Finite Group Databasebeta

← #237925

Subgroups — #237925

Order 512.

828 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 186 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#37642C211yesmaximal, characteristic
2#21870C212yesmaximal
3#21321C211yesmaximal, characteristic
4#10359C218yesmaximal, complemented
5#8034C212yesmaximal
6#8031C211yesmaximal, characteristic
7C2×#533C2211yescharacteristic
8C2×#432C2212yes—
9C22×#97C2211yescharacteristic
10#2844C2214yes—
11#2628—22no—
12#2628—24no—
13#2571—22no—
14#2571—21no—
15#2551—22no—
16#2551—21no—
17#2285C2211yescharacteristic
18#2285C2212yes—
19#1993C2214yes—
20#1993C2214yes—
21#1992—21no—
22#1992—21no—
23#1960C2214yes—
24#1873—24no—
25#1803—22no—
26#1803—22no—
27#1735—22no—
28#1295C2214yes—
29#1295C2218yes—
30C23×C8—21no—
31#533—22no—
32#533—24no—
33#533Q814yescomplemented
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8C2312yes—
37C2×#125—21no—
38C2×C22⋊Q8—22no—
39C2×C4×Q8—22no—
40C2×#117—21no—
41C2×#117D412yes—
42C2×#117D414yes—
43C22×C42C2311yescharacteristic
44#432—24no—
45#432—24no—
46C2×C4⋊C8—22no—
47C2×#103—24no—
48C2×#103—24no—
49C2×C22⋊C8—22no—
50C2×#97—24no—
51C2×#97—21no—
52C2×#97C2312yes—
53C2×#97C2314yes—
54#400—24no—
55#400—28no—
56#399—24no—
57#395—24no—
58#395—28no—
59#393—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#384D412yes—
64#384D412yes—
65#384D411yescharacteristic
66#384D411yescharacteristic
67#382—24no—
68#382—22no—
69#382C2314yes—
70#382C2312yes—
71#380—22no—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81C22×Q8—24no—
82C22×Q8—22no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C22×C8—24no—
86C22×C8—22no—
87C22×C8—24no—
88C22×C8—42no—
89C22×C8—42no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—22no—
93C4⋊Q8—88no—
94C4⋊Q8—84no—
95C4⋊Q8C2×Q814yes—
96#125—24no—
97#125—42no—
98C22⋊Q8—28no—
99C22⋊Q8—44no—
100C4×Q8—28no—
101C4×Q8—44no—
102#117—24no—
103#117—42no—
104#117—44no—
105#117—84no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—41no—
124C2×C42—24no—
125C2×C42—22no—
126C2×C42—41no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×Q812yes—
130C2×C42#4114yes—
131C2×C42#4112yes—
132C2×C42C2411yescharacteristic
133C4⋊C8—44no—
134#103—84no—
135#103—84no—
136C22⋊C8—44no—
137#97—24no—
138#97—24no—
139#97—24no—
140#97—42no—
141#95—24no—
142#95—28no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147C24#12011yescharacteristic
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—84no—
155C2×Q8—88no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#12012yes—
166C22×C4#12011yescharacteristic
167C22×C4C22⋊Q812yes—
168C22×C4C22⋊Q814yes—
169C22×C4#12412yes—
170C22×C4#12414yes—
171C22×C4C22×D411yescharacteristic
172C22×C4C22×D412yes—
173C2×C8—24no—
174C2×C8—42no—
175C2×C8—48no—
176C2×C8—42no—
177C2×C8—44no—
178C2×C8—44no—
179C2×C8—44no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—48no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—84no—
196C4⋊C4—88no—
197C22⋊C4—44no—
198C22⋊C4—82no—
199C22⋊C4—84no—
200C42—48no—
201C42—44no—
202C42—44no—
203C42C22⋊Q814yes—
204C42C22⋊Q812yes—
205C42#12412yes—
206C42#12414yes—
207C42C22×Q811yescharacteristic
208C42C2×#4111yescharacteristic
209C42C2×#4112yes—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23#39312yes—
215C23C2×#12011yescharacteristic
216Q8—44no—
217Q8—44no—
218Q8—82no—
219Q8—82no—
220Q8—168no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#39314yes—
245C2×C4#39312yes—
246C2×C4#45512yes—
247C2×C4#45612yes—
248C2×C4C2×C22⋊Q811yescharacteristic
249C2×C4C2×C22⋊Q812yes—
250C2×C4C2×#12411yescharacteristic
251C2×C4C2×#12412yes—
252C8—44no—
253C8—84no—
254C22—22no—
255C22—24no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22C2×#39311yescharacteristic
260C22C2×#45511yescharacteristic
261C22C2×#45611yescharacteristic
262C22#301912yes—
263C22#302012yes—
264C4—21no—
265C4—21no—
266C4—22no—
267C4—22no—
268C4—24no—
269C4—82no—
270C4—84no—
271C4—82no—
272C4—84no—
273C4—84no—
274C4#202814yes—
275C2—42no—
276C2#1905611yescharacteristic
277C2#1905811yescharacteristic
278C2#1906612yes—
279C2C2×#202811yescharacteristic
280C2#2346212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10359C2semidirect
33. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#218702
#213212
#103592
#80342
#80312