Finite Group Databasebeta

← #7674571

Subgroups — #7674571

Order 512.

5454 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#43265C218yesmaximal, complemented
2#43198C218yesmaximal, complemented
3#38942C218yesmaximal, complemented
4#38936C218yesmaximal, complemented
5#38288C218yesmaximal, complemented
6#38286C218yesmaximal, complemented
7#35649C211yesmaximal, characteristic, complemented
8#35640C211yesmaximal, characteristic, complemented
9#35528C212yesmaximal, complemented
10#35264C212yesmaximal, complemented
11#35256C212yesmaximal, complemented
12C2×#2641C211yesmaximal, characteristic, complemented
13#34269C212yesmaximal, complemented
14#34255C212yesmaximal, complemented
15#34056C212yesmaximal, complemented
16C2×#544C2211yescharacteristic
17C2×#537C2211yescharacteristic
18C2×#535C2212yes—
19C2×#535C2212yes—
20C22×#127C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#518C2212yes—
23C2×C4×#41C2212yes—
24#2861C2214yescomplemented
25#2858C2214yescomplemented
26#2856C2214yescomplemented
27#2850C2214yescomplemented
28#2803C22116yescomplemented
29#2795C2218yescomplemented
30#2785C2218yescomplemented
31#2781C22116yescomplemented
32#2756C22116yescomplemented
33#2756C22116yescomplemented
34#2733C2218yescomplemented
35#2699C2218yescomplemented
36#2699C22116yescomplemented
37#2686C22116yescomplemented
38#2641C2218yes—
39#2641C2218yes—
40#2641C2214yescomplemented
41#2640C2214yescomplemented
42#2620C2214yescomplemented
43#2619C2214yescomplemented
44#2609C2218yescomplemented
45#2591C2214yes—
46#2583C2214yes—
47#2572C2218yescomplemented
48#2555C2218yescomplemented
49#2552C2218yescomplemented
50#2550C2218yescomplemented
51#2544C2218yes—
52#2541C2218yes—
53#2537C2218yescomplemented
54#2518C22116yescomplemented
55#2504C2214yes—
56#2502C2214yes—
57#2500C22116yescomplemented
58#2491C2218yescomplemented
59#2490C2218yescomplemented
60#2490C2218yescomplemented
61#2489C2218yescomplemented
62#2476C22116yescomplemented
63#2472C22116yes—
64#2459C22116yescomplemented
65#2428C2218yes—
66#2428C2218yes—
67#2428C2218yescomplemented
68#2426C2218yescomplemented
69#2424C22116yes—
70#2420C22116yescomplemented
71#2414C22116yescomplemented
72#2410C2218yes—
73#2410C2218yes—
74C2×#394C2214yes—
75C2×#394C2214yes—
76#2367C2218yescomplemented
77#2366C2218yescomplemented
78#2358C22116yes—
79#2357C2218yescomplemented
80#2354C2218yescomplemented
81#2342C22116yes—
82#2324C2218yescomplemented
83#2320C2214yescomplemented
84#2320C2214yescomplemented
85#2319C2214yescomplemented
86#2319C2214yescomplemented
87#2319C2218yescomplemented
88#2318C2218yes—
89#2318C2218yes—
90#2317C2212yes—
91#2311C2212yes—
92#2309C2212yes—
93C4×#121C2218yescomplemented
94#2298C2218yescomplemented
95#2297C2212yes—
96#2297C2212yes—
97#2296C2212yes—
98#2296C2212yes—
99#2293C2218yescomplemented
100C2×#390C2212yes—
101C2×#390C2212yes—
102#2283C2218yescomplemented
103#2274C2211yescharacteristic
104#2274C2211yescharacteristic
105C4×#117C2212yes—
106C2×#376C2211yescharacteristic
107C22×#41C2312yes—
108C23×D4C2312yes—
109C23×D4C2312yes—
110C23×D4C2312yes—
111#544—44no—
112#537—44no—
113#535—216no—
114#535—48no—
115C2×C4⋊Q8C2314yes—
116C2×#127—44no—
117C2×#127C2314yes—
118C2×#127C2314yes—
119C2×#127C2318yes—
120C2×#125—44no—
121C2×#125C2314yes—
122C2×#125C2318yes—
123C2×#124C2314yes—
124C2×#124C2314yes—
125C2×#124C23116yes—
126C2×C22⋊Q8C2314yes—
127C2×C22⋊Q8C2314yes—
128C2×#121C2318yes—
129C2×#121C2314yes—
130C2×#121C2314yes—
131C2×#121C2314yes—
132C2×#121C2314yes—
133C2×#121C23116yes—
134C2×#121C23116yes—
135C2×#121C23116yes—
136C2×#120C2318yes—
137C2×#120C2318yes—
138#518—216no—
139C4×#41—216no—
140C2×C4×Q8C2314yes—
141C2×C4×D4C2318yes—
142C2×C4×D4C2314yes—
143C2×C4×D4C2318yes—
144C2×C4×D4C2318yes—
145C2×C4×D4C23116yes—
146C2×#117C2312yes—
147C2×#117C2312yes—
148C2×#117C2314yes—
149C2×#117C23116yes—
150C2×#117C2312yes—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2312yes—
154C22×C4⋊C4C2312yes—
155C22×C22⋊C4C2318yes—
156C22×C22⋊C4C2318yes—
157C22×C22⋊C4C2318yes—
158C22×C42C2314yes—
159C22×C42C2312yes—
160C22×C42C2311yescharacteristic
161#400C2318yes—
162#398C2318yes—
163#398C23116yes—
164#397C23116yes—
165#397C23116yes—
166#396C23116yes—
167#395C2318yes—
168#394C2318yes—
169#394C2318yes—
170#394C2318yes—
171#394C23116yes—
172#394C23116yes—
173#394C23116yes—
174#394C2318yes—
175#394C23116yes—
176#394C2318yes—
177#394C23116yes—
178#394C23116yes—
179#394C23116yes—
180#394C23116yes—
181#393C23116yes—
182#392C23116yes—
183#392C23116yes—
184#392C23116yes—
185#391C2314yes—
186#391C2318yes—
187#391C2312yes—
188#391C2312yes—
189#391C2314yes—
190#390C2318yes—
191#390C2314yes—
192#390C2314yes—
193#390C2314yes—
194#390C2314yes—
195#390C2312yes—
196#390C2312yes—
197#390C2314yes—
198#390C2314yes—
199#390C23116yes—
200#390C23116yes—
201#390C23116yes—
202#390C2318yes—
203#390C2314yes—
204#390C2318yes—
205#390C2314yes—
206#390C2318yes—
207#389C2318yes—
208#389C2314yes—
209#389C2314yes—
210#389C2314yes—
211#389C2314yes—
212#389C23116yes—
213#388C23116yes—
214#388C2318yes—
215#388C2318yes—
216#388C2318yes—
217#388C2318yes—
218#388C23116yes—
219#388C23116yes—
220#388C23116yes—
221#388C23116yes—
222#388C23116yes—
223#388C23116yes—
224#388C23116yes—
225#388C23116yes—
226#388C23116yes—
227#387C2318yes—
228#387C2318yes—
229#387C2318yes—
230#387C2318yes—
231#386C2318yes—
232#386C23116yes—
233#386C2314yes—
234#386C2314yes—
235#386C2318yes—
236#386C23116yes—
237#386C23116yes—
238#386C23116yes—
239#386C23116yes—
240#385C2318yes—
241#385C2314yes—
242#385C2318yes—
243#385C2314yes—
244#385C23116yes—
245#384C2314yes—
246#384C2318yes—
247#383C2318yes—
248#383C2318yes—
249#382C2314yes—
250#382C2318yes—
251#382C2318yes—
252#382C2318yes—
253#381C23116yes—
254#381C2318yes—
255#381C2318yes—
256#380C2318yes—
257#380C2318yes—
258#380C2318yes—
259#380C23116yes—
260#379C2318yes—
261C4×C4⋊C4C2314yes—
262C4×C4⋊C4C2314yes—
263C4×C4⋊C4C2318yes—
264C4×C4⋊C4C2318yes—
265C4×C4⋊C4C2318yes—
266C4×C4⋊C4C2318yes—
267C4×C22⋊C4C23116yes—
268C4×C22⋊C4C2318yes—
269C4×C22⋊C4C2314yes—
270C4×C22⋊C4C2314yes—
271C4×C22⋊C4C23116yes—
272C4×C22⋊C4C2318yes—
273C4×C22⋊C4C2318yes—
274#376C2314yes—
275#376C2314yes—
276#376C2314yes—
277C2×#95C2312yes—
278C2×#95C2312yes—
279C43C2314yes—
280C25C2414yes—
281C2×#41—28no—
282C2×#41—28no—
283C2×#41—28no—
284C22×Q8C2412yes—
285C22×Q8C2414yes—
286C22×D4—28no—
287C22×D4—28no—
288C22×D4—28no—
289C22×D4—44no—
290C22×D4—44no—
291C22×D4—84no—
292C22×D4—84no—
293C22×D4C2414yes—
294C22×D4C2414yes—
295C22×D4C2414yes—
296C22×D4C2412yes—
297C22×D4C24116yes—
298C22×D4C2418yes—
299C22×D4C2418yes—
300C22×D4C2414yes—
301C22×D4C2418yes—
302C22×D4C2418yes—
303C23×C4C2414yes—
304C23×C4C2412yes—
305C23×C4C2412yes—
306C23×C4C2414yes—
307C23×C4C24116yes—
308C23×C4C2411yescharacteristic
309C23×C4C2411yescharacteristic
310C23×C4C2418yes—
311C23×C4C2411yescharacteristic
312C4⋊Q8—48no—
313#127—216no—
314#127—416no—
315#127—48no—
316#125—416no—
317#125—48no—
318#124—216no—
319#124—48no—
320#124—816no—
321C22⋊Q8—48no—
322C22⋊Q8—48no—
323#121—232no—
324#121—48no—
325#121—48no—
326#121—48no—
327#121—48no—
328#121—816no—
329#121—816no—
330#121—816no—
331#120—232no—
332#120—416no—
333C4×Q8—216no—
334C4×D4—216no—
335C4×D4—232no—
336C4×D4—232no—
337C4×D4—232no—
338C4×D4—816no—
339#117—28no—
340#117—28no—
341#117—216no—
342#117—816no—
343#117#41116yes—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2418yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2418yes—
356C2×C4⋊C4C2418yes—
357C2×C4⋊C4C24116yes—
358C2×C4⋊C4C2418yes—
359C2×C4⋊C4C2418yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C2414yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2418yes—
364C2×C4⋊C4C2414yes—
365C2×C22⋊C4—232no—
366C2×C22⋊C4—88no—
367C2×C22⋊C4—88no—
368C2×C22⋊C4C2414yes—
369C2×C22⋊C4C24116yes—
370C2×C22⋊C4C2418yes—
371C2×C22⋊C4C2414yes—
372C2×C22⋊C4C24116yes—
373C2×C22⋊C4C24116yes—
374C2×C22⋊C4C24116yes—
375C2×C22⋊C4C24116yes—
376C2×C22⋊C4C24116yes—
377C2×C22⋊C4C24116yes—
378C2×C22⋊C4C24116yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C2418yes—
382C2×C22⋊C4C2418yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C2418yes—
385C2×C22⋊C4C24116yes—
386C2×C22⋊C4C2418yes—
387C2×C22⋊C4C2414yes—
388C2×C42—28no—
389C2×C42—24no—
390C2×C42—216no—
391C2×C42C2412yes—
392C2×C42C2414yes—
393C2×C42C2414yes—
394C2×C42C2414yes—
395C2×C42C2418yes—
396C2×C42C2414yes—
397C2×C42C2412yes—
398C2×C42C2418yes—
399C2×C42C2418yes—
400C2×C42C2418yes—
401C2×C42C2418yes—
402C2×C42C2414yes—
403#95C2412yes—
404#95C2414yes—
405#95C2412yes—
406#95C2414yes—
407#95C2412yes—
408#95C2412yes—
409#95C2418yes—
410#95C2418yes—
411#95C2418yes—
412#95C2418yes—
413#95C2418yes—
414#95C2418yes—
415#95C24116yes—
416#95C24116yes—
417#95C2418yes—
418#95C2418yes—
419#95C2418yes—
420#95C2418yes—
421#95C2418yes—
422#95C2418yes—
423C24—28no—
424C24—28no—
425C24—28no—
426C24—48no—
427C24—48no—
428C24C2518yes—
429C24C2514yes—
430C24C2511yescharacteristic
431#41—432no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×D4—28no—
438C2×D4—28no—
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—232no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—28no—
450C2×D4—416no—
451C2×D4—416no—
452C2×D4—416no—
453C2×D4—416no—
454C2×D4—48no—
455C2×D4—48no—
456C2×D4—48no—
457C2×D4—48no—
458C2×D4—44no—
459C2×D4—48no—
460C2×D4—416no—
461C2×D4—416no—
462C2×D4—432no—
463C2×D4—816no—
464C2×D4—816no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—216no—
474C22×C4—232no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—416no—
482C22×C4—48no—
483C22×C4—416no—
484C22×C4—48no—
485C22×C4C2×#4118yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4C2514yes—
488C22×C4C2514yes—
489C22×C4C2512yes—
490C22×C4C2512yes—
491C22×C4C2518yes—
492C22×C4C2514yes—
493C22×C4C2514yes—
494C22×C4C2514yes—
495C22×C4C2514yes—
496C22×C4C2514yes—
497C22×C4C2518yes—
498C22×C4C2512yes—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4C2×#41116yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—232no—
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—416no—
534C22⋊C4—48no—
535C22⋊C4C2×#41116yes—
536C42—24no—
537C42—24no—
538C42—28no—
539C42—216no—
540C42—28no—
541C42—28no—
542C42—28no—
543C42—48no—
544C42—48no—
545C42—48no—
546C42—48no—
547C42C2×#41116yes—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—216no—
558C23—216no—
559C23—28no—
560C23—416no—
561C23—88no—
562C23—88no—
563C23—816no—
564C23—816no—
565C23C22×#4114yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2611yescharacteristic
568Q8—48no—
569Q8—164no—
570D4—416no—
571D4—416no—
572D4—416no—
573D4—48no—
574D4—832no—
575D4—816no—
576D4—816no—
577D4—164no—
578D4—168no—
579D4—168no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—416no—
609C2×C4—832no—
610C2×C4—816no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#555116yes—
615C2×C4C22×#41116yes—
616C2×C4C22×#4114yes—
617C2×C4C22×#4114yes—
618C2×C4C2×H(5,𝔽2)14yes—
619C22—28no—
620C22—22no—
621C22—22no—
622C22—48no—
623C22—416no—
624C22—48no—
625C22—416no—
626C22—48no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—1616no—
633C22C2×#55514yes—
634C22C23×#4112yes—
635C22C22×H(5,𝔽2)11yescharacteristic
636C22H(7,𝔽2)12yes—
637C22H(7,𝔽2)12yes—
638C4—216no—
639C4—44no—
640C4—48no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4C2×#55518yes—
650C2—24no—
651C2—84no—
652C2—88no—
653C2C22×#55511yescharacteristic
654C2#6285814yes—
655C2#6287611yescharacteristic
656C2#6287611yescharacteristic
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. #43265C2semidirect
2. #43198C2semidirect
3. #38942C2semidirect
4. #38936C2semidirect
5. #38288C2semidirect
6. #38286C2semidirect
7. #35649C2semidirect
8. #35640C2semidirect
9. #35528C2semidirect
10. #35264C2semidirect
11. #35256C2semidirect
12. C2×#2641C2semidirect
13. #34269C2semidirect
14. #34255C2semidirect
15. #34056C2semidirect
24. #2861C22semidirect
25. #2858C22semidirect
26. #2856C22semidirect
27. #2850C22semidirect
28. #2803C22semidirect
29. #2795C22semidirect
30. #2785C22semidirect
31. #2781C22semidirect
32. #2756C22semidirect
33. #2756C22semidirect
34. #2733C22semidirect
35. #2699C22semidirect
36. #2699C22semidirect
37. #2686C22semidirect
40. #2641C22semidirect
41. #2640C22semidirect
42. #2620C22semidirect
43. #2619C22semidirect
44. #2609C22semidirect
47. #2572C22semidirect
48. #2555C22semidirect
49. #2552C22semidirect
50. #2550C22semidirect
53. #2537C22semidirect
54. #2518C22semidirect
57. #2500C22semidirect
58. #2491C22semidirect
59. #2490C22semidirect
60. #2490C22semidirect
61. #2489C22semidirect
62. #2476C22semidirect
64. #2459C22semidirect
67. #2428C22semidirect
68. #2426C22semidirect
70. #2420C22semidirect
71. #2414C22semidirect
76. #2367C22semidirect
77. #2366C22semidirect
79. #2357C22semidirect
80. #2354C22semidirect
82. #2324C22semidirect
83. #2320C22semidirect
84. #2320C22semidirect
85. #2319C22semidirect
86. #2319C22semidirect
87. #2319C22semidirect
93. C4×#121C22semidirect
94. #2298C22semidirect
99. #2293C22semidirect
102. #2283C22semidirect

Maximal subgroups

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

subgroupindex
#432652
#431982
#389422
#389362
#382882
#382862
#356492
#356402
#355282
#352642
#352562
C2×#26412
#342692
#342552
#340562