Finite Group Databasebeta

← #6561541

Subgroups — #6561541

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36497C211yesmaximal, characteristic
3#30306C212yesmaximal
4#30305C211yesmaximal, characteristic
5#29319C211yesmaximal, characteristic
6#29317C212yesmaximal
7#29213C211yesmaximal, characteristic
8#29211C212yesmaximal
9#28836C212yesmaximal
10#28834C214yesmaximal
11#28799C212yesmaximal
12#28796C214yesmaximal
13#23838C211yesmaximal, characteristic
14#23837C212yesmaximal
15#22741C212yesmaximal
16#22725C212yesmaximal
17#20572C211yesmaximal, characteristic
18C2×C8×Q8C2212yes—
19#2868C2211yescharacteristic
20#2864C2212yes—
21#2862C2212yes—
22#2859C2211yescharacteristic
23#2582—21no—
24#2571C2212yes—
25#2570C2214yes—
26#2570C2212yes—
27#2568—21no—
28#2552C2211yescharacteristic
29#2551C2211yescharacteristic
30#2550C2212yes—
31#2520—22no—
32#2511—22no—
33#2462—22no—
34#2453—21no—
35#2451—21no—
36#2446—22no—
37#2370—21no—
38#2365—21no—
39#2365—21no—
40#2355—21no—
41#2355C2211yescharacteristic
42#2355C2211yescharacteristic
43#2355C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C42×Q8C2211yescharacteristic
49#2085C2212yes—
50#2085C2214yes—
51#2085C2214yes—
52#2084C2211yescharacteristic
53#2084C2212yes—
54#2084C2214yes—
55#2084C2214yes—
56#2083C2211yescharacteristic
57#2059C2212yes—
58#2058C2212yes—
59#2058C2212yes—
60#2058C2212yes—
61#2058C2214yes—
62#2058C2211yescharacteristic
63#2057C2211yescharacteristic
64#2057C2212yes—
65#2057C2212yes—
66#1993C2214yes—
67#1992C2212yes—
68#1949C2214yes—
69#1949C2214yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1943C2214yes—
78#1943C2214yes—
79#1847C2214yes—
80#1847C2212yes—
81#1846C2214yes—
82#1844C2212yes—
83#1835C2214yes—
84#1834C2212yes—
85C4×#107C2212yes—
86C4×#106C2214yes—
87C4×#106C2214yes—
88C4×#106C2212yes—
89#1770C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1744C2212yes—
94#1744C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125—21no—
103C2×#125C2311yescharacteristic
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C8×Q8Q8116yescomplemented
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#107C2312yes—
114C2×#107C2312yes—
115C2×#106C2312yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#97C2312yes—
124C2×#97C2314yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2314yes—
127#400—24no—
128#400—22no—
129#400—24no—
130#398—24no—
131#398—22no—
132#398—22no—
133#398—22no—
134#395—22no—
135#395—22no—
136#391—22no—
137#391—21no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—21no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389C2311yescharacteristic
157#389C2311yescharacteristic
158#389C2312yes—
159#389C2314yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#387—24no—
165#387—22no—
166#387—22no—
167#387—21no—
168#387—21no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#387—22no—
176#387—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2312yes—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—22no—
186#382—21no—
187#382C2311yescharacteristic
188#382C2314yes—
189#382C2314yes—
190#382C2312yes—
191#382C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#336C2312yes—
225#336C2311yescharacteristic
226#336C2311yescharacteristic
227#336C2314yes—
228#336C2312yes—
229C22×Q8—22no—
230C22×Q8—21no—
231C22×Q8—22no—
232C22×Q8C2×D411yescharacteristic
233C22×C8C2412yes—
234C22×C8C2414yes—
235C22×C8C2412yes—
236C4⋊Q8—24no—
237C4⋊Q8—42no—
238C4⋊Q8—44no—
239C4⋊Q8—42no—
240C4⋊Q8—44no—
241C4⋊Q8—44no—
242C4⋊Q8—82no—
243#125—22no—
244#125—41no—
245#125—44no—
246#125—82no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—44no—
250C4×Q8—84no—
251C4×Q8—84no—
252C4×Q8C2×Q818yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
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—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2412yes—
310C2×C42C2412yes—
311C2×C42C2411yescharacteristic
312#107—42no—
313#107—42no—
314#107—44no—
315#107—44no—
316#106—24no—
317#106—44no—
318#106—44no—
319#106—44no—
320#106—44no—
321#106—42no—
322C4⋊C8C2×Q818yes—
323C4⋊C8C2×Q8116yes—
324#97—42no—
325#97—44no—
326C4×C8C2×Q818yes—
327C4×C8C2×Q8116yes—
328#95—21no—
329#95—22no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—82no—
366C2×Q8C4⋊Q814yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2511yescharacteristic
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8C22×Q814yes—
394C2×C8C22×Q818yes—
395C2×C8C22×Q814yes—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—84no—
433C4⋊C4—82no—
434C4⋊C4—81no—
435C4⋊C4—81no—
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—24no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—22no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—42no—
457C42—42no—
458C42—82no—
459C42—82no—
460C42—84no—
461C42C22×Q818yes—
462C42C22×Q814yes—
463C23C23×D411yescharacteristic
464Q8—28no—
465Q8—84no—
466Q8—88no—
467Q8—88no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4C2×C4⋊Q814yes—
518C2×C4C2×C4⋊Q812yes—
519C2×C4C2×C4⋊Q814yes—
520C2×C4C2×C4⋊Q812yes—
521C2×C4#57512yes—
522C2×C4#57512yes—
523C2×C4#57712yes—
524C2×C4#57712yes—
525C2×C4C23×Q812yes—
526C2×C4C2×2−1+411yescharacteristic
527C2×C4#58512yes—
528C2×C4#58511yescharacteristic
529C8—24no—
530C8—28no—
531C8—24no—
532C22C22×C4⋊Q811yescharacteristic
533C22#348711yescharacteristic
534C22#348911yescharacteristic
535C22C2×#57511yescharacteristic
536C22C2×#57511yescharacteristic
537C22C2×#57711yescharacteristic
538C22C2×#57711yescharacteristic
539C4—22no—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—22no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—82no—
550C4—81no—
551C4—82no—
552C4—81no—
553C4—82no—
554C4—82no—
555C4—81no—
556C4—84no—
557C4#316214yes—
558C4#350212yes—
559C4#350412yes—
560C2#3595711yescharacteristic
561C2C2×#316211yescharacteristic
562C2#6066511yescharacteristic
563C2#6189911yescharacteristic
564C2#6190411yescharacteristic
565C2#6214411yescharacteristic
566C2#6214511yescharacteristic
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. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#364972
#303062
#303052
#293192
#293172
#292132
#292112
#288362
#288342
#287992
#287962
#238382
#238372
#227412
#227252
#205722