Finite Group Databasebeta

← #237973

Subgroups — #237973

Order 512.

766 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 182 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#35345C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#17584C212yesmaximal
4#10369C214yesmaximal, complemented
5#10365C214yesmaximal, complemented
6#7962C212yesmaximal
7#7868C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2212yes—
10C2×#423C2211yescharacteristic
11#2864C2212yes—
12#2859C2212yes—
13#2629—22no—
14#2608—24no—
15#2606—24no—
16#2330—22no—
17#2283C2212yes—
18C4×#117C2211yescharacteristic
19#1995—22no—
20#1994C2212yes—
21#1994C2212yes—
22#1733—22no—
23#1518C2218yes—
24#1515—24no—
25#1508—24no—
26#1286C2218yes—
27#1276C2212yes—
28#1276C2212yes—
29C22×M16—21no—
30#533—21no—
31#533—22no—
32#533—21no—
33#533—22no—
34#533Q812yes—
35#533C2312yes—
36C2×C4⋊Q8Q812yes—
37C2×#125—22no—
38C2×#125—21no—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×#117—21no—
42C2×#117D412yes—
43C22×C42C2311yescharacteristic
44#432—22no—
45#432—22no—
46#431—22no—
47#431C2314yes—
48#423—22no—
49#423—22no—
50#423—21no—
51#423C2312yes—
52C2×C4⋊C8—22no—
53#407—22no—
54C4×M16—22no—
55#398—28no—
56#396—28no—
57#389—28no—
58#389—24no—
59#389—28no—
60#387—28no—
61#387—28no—
62#384—22no—
63#383D414yes—
64#382C2314yes—
65#381—22no—
66#380—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4D414yes—
71#376—22no—
72#376D412yes—
73C43C2311yescharacteristic
74C43C2311yescharacteristic
75#339—24no—
76#339—24no—
77#328—216no—
78C22×Q8—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D412yes—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—42no—
84C2×M16—44no—
85C22×C8—41no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8—22no—
89C4⋊Q8C2×Q814yes—
90#125—22no—
91#125—22no—
92#125—44no—
93#125—42no—
94#125—44no—
95C22⋊Q8—24no—
96C22⋊Q8—24no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C4×Q8—24no—
100C4×Q8—24no—
101C4×Q8—42no—
102C4×Q8—42no—
103#117—24no—
104#117—22no—
105#117—22no—
106#117—42no—
107#117—41no—
108#117—41no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—42no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—41no—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—41no—
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×Q811yescharacteristic
129C2×C42C2×Q811yescharacteristic
130C2×C42#4112yes—
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4112yes—
134C2×C42C2411yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—24no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C22⋊C8—24no—
140C22⋊C8—42no—
141#97—42no—
142C4×C8—24no—
143#95—28no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95C2×D418yes—
149C24#12011yescharacteristic
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—84no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—28no—
158C22×C4—42no—
159C22×C4—82no—
160C22×C4#12011yescharacteristic
161C22×C4#12012yes—
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164C22×C4C22⋊Q812yes—
165C22×C4C22⋊Q812yes—
166C22×C4#12411yescharacteristic
167C22×C4#12411yescharacteristic
168C22×C4#12412yes—
169C22×C4#12412yes—
170C22×C4C22×D412yes—
171C22×C4C22×D411yescharacteristic
172M16—44no—
173M16—82no—
174M16—84no—
175C2×C8—44no—
176C2×C8—42no—
177C2×C8—82no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—44no—
181C4⋊C4—416no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—88no—
189C4⋊C4—88no—
190C22⋊C4—44no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—84no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—42no—
198C42—42no—
199C42—84no—
200C42—82no—
201C42—82no—
202C42C22⋊Q812yes—
203C42C22⋊Q811yescharacteristic
204C42C22⋊Q812yes—
205C42C22⋊Q811yescharacteristic
206C42#12412yes—
207C42#12412yes—
208C42#12412yes—
209C42#13714yes—
210C42C22×Q811yescharacteristic
211C42C2×#4111yescharacteristic
212C42C2×#4112yes—
213C23—22no—
214C23—42no—
215C23#39311yescharacteristic
216C23#39311yescharacteristic
217C23C2×#12011yescharacteristic
218Q8—84no—
219Q8—84no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—48no—
241C2×C4—84no—
242C2×C4#39311yescharacteristic
243C2×C4#39311yescharacteristic
244C2×C4#39312yes—
245C2×C4#39312yes—
246C2×C4C2×C22⋊Q812yes—
247C2×C4C2×C22⋊Q811yescharacteristic
248C2×C4C2×#12412yes—
249C2×C4C2×#12411yescharacteristic
250C2×C4C2×#13712yes—
251C2×C4#57512yes—
252C8—84no—
253C8—82no—
254C22—22no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—22no—
259C22—82no—
260C22C2×#39311yescharacteristic
261C22#300712yes—
262C4—21no—
263C4—21no—
264C4—22no—
265C4—22no—
266C4—22no—
267C4—22no—
268C4—44no—
269C4—82no—
270C4—88no—
271C4—84no—
272C4#307812yes—
273C4#313911yescharacteristic
274C4#314011yescharacteristic
275C2—22no—
276C2—41no—
277C2—41no—
278C2#2345511yescharacteristic
279C2#2347212yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10369C2semidirect
5. #10365C2semidirect

Maximal subgroups

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

subgroupindex
#353452
#213292
#175842
#103692
#103652
#79622
#78682