Finite Group Databasebeta

← #6528551

Subgroups — #6528551

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37643C211yesmaximal, characteristic, complemented
2#35279C211yesmaximal, characteristic
3#29119C214yesmaximal
4#27970C214yesmaximal, complemented
5#27824C212yesmaximal, complemented
6#27822C212yesmaximal, complemented
7#27816C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#27737C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27729C212yesmaximal, complemented
12#27727C212yesmaximal, complemented
13#21834C211yesmaximal, characteristic, complemented
14#21833C211yesmaximal, characteristic, complemented
15#20737C211yesmaximal, characteristic
16#20737C211yesmaximal, characteristic
17#20538C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857—21no—
23#2857—21no—
24#2845—22no—
25#2629—22no—
26#2629—22no—
27#2629C2214yes—
28#2628—22no—
29#2568C2212yes—
30#2567C2212yes—
31#2563C2212yes—
32#2552—21no—
33#2552—21no—
34#2552C2212yes—
35#2552C2212yes—
36#2552C2212yes—
37#2552C2212yes—
38#2542C2212yes—
39#2487—22no—
40#2409—22no—
41#2317C2212yes—
42#2311C2212yes—
43#2285C2211yescharacteristic
44#2285C2211yescharacteristic
45#2285C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2100C2214yes—
49#2099C2214yes—
50#2072C2214yes—
51#2071C2214yes—
52#2026C2214yes—
53#2025C2214yes—
54#1992C2214yes—
55#1991C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1987C2212yes—
63#1987C2212yes—
64#1936C2214yes—
65#1935C2214yes—
66#1934C2214yes—
67#1933C2214yes—
68#1836C2214yes—
69#1834C2214yes—
70#1825C2212yes—
71#1825C2212yes—
72#1824C2214yes—
73#1823C2214yes—
74#1803C2211yescharacteristic
75#1803C2211yescharacteristic
76#1803C2211yescharacteristic
77#1803C2211yescharacteristic
78#1802C2211yescharacteristic
79#1802C2211yescharacteristic
80#1802C2211yescharacteristic
81#1802C2211yescharacteristic
82#1788C2212yes—
83#1788C2212yes—
84#1770C2212yes—
85#1770C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92#1750C2214yes—
93C22×#41C2311yescharacteristic
94C23×Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125C2314yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—21no—
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123#431—24no—
124#431—24no—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—24no—
144#399—24no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#398—24no—
150#398—24no—
151#396—24no—
152#396—24no—
153#395—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#386—24no—
170#385—24no—
171#384—22no—
172#384—22no—
173#384D411yescharacteristic
174#384D411yescharacteristic
175#384C2314yes—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2314yes—
181#382—21no—
182#382—21no—
183#382D412yes—
184#382C2314yes—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380D412yes—
192#380D412yes—
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201#340C2312yes—
202#340C2312yes—
203#340C2312yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—41no—
224C2×#41—41no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—44no—
228C22×Q8—82no—
229C22×Q8—82no—
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250#125—44no—
251#125—44no—
252#125—84no—
253#125—84no—
254C22⋊Q8—84no—
255C22⋊Q8—84no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—44no—
265C4×D4—44no—
266#117—22no—
267#117—22no—
268#117—42no—
269#117—42no—
270#117—44no—
271#117—82no—
272#117—82no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4C2×D414yes—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42#4114yes—
327C2×C42#4114yes—
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335#103—24no—
336#103—24no—
337#103—42no—
338#103—42no—
339#102—24no—
340#102—24no—
341#102—42no—
342#102—42no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345#97—44no—
346C4×C8—24no—
347C4×C8—24no—
348C4×C8—44no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95C2×D414yes—
360#95C2×D414yes—
361C24—21no—
362C24C22×D411yescharacteristic
363#41—44no—
364#41—44no—
365#41—44no—
366#41—44no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—816no—
379C2×Q82−1+412yes—
380C2×D4—24no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D42−1+412yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4#12414yes—
418C22×C4#12414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2511yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—24no—
491C42—24no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—44no—
496C42—44no—
497C42#12414yes—
498C42#12414yes—
499C42C2×#4112yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C42C2×#4112yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×#12412yes—
517C23C23×D411yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—88no—
523Q8—88no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—88no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4C2×#12412yes—
576C2×C4C2×#12412yes—
577C2×C4C2×#12412yes—
578C2×C4C2×#12412yes—
579C2×C4C2×#12412yes—
580C2×C4#57612yes—
581C2×C4#57712yes—
582C2×C4#57712yes—
583C2×C4#57812yes—
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C2×C4#58511yescharacteristic
589C2×C4#58511yescharacteristic
590C8—42no—
591C8—42no—
592C8—42no—
593C8—42no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22C22×#12411yescharacteristic
609C22#348511yescharacteristic
610C22#348611yescharacteristic
611C22C2×#57611yescharacteristic
612C22C2×#57711yescharacteristic
613C22C2×#57711yescharacteristic
614C22C2×#57811yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4—84no—
627C4—81no—
628C4#313614yes—
629C4#313714yes—
630C4#349511yescharacteristic
631C4#349511yescharacteristic
632C4#349511yescharacteristic
633C4#349511yescharacteristic
634C2—42no—
635C2—82no—
636C2#3564411yescharacteristic
637C2C2×#313611yescharacteristic
638C2C2×#313711yescharacteristic
639C2#6179111yescharacteristic
640C2#6179211yescharacteristic
641C2#6186811yescharacteristic
642C2#6186911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37643C2semidirect
4. #27970C2semidirect
5. #27824C2semidirect
6. #27822C2semidirect
7. #27816C2semidirect
8. #27814C2semidirect
9. #27737C2semidirect
10. #27735C2semidirect
11. #27729C2semidirect
12. #27727C2semidirect
13. #21834C2semidirect
14. #21833C2semidirect
17. #20538C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#352792
#291192
#279702
#278242
#278222
#278162
#278142
#277372
#277352
#277292
#277272
#218342
#218332
#207372
#205382