Finite Group Databasebeta

← #7104842

Subgroups — #7104842

Order 512.

1229 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59242C211yesmaximal, characteristic
2#39751C211yesmaximal, characteristic
3#32578C212yesmaximal, complemented
4#32335C212yesmaximal, complemented
5#32328C212yesmaximal, complemented
6#32245C212yesmaximal, complemented
7#29166C211yesmaximal, characteristic
8#28808C212yesmaximal, complemented
9#28045C212yesmaximal, complemented
10#27968C211yesmaximal, characteristic
11#27724C212yesmaximal
12#27378C212yesmaximal, complemented
13#26287C212yesmaximal, complemented
14#26015C212yesmaximal
15#24590C212yesmaximal
16#23657C211yesmaximal, characteristic
17#23242C211yesmaximal, characteristic
18#22623C212yesmaximal
19#22084C211yesmaximal, characteristic
20C2×#483C2211yescharacteristic
21C2×#428C2212yes—
22#2857C2211yescharacteristic
23#2805—21no—
24#2787—22no—
25#2755—22no—
26#2726—21no—
27#2659—21no—
28#2584C2212yes—
29#2582C2211yescharacteristic
30#2571C2212yes—
31#2552C2211yescharacteristic
32#2550C2212yes—
33#2524—22no—
34#2514—21no—
35#2514C2212yes—
36#2508C2212yes—
37#2506C2212yes—
38#2499C2212yes—
39#2495—21no—
40#2489—21no—
41#2442—22no—
42#2442C2212yes—
43#2370—21no—
44#2355C2211yescharacteristic
45#2355C2211yescharacteristic
46#2347C2212yes—
47#2345C2212yes—
48#2329—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2211yescharacteristic
51C2×#389C2211yescharacteristic
52#2285C2211yescharacteristic
53#2101C2212yes—
54#2101C2212yes—
55#2100C2212yes—
56#2100C2212yes—
57#2100C2211yescharacteristic
58#2099C2212yes—
59#2095C2211yescharacteristic
60#2087C2212yes—
61#2087C2212yes—
62#2085C2212yes—
63#2085C2212yes—
64#2085C2212yes—
65#2084C2212yes—
66#2084C2212yes—
67#2083C2212yes—
68#2074C2212yes—
69#2068C2212yes—
70#2066C2212yes—
71#2059C2212yes—
72#2058C2211yescharacteristic
73#2026C2212yes—
74#2024C2211yescharacteristic
75#1993C2212yes—
76#1991C2212yes—
77#1989C2212yes—
78#1984C2212yes—
79#1949C2212yes—
80#1949C2212yes—
81#1948C2212yes—
82#1944C2212yes—
83#1944C2212yes—
84#1941C2212yes—
85#1935C2212yes—
86#1935C2212yes—
87#1933C2212yes—
88#1928C2212yes—
89#1927C2212yes—
90#1921C2212yes—
91#1920C2212yes—
92#1920C2212yes—
93#1919C2212yes—
94#1918C2212yes—
95#1918C2212yes—
96#1918C2212yes—
97#1918C2212yes—
98#1917C2212yes—
99#1917C2212yes—
100#1894C2212yes—
101#1872C2211yescharacteristic
102#1865C2212yes—
103#1863C2212yes—
104#1852C2211yescharacteristic
105#1851C2212yes—
106#1836C2212yes—
107#1836C2212yes—
108#1825C2212yes—
109#1825C2212yes—
110#1823C2211yescharacteristic
111#1809C2212yes—
112#1776C2212yes—
113C4×#107C2212yes—
114#1771C2212yes—
115#1753C2212yes—
116#1749C2211yescharacteristic
117#1736C2212yes—
118#1728C2211yescharacteristic
119C22×M16C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×#125—21no—
122C2×#125—22no—
123C2×#125C2312yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8D411yescharacteristic
126C2×C22⋊Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#483—24no—
135#428—28no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155#401—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#395—22no—
172#395—22no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—22no—
177#391D411yescharacteristic
178#391C2312yes—
179#391C2312yes—
180#391C2311yescharacteristic
181#389—22no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389D411yescharacteristic
191#389C2311yescharacteristic
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—21no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387D412yes—
214#385—21no—
215#385—22no—
216#385D412yes—
217#384C2312yes—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#383—22no—
222#383—22no—
223#382—22no—
224#382—21no—
225#382—22no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—22no—
233#380—22no—
234#380D411yescharacteristic
235#380C2311yescharacteristic
236#380C2312yes—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2311yescharacteristic
250C4×C22⋊C4—21no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4C2312yes—
253C2×#95—21no—
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#336C2311yescharacteristic
275#336C2312yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2311yescharacteristic
280#336C2312yes—
281#336C2312yes—
282C22×Q8—21no—
283C22×Q8C2×D411yescharacteristic
284C22×Q8C2411yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C2×M16—24no—
291C2×M16—24no—
292C2×M16—42no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—42no—
298#125—42no—
299#125—42no—
300#125—44no—
301#125—84no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—42no—
305C22⋊Q8—44no—
306C4×Q8—42no—
307C4×Q8—42no—
308#117—24no—
309#117—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
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—42no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4#4114yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—42no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2412yes—
374#107—24no—
375#107—24no—
376#107—24no—
377#107—42no—
378#106—24no—
379#106—24no—
380#106—24no—
381#106—42no—
382C4⋊C8—24no—
383C4⋊C8—42no—
384#103—24no—
385#103—24no—
386#103—42no—
387#103—42no—
388C22⋊C8—22no—
389C22⋊C8—22no—
390#97—24no—
391#97—42no—
392C4×C8—42no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—21no—
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#95C2×D412yes—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427C24C22×D411yescharacteristic
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
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
474M16—48no—
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—
486C2×C82−1+414yes—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—81no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—81no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—41no—
547C42—41no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—82no—
552C42—82no—
553C42—82no—
554C42—82no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23#54812yes—
562C23C23×D411yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—82no—
566Q8—162no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—82no—
627C2×C4—162no—
628C2×C4#54812yes—
629C2×C4#57512yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58512yes—
634C8—24no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#54811yescharacteristic
646C22#348511yescharacteristic
647C22#349911yescharacteristic
648C22C2×#57511yescharacteristic
649C22#358511yescharacteristic
650C22#358611yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—44no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4#349512yes—
672C4#356312yes—
673C2—42no—
674C2#4444811yescharacteristic
675C2#6132011yescharacteristic
676C2#6133311yescharacteristic
677C2#6171211yescharacteristic
678C2#6177611yescharacteristic
679C2#6250211yescharacteristic
680C2#6251211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32578C2semidirect
4. #32335C2semidirect
5. #32328C2semidirect
6. #32245C2semidirect
8. #28808C2semidirect
9. #28045C2semidirect
12. #27378C2semidirect
13. #26287C2semidirect

Maximal subgroups

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

subgroupindex
#592422
#397512
#325782
#323352
#323282
#322452
#291662
#288082
#280452
#279682
#277242
#273782
#262872
#260152
#245902
#236572
#232422
#226232
#220842