Finite Group Databasebeta

← #6594130

Subgroups — #6594130

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36191C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33059C211yesmaximal, characteristic, complemented
5#33053C211yesmaximal, characteristic, complemented
6#29225C211yesmaximal, characteristic, complemented
7#29147C211yesmaximal, characteristic, complemented
8#27983C211yesmaximal, characteristic, complemented
9#27980C211yesmaximal, characteristic, complemented
10#27966C211yesmaximal, characteristic, complemented
11#27965C211yesmaximal, characteristic, complemented
12#27746C214yesmaximal, complemented
13#27721C214yesmaximal, complemented
14#27697C214yesmaximal, complemented
15#22497C214yesmaximal
16#21657C211yesmaximal, characteristic, complemented
17#20764C211yesmaximal, characteristic, complemented
18C2×#1836C211yesmaximal, characteristic, complemented
19#20491C211yesmaximal, characteristic
20C2×#554C2212yes—
21C2×#491C2211yescharacteristic
22C2×#435C2212yes—
23C2×C4×M16C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2851C2211yescharacteristic
30#2851C2211yescharacteristic
31#2574—22no—
32#2570C2212yes—
33#2567C2212yes—
34#2552—22no—
35#2520—22no—
36#2502C2212yes—
37#2502C2212yes—
38#2450—22no—
39#2440—22no—
40#2440—22no—
41#2440C2212yes—
42#2439C2212yes—
43#2368—22no—
44#2352C2212yes—
45D4×C4⋊C4—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2100C2214yes—
55#2099C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2024C2214yes—
59#1993C2214yes—
60#1992C2211yescharacteristic
61#1992C2211yescharacteristic
62#1992C2211yescharacteristic
63#1990C2214yes—
64#1988C2214yes—
65#1960C2211yescharacteristic
66#1960C2211yescharacteristic
67#1960C2211yescharacteristic
68#1935C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1894C2214yes—
72#1892C2214yes—
73#1873C2214yes—
74#1872C2214yes—
75#1836C2211yescharacteristic
76#1836C2211yescharacteristic
77#1836C2211yescharacteristic
78#1836C2211yescharacteristic
79#1836C2211yescharacteristic
80#1836C2211yescharacteristic
81#1836C2211yescharacteristic
82#1836C2214yes—
83#1836C2211yescharacteristic
84#1836C2211yescharacteristic
85#1834C2214yes—
86#1824C2214yes—
87#1823C2214yes—
88#1805C2214yes—
89#1771C2212yes—
90#1767C2211yescharacteristic
91#1763C2214yes—
92#1753C2212yes—
93C8×C22⋊C4C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1736C2214yes—
97C2×#340C2214yes—
98C4×#97C2211yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101#554—216no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125C2311yescharacteristic
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4D412yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117C2312yes—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#491—24no—
126#435—28no—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C4×M16—24no—
135C2×#97—24no—
136C2×#97C2312yes—
137C2×#97C2311yescharacteristic
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×#97C2312yes—
141C2×#97C2311yescharacteristic
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—24no—
148#398—24no—
149#396—22no—
150#395—24no—
151#395—22no—
152#393—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#389—24no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#386—22no—
175#385—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#385—22no—
180#384—22no—
181#384—22no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#382—22no—
191#382C2311yescharacteristic
192#382C2312yes—
193#382C2312yes—
194#382C2311yescharacteristic
195#382C2311yescharacteristic
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—22no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2311yescharacteristic
211C4×C22⋊C4C2311yescharacteristic
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2312yes—
227#336C2312yes—
228#336C2312yes—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8C2412yes—
232C22×D4C2×D411yescharacteristic
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—42no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—42no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—28no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—84no—
255#125—22no—
256#125—22no—
257#125—48no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—48no—
261C22⋊Q8—48no—
262C22⋊Q8—48no—
263C4×Q8—28no—
264C4×D4—28no—
265C4×D4—48no—
266C4×D4#4118yes—
267#117—44no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—84no—
293C2×C4⋊C4—84no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D411yescharacteristic
310C2×C22⋊C4C2×D411yescharacteristic
311C2×C22⋊C4C2×D412yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C4⋊C8—24no—
329#103—28no—
330#103—44no—
331#103—44no—
332#103—44no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335#97—24no—
336#97—22no—
337#97—22no—
338#97—24no—
339#97—22no—
340#97#4114yes—
341C4×C8—24no—
342C4×C8—42no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—82no—
371C2×Q8—84no—
372C2×Q8—82no—
373C2×D4—24no—
374C2×D4#12414yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—82no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—82no—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4114yes—
412C22×C4C2511yescharacteristic
413M16—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—48no—
426C2×C8C2×#4114yes—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4#12414yes—
458C22⋊C4—24no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4#12414yes—
465C22⋊C4#12414yes—
466C22⋊C4#13718yes—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—42no—
476C42—44no—
477C42—82no—
478C42—82no—
479C42C2×#4114yes—
480C42C2×#4112yes—
481C422−1+412yes—
482C23—22no—
483C23—21no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23C2×#12412yes—
490C23C2×#12412yes—
491C23C2×#13714yes—
492C23C23×D411yescharacteristic
493Q8—44no—
494Q8—44no—
495Q8—164no—
496Q8—164no—
497D4—28no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—164no—
541C2×C4—164no—
542C2×C4—164no—
543C2×C4C2×#12412yes—
544C2×C4C2×#12412yes—
545C2×C4C2×#12412yes—
546C2×C4C2×#12412yes—
547C2×C4#54714yes—
548C2×C4C2×#13714yes—
549C2×C4C2×#13714yes—
550C2×C4C22×#4111yescharacteristic
551C2×C4C22×#4112yes—
552C2×C4C2×2−1+411yescharacteristic
553C2×C4#58512yes—
554C8—24no—
555C8—42no—
556C8—42no—
557C8—42no—
558C22—22no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22#313914yes—
568C22C22×#12411yescharacteristic
569C22C2×#54711yescharacteristic
570C22#350011yescharacteristic
571C22C22×#13712yes—
572C22#358512yes—
573C4—22no—
574C4—22no—
575C4—24no—
576C4—24no—
577C4—24no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4#347512yes—
590C4#351512yes—
591C2—24no—
592C2—42no—
593C2#3616411yescharacteristic
594C2C2×#313911yescharacteristic
595C2#6062611yescharacteristic
596C2#6125412yes—
597C2#6258212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33293C2semidirect
4. #33059C2semidirect
5. #33053C2semidirect
6. #29225C2semidirect
7. #29147C2semidirect
8. #27983C2semidirect
9. #27980C2semidirect
10. #27966C2semidirect
11. #27965C2semidirect
12. #27746C2semidirect
13. #27721C2semidirect
14. #27697C2semidirect
16. #21657C2semidirect
17. #20764C2semidirect
18. C2×#1836C2semidirect

Maximal subgroups

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

subgroupindex
#361912
#361802
#332932
#330592
#330532
#292252
#291472
#279832
#279802
#279662
#279652
#277462
#277212
#276972
#224972
#216572
#207642
C2×#18362
#204912