Finite Group Databasebeta

← #6372213

Subgroups — #6372213

Order 512.

3418 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C211yesmaximal, characteristic
2C2×#2875C211yesmaximal, characteristic
3#34183C211yesmaximal, characteristic
4#21326C2112yesmaximal, complemented
5#21325C213yesmaximal, complemented
6#21322C213yesmaximal, complemented
7#21321C213yesmaximal, complemented
8C2×#1992C213yesmaximal, complemented
9#20348C214yesmaximal
10C22×#131C2212yes—
11#3504—28no—
12#3504C4116yescomplemented
13C2×#557—22no—
14C2×#554—26no—
15C2×#544C2216yes—
16C2×#533C2213yes—
17C22×#125C2213yes—
18C2×#520C2212yes—
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21#2875—28no—
22C2×#405C2216yes—
23C2×C4×M16C2213yes—
24C22×#97C2213yes—
25Q8×C4⋊C4—22no—
26D4×C4⋊C4—26no—
27#2317C2216yes—
28#2285C2213yes—
29C2×#384C2213yes—
30#1993C22112yes—
31#1992C2213yes—
32#1992C2216yes—
33#1992C22112yes—
34#1992C2211yescharacteristic
35#1992C2213yes—
36#1992C2213yes—
37#1991C22112yes—
38#1990C22112yes—
39#1736C22112yes—
40#1736C22112yes—
41#1735C22112yes—
42C2×#340C22112yes—
43C22×#41C2311yescharacteristic
44C2×#131—28no—
45C2×#131—28no—
46C2×#131Q8116yescomplemented
47C22×M16C2316yes—
48C23×C8C2316yes—
49#557—232no—
50#554—296no—
51#544—224no—
52#544C2×C4148yescomplemented
53#533—212no—
54#533C2×C4124yescomplemented
55C2×C4⋊Q8—26no—
56C2×C4⋊Q8C2316yes—
57C2×#125—26no—
58C2×#125—26no—
59C2×#125—212no—
60C2×#125C2×C4124yescomplemented
61C2×#125C2313yes—
62C2×#125C2316yes—
63C2×#125C2311yescharacteristic
64C2×C22⋊Q8—26no—
65C2×C22⋊Q8—212no—
66#520—28no—
67#520—216no—
68#520C2×C4116yes—
69C4×#41C2×C418yes—
70C4×#41D418yes—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—26no—
74C2×C4×Q8D412yes—
75C2×C4×D4—26no—
76C2×C4×D4—26no—
77C2×C4×D4D416yes—
78C2×#117C2316yes—
79C2×#117C2316yes—
80C2×#117C2313yes—
81C22×C4⋊C4—26no—
82C22×C4⋊C4—26no—
83C22×C4⋊C4C2316yes—
84C22×C4⋊C4C2316yes—
85C22×C42C2313yes—
86#405—224no—
87C4×M16—212no—
88C2×#97—212no—
89C2×#97C2311yescharacteristic
90C2×#97C2313yes—
91C2×#97C2316yes—
92C2×C4×C8C2316yes—
93#389—212no—
94#385—212no—
95#384—26no—
96#384C2316yes—
97#384C2313yes—
98#384C2311yescharacteristic
99#382C2316yes—
100#380—26no—
101C4×C4⋊C4—26no—
102#340C23112yes—
103#340C23124yes—
104#340C23112yes—
105#340C2314yes—
106#340C23112yes—
107C2×#41C22×C414yes—
108C2×#41C2×D414yes—
109C2×#41C2×Q818yes—
110C2×#41#4118yes—
111C22×Q8—26no—
112C22×Q8C2×D411yescharacteristic
113C22×D4C2×D413yes—
114C23×C4—26no—
115C23×C4—26no—
116C23×C4C2×D416yes—
117C23×C4C2413yes—
118#131—264no—
119C2×M16—212no—
120C2×M16—212no—
121C2×M16C2×Q8124yes—
122C22×C8—212no—
123C22×C8—212no—
124C22×C8C2×Q8124yes—
125C22×C8C2412yes—
126C22×C8C2416yes—
127C4⋊Q8—248no—
128C4⋊Q8—212no—
129C4⋊Q8C22×C4124yes—
130#125—248no—
131#125—248no—
132#125—26no—
133#125—212no—
134#125—22no—
135#125C22×C4112yes—
136#125C22×C4124yes—
137#125C22×C414yes—
138C22⋊Q8—248no—
139C22⋊Q8—296no—
140C4×Q8—216no—
141C4×Q8—248no—
142C4×Q8—48no—
143C4×Q8C22⋊C4116yes—
144C4×D4—248no—
145C4×D4—424no—
146C4×D4C22⋊C4148yes—
147#117—212no—
148#117—224no—
149#117C22×C4124yes—
150#117C22×C4112yes—
151#117C2×D4112yes—
152C2×C4⋊C4—248no—
153C2×C4⋊C4—26no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—26no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—224no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—812no—
165C2×C4⋊C4C22×C4124yes—
166C2×C4⋊C4C2×D416yes—
167C2×C4⋊C4C2416yes—
168C2×C4⋊C4C2412yes—
169C2×C4⋊C4C2416yes—
170C2×C4⋊C4C2412yes—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4C2×D416yes—
174C2×C42—26no—
175C2×C42—26no—
176C2×C42C22×C4112yes—
177C2×C42C2×D416yes—
178C2×C42C2×D4112yes—
179C2×C42C2413yes—
180C2×C42C2411yescharacteristic
181#97—22no—
182#97—26no—
183#97—212no—
184C4×C8—212no—
185#95—212no—
186C24C22×D413yes—
187#41C4×Q8132yes—
188#41C22⋊Q8132yes—
189C2×Q8—224no—
190C2×Q8—86no—
191C2×Q8—86no—
192C2×Q8C2×C22⋊C414yes—
193C2×Q8#12414yes—
194C2×Q8C4⋊Q814yes—
195C2×D4C2×C22⋊C4112yes—
196C2×D4#124112yes—
197C2×D4C4⋊Q8112yes—
198C22×C4—22no—
199C22×C4—26no—
200C22×C4—224no—
201C22×C4—22no—
202C22×C4—26no—
203C22×C4—46no—
204C22×C4—46no—
205C22×C4—412no—
206C22×C4—86no—
207C22×C4—86no—
208C22×C4—812no—
209C22×C4C2×C22⋊C4124yes—
210C22×C4C23×C416yes—
211C22×C4C22×D416yes—
212C22×C4C22×D412yes—
213C22×C4C22×D413yes—
214C22×C4C22×D416yes—
215C22×C4C22×Q8112yes—
216C22×C4C2×#41112yes—
217C22×C4C2511yescharacteristic
218M16—248no—
219C2×C8—248no—
220C2×C8—26no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—26no—
224C2×C8C22×Q814yes—
225C2×C8C22×Q8112yes—
226C4⋊C4—224no—
227C4⋊C4—248no—
228C4⋊C4—224no—
229C4⋊C4—28no—
230C4⋊C4—224no—
231C4⋊C4—26no—
232C4⋊C4—22no—
233C4⋊C4—212no—
234C4⋊C4—24no—
235C4⋊C4—44no—
236C4⋊C4—412no—
237C4⋊C4—412no—
238C4⋊C4—812no—
239C4⋊C4C2×C22⋊C4124yes—
240C4⋊C4C23×C4112yes—
241C4⋊C4C23×C414yes—
242C22⋊C4—224no—
243C22⋊C4—412no—
244C22⋊C4C2×C22⋊C4124yes—
245C42—224no—
246C42—412no—
247C42C2×C22⋊C4124yes—
248C42C23×C412yes—
249C42C23×C416yes—
250C42C22×D416yes—
251C42C22×D412yes—
252C23—26no—
253C23—26no—
254C23C22×C22⋊C416yes—
255C23C2×#12416yes—
256C23C2×C4⋊Q816yes—
257C23C23×D411yescharacteristic
258Q8—824no—
259Q8#391116yes—
260D4#391148yes—
261C2×C4—22no—
262C2×C4—26no—
263C2×C4—22no—
264C2×C4—26no—
265C2×C4—24no—
266C2×C4—212no—
267C2×C4—424no—
268C2×C4—46no—
269C2×C4—42no—
270C2×C4—46no—
271C2×C4—42no—
272C2×C4—412no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—46no—
276C2×C4—824no—
277C2×C4—1612no—
278C2×C4C22×C22⋊C414yes—
279C2×C4C22×C22⋊C4112yes—
280C2×C4C22×C22⋊C416yes—
281C2×C4C2×C4×Q8124yes—
282C2×C4C2×C22⋊Q8124yes—
283C2×C4C2×#12416yes—
284C2×C4C2×C4⋊Q816yes—
285C2×C4C24×C411yescharacteristic
286C2×C4C23×D411yescharacteristic
287C2×C4C23×Q812yes—
288C2×C4C22×#4112yes—
289C8—24no—
290C8—212no—
291C22—212no—
292C22—26no—
293C22—26no—
294C22—212no—
295C22C2×#391112yes—
296C22C23×C22⋊C411yescharacteristic
297C22C22×#12411yescharacteristic
298C22C22×C4⋊Q811yescharacteristic
299C22#358512yes—
300C22#358612yes—
301C4—24no—
302C4—212no—
303C4—412no—
304C4—44no—
305C4—86no—
306C4—82no—
307C4C2×#391112yes—
308C4C22×C4×Q812yes—
309C4C22×C22⋊Q812yes—
310C2—212no—
311C2C22×#39111yescharacteristic
312C2#6008912yes—
313C2#6062611yescharacteristic
314C2#6062711yescharacteristic
315C2#6066812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21326C2semidirect
5. #21325C2semidirect
6. #21322C2semidirect
7. #21321C2semidirect
8. C2×#1992C2semidirect
12. #3504C4semidirect
46. C2×#131Q8semidirect
52. #544C2×C4semidirect
54. #533C2×C4semidirect
60. C2×#125C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35042
C2×#28752
#341832
#213262
#213252
#213222
#213212
C2×#19922
#203482