Finite Group Databasebeta

← #141113

Subgroups — #141113

Order 512.

414 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 96 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#33215C211yesmaximal, characteristic
2#11724C212yesmaximal
3#11718C212yesmaximal
4#10711C211yesmaximal, characteristic
5#8601C211yesmaximal, characteristic
6C2×#500C411yescharacteristic, complemented
7C2×#405—21no—
8C2×#405C2211yescharacteristic
9#2628C2212yes—
10#1993C412yescomplemented
11#1944—21no—
12#1942—21no—
13#1846C411yescharacteristic, complemented
14#1809—21no—
15#1808—21no—
16C2×#344—21no—
17#1742C2211yescharacteristic
18#1740C2211yescharacteristic
19#1564—21no—
20#1458C2212yes—
21#1354—24no—
22#1352—24no—
23#1275—21no—
24C22×M16—21no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×#117—21no—
29C2×#117C2311yescharacteristic
30#500—41no—
31#500—41no—
32#431—22no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#106—21no—
37#405—24no—
38#405—41no—
39#405—41no—
40#405—42no—
41C2×#97—41no—
42C2×#97C2×C411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8—41no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#400—22no—
48#399—22no—
49#395—22no—
50#393—22no—
51#389—22no—
52#382C2×C412yes—
53#380—22no—
54#344—24no—
55#343—21no—
56#343D412yes—
57#342—21no—
58#342D412yes—
59#340—22no—
60#340—22no—
61#335—41no—
62#334—41no—
63#324—22no—
64#323—22no—
65C22×Q8—22no—
66C23×C4C2×D411yescharacteristic
67C2×M16—21no—
68C2×M16—21no—
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8SD1612yes—
86C22×C8Q1612yes—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89#117—22no—
90#117—22no—
91#117—41no—
92#117—41no—
93#117—41no—
94#117—41no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C2×D411yescharacteristic
101C2×C4⋊C4#4111yescharacteristic
102C2×C22⋊C4—22no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4C2×D411yescharacteristic
105C2×C22⋊C4#4111yescharacteristic
106C2×C42—41no—
107C2×C42C22×C411yescharacteristic
108C2×C42C2×D411yescharacteristic
109#107—42no—
110#107—41no—
111#107—41no—
112#107—42no—
113#106—42no—
114C4⋊C8—42no—
115C22⋊C8—42no—
116#97—41no—
117#97—41no—
118#97—44no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—42no—
124C4×C8—44no—
125#95—22no—
126#95—22no—
127#95—42no—
128#95—42no—
129C24#12111yescharacteristic
130C2×Q8—82no—
131C2×Q8—162no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C4×D411yescharacteristic
141C22×C4C4×D411yescharacteristic
142C22×C4#12111yescharacteristic
143C22×C4#12411yescharacteristic
144C22×C4#12711yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146C22×C4C2×Q1611yescharacteristic
147C22×C4#13711yescharacteristic
148C22×C4#13711yescharacteristic
149M16—24no—
150M16—42no—
151M16—42no—
152M16—42no—
153M16—42no—
154M16—42no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—82no—
170C2×C8#9911yescharacteristic
171C2×C8#9911yescharacteristic
172C2×C8#10312yes—
173C2×C8#10312yes—
174C2×C8#10412yes—
175C2×C8#10412yes—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4—162no—
186C22⋊C4—24no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—42no—
190C22⋊C4—162no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42—42no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23#39011yescharacteristic
202C23#46111yescharacteristic
203C23#46211yescharacteristic
204Q8—164no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4C2×#9911yescharacteristic
233C2×C4#41011yescharacteristic
234C2×C4C2×#10311yescharacteristic
235C2×C4#41911yescharacteristic
236C2×C4C2×#10411yescharacteristic
237C2×C4C2×#10411yescharacteristic
238C2×C4#46111yescharacteristic
239C2×C4#46211yescharacteristic
240C8—22no—
241C8—22no—
242C8—44no—
243C8—41no—
244C8—162no—
245C8#32812yes—
246C8#32812yes—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22#196211yescharacteristic
257C22#196411yescharacteristic
258C22#198111yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—84no—
266C4—162no—
267C4—162no—
268C4C2×#32811yescharacteristic
269C4#150311yescharacteristic
270C4#150611yescharacteristic
271C4#150811yescharacteristic
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2#971111yescharacteristic
277C2#1323111yescharacteristic
278C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#500C4semidirect
10. #1993C4semidirect
13. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332152
#117242
#117182
#107112
#86012