Finite Group Databasebeta

← #6624356

Subgroups — #6624356

Order 512.

1565 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#33146C212yesmaximal
4#32957C211yesmaximal, characteristic
5#29111C212yesmaximal
6#28124C212yesmaximal
7#27960C211yesmaximal, characteristic
8#27647C211yesmaximal, characteristic
9#27441C214yesmaximal, complemented
10#27377C214yesmaximal, complemented
11#26022C212yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#23069C212yesmaximal, complemented
14#22763C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21579C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#483C2212yes—
20C2×#477C2212yes—
21C2×#451C2211yescharacteristic
22C2×C4×Q16C2212yes—
23#2857C2211yescharacteristic
24#2857C2212yes—
25#2586—22no—
26#2567C2211yescharacteristic
27#2551C2211yescharacteristic
28#2549—22no—
29#2507—21no—
30#2507C2212yes—
31#2451C2212yes—
32#2440—21no—
33#2369—21no—
34#2369C2211yescharacteristic
35#2355—22no—
36#2355—21no—
37#2355—21no—
38#2355C2212yes—
39#2355C2214yes—
40#2355C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42#2345—21no—
43C4×C22⋊Q8C2212yes—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#391C2212yes—
46C2×#387—22no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yes—
53#2084C2212yes—
54#2072C2212yes—
55#2072C2214yes—
56#2066C2214yes—
57#2056C2214yes—
58#2024C2214yes—
59#1990C2212yes—
60#1988C2214yes—
61#1948C2212yes—
62#1947C2214yes—
63#1939C2212yes—
64#1935C2214yes—
65#1935C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1919C2214yes—
69#1918C2214yes—
70#1918C2214yes—
71#1917C2214yes—
72#1878C2212yes—
73#1878C2214yes—
74#1872C2211yescharacteristic
75#1863C2214yes—
76#1863C2212yes—
77#1852C2212yes—
78#1852C2211yescharacteristic
79#1834C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C4×Q8—28no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42—22no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#483—28no—
122#477—28no—
123#451—24no—
124C4×Q16—28no—
125C2×#107C2314yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2311yescharacteristic
139C2×C22⋊C8C2311yescharacteristic
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#398—22no—
143#398—22no—
144#398—22no—
145#395—22no—
146#395—22no—
147#395—22no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2314yes—
156#391C2311yescharacteristic
157#391C2314yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—21no—
174#387—22no—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#385—22no—
180#385—21no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382—22no—
189#382—24no—
190#382—22no—
191#380—21no—
192#380—22no—
193#380C2312yes—
194#380C2311yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C2×Q16—24no—
251C2×Q16—24no—
252C2×Q16—42no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C22⋊Q8—22no—
260C22⋊Q8—84no—
261C22⋊Q8—82no—
262C22⋊Q8#4114yes—
263C4×Q8—28no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—44no—
270C4×Q8#4118yes—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—84no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337#107—24no—
338#107—28no—
339#106—28no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—24no—
343#103—28no—
344#103—28no—
345#103—24no—
346#103—24no—
347#103—44no—
348#103—42no—
349C22⋊C8—22no—
350C22⋊C8—24no—
351C22⋊C8—22no—
352#97—44no—
353C4×C8—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—81no—
395C2×Q8—82no—
396C2×Q8—81no—
397C2×Q8C2×#4112yes—
398C2×Q8C2×#4114yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4#13714yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4112yes—
441C22×C42−1+412yes—
442C22×C4C2511yescharacteristic
443Q16—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4#13714yes—
489C4⋊C4C2×#4112yes—
490C4⋊C4C2×#4114yes—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C42—28no—
495C42—28no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—82no—
506C42—84no—
507C42—84no—
508C42—82no—
509C42C2×#4114yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#54812yes—
515C23C2×#13712yes—
516C23#57612yes—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—24no—
521Q8—48no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—164no—
526Q8—162no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4#54812yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13714yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4112yes—
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58512yes—
594C8—44no—
595C8—42no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22#337514yes—
604C22C2×#52511yescharacteristic
605C22C2×#54811yescharacteristic
606C22#348511yescharacteristic
607C22C22×#13711yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57611yescharacteristic
610C22#358611yescharacteristic
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—41no—
622C4—44no—
623C4—82no—
624C4—81no—
625C4—84no—
626C4—81no—
627C4—82no—
628C4#311314yes—
629C4#347512yes—
630C4#349512yes—
631C2—24no—
632C2#3651611yescharacteristic
633C2C2×#311311yescharacteristic
634C2#6058411yescharacteristic
635C2C2×#337511yescharacteristic
636C2#6132211yescharacteristic
637C2#6172311yescharacteristic
638C2#6172611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #27441C2semidirect
10. #27377C2semidirect
11. #26022C2semidirect
12. #24682C2semidirect
13. #23069C2semidirect
14. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#23552
#331462
#329572
#291112
#281242
#279602
#276472
#274412
#273772
#260222
#246822
#230692
#227632
#226292
#215792
C4×#4512