Finite Group Databasebeta

← #6971068

Subgroups — #6971068

Order 512.

1713 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic, complemented
3#32772C212yesmaximal, complemented
4#31939C214yesmaximal, complemented
5#30998C214yesmaximal, complemented
6#30215C211yesmaximal, characteristic, complemented
7#29953C212yesmaximal, complemented
8#29668C212yesmaximal, complemented
9#29142C211yesmaximal, characteristic, complemented
10#28373C211yesmaximal, characteristic, complemented
11#24832C212yesmaximal, complemented
12#24743C212yesmaximal, complemented
13#24255C212yesmaximal, complemented
14#24132C212yesmaximal
15#23934C212yesmaximal, complemented
16#23807C211yesmaximal, characteristic, complemented
17#22010C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#533C2212yes—
21C2×#477C2212yes—
22C2×#464C2214yes—
23C2×#462C2212yes—
24C2×#451C2214yes—
25C2×#441C2212yes—
26C2×#440C2212yes—
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2857C2212yes—
30#2850C2212yes—
31#2568C2211yescharacteristic
32#2547C2211yescharacteristic
33#2536C2212yes—
34#2503C2212yes—
35#2492—22no—
36#2489—22no—
37#2488—22no—
38#2488—22no—
39#2453—22no—
40#2436C2212yes—
41#2421C2214yes—
42#2415—22no—
43#2362—22no—
44#2351C2212yes—
45#2339—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C2×#391C2212yes—
49C42×D4C2211yescharacteristic
50#2097C2212yes—
51#2095C2211yescharacteristic
52#2095C2212yes—
53#2094C2211yescharacteristic
54#2085C2214yes—
55#2074C2214yes—
56#2068C2214yes—
57#2068C2214yes—
58#2042C2214yes—
59#2035C2214yes—
60#2033C2214yes—
61#2024C2212yes—
62#2024C2212yes—
63#2024C2212yes—
64#2023C2212yes—
65#2016C2211yescharacteristic
66#2016C2212yes—
67#2015C2212yes—
68#2013C2211yescharacteristic
69#2001C2214yes—
70#2000C2214yes—
71#1993C2212yes—
72#1969C2214yes—
73#1960C2212yes—
74#1958C2212yes—
75#1955C2212yes—
76#1940C2212yes—
77#1923C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1872C2214yes—
81#1852C2214yes—
82#1835C2212yes—
83#1817C2212yes—
84#1805C2212yes—
85#1805C2214yes—
86#1804C2212yes—
87#1797C2212yes—
88#1787C2214yes—
89C4×C4⋊C8C2211yescharacteristic
90#1745C2211yescharacteristic
91C22×#41—21no—
92C23×Q8C2312yes—
93C22×Q16C2312yes—
94C22×Q16C2314yes—
95C22×SD16C2312yes—
96D4×Q8—44no—
97#547—44no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#124C2311yescharacteristic
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2314yes—
113C2×C22⋊Q8C2312yes—
114C2×#121C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2314yes—
119C2×C4×D4—21no—
120C2×C4×D4D412yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126#477—44no—
127#464—48no—
128#462—44no—
129#451—48no—
130#441—44no—
131#440—44no—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151#400—21no—
152#400—21no—
153#400—24no—
154#400—22no—
155#400—22no—
156#398—22no—
157#397—24no—
158#397—21no—
159#397—21no—
160#394—22no—
161#393—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2311yescharacteristic
176#390C2311yescharacteristic
177#389—24no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#388—24no—
182#388—24no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#382C2312yes—
195#382C2312yes—
196#382C2312yes—
197#382C2311yescharacteristic
198#381—22no—
199#381—22no—
200#380—22no—
201#380—22no—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—22no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C2×#95—22no—
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2311yescharacteristic
219#336C2312yes—
220#336C2311yescharacteristic
221#336C2314yes—
222#336C2312yes—
223C2×#41—44no—
224C2×#41—48no—
225C22×Q8—22no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—24no—
229C22×Q8—44no—
230C22×Q8—42no—
231C22×Q8—44no—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×Q8C2411yescharacteristic
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—28no—
247C2×Q16—42no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×SD16—24no—
251C2×SD16—42no—
252C2×SD16—42no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#125—48no—
262#124—22no—
263#124—41no—
264#124—44no—
265#123—48no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273#121—44no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284#117—48no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—44no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—42no—
331#107—44no—
332#106—42no—
333C4⋊C8—24no—
334C4⋊C8—44no—
335#103—42no—
336#103—42no—
337#103—42no—
338#103—44no—
339#103—44no—
340#103—44no—
341#103—44no—
342#103—44no—
343#102—42no—
344#102—44no—
345C22⋊C8—44no—
346C22⋊C8—44no—
347#97—24no—
348C4×C8—42no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95C2×D411yescharacteristic
368#95C2×D411yescharacteristic
369#95C2×D412yes—
370#95C2×D412yes—
371C24—21no—
372C24C22×D412yes—
373#41—816no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—88no—
395C2×Q8—88no—
396C2×Q8H(5,𝔽2)14yes—
397C2×Q82−1+412yes—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C2511yescharacteristic
441Q16—48no—
442Q16—88no—
443SD16—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C4⋊C4—28no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—48no—
496C42—42no—
497C42—44no—
498C42H(5,𝔽2)14yes—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23#57514yes—
511C23C23×D411yescharacteristic
512Q8—24no—
513Q8—24no—
514Q8—24no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—44no—
519Q8—44no—
520Q8—88no—
521Q8—84no—
522Q8—88no—
523D4—28no—
524D4—42no—
525D4—42no—
526D4—88no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4#53512yes—
572C2×C4#53512yes—
573C2×C4#57814yes—
574C2×C4C2×H(5,𝔽2)12yes—
575C2×C4C2×H(5,𝔽2)12yes—
576C2×C4C2×2−1+411yescharacteristic
577C2×C4#58511yescharacteristic
578C8—42no—
579C8—42no—
580C8—44no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22—81no—
593C22C2×#53511yescharacteristic
594C22#348411yescharacteristic
595C22#348911yescharacteristic
596C22C2×#57512yes—
597C22C2×#57812yes—
598C4—22no—
599C4—22no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4—81no—
611C4—84no—
612C4—82no—
613C4#353612yes—
614C4#354212yes—
615C2—44no—
616C2—161no—
617C2#4247811yescharacteristic
618C2#6079911yescharacteristic
619C2#6082111yescharacteristic
620C2#6160312yes—
621C2#6213012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36151C2semidirect
3. #32772C2semidirect
4. #31939C2semidirect
5. #30998C2semidirect
6. #30215C2semidirect
7. #29953C2semidirect
8. #29668C2semidirect
9. #29142C2semidirect
10. #28373C2semidirect
11. #24832C2semidirect
12. #24743C2semidirect
13. #24255C2semidirect
15. #23934C2semidirect
16. #23807C2semidirect
17. #22010C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#361512
#327722
#319392
#309982
#302152
#299532
#296682
#291422
#283732
#248322
#247432
#242552
#241322
#239342
#238072
#220102