Finite Group Databasebeta

← #6561558

Subgroups — #6561558

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic
3#30310C212yesmaximal
4#30304C211yesmaximal, characteristic
5#29320C211yesmaximal, characteristic
6#29312C212yesmaximal
7#29213C211yesmaximal, characteristic
8#29209C212yesmaximal
9#28839C214yesmaximal
10#28828C212yesmaximal
11#28807C214yesmaximal
12#28792C212yesmaximal
13#23841C212yesmaximal
14#23838C211yesmaximal, characteristic
15#22743C212yesmaximal
16#22728C212yesmaximal
17#20573C211yesmaximal, characteristic
18C2×#446C2212yes—
19#2868C2211yescharacteristic
20#2862C2212yes—
21#2859C2211yescharacteristic
22#2859C2212yes—
23#2582—21no—
24#2571C2211yescharacteristic
25#2571C2212yes—
26#2571C2212yes—
27#2570C2214yes—
28#2568—21no—
29#2552C2211yescharacteristic
30#2552C2212yes—
31#2511—22no—
32#2508—21no—
33#2508—22no—
34#2506—21no—
35#2506—22no—
36#2446—22no—
37#2365—21no—
38#2355—21no—
39#2355C2212yes—
40#2351—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2211yescharacteristic
43Q8×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×#125C2212yes—
47C4×#125C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2085C2212yes—
50#2085C2214yes—
51#2085C2214yes—
52#2085C2214yes—
53#2084C2211yescharacteristic
54#2083C2211yescharacteristic
55#2083C2212yes—
56#2083C2214yes—
57#2059C2212yes—
58#2059C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2058C2211yescharacteristic
62#2057C2211yescharacteristic
63#2057C2212yes—
64#2057C2212yes—
65#2057C2212yes—
66#1992C2212yes—
67#1991C2214yes—
68#1949C2214yes—
69#1949C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2214yes—
75#1947C2214yes—
76#1944C2214yes—
77#1942C2214yes—
78#1942C2214yes—
79#1847C2212yes—
80#1847C2214yes—
81#1846C2214yes—
82#1845C2212yes—
83#1836C2214yes—
84#1834C2212yes—
85#1776C2214yes—
86#1776C2214yes—
87C4×#107C2212yes—
88C4×#106C2212yes—
89#1771C2214yes—
90C8×C4⋊C4C2212yes—
91#1767C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—22no—
100C2×#125—21no—
101C2×#125C2312yes—
102C2×#125C2311yescharacteristic
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110#446Q8116yescomplemented
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#107C2312yes—
114C2×#107C2314yes—
115C2×#107C2312yes—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#97C2312yes—
124C2×#97C2314yes—
125C2×C4×C8C2314yes—
126C2×C4×C8C2312yes—
127#400—24no—
128#400—24no—
129#400—22no—
130#400—22no—
131#400—22no—
132#398—24no—
133#398—22no—
134#398—22no—
135#398—22no—
136#391—22no—
137#391—22no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#389—22no—
144#389—21no—
145#389—22no—
146#389—21no—
147#389—22no—
148#389—21no—
149#389—22no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—21no—
157#389—22no—
158#389C2311yescharacteristic
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2314yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—24no—
167#387—22no—
168#387—21no—
169#387—22no—
170#387—21no—
171#387—24no—
172#387—24no—
173#384—21no—
174#384—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2312yes—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2312yes—
184#382—22no—
185#382—21no—
186#382—22no—
187#382—22no—
188#382—21no—
189#382C2312yes—
190#382C2311yescharacteristic
191#382C2314yes—
192#382C2314yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C43C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#336C2311yescharacteristic
225#336C2311yescharacteristic
226#336C2312yes—
227#336C2312yes—
228#336C2314yes—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8C2×D411yescharacteristic
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2414yes—
235C4⋊Q8—42no—
236C4⋊Q8—44no—
237C4⋊Q8—44no—
238C4⋊Q8—82no—
239#125—24no—
240#125—22no—
241#125—42no—
242#125—41no—
243#125—84no—
244#125—82no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—84no—
251C4×Q8C2×Q818yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—24no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2411yescharacteristic
314#107—42no—
315#107—44no—
316#107—44no—
317#107—44no—
318#107—42no—
319#107—44no—
320#106—24no—
321#106—44no—
322#106—44no—
323#106—42no—
324C4⋊C8C2×Q818yes—
325C4⋊C8C2×Q8116yes—
326#97—42no—
327#97C2×Q8116yes—
328C4×C8—44no—
329C4×C8C2×Q818yes—
330#95—21no—
331#95—22no—
332#95—21no—
333#95—21no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—21no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8C4⋊Q814yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2511yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8C22×Q814yes—
391C2×C8C22×Q814yes—
392C2×C8C22×Q818yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4C4⋊Q818yes—
440C4⋊C4C4⋊Q814yes—
441C4⋊C42−1+412yes—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—22no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—22no—
451C42—21no—
452C42—22no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—44no—
457C42—42no—
458C42—42no—
459C42—44no—
460C42—84no—
461C42C22×Q814yes—
462C42C22×Q818yes—
463C23C23×D411yescharacteristic
464Q8—28no—
465Q8—88no—
466Q8—84no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4C2×C4⋊Q812yes—
517C2×C4C2×C4⋊Q814yes—
518C2×C4C2×C4⋊Q814yes—
519C2×C4C2×C4⋊Q812yes—
520C2×C4#57512yes—
521C2×C4#57512yes—
522C2×C4#57612yes—
523C2×C4#57812yes—
524C2×C4C23×Q812yes—
525C2×C4C2×2−1+411yescharacteristic
526C2×C4#58512yes—
527C2×C4#58511yescharacteristic
528C8—24no—
529C8—24no—
530C8—28no—
531C22C22×C4⋊Q811yescharacteristic
532C22#348711yescharacteristic
533C22#348911yescharacteristic
534C22C2×#57511yescharacteristic
535C22C2×#57511yescharacteristic
536C22C2×#57611yescharacteristic
537C22C2×#57811yescharacteristic
538C4—22no—
539C4—22no—
540C4—24no—
541C4—24no—
542C4—22no—
543C4—41no—
544C4—42no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—82no—
549C4—81no—
550C4—82no—
551C4—82no—
552C4—81no—
553C4—84no—
554C4—81no—
555C4—82no—
556C4#316614yes—
557C4#350212yes—
558C4#350412yes—
559C2#3595711yescharacteristic
560C2C2×#316611yescharacteristic
561C2#6066711yescharacteristic
562C2#6189811yescharacteristic
563C2#6190511yescharacteristic
564C2#6213311yescharacteristic
565C2#6213611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
110. #446Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#364472
#303102
#303042
#293202
#293122
#292132
#292092
#288392
#288282
#288072
#287922
#238412
#238382
#227432
#227282
#205732