Finite Group Databasebeta

← #512665

Subgroups — #512665

Order 512.

811 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60809C211yesmaximal, characteristic
2#33358C212yesmaximal
3#13610C218yesmaximal, complemented
4#13596C212yesmaximal, complemented
5#13586C212yesmaximal, complemented
6C2×#541C411yescharacteristic, complemented
7#3449C2211yescharacteristic
8#3223—22no—
9#3223C414yescomplemented
10#3222—22no—
11#3222C2214yes—
12#3152C411yescharacteristic, complemented
13#3152C2211yescharacteristic
14#3142C411yescharacteristic, complemented
15#3142C2211yescharacteristic
16#3037—22no—
17#3007—22no—
18#2921C411yescharacteristic, complemented
19C2×#410—22no—
20#2129—22no—
21#2128C2214yes—
22#2127—22no—
23#2127C2214yes—
24#2126—22no—
25#2126C2214yes—
26#1412C2218yescomplemented
27#1409C2218yescomplemented
28C22×#41D412yes—
29C2×#137—22no—
30C2×#137—22no—
31#541—22no—
32#541—42no—
33#541C2×C414yescomplemented
34#537—41no—
35#536C2×C412yes—
36#536D412yes—
37#535D412yes—
38#535C2312yes—
39#534—41no—
40#532C2×C411yescharacteristic
41#532D412yes—
42#532C2311yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic, complemented
44C2×#126—41no—
45C2×#125—41no—
46C2×#125C2×C411yescharacteristic, complemented
47#525—42no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—42no—
51C2×#117C2×C411yescharacteristic
52#497—22no—
53#497C2×C414yescomplemented
54#494—21no—
55#494C2×C412yescomplemented
56#492—21no—
57#492C2312yes—
58#491—21no—
59#491C2×C412yescomplemented
60#490—21no—
61#490C2312yes—
62#488—22no—
63#488C2×C414yescomplemented
64#464—24no—
65#464—24no—
66#462—24no—
67#460—24no—
68#452—24no—
69#452—24no—
70#451—24no—
71#450—24no—
72#433—22no—
73#433C2×C414yescomplemented
74#423—21no—
75#417—22no—
76#417—22no—
77#410—24no—
78#410—24no—
79#410—42no—
80#410—42no—
81C2×#99—24no—
82C2×#99—42no—
83#407—21no—
84C4×M16C2×C411yescharacteristic, complemented
85C4×M16C2×C411yescharacteristic, complemented
86#354—24no—
87#354—24no—
88#353—24no—
89#353C2318yes—
90C2×#41—22no—
91C2×#41—22no—
92C2×#41—82no—
93C2×#41—82no—
94C2×#41C22⋊C412yes—
95C2×#41C22⋊C412yes—
96C2×#41C2×D412yes—
97C2×#41C2×D412yes—
98C22×Q8—21no—
99C22×Q8C22⋊C412yes—
100C22×D4—22no—
101C22×D4C2×D412yes—
102C23×C4—22no—
103C23×C4C22⋊C411yescharacteristic
104#137—82no—
105#137—82no—
106#137—82no—
107#137—82no—
108C2×Q16—24no—
109C2×Q16—24no—
110C2×SD16—24no—
111C2×SD16—24no—
112C2×M16—21no—
113C2×M16—21no—
114C4⋊Q8—21no—
115C4⋊Q8—41no—
116C4⋊Q8C22⋊C411yescharacteristic
117C4⋊Q8C22⋊C412yes—
118C4⋊Q8C22×C412yes—
119C4⋊Q8C22×C411yescharacteristic
120#127C22⋊C411yescharacteristic
121#127C2×D412yes—
122#127C2411yescharacteristic
123#126—44no—
124#126—44no—
125#125—21no—
126#125—41no—
127#125—44no—
128#125—44no—
129#125C22×C412yes—
130#124C22⋊C412yes—
131#124C22×C412yes—
132#124C2×D412yes—
133#124C2×D412yes—
134#123—42no—
135#123—42no—
136C22⋊Q8—24no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139C22⋊Q8—42no—
140#121—42no—
141#121—42no—
142#121—41no—
143#121—42no—
144#121—41no—
145#120—42no—
146#120—42no—
147C4×Q8—24no—
148C4×Q8—42no—
149C4×D4—41no—
150C4×D4—42no—
151C4×D4—41no—
152#117—22no—
153#117—44no—
154#117—44no—
155#117—82no—
156#117—82no—
157#117C22⋊C412yes—
158#117C22×C412yes—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—82no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—44no—
169C2×C42—41no—
170C2×C42—82no—
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C4⋊C8—24no—
174#103—24no—
175#103—24no—
176#102—24no—
177#102—24no—
178#99—28no—
179#99—44no—
180#99—44no—
181#99—44no—
182C22⋊C8—24no—
183#97—21no—
184#97C22×C412yes—
185C4×C8—21no—
186C4×C8C22×C412yes—
187C24—22no—
188C24#9911yescharacteristic
189#41—42no—
190#41—42no—
191#41—42no—
192#41—42no—
193#41—82no—
194#41—82no—
195#41—82no—
196#41—82no—
197#41—82no—
198#41—82no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—41no—
204C2×Q8—82no—
205C2×Q8C2×C22⋊C412yes—
206C2×Q8C2×C22⋊C412yes—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—41no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—82no—
216C2×D4—82no—
217C2×D4—82no—
218C2×D4C2×C22⋊C412yes—
219C2×D4C22×D412yes—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—84no—
235C22×C4#9912yes—
236C22×C4#9911yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239Q16—84no—
240Q16—84no—
241SD16—84no—
242SD16—84no—
243M16—42no—
244M16—42no—
245C2×C8—22no—
246C2×C8—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—84no—
261C4⋊C4—84no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—48no—
270C22⋊C4—84no—
271C42—21no—
272C42—21no—
273C42—44no—
274C42—84no—
275C42—84no—
276C42C2×C22⋊C411yescharacteristic
277C42C2×C22⋊C412yes—
278C42C23×C411yescharacteristic
279C23—22no—
280C23—22no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23—81no—
285C23—162no—
286C23C2×#9912yes—
287C23C2×#9911yescharacteristic
288Q8—42no—
289Q8—42no—
290Q8—44no—
291Q8—82no—
292Q8—82no—
293D4—42no—
294D4—42no—
295D4—82no—
296D4—82no—
297D4—82no—
298D4—82no—
299D4—82no—
300D4—161no—
301D4—162no—
302D4—161no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—88no—
331C2×C4—168no—
332C2×C4C2×#9912yes—
333C2×C4C2×#9911yescharacteristic
334C2×C4C22×C22⋊C411yescharacteristic
335C8—42no—
336C8—42no—
337C22—22no—
338C22—21no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—84no—
346C22—161no—
347C22—162no—
348C22C22×#9911yescharacteristic
349C4—21no—
350C4—22no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—82no—
357C4—82no—
358C4—168no—
359C4#212712yes—
360C2—21no—
361C2—42no—
362C2—41no—
363C2—82no—
364C2—82no—
365C2C2×#212711yescharacteristic
subgroupG/H
center, socleC2C2×#2127
commutatorC42C23×C4
Frattini, cosocle#127C24
prime radical#3449C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C23×C4 ◁ #3449 ◁ G
LECSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13610C2semidirect
4. #13596C2semidirect
5. #13586C2semidirect
6. C2×#541C4semidirect
9. #3223C4semidirect
12. #3152C4semidirect
14. #3142C4semidirect
18. #2921C4semidirect
26. #1412C22semidirect
27. #1409C22semidirect
33. #541C2×C4semidirect
43. C2×C4⋊Q8C2×C4semidirect
46. C2×#125C2×C4semidirect
53. #497C2×C4semidirect
55. #494C2×C4semidirect
59. #491C2×C4semidirect
63. #488C2×C4semidirect
73. #433C2×C4semidirect
84. C4×M16C2×C4semidirect
85. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#608092
#333582
#136102
#135962
#135862