Finite Group Databasebeta

← #7632430

Subgroups — #7632430

Order 512.

8412 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic, complemented
2C2×C4×2−1+4C211yesmaximal, characteristic, complemented
3#36580C218yesmaximal, complemented
4#36493C218yesmaximal, complemented
5#36041C218yesmaximal, complemented
6#35844C218yesmaximal, complemented
7#35679C212yesmaximal, complemented
8#35676C212yesmaximal, complemented
9#35356C218yesmaximal, complemented
10#35272C212yesmaximal, complemented
11#35129C214yesmaximal, complemented
12#34963C214yesmaximal, complemented
13#34952C212yesmaximal, complemented
14#34265C214yesmaximal, complemented
15#34187C211yesmaximal, characteristic, complemented
16C22×2−1+4C2211yescharacteristic
17C2×#559—216no—
18C2×#559C2218yes—
19C2×#559C2218yes—
20C2×#535C2214yes—
21C22×#127C2211yescharacteristic
22C22×#124C2212yes—
23C22×#124C2212yes—
24C22×#123C2214yes—
25C4×2−1+4—216no—
26C2×#520C2214yes—
27C2×#520C2212yes—
28C2×#520C2214yescomplemented
29C2×#519C2211yescharacteristic
30C2×#519C2214yescomplemented
31C2×C4×#41C2214yes—
32C2×C4×#41C2212yes—
33C2×C4×#41C2214yescomplemented
34C22×C4×Q8C2211yescharacteristic
35C22×C4×Q8C2211yescharacteristic
36C22×C4×Q8C2214yescomplemented
37C22×C4×D4C2212yes—
38C22×C4×D4C2212yes—
39#2856C2218yes—
40#2850C22116yescomplemented
41#2632C2218yescomplemented
42#2631C2218yescomplemented
43#2619C2218yes—
44#2610C22116yescomplemented
45#2598C22116yescomplemented
46#2592C2218yescomplemented
47#2582C2214yescomplemented
48#2581C2214yescomplemented
49#2580C2214yescomplemented
50#2580C2218yescomplemented
51#2579C2218yes—
52#2548C2214yes—
53#2548C2214yes—
54#2541C2214yes—
55#2541C2214yes—
56#2540C2218yescomplemented
57#2538C2214yescomplemented
58#2535C2218yescomplemented
59#2487C2214yes—
60#2482C2218yes—
61#2482C2218yes—
62#2471C22132yescomplemented
63#2458C22132yescomplemented
64#2456C22132yescomplemented
65#2448C22116yescomplemented
66#2448C22116yescomplemented
67#2443C22132yescomplemented
68#2436C22116yescomplemented
69#2431C22116yescomplemented
70#2409C2214yes—
71#2408C2218yes—
72#2408C2218yes—
73#2404C2218yes—
74#2404C2218yes—
75#2398C2218yescomplemented
76#2397C2218yescomplemented
77#2396C2218yescomplemented
78#2367C22116yescomplemented
79#2366C22116yescomplemented
80#2363C22116yescomplemented
81#2361C22116yescomplemented
82#2356C2218yescomplemented
83#2355C2218yescomplemented
84#2321C22116yes—
85#2313C2214yes—
86#2310C2212yes—
87#2309C2214yes—
88C4×C22⋊Q8C2218yescomplemented
89C4×#121C22116yescomplemented
90C4×#120C22116yes—
91#2297C2214yes—
92#2296C2214yes—
93#2285C2212yes—
94#2283C2214yes—
95#2274C2212yes—
96C2×2−1+4—216no—
97C2×2−1+4—28no—
98C22×#41C2314yes—
99C22×#41C2314yes—
100C22×#41C2312yes—
101C23×Q8C2314yes—
102C23×Q8C2311yescharacteristic
103C23×Q8C2311yescharacteristic
104C23×D4C2312yes—
105C23×D4C2312yes—
106C23×D4C2312yes—
107C24×C4C2314yes—
108#559—264no—
109#559—264no—
110#535—232no—
111C2×#127—28no—
112C2×#127C2314yes—
113C2×#127C2314yes—
114C2×#127C2318yes—
115C2×#124—216no—
116C2×#124—216no—
117C2×#124C2318yes—
118C2×#124C2318yes—
119C2×#124C2318yes—
120C2×#124C23116yes—
121C2×#123—232no—
122C2×#123C23132yes—
123C2×C22⋊Q8C2318yes—
124C2×#121C23116yes—
125C2×#121C23116yes—
126C2×#121C23116yes—
127C2×#120C23116yes—
128#520—232no—
129#520—216no—
130#520—232no—
131#519—28no—
132#519—232no—
133C4×#41—232no—
134C4×#41—232no—
135C4×#41—216no—
136C2×C4×Q8—232no—
137C2×C4×Q8—28no—
138C2×C4×Q8D4116yescomplemented
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2314yes—
144C2×C4×Q8C2314yes—
145C2×C4×Q8C2314yes—
146C2×C4×Q8C2314yes—
147C2×C4×Q8C2318yes—
148C2×C4×Q8C2314yes—
149C2×C4×Q8C2314yes—
150C2×C4×Q8C2314yes—
151C2×C4×D4—216no—
152C2×C4×D4—216no—
153C2×C4×D4C2318yes—
154C2×C4×D4C2314yes—
155C2×C4×D4C2312yes—
156C2×C4×D4C2318yes—
157C2×C4×D4C2318yes—
158C2×C4×D4C23116yes—
159C2×C4×D4C2314yes—
160C2×C4×D4C2314yes—
161C2×C4×D4C2314yes—
162C2×C4×D4C2314yes—
163C2×C4×D4C2312yes—
164C2×C4×D4C2318yes—
165C2×C4×D4C23132yes—
166C2×#117C2312yes—
167C2×#117C2314yes—
168C2×#117C2318yes—
169C2×#117C2314yes—
170C2×#117C2314yes—
171C2×#117C2314yes—
172C2×#117C2318yes—
173C22×C4⋊C4C2312yes—
174C22×C4⋊C4C2314yes—
175C22×C4⋊C4C2311yescharacteristic
176C22×C4⋊C4C2318yes—
177C22×C4⋊C4C2312yes—
178C22×C4⋊C4C2311yescharacteristic
179C22×C22⋊C4C2314yes—
180C22×C22⋊C4C2314yes—
181C22×C22⋊C4C2314yes—
182C22×C42C2318yes—
183C22×C42C2314yes—
184C22×C42C2312yes—
185C22×C42C2312yes—
186C22×C42C2311yescharacteristic
187#400C2318yes—
188#399C2318yes—
189#399C23116yes—
190#399C23116yes—
191#398C23116yes—
192#397C23116yes—
193#397C23116yes—
194#396C23116yes—
195#395C2318yes—
196#394C23116yes—
197#394C23116yes—
198#393C2318yes—
199#393C23116yes—
200#393C23116yes—
201#392C23116yes—
202#392C23116yes—
203#392C23116yes—
204#392C23116yes—
205#391C23116yes—
206#391C2318yes—
207#391C2318yes—
208#390C23116yes—
209#390C2318yes—
210#390C2318yes—
211#390C23132yes—
212#390C23116yes—
213#390C2318yes—
214#390C23116yes—
215#389C2318yes—
216#389C2318yes—
217#389C2318yes—
218#388C23132yes—
219#388C23132yes—
220#388C23132yes—
221#388C23132yes—
222#387C23132yes—
223#387C23116yes—
224#386C2318yes—
225#386C23132yes—
226#386C2318yes—
227#386C23132yes—
228#386C23132yes—
229#386C23132yes—
230#385C2318yes—
231#385C23116yes—
232#385C2318yes—
233#385C23132yes—
234#384C2314yes—
235#383C2318yes—
236#382C2314yes—
237#382C2318yes—
238#381C2318yes—
239#380C2314yes—
240#380C2314yes—
241#380C2318yes—
242#379C23116yes—
243C4×C4⋊C4C2318yes—
244C4×C4⋊C4C2314yes—
245C4×C4⋊C4C23116yes—
246C4×C4⋊C4C2314yes—
247C4×C4⋊C4C23116yes—
248C4×C22⋊C4C2318yes—
249C4×C22⋊C4C23116yes—
250C4×C22⋊C4C23132yes—
251#376C2318yes—
252C25C2414yes—
2532−1+4—2128no—
254C2×#41—216no—
255C2×#41—232no—
256C2×#41—216no—
257C2×#41—216no—
258C2×#41—28no—
259C2×#41—216no—
260C2×#41—28no—
261C2×#41#41116yes—
262C22×Q8—216no—
263C22×Q8—216no—
264C22×Q8—24no—
265C22×Q8—216no—
266C22×Q8—28no—
267C22×Q8—44no—
268C22×Q8C2×D418yes—
269C22×Q8C2414yes—
270C22×Q8C2412yes—
271C22×Q8C2414yes—
272C22×Q8C2412yes—
273C22×Q8C2412yes—
274C22×D4—28no—
275C22×D4—28no—
276C22×D4—28no—
277C22×D4—216no—
278C22×D4—48no—
279C22×D4—84no—
280C22×D4C2414yes—
281C22×D4C2414yes—
282C22×D4C2418yes—
283C22×D4C2414yes—
284C22×D4C2412yes—
285C22×D4C2414yes—
286C22×D4C24132yes—
287C22×D4C2418yes—
288C23×C4—216no—
289C23×C4—88no—
290C23×C4C2418yes—
291C23×C4C2414yes—
292C23×C4C2412yes—
293C23×C4C2411yescharacteristic
294C23×C4C2414yes—
295C23×C4C2412yes—
296C23×C4C2412yes—
297C23×C4C2418yes—
298C23×C4C2414yes—
299C23×C4C24116yes—
300C23×C4C2412yes—
301C23×C4C2411yescharacteristic
302C23×C4C2411yescharacteristic
303#127—232no—
304#127—216no—
305#127—216no—
306#124—264no—
307#124—232no—
308#124—232no—
309#124—232no—
310#123—2128no—
311C22⋊Q8—416no—
312#121—264no—
313#121—432no—
314#121—816no—
315#120—264no—
316C4×Q8—216no—
317C4×Q8—216no—
318C4×Q8—216no—
319C4×Q8—232no—
320C4×Q8—216no—
321C4×Q8—28no—
322C4×Q8—216no—
323C4×Q8—216no—
324C4×Q8—216no—
325C4×Q8—28no—
326C4×Q8C2×D4132yes—
327C4×Q8C2×D4132yes—
328C4×D4—216no—
329C4×D4—232no—
330C4×D4—216no—
331C4×D4—264no—
332C4×D4—216no—
333C4×D4—232no—
334C4×D4—232no—
335C4×D4—232no—
336C4×D4—216no—
337C4×D4—28no—
338C4×D4—216no—
339C4×D4—28no—
340C4×D4—832no—
341#117—216no—
342#117—28no—
343#117—216no—
344#117—216no—
345#117—232no—
346#117—232no—
347#117—216no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—216no—
351C2×C4⋊C4—232no—
352C2×C4⋊C4C2×D418yes—
353C2×C4⋊C4C2×D4116yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C24116yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C24116yes—
369C2×C4⋊C4C2414yes—
370C2×C4⋊C4C2418yes—
371C2×C22⋊C4—216no—
372C2×C22⋊C4—216no—
373C2×C22⋊C4—216no—
374C2×C22⋊C4C24116yes—
375C2×C22⋊C4C24116yes—
376C2×C22⋊C4C24116yes—
377C2×C22⋊C4C2414yes—
378C2×C22⋊C4C2418yes—
379C2×C22⋊C4C2414yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C2418yes—
382C2×C22⋊C4C24132yes—
383C2×C22⋊C4C24116yes—
384C2×C22⋊C4C24132yes—
385C2×C22⋊C4C24132yes—
386C2×C22⋊C4C24116yes—
387C2×C42—28no—
388C2×C42—232no—
389C2×C42—216no—
390C2×C42C2×D4116yes—
391C2×C42C2×D418yes—
392C2×C42C2418yes—
393C2×C42C2414yes—
394C2×C42C2414yes—
395C2×C42C2418yes—
396C2×C42C2418yes—
397C2×C42C2418yes—
398C2×C42C2414yes—
399C2×C42C24116yes—
400C2×C42C2414yes—
401C2×C42C2412yes—
402C2×C42C2414yes—
403C2×C42C2412yes—
404#95C2418yes—
405#95C24116yes—
406#95C24116yes—
407#95C24116yes—
408#95C24116yes—
409#95C24116yes—
410#95C2418yes—
411#95C24116yes—
412#95C24116yes—
413C24—28no—
414C24—216no—
415C24—48no—
416C24—48no—
417C24C2518yes—
418C24C2514yes—
419C24C2511yescharacteristic
420#41—2128no—
421#41—2128no—
422#41—264no—
423C2×Q8—2128no—
424C2×Q8—232no—
425C2×Q8—28no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—216no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—24no—
433C2×Q8—28no—
434C2×Q8—44no—
435C2×Q8—416no—
436C2×Q8C22×D4116yes—
437C2×Q8C2×#4118yes—
438C2×D4—28no—
439C2×D4—216no—
440C2×D4—28no—
441C2×D4—264no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—216no—
446C2×D4—28no—
447C2×D4—216no—
448C2×D4—24no—
449C2×D4—216no—
450C2×D4—216no—
451C2×D4—216no—
452C2×D4—48no—
453C2×D4—432no—
454C2×D4—432no—
455C2×D4—432no—
456C2×D4—416no—
457C2×D4—432no—
458C2×D4—816no—
459C2×D4C2×#41116yes—
460C2×D4C2×#4118yes—
461C22×C4—216no—
462C22×C4—28no—
463C22×C4—216no—
464C22×C4—28no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—216no—
472C22×C4—216no—
473C22×C4—28no—
474C22×C4—232no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—416no—
480C22×C4—432no—
481C22×C4—48no—
482C22×C4—832no—
483C22×C4C22×D414yes—
484C22×C4C22×D418yes—
485C22×C4C22×D418yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C2×#41116yes—
489C22×C4C2×#4118yes—
490C22×C4C25116yes—
491C22×C4C2514yes—
492C22×C4C2514yes—
493C22×C4C2514yes—
494C22×C4C2514yes—
495C22×C4C2511yescharacteristic
496C22×C4C2514yes—
497C22×C4C2514yes—
498C22×C4C2512yes—
499C22×C4C2514yes—
500C22×C4C2512yes—
501C22×C4C2511yescharacteristic
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—216no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—416no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4C22×D4116yes—
518C4⋊C4C22×D4132yes—
519C22⋊C4—28no—
520C22⋊C4—216no—
521C22⋊C4—232no—
522C22⋊C4—232no—
523C22⋊C4—232no—
524C22⋊C4—216no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—264no—
528C22⋊C4—416no—
529C22⋊C4—432no—
530C22⋊C4—432no—
531C22⋊C4—416no—
532C42—28no—
533C42—216no—
534C42—28no—
535C42—216no—
536C42—28no—
537C42—216no—
538C42—216no—
539C42—416no—
540C42C22×D4116yes—
541C42C22×D418yes—
542C42C22×D4116yes—
543C42C22×D418yes—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—216no—
552C23—216no—
553C23—216no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—832no—
558C23C23×D412yes—
559C23C22×#4118yes—
560C23C22×#4114yes—
561C23C2611yescharacteristic
562Q8—232no—
563Q8—232no—
564Q8—216no—
565Q8—48no—
566Q8—48no—
567D4—232no—
568D4—232no—
569D4—216no—
570D4—416no—
571D4—432no—
572D4—416no—
573D4—816no—
574D4—864no—
575C2×C4—264no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—216no—
583C2×C4—232no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—216no—
590C2×C4—28no—
591C2×C4—216no—
592C2×C4—28no—
593C2×C4—216no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—416no—
601C2×C4—816no—
602C2×C4—88no—
603C2×C4—832no—
604C2×C4#525116yes—
605C2×C4C23×D418yes—
606C2×C4C23×D418yes—
607C2×C4C23×D414yes—
608C2×C4C23×D412yes—
609C2×C4C23×D414yes—
610C2×C4C23×D412yes—
611C2×C4C22×#4118yes—
612C2×C4C22×#4114yes—
613C2×C4C22×#4114yes—
614C2×C4#58514yes—
615C22—28no—
616C22—22no—
617C22—22no—
618C22—216no—
619C22—28no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—432no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—832no—
628C22C2×#52514yes—
629C22C24×D411yescharacteristic
630C22C23×#4112yes—
631C22C2×#58512yes—
632C22H(7,𝔽2)12yes—
633C4—22no—
634C4—28no—
635C4—24no—
636C4—28no—
637C4—28no—
638C4—416no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4C2×#52518yes—
644C4#348714yes—
645C2—24no—
646C2—48no—
647C2—88no—
648C2C22×#52511yescharacteristic
649C2C2×#348711yescharacteristic
650C2#6285211yescharacteristic
651C2#6301014yes—
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. C22×#559C2semidirect
2. C2×C4×2−1+4C2semidirect
3. #36580C2semidirect
4. #36493C2semidirect
5. #36041C2semidirect
6. #35844C2semidirect
7. #35679C2semidirect
8. #35676C2semidirect
9. #35356C2semidirect
10. #35272C2semidirect
11. #35129C2semidirect
12. #34963C2semidirect
13. #34952C2semidirect
14. #34265C2semidirect
15. #34187C2semidirect
28. C2×#520C22semidirect
30. C2×#519C22semidirect
33. C2×C4×#41C22semidirect
36. C22×C4×Q8C22semidirect
40. #2850C22semidirect
41. #2632C22semidirect
42. #2631C22semidirect
44. #2610C22semidirect
45. #2598C22semidirect
46. #2592C22semidirect
47. #2582C22semidirect
48. #2581C22semidirect
49. #2580C22semidirect
50. #2580C22semidirect
56. #2540C22semidirect
57. #2538C22semidirect
58. #2535C22semidirect
62. #2471C22semidirect
63. #2458C22semidirect
64. #2456C22semidirect
65. #2448C22semidirect
66. #2448C22semidirect
67. #2443C22semidirect
68. #2436C22semidirect
69. #2431C22semidirect
75. #2398C22semidirect
76. #2397C22semidirect
77. #2396C22semidirect
78. #2367C22semidirect
79. #2366C22semidirect
80. #2363C22semidirect
81. #2361C22semidirect
82. #2356C22semidirect
83. #2355C22semidirect
88. C4×C22⋊Q8C22semidirect
89. C4×#121C22semidirect
138. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5592
C2×C4×2−1+42
#365802
#364932
#360412
#358442
#356792
#356762
#353562
#352722
#351292
#349632
#349522
#342652
#341872