Finite Group Databasebeta

← #6594218

Subgroups — #6594218

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36191C211yesmaximal, characteristic, complemented
2#36182C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#33058C211yesmaximal, characteristic, complemented
5#33058C211yesmaximal, characteristic, complemented
6#29227C211yesmaximal, characteristic, complemented
7#29145C211yesmaximal, characteristic, complemented
8#27982C211yesmaximal, characteristic, complemented
9#27981C211yesmaximal, characteristic, complemented
10#27967C211yesmaximal, characteristic, complemented
11#27966C211yesmaximal, characteristic, complemented
12#27731C214yesmaximal, complemented
13#27704C214yesmaximal, complemented
14#27703C214yesmaximal, complemented
15#22501C214yesmaximal, complemented
16C4×#490C211yesmaximal, characteristic, complemented
17#20765C211yesmaximal, characteristic, complemented
18#20764C211yesmaximal, characteristic, complemented
19#20494C211yesmaximal, characteristic, complemented
20C2×#553C2212yes—
21C2×#490C2211yescharacteristic
22C2×#436C2212yes—
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2856C2211yescharacteristic
28#2856C2211yescharacteristic
29#2851C2211yescharacteristic
30#2851C2211yescharacteristic
31#2574—22no—
32#2570C2212yes—
33#2566C2212yes—
34#2520—22no—
35#2512—22no—
36#2504C2212yes—
37#2500C2212yes—
38#2450—22no—
39#2440C2212yes—
40#2439C2212yes—
41#2422—22no—
42#2416—22no—
43#2368—22no—
44#2361—22no—
45#2352C2212yes—
46C4×#127C2211yescharacteristic
47C4×#125C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2073C2214yes—
57#2071C2214yes—
58#2023C2214yes—
59#1993C2214yes—
60#1992C2211yescharacteristic
61#1992C2211yescharacteristic
62#1992C2211yescharacteristic
63#1991C2214yes—
64#1989C2214yes—
65#1961C2211yescharacteristic
66#1961C2211yescharacteristic
67#1961C2211yescharacteristic
68#1936C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1895C2214yes—
72#1895C2214yes—
73#1875C2214yes—
74#1874C2214yes—
75#1836C2211yescharacteristic
76#1836C2211yescharacteristic
77#1836C2211yescharacteristic
78#1836C2211yescharacteristic
79#1836C2211yescharacteristic
80#1836C2211yescharacteristic
81#1836C2211yescharacteristic
82#1836C2211yescharacteristic
83#1836C2211yescharacteristic
84#1835C2214yes—
85#1834C2214yes—
86#1825C2214yes—
87#1823C2214yes—
88#1804C2214yes—
89C8×C4⋊C4C2212yes—
90#1767C2211yescharacteristic
91C4×#102C2214yes—
92#1753C2212yes—
93#1753C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1736C2214yes—
97#1736C2214yes—
98C4×#97C2211yescharacteristic
99C22×M16C2312yes—
100C22×M16C2312yes—
101#553—216no—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#126—22no—
105C2×#125C2311yescharacteristic
106C2×#125C2312yes—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4D412yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#490—24no—
127#436—28no—
128C2×C4⋊C8C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C4×M16—24no—
136C4×M16—24no—
137C2×#97C2311yescharacteristic
138C2×#97C2311yescharacteristic
139C2×#97C2311yescharacteristic
140C2×#97C2312yes—
141C2×#97C2311yescharacteristic
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#398—22no—
147#398—22no—
148#397—24no—
149#396—24no—
150#396—22no—
151#395—22no—
152#394—24no—
153#393—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390C2312yes—
161#390C2312yes—
162#390C2312yes—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—24no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#386—22no—
178#386—24no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—24no—
183#384—22no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382C2311yescharacteristic
191#382C2312yes—
192#382C2312yes—
193#382C2312yes—
194#382C2311yescharacteristic
195#382C2311yescharacteristic
196#382C2312yes—
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×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4C2×D411yescharacteristic
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—42no—
247C2×M16—42no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—44no—
253#127—22no—
254#127—22no—
255#126—48no—
256#125—28no—
257#125—22no—
258#125—22no—
259#123—48no—
260C22⋊Q8—48no—
261#121—28no—
262#121—28no—
263#121—48no—
264#121—48no—
265#121—84no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—48no—
269C4×D4—84no—
270C4×D4#4118yes—
271#117—28no—
272#117—44no—
273C2×C4⋊C4—28no—
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—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—84no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D411yescharacteristic
313C2×C22⋊C4C2×D411yescharacteristic
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C4⋊C8—24no—
331#102—28no—
332#102—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#97—22no—
338#97—22no—
339#97—22no—
340#97—42no—
341#97#4114yes—
342C4×C8—24no—
343C4×C8—24no—
344C4×C8—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24—22no—
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C2×Q8—44no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4#12414yes—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—84no—
405C22×C4—82no—
406C22×C4—84no—
407C22×C4—82no—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2511yescharacteristic
419M16—48no—
420M16—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8C2×#4114yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
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—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4#12414yes—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4#12414yes—
471C22⋊C4#12414yes—
472C42—24no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—42no—
480C42—42no—
481C42—44no—
482C42—82no—
483C42—82no—
484C42C2×#4114yes—
485C42C2×#4112yes—
486C422−1+412yes—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23C2×#12412yes—
501C23C2×#12412yes—
502C23C23×D411yescharacteristic
503Q8—164no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—88no—
508D4—88no—
509D4—164no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—84no—
550C2×C4—164no—
551C2×C4—164no—
552C2×C4—164no—
553C2×C4C2×#12412yes—
554C2×C4C2×#12412yes—
555C2×C4C2×#12412yes—
556C2×C4C2×#12412yes—
557C2×C4#54714yes—
558C2×C4#57514yes—
559C2×C4C22×#4111yescharacteristic
560C2×C4C22×#4112yes—
561C2×C4C2×2−1+411yescharacteristic
562C2×C4#58512yes—
563C8—24no—
564C8—42no—
565C8—42no—
566C8—42no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—82no—
578C22—84no—
579C22—84no—
580C22—82no—
581C22—82no—
582C22C22×#12411yescharacteristic
583C22C2×#54711yescharacteristic
584C22#350011yescharacteristic
585C22C2×#57512yes—
586C22#358512yes—
587C4—22no—
588C4—22no—
589C4—24no—
590C4—24no—
591C4—22no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#347512yes—
603C4#351512yes—
604C2—42no—
605C2—42no—
606C2—162no—
607C2#3616411yescharacteristic
608C2#6061811yescharacteristic
609C2#6062611yescharacteristic
610C2#6125512yes—
611C2#6259212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36182C2
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
1. #36191C2semidirect
3. #33296C2semidirect
4. #33058C2semidirect
5. #33058C2semidirect
6. #29227C2semidirect
7. #29145C2semidirect
8. #27982C2semidirect
9. #27981C2semidirect
10. #27967C2semidirect
11. #27966C2semidirect
12. #27731C2semidirect
13. #27704C2semidirect
14. #27703C2semidirect
15. #22501C2semidirect
16. C4×#490C2semidirect
17. #20765C2semidirect
18. #20764C2semidirect
19. #20494C2semidirect

Maximal subgroups

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

subgroupindex
#361912
#361822
#332962
#330582
#292272
#291452
#279822
#279812
#279672
#279662
#277312
#277042
#277032
#225012
C4×#4902
#207652
#207642
#204942