Finite Group Databasebeta

← #6484970

Subgroups — #6484970

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33059C211yesmaximal, characteristic, complemented
4#33058C211yesmaximal, characteristic
5#33053C211yesmaximal, characteristic, complemented
6#33052C211yesmaximal, characteristic
7#30191C214yesmaximal, complemented
8#29363C214yesmaximal, complemented
9#29226C211yesmaximal, characteristic, complemented
10#29225C211yesmaximal, characteristic, complemented
11#29150C211yesmaximal, characteristic, complemented
12#29149C211yesmaximal, characteristic, complemented
13#29134C214yesmaximal, complemented
14#28757C211yesmaximal, characteristic, complemented
15#28757C211yesmaximal, characteristic, complemented
16#28723C211yesmaximal, characteristic, complemented
17#28722C211yesmaximal, characteristic, complemented
18#21896C214yesmaximal, complemented
19#20427C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#491C2211yescharacteristic
22C2×#491C2211yescharacteristic
23C2×#490C2211yescharacteristic
24C2×#490C2211yescharacteristic
25C2×#424C2212yes—
26C2×C4×M16C2211yescharacteristic
27C22×#97C2211yescharacteristic
28#2868C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2851C2211yescharacteristic
35#2851C2211yescharacteristic
36#2647C2212yes—
37#2640C2212yes—
38#2629C2212yes—
39#2620C2212yes—
40#2590—22no—
41#2590—22no—
42#2587—22no—
43#2587—22no—
44#2512—22no—
45#2491—22no—
46#2452—22no—
47#2422—22no—
48#2330C2212yes—
49#2326C2212yes—
50#2314C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#125C2211yescharacteristic
54C4×#125C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2100C2214yes—
57#2099C2214yes—
58#2072C2214yes—
59#2071C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2026C2214yes—
63#2025C2214yes—
64#1992C2211yescharacteristic
65#1992C2211yescharacteristic
66#1992C2211yescharacteristic
67#1992C2211yescharacteristic
68#1988C2214yes—
69#1987C2214yes—
70#1983C2214yes—
71#1983C2214yes—
72#1961C2211yescharacteristic
73#1961C2211yescharacteristic
74#1960C2211yescharacteristic
75#1960C2211yescharacteristic
76#1936C2214yes—
77#1935C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1895C2214yes—
81#1894C2214yes—
82#1893C2214yes—
83#1892C2214yes—
84#1836C2211yescharacteristic
85#1836C2211yescharacteristic
86#1836C2211yescharacteristic
87#1836C2211yescharacteristic
88#1805C2214yes—
89#1804C2214yes—
90#1803C2214yes—
91#1802C2214yes—
92#1768C2212yes—
93#1768C2212yes—
94#1767C2211yescharacteristic
95#1767C2211yescharacteristic
96#1767C2211yescharacteristic
97#1767C2211yescharacteristic
98#1729C2212yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101#542—216no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—22no—
107C2×#125C2311yescharacteristic
108C2×#125C2311yescharacteristic
109C2×#125C2312yes—
110C2×#124—22no—
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#491—42no—
125#491—42no—
126#490—42no—
127#490—42no—
128#424—28no—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C4×M16—42no—
142C2×#97—42no—
143C2×#97C2311yescharacteristic
144C2×#97C2311yescharacteristic
145C2×#97C2311yescharacteristic
146C2×#97C2311yescharacteristic
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#397—24no—
156#396—24no—
157#396—22no—
158#396—22no—
159#394—24no—
160#393—24no—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—24no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—22no—
194#385—24no—
195#384—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#382C2311yescharacteristic
202#382C2311yescharacteristic
203#382C2312yes—
204#380—22no—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376C2311yescharacteristic
217#376C2311yescharacteristic
218C43C2311yescharacteristic
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232C22×Q8—22no—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2412yes—
237C23×C4—22no—
238C23×C4C2×D412yes—
239C23×C4C2411yescharacteristic
240C2×M16—24no—
241C2×M16—42no—
242C2×M16—42no—
243C22×C8—24no—
244C22×C8—42no—
245C22×C8—42no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—28no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—84no—
254#127—22no—
255#127—22no—
256#126—48no—
257#125—22no—
258#125—22no—
259#125—22no—
260#125—22no—
261#125—44no—
262#124—28no—
263#124—44no—
264#124—48no—
265#123—48no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268#121—28no—
269#121—84no—
270C4×Q8—28no—
271C4×D4—28no—
272C4×D4—84no—
273#117—42no—
274#117#4118yes—
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—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D411yescharacteristic
315C2×C22⋊C4C2×D411yescharacteristic
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×C42C2×D412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8—24no—
335C4⋊C8—24no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#102—44no—
341#102—44no—
342#102—44no—
343#102—44no—
344#97—22no—
345#97—22no—
346#97—22no—
347#97—22no—
348#97—42no—
349C4×C8—42no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—22no—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—44no—
375C2×Q8—84no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—84no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—84no—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C2×#4114yes—
410C22×C4C2511yescharacteristic
411M16—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—48no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4#12414yes—
454C4⋊C4#12414yes—
455C22⋊C4—42no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4#12414yes—
467C22⋊C4#12414yes—
468C42—24no—
469C42—21no—
470C42—21no—
471C42—22no—
472C42—22no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—44no—
478C42—82no—
479C42—82no—
480C42C2×#4114yes—
481C42C2×#4114yes—
482C42H(5,𝔽2)12yes—
483C422−1+412yes—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23C2×#12412yes—
494C23C23×D411yescharacteristic
495Q8—44no—
496Q8—44no—
497Q8—164no—
498D4—44no—
499D4—44no—
500D4—88no—
501D4—164no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—84no—
541C2×C4—164no—
542C2×C4C2×#12412yes—
543C2×C4C2×#12412yes—
544C2×C4C2×#12412yes—
545C2×C4C2×#12412yes—
546C2×C4C2×#12412yes—
547C2×C4#53714yes—
548C2×C4C22×#4112yes—
549C2×C4C2×H(5,𝔽2)11yescharacteristic
550C2×C4C2×2−1+411yescharacteristic
551C2×C4#58512yes—
552C8—42no—
553C8—42no—
554C8—42no—
555C8—42no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—84no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—84no—
566C22C22×#12411yescharacteristic
567C22C2×#53711yescharacteristic
568C22#344911yescharacteristic
569C22#358512yes—
570C22#358612yes—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—24no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4#350612yes—
588C4#350712yes—
589C2—42no—
590C2—162no—
591C2#3518011yescharacteristic
592C2#6062611yescharacteristic
593C2#6062711yescharacteristic
594C2#6091912yes—
595C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35336C2semidirect
3. #33059C2semidirect
5. #33053C2semidirect
7. #30191C2semidirect
8. #29363C2semidirect
9. #29226C2semidirect
10. #29225C2semidirect
11. #29150C2semidirect
12. #29149C2semidirect
13. #29134C2semidirect
14. #28757C2semidirect
15. #28757C2semidirect
16. #28723C2semidirect
17. #28722C2semidirect
18. #21896C2semidirect
19. #20427C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#352772
#330592
#330582
#330532
#330522
#301912
#293632
#292262
#292252
#291502
#291492
#291342
#287572
#287232
#287222
#218962
#204272