Finite Group Databasebeta

← #7641993

Subgroups — #7641993

Order 512.

8550 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3444C211yesmaximal, characteristic, complemented
2C22×#532C211yesmaximal, characteristic, complemented
3#37616C218yesmaximal, complemented
4#37544C218yesmaximal, complemented
5#37448C218yesmaximal, complemented
6#35327C214yesmaximal, complemented
7#35320C211yesmaximal, characteristic, complemented
8#35317C211yesmaximal, characteristic, complemented
9#35298C214yesmaximal, complemented
10#35293C214yesmaximal, complemented
11#35285C211yesmaximal, characteristic, complemented
12#35274C211yesmaximal, characteristic, complemented
13#35233C218yesmaximal, complemented
14#35190C218yesmaximal, complemented
15#35149C214yesmaximal, complemented
16#34012C211yesmaximal, characteristic, complemented
17C23×#41C2212yes—
18C2×#536C2214yes—
19C2×#535C2214yes—
20#3444—216no—
21C2×#533C2211yescharacteristic
22C2×#533C2212yescomplemented
23C2×#532—216no—
24C2×#532C2214yes—
25C2×#532C2214yes—
26C2×#532C2211yescharacteristic
27C2×#532C2214yes—
28C2×#532C2214yes—
29C2×#532C2212yescomplemented
30C22×C4⋊Q8C2211yescharacteristic
31C22×#127C2211yescharacteristic
32C2×#529C2214yes—
33C2×#529C2214yescomplemented
34C2×#529C2214yescomplemented
35C22×#124C2212yes—
36C2×#525C2214yes—
37C2×#525C2212yes—
38C22×#121C2214yes—
39C2×C4×#41C2212yes—
40C2×C4×#41C2214yescomplemented
41C22×C4×D4C2214yes—
42C23×C42C2211yescharacteristic
43#2868C2212yescomplemented
44#2867C2212yescomplemented
45#2862C2214yescomplemented
46#2858C2212yescomplemented
47#2858C2214yescomplemented
48#2858C2212yescomplemented
49#2850C22116yescomplemented
50#2849C22116yes—
51#2813C22132yescomplemented
52#2658C2214yescomplemented
53#2657C2214yescomplemented
54#2642C2214yescomplemented
55#2641C2214yescomplemented
56#2636C22132yescomplemented
57#2635C22132yescomplemented
58#2620C22116yescomplemented
59#2620C22116yescomplemented
60#2619C22116yescomplemented
61#2613C22116yes—
62#2598C22132yescomplemented
63#2568C2218yescomplemented
64#2566C22116yescomplemented
65#2548C2218yescomplemented
66#2545C22116yescomplemented
67#2544C22116yescomplemented
68#2544C22116yescomplemented
69#2542C2218yescomplemented
70#2541C2218yescomplemented
71#2534C22132yes—
72#2503C2218yes—
73#2501C2218yes—
74#2500C22116yescomplemented
75#2435C2218yes—
76#2433C2218yes—
77#2432C22116yescomplemented
78#2431C22116yescomplemented
79#2431C22116yescomplemented
80#2407C22132yescomplemented
81#2328C22116yescomplemented
82#2326C22116yescomplemented
83#2310C2214yes—
84#2309C2214yes—
85#2285C2212yes—
86#2285C2214yescomplemented
87#2283C2214yes—
88#2283C2214yescomplemented
89#2283C2214yescomplemented
90#2280C2214yes—
91#2280C2212yes—
92C4×#117C2211yescharacteristic
93C4×#117C2211yescharacteristic
94C4×#117C2214yescomplemented
95C22×#41—216no—
96C22×#41—416no—
97C22×#41C2312yes—
98C22×#41C2318yes—
99C22×#41C2318yes—
100C22×#41C2314yes—
101C23×Q8C2312yes—
102C23×D4C2312yes—
103C23×D4C2314yes—
104C23×D4C2314yes—
105C24×C4C2311yescharacteristic
106C24×C4C2314yes—
107C24×C4C2312yes—
108#536—416no—
109#535—416no—
110#533—28no—
111#533—216no—
112#532—232no—
113#532—232no—
114#532—232no—
115#532—232no—
116#532—28no—
117#532—216no—
118C2×C4⋊Q8—28no—
119C2×C4⋊Q8C2314yes—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2318yes—
125C2×#127—28no—
126C2×#127C2312yes—
127C2×#127C2311yescharacteristic
128C2×#127C2312yes—
129C2×#127C2314yes—
130C2×#127C23116yes—
131C2×#127C2318yes—
132#529—232no—
133#529—232no—
134#529—232no—
135C2×#126C2318yes—
136C2×#125C2314yes—
137C2×#125C2312yes—
138C2×#124—216no—
139C2×#124C2314yes—
140C2×#124C2314yes—
141C2×#124C2312yes—
142C2×#124C2318yes—
143C2×#124C2314yes—
144C2×#124C23116yes—
145C2×#124C2318yes—
146C2×#124C2318yes—
147#525—216no—
148#525—232no—
149C2×#123C2314yes—
150C2×#123C2318yes—
151C2×#123C23116yes—
152C2×C22⋊Q8C2314yes—
153C2×C22⋊Q8C2314yes—
154C2×C22⋊Q8C2314yes—
155C2×C22⋊Q8C23116yes—
156C2×#121—232no—
157C2×#121C2314yes—
158C2×#121C2318yes—
159C2×#121C2318yes—
160C2×#121C2318yes—
161C2×#121C2318yes—
162C2×#121C2314yes—
163C2×#121C2314yes—
164C2×#121C23116yes—
165C2×#120C2314yes—
166C2×#120C23116yes—
167C4×#41—216no—
168C4×#41—232no—
169C2×C4×Q8C2314yes—
170C2×C4×Q8C2314yes—
171C2×C4×Q8C2314yes—
172C2×C4×D4—232no—
173C2×C4×D4C2318yes—
174C2×C4×D4C2318yes—
175C2×C4×D4C2314yes—
176C2×C4×D4C2318yes—
177C2×C4×D4C2318yes—
178C2×C4×D4C2318yes—
179C2×C4×D4C2318yes—
180C2×C4×D4C2314yes—
181C2×C4×D4C2318yes—
182C2×C4×D4C2314yes—
183C2×#117C2312yes—
184C2×#117C2312yes—
185C2×#117C2312yes—
186C2×#117C2318yes—
187C2×#117C2314yes—
188C2×#117C2314yes—
189C2×#117C2318yes—
190C22×C4⋊C4C2314yes—
191C22×C22⋊C4C2318yes—
192C22×C42D4116yescomplemented
193C22×C42C2314yes—
194C22×C42C2312yes—
195C22×C42C2311yescharacteristic
196C22×C42C2312yes—
197C22×C42C2312yes—
198C22×C42C2312yes—
199C22×C42C2311yescharacteristic
200C22×C42C2312yes—
201#400C23132yes—
202#399C23132yes—
203#396C23164yes—
204#394C23132yes—
205#394C23164yes—
206#394C23164yes—
207#392C23132yes—
208#392C23164yes—
209#391C23116yes—
210#391C23116yes—
211#391C23116yes—
212#390C23116yes—
213#390C23116yes—
214#390C23116yes—
215#390C23116yes—
216#390C23116yes—
217#388C23164yes—
218#386C23164yes—
219#386C23164yes—
220#385C23164yes—
221#384C23116yes—
222#384C2314yes—
223#383C2318yes—
224#382C23116yes—
225#382C2314yes—
226#382C2318yes—
227#381C23116yes—
228#381C2314yes—
229#381C2318yes—
230#380C2318yes—
231#380C2318yes—
232#380C2318yes—
233#379C23116yes—
234#379C2314yes—
235C4×C4⋊C4C2318yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2314yes—
238C4×C4⋊C4C2318yes—
239C4×C22⋊C4C2318yes—
240C4×C22⋊C4C2318yes—
241C4×C22⋊C4C2314yes—
242C4×C22⋊C4C2314yes—
243#376C2318yes—
244#376C2312yes—
245#376C2312yes—
246C43C2312yes—
247C43C2312yes—
248C25C2414yes—
249C25C2411yescharacteristic
250C2×#41—216no—
251C2×#41—28no—
252C2×#41—232no—
253C2×#41—232no—
254C2×#41—216no—
255C2×#41—432no—
256C2×#41—432no—
257C2×#41—416no—
258C2×#41—4128no—
259C22×Q8—28no—
260C22×Q8—216no—
261C22×Q8C2412yes—
262C22×Q8C2414yes—
263C22×Q8C2414yes—
264C22×Q8C2414yes—
265C22×D4—216no—
266C22×D4—216no—
267C22×D4—28no—
268C22×D4—216no—
269C22×D4—416no—
270C22×D4—416no—
271C22×D4C2414yes—
272C22×D4C2412yes—
273C22×D4C2418yes—
274C22×D4C2414yes—
275C22×D4C2414yes—
276C22×D4C2414yes—
277C22×D4C2418yes—
278C22×D4C2418yes—
279C22×D4C24116yes—
280C22×D4C24132yes—
281C23×C4—28no—
282C23×C4—216no—
283C23×C4—48no—
284C23×C4—416no—
285C23×C4C2×D418yes—
286C23×C4C2×D4116yes—
287C23×C4C2414yes—
288C23×C4C2414yes—
289C23×C4C2412yes—
290C23×C4C2412yes—
291C23×C4C2412yes—
292C23×C4C2412yes—
293C23×C4C2412yes—
294C23×C4C2411yescharacteristic
295C23×C4C2418yes—
296C23×C4C24116yes—
297C23×C4C2412yes—
298C23×C4C2412yes—
299C23×C4C2412yes—
300C23×C4C2411yescharacteristic
301C4⋊Q8—216no—
302C4⋊Q8—28no—
303C4⋊Q8—28no—
304C4⋊Q8—28no—
305C4⋊Q8—24no—
306C4⋊Q8—416no—
307#127—216no—
308#127—28no—
309#127—28no—
310#127—24no—
311#127—432no—
312#127—416no—
313#126—232no—
314#125—216no—
315#125—28no—
316#124—216no—
317#124—216no—
318#124—232no—
319#124—216no—
320#124—28no—
321#124—432no—
322#124—416no—
323#124—416no—
324#123—216no—
325#123—232no—
326#123—432no—
327C22⋊Q8—216no—
328C22⋊Q8—216no—
329C22⋊Q8—216no—
330C22⋊Q8—432no—
331#121—232no—
332#121—232no—
333#121—232no—
334#121—232no—
335#121—216no—
336#121—216no—
337#121—216no—
338#121—432no—
339#120—216no—
340#120—432no—
341C4×Q8—216no—
342C4×Q8—216no—
343C4×Q8—216no—
344C4×D4—232no—
345C4×D4—232no—
346C4×D4—232no—
347C4×D4—232no—
348C4×D4—232no—
349C4×D4—232no—
350C4×D4—232no—
351C4×D4—216no—
352C4×D4—216no—
353C4×D4—216no—
354#117—216no—
355#117—28no—
356#117—28no—
357#117—28no—
358#117—232no—
359#117—216no—
360#117—416no—
361C2×C4⋊C4—216no—
362C2×C4⋊C4C2418yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2414yes—
366C2×C4⋊C4C2414yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C2418yes—
369C2×C4⋊C4C2418yes—
370C2×C4⋊C4C2414yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C24116yes—
373C2×C4⋊C4C24132yes—
374C2×C22⋊C4—232no—
375C2×C22⋊C4C24116yes—
376C2×C22⋊C4C24116yes—
377C2×C22⋊C4C2414yes—
378C2×C22⋊C4C2418yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C2418yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C2414yes—
384C2×C22⋊C4C24116yes—
385C2×C22⋊C4C24132yes—
386C2×C22⋊C4C24132yes—
387C2×C22⋊C4C24164yes—
388C2×C42—216no—
389C2×C42C2×D4116yes—
390C2×C42C2×D418yes—
391C2×C42C2×D4116yes—
392C2×C42C2×D4116yes—
393C2×C42C2×D4116yes—
394C2×C42C2×D418yes—
395C2×C42C2×D4116yes—
396C2×C42C2418yes—
397C2×C42C2414yes—
398C2×C42C2414yes—
399C2×C42C2414yes—
400C2×C42C2418yes—
401C2×C42C2418yes—
402C2×C42C2412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2412yes—
405C2×C42C2412yes—
406C2×C42C2412yes—
407C2×C42C2412yes—
408C2×C42C2412yes—
409C2×C42C2411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#95C24116yes—
413#95C24116yes—
414#95C2418yes—
415#95C2418yes—
416#95C24164yes—
417C24—24no—
418C24—28no—
419C24—216no—
420C24—44no—
421C24—416no—
422C24C22×D414yes—
423C24C2518yes—
424C24C2514yes—
425C24C2512yes—
426C24C2511yescharacteristic
427#41—464no—
428#41—4128no—
429#41—4128no—
430#41—432no—
431C2×Q8—24no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8—28no—
435C2×Q8—216no—
436C2×Q8—28no—
437C2×Q8—216no—
438C2×Q8—216no—
439C2×Q8—432no—
440C2×D4—216no—
441C2×D4—28no—
442C2×D4—232no—
443C2×D4—216no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—216no—
447C2×D4—216no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—216no—
451C2×D4—28no—
452C2×D4—216no—
453C2×D4—232no—
454C2×D4—216no—
455C2×D4—416no—
456C2×D4—432no—
457C2×D4—416no—
458C2×D4—464no—
459C2×D4—432no—
460C2×D4—464no—
461C2×D4—464no—
462C2×D4—432no—
463C22×C4—24no—
464C22×C4—232no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—216no—
472C22×C4—216no—
473C22×C4—28no—
474C22×C4—232no—
475C22×C4—216no—
476C22×C4—432no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—432no—
481C22×C4—416no—
482C22×C4—464no—
483C22×C4C22×D418yes—
484C22×C4C22×D418yes—
485C22×C4C22×D418yes—
486C22×C4C22×D418yes—
487C22×C4C22×D418yes—
488C22×C4C22×D414yes—
489C22×C4C22×D418yes—
490C22×C4C22×D418yes—
491C22×C4C22×D418yes—
492C22×C4C22×D414yes—
493C22×C4H(5,𝔽2)18yes—
494C22×C4C2518yes—
495C22×C4C2518yes—
496C22×C4C2518yes—
497C22×C4C2518yes—
498C22×C4C2514yes—
499C22×C4C2512yes—
500C22×C4C2512yes—
501C22×C4C2512yes—
502C22×C4C2512yes—
503C22×C4C2511yescharacteristic
504C22×C4C2511yescharacteristic
505C22×C4C2512yes—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—416no—
517C4⋊C4—432no—
518C22⋊C4—216no—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—232no—
523C22⋊C4—232no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—416no—
528C22⋊C4—432no—
529C22⋊C4—432no—
530C22⋊C4—464no—
531C42—216no—
532C42—28no—
533C42—28no—
534C42—28no—
535C42—216no—
536C42—216no—
537C42C22×D414yes—
538C42C22×D418yes—
539C42C22×D414yes—
540C42C22×D418yes—
541C42C22×D418yes—
542C42C22×D418yes—
543C42C22×D418yes—
544C42C22×D418yes—
545C42C22×D414yes—
546C42C22×D414yes—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—28no—
555C23—216no—
556C23—232no—
557C23—216no—
558C23—48no—
559C23—416no—
560C23—432no—
561C23—416no—
562C23—832no—
563C23C23×D414yes—
564C23C23×D412yes—
565C23C2×H(5,𝔽2)14yes—
566C23C2611yescharacteristic
567Q8—416no—
568Q8—416no—
569Q8—48no—
570Q8—416no—
571D4—432no—
572D4—464no—
573D4—432no—
574D4—416no—
575D4—416no—
576D4—416no—
577D4—48no—
578D4—432no—
579D4—416no—
580D4—1632no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—216no—
587C2×C4—216no—
588C2×C4—216no—
589C2×C4—216no—
590C2×C4—216no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—216no—
598C2×C4—216no—
599C2×C4—216no—
600C2×C4—216no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—464no—
605C2×C4—432no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4#535116yes—
609C2×C4C23×D414yes—
610C2×C4C23×D414yes—
611C2×C4C23×D414yes—
612C2×C4C23×D414yes—
613C2×C4C23×D412yes—
614C2×C4C23×D412yes—
615C2×C4C23×D414yes—
616C2×C4C2×H(5,𝔽2)14yes—
617C2×C4C2×H(5,𝔽2)14yes—
618C2×C4#58518yes—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—416no—
625C22—48no—
626C22—416no—
627C22—432no—
628C22—48no—
629C22—44no—
630C22—48no—
631C22—416no—
632C22—832no—
633C22C2×#53514yes—
634C22C24×D411yescharacteristic
635C22C22×H(5,𝔽2)12yes—
636C22C2×#58514yes—
637C4—24no—
638C4—24no—
639C4—24no—
640C4—24no—
641C4—48no—
642C4—48no—
643C4—48no—
644C4—44no—
645C4—48no—
646C4C2×#53514yes—
647C4C2×#53514yes—
648C4#344918yes—
649C2—24no—
650C2—44no—
651C2—84no—
652C2—88no—
653C2C22×#53511yescharacteristic
654C2C2×#344912yes—
655C2#6283614yes—
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. C2×#3444C2semidirect
2. C22×#532C2semidirect
3. #37616C2semidirect
4. #37544C2semidirect
5. #37448C2semidirect
6. #35327C2semidirect
7. #35320C2semidirect
8. #35317C2semidirect
9. #35298C2semidirect
10. #35293C2semidirect
11. #35285C2semidirect
12. #35274C2semidirect
13. #35233C2semidirect
14. #35190C2semidirect
15. #35149C2semidirect
16. #34012C2semidirect
22. C2×#533C22semidirect
29. C2×#532C22semidirect
33. C2×#529C22semidirect
34. C2×#529C22semidirect
40. C2×C4×#41C22semidirect
43. #2868C22semidirect
44. #2867C22semidirect
45. #2862C22semidirect
46. #2858C22semidirect
47. #2858C22semidirect
48. #2858C22semidirect
49. #2850C22semidirect
51. #2813C22semidirect
52. #2658C22semidirect
53. #2657C22semidirect
54. #2642C22semidirect
55. #2641C22semidirect
56. #2636C22semidirect
57. #2635C22semidirect
58. #2620C22semidirect
59. #2620C22semidirect
60. #2619C22semidirect
62. #2598C22semidirect
63. #2568C22semidirect
64. #2566C22semidirect
65. #2548C22semidirect
66. #2545C22semidirect
67. #2544C22semidirect
68. #2544C22semidirect
69. #2542C22semidirect
70. #2541C22semidirect
74. #2500C22semidirect
77. #2432C22semidirect
78. #2431C22semidirect
79. #2431C22semidirect
80. #2407C22semidirect
81. #2328C22semidirect
82. #2326C22semidirect
86. #2285C22semidirect
88. #2283C22semidirect
89. #2283C22semidirect
94. C4×#117C22semidirect
192. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34442
C22×#5322
#376162
#375442
#374482
#353272
#353202
#353172
#352982
#352932
#352852
#352742
#352332
#351902
#351492
#340122