Finite Group Databasebeta

← #6800401

Subgroups — #6800401

Order 512.

1237 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39609C211yesmaximal, characteristic
2#35181C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic
4#32330C212yesmaximal, complemented
5#32323C212yesmaximal, complemented
6#31485C212yesmaximal, complemented
7#31460C212yesmaximal, complemented
8#29976C212yesmaximal
9#28801C212yesmaximal, complemented
10#27971C212yesmaximal
11#27968C211yesmaximal, characteristic
12#27742C212yesmaximal, complemented
13#27657C212yesmaximal
14#27652C211yesmaximal, characteristic
15#26521C212yesmaximal, complemented
16#26120C212yesmaximal, complemented
17#22628C211yesmaximal, characteristic
18#22094C212yesmaximal
19#21579C211yesmaximal, characteristic
20C2×#485C2212yes—
21C2×#483C2211yescharacteristic
22#2857C2211yescharacteristic
23#2851—22no—
24#2774—22no—
25#2742—22no—
26#2646—21no—
27#2646—21no—
28#2628C2212yes—
29#2571—21no—
30#2571C2212yes—
31#2570—22no—
32#2570C2212yes—
33#2570C2212yes—
34#2568C2211yescharacteristic
35#2567—21no—
36#2552—21no—
37#2552C2212yes—
38#2551C2211yescharacteristic
39#2511—21no—
40#2509C2212yes—
41#2492—21no—
42#2451C2212yes—
43#2423—21no—
44#2365C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2323C2212yes—
47#2317C2212yes—
48#2317C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2101C2212yes—
56#2101C2212yes—
57#2100C2212yes—
58#2100C2212yes—
59#2099C2212yes—
60#2099C2212yes—
61#2087C2212yes—
62#2087C2212yes—
63#2086C2212yes—
64#2086C2212yes—
65#2085C2212yes—
66#2084C2211yescharacteristic
67#2084C2212yes—
68#2083C2211yescharacteristic
69#2074C2212yes—
70#2072C2211yescharacteristic
71#2072C2211yescharacteristic
72#2072C2212yes—
73#2057C2212yes—
74#2057C2212yes—
75#1993C2212yes—
76#1993C2212yes—
77#1990C2212yes—
78#1987C2212yes—
79#1949C2212yes—
80#1949C2212yes—
81#1948C2212yes—
82#1947C2212yes—
83#1944C2212yes—
84#1942C2212yes—
85#1940C2212yes—
86#1935C2212yes—
87#1933C2212yes—
88#1928C2212yes—
89#1921C2212yes—
90#1921C2212yes—
91#1919C2212yes—
92#1919C2212yes—
93#1918C2212yes—
94#1917C2212yes—
95#1916C2212yes—
96#1872C2211yescharacteristic
97#1863C2212yes—
98#1863C2212yes—
99#1852C2211yescharacteristic
100#1852C2212yes—
101#1836C2212yes—
102#1836C2212yes—
103#1835C2212yes—
104#1825C2212yes—
105#1825C2212yes—
106#1825C2212yes—
107#1824C2212yes—
108#1823C2211yescharacteristic
109#1805C2212yes—
110#1805C2212yes—
111#1803C2212yes—
112#1798C2212yes—
113#1797C2211yescharacteristic
114#1776C2212yes—
115#1771C2212yes—
116#1763C2212yes—
117#1750C2212yes—
118#1749C2211yescharacteristic
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2312yes—
122C2×#125—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C2311yescharacteristic
125C2×C4×Q8—21no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134#485—28no—
135#483—24no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#399—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#395—22no—
170#395—22no—
171#395—22no—
172#393—22no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—22no—
178#391C2312yes—
179#391C2311yescharacteristic
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2311yescharacteristic
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#387—21no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#385—21no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#384—21no—
215#384—22no—
216#384D411yescharacteristic
217#384D411yescharacteristic
218#384C2312yes—
219#384C2312yes—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#382—22no—
224#382—22no—
225#382D412yes—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380—21no—
233#380D412yes—
234#380D411yescharacteristic
235#380D411yescharacteristic
236#380C2312yes—
237#380C2311yescharacteristic
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4—21no—
252C2×#95—21no—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#340C2312yes—
257#340C2312yes—
258#340C2312yes—
259#340C2312yes—
260#340C2312yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#340C2312yes—
266#340C2312yes—
267#340C2312yes—
268#340C2312yes—
269#340C2312yes—
270#340C2312yes—
271#340C2312yes—
272#340C2312yes—
273#340C2312yes—
274#340C2312yes—
275#336C2311yescharacteristic
276#336C2311yescharacteristic
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280#336C2312yes—
281#336C2312yes—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8C2411yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—24no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—82no—
298#125—84no—
299C22⋊Q8—22no—
300C22⋊Q8—82no—
301C22⋊Q8#4114yes—
302C4×Q8—24no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—82no—
306#117—24no—
307#117—24no—
308#117—42no—
309#117—42no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—82no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—24no—
378#107—42no—
379#107—42no—
380#107—42no—
381#106—24no—
382#106—24no—
383C4⋊C8—24no—
384C4⋊C8—42no—
385C4⋊C8—42no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—42no—
390C22⋊C8—22no—
391C22⋊C8—24no—
392C22⋊C8—22no—
393#97—24no—
394#97—42no—
395C4×C8—42no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427C24C22×D411yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—82no—
435C2×Q8—82no—
436C2×Q8—82no—
437C2×Q8C2×#4112yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—41no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4C2×#4112yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—81no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—44no—
552C42—82no—
553C42—82no—
554C42—82no—
555C422−1+414yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23#54812yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—42no—
566Q8—84no—
567Q8—164no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4#53612yes—
628C2×C4#53612yes—
629C2×C4#54812yes—
630C2×C4#57812yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×2−1+412yes—
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58512yes—
635C8—42no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22C2×#53611yescharacteristic
647C22C2×#54811yescharacteristic
648C22#348511yescharacteristic
649C22C2×#57811yescharacteristic
650C22#358511yescharacteristic
651C22#358511yescharacteristic
652C22#358611yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4#351512yes—
673C4#356312yes—
674C2—42no—
675C2#3912211yescharacteristic
676C2#6087011yescharacteristic
677C2#6087411yescharacteristic
678C2#6133411yescharacteristic
679C2#6134811yescharacteristic
680C2#6179511yescharacteristic
681C2#6180611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #32330C2semidirect
5. #32323C2semidirect
6. #31485C2semidirect
7. #31460C2semidirect
9. #28801C2semidirect
12. #27742C2semidirect
15. #26521C2semidirect
16. #26120C2semidirect

Maximal subgroups

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

subgroupindex
#396092
#351812
#331492
#323302
#323232
#314852
#314602
#299762
#288012
#279712
#279682
#277422
#276572
#276522
#265212
#261202
#226282
#220942
#215792