22×C4C2511yescharacteristic
497C22×C4C2511yescharacteristic
498C22×C4C2511yescharacteristic
499C22×C4C2512yes—
500C22×C4C2512yes—
501C22×C4C2512yes—
502C22×C4C2511yescharacteristic
503C22×C4C2511yescharacteristic
504C22×C4C2511yescharacteristic
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—216no—
516C4⋊C4—216no—
517C4⋊C4—28no—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—28no—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—416no—
528C4⋊C4—416no—
529C4⋊C4—416no—
530C4⋊C4—416no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C22⋊C4—216no—
536C22⋊C4—28no—
537C22⋊C4—28no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—416no—
544C22⋊C4—416no—
545C22⋊C4—416no—
546C22⋊C4—416no—
547C42—216no—
548C42—22no—
549C42—24no—
550C42—28no—
551C42—24no—
552C42—24no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—24no—
558C42—216no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—48no—
563C42—416no—
564C42—416no—
565C42C2×#4114yes—
566C42C2×#4114yes—
567C42C2×#4114yes—
568C42C2×#4114yes—
569C42H(5,𝔽2)14yes—
570C422−1+418yes—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—21no—
575C23—24no—
576C23—24no—
577C23—28no—
578C23—28no—
579C23—24no—
580C23C22×#4112yes—
581C23C2611yescharacteristic
582Q8—48no—
583Q8—416no—
584Q8—416no—
585D4—48no—
586D4—416no—
587D4—48no—
588D4—48no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—416no—
626C2×C4—816no—
627C2×C4C22×#4112yes—
628C2×C4C22×#4112yes—
629C2×C4C22×#4112yes—
630C2×C4C22×#4112yes—
631C2×C4C22×#4112yes—
632C2×C4C22×#4112yes—
633C2×C4C2×H(5,𝔽2)14yes—
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4C2×2−1+414yes—
636C2×C4C2×2−1+414yes—
637C2×C4C2×2−1+414yes—
638C2×C4#58512yes—
639C2×C4#58514yes—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—42no—
644C22—42no—
645C22—48no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22C23×#4111yescharacteristic
651C22C22×H(5,𝔽2)11yescharacteristic
652C22C22×2−1+412yes—
653C22C2×#58511yescharacteristic
654C22C2×#58512yes—
655C4—24no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—24no—
660C4—24no—
661C4—48no—
662C4—42no—
663C4—44no—
664C4—48no—
665C4—48no—
666C4—48no—
667C4—48no—
668C4#350614yes—
669C4#350914yes—
670C4#351214yes—
671C2—42no—
672C2—84no—
673C2#6276112yes—
674C2C2×#350611yescharacteristic
675C2C2×#350911yescharacteristic
676C2C2×#351211yescharacteristic
677C2#6283512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#532C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59679C2semidirect
2. #59445C2semidirect
3. #46159C2semidirect
4. #39816C2semidirect
5. #39513C2semidirect
6. #39506C2semidirect
7. #39474C2semidirect
8. #39385C2semidirect
9. #39379C2semidirect
11. #37393C2semidirect
12. #36645C2semidirect
13. #36640C2semidirect
14. #36477C2semidirect
15. #36454C2semidirect
16. #35309C2semidirect
17. #34374C2semidirect

Maximal subgroups

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

subgroupindex
#596792
#594452
#461592
#398162
#395132
#395062
#394742
#393852
#393792
#376402
#373932
#366452
#366402
#364772
#364542
#353092
#343742
C4×#5322