Finite Group Databasebeta

← #237972

Subgroups — #237972

Order 512.

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

normal subgroups only

This group has 774 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#35344C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#17584C212yesmaximal
4#10367C214yesmaximal, complemented
5#10363C214yesmaximal, complemented
6#7962C212yesmaximal
7#7868C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2212yes—
10C2×#423C2211yescharacteristic
11#2864C2212yes—
12#2862C2212yes—
13#2628—22no—
14#2607—24no—
15#2605—24no—
16#2331—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#533Q812yescomplemented
35#533C2312yes—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8Q812yescomplemented
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#400—24no—
56#399—24no—
57#395—24no—
58#393—24no—
59#391—28no—
60#391—28no—
61#389—24no—
62#387—28no—
63#387—28no—
64#383D414yes—
65#382—22no—
66#382C2314yes—
67#381—22no—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4D414yes—
73#376—22no—
74#376D412yes—
75C43C2311yescharacteristic
76C43C2311yescharacteristic
77#339—24no—
78#339—24no—
79#328—216no—
80C22×Q8—24no—
81C22×Q8—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—22no—
85C2×M16—41no—
86C2×M16—42no—
87C2×M16—44no—
88C22×C8—41no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8C2×Q814yes—
95#125—22no—
96#125—22no—
97#125—42no—
98C22⋊Q8—24no—
99C22⋊Q8—24no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C4×Q8—24no—
103C4×Q8—24no—
104C4×Q8—42no—
105C4×Q8—42no—
106#117—24no—
107#117—22no—
108#117—22no—
109#117—42no—
110#117—41no—
111#117—41no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—42no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—41no—
122C2×C42—24no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—41no—
126C2×C42C2×D412yes—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×Q811yescharacteristic
131C2×C42C2×Q811yescharacteristic
132C2×C42#4112yes—
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4112yes—
136C2×C42C2411yescharacteristic
137C4⋊C8—24no—
138C4⋊C8—24no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141C22⋊C8—24no—
142C22⋊C8—42no—
143#97—42no—
144C4×C8—24no—
145#95—24no—
146#95—24no—
147#95—28no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95C2×D418yes—
153C24#12011yescharacteristic
154C2×Q8—24no—
155C2×Q8—48no—
156C2×Q8—42no—
157C2×Q8—88no—
158C2×Q8—84no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—28no—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—82no—
166C22×C4#12011yescharacteristic
167C22×C4#12012yes—
168C22×C4C22⋊Q811yescharacteristic
169C22×C4C22⋊Q811yescharacteristic
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q812yes—
172C22×C4#12411yescharacteristic
173C22×C4#12411yescharacteristic
174C22×C4#12412yes—
175C22×C4#12412yes—
176C22×C4C22×D412yes—
177C22×C4C22×D411yescharacteristic
178M16—44no—
179M16—82no—
180M16—84no—
181C2×C8—44no—
182C2×C8—42no—
183C2×C8—82no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—44no—
187C4⋊C4—48no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—88no—
195C22⋊C4—44no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—84no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—42no—
203C42—42no—
204C42—84no—
205C42—82no—
206C42—82no—
207C42C22⋊Q812yes—
208C42C22⋊Q811yescharacteristic
209C42C22⋊Q812yes—
210C42C22⋊Q811yescharacteristic
211C42#12412yes—
212C42#12412yes—
213C42#12412yes—
214C42#13714yes—
215C42C22×Q811yescharacteristic
216C42C2×#4111yescharacteristic
217C42C2×#4112yes—
218C23—22no—
219C23—42no—
220C23#39311yescharacteristic
221C23#39311yescharacteristic
222C23C2×#12011yescharacteristic
223Q8—84no—
224Q8—84no—
225Q8—168no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—48no—
247C2×C4—84no—
248C2×C4#39311yescharacteristic
249C2×C4#39311yescharacteristic
250C2×C4#39312yes—
251C2×C4#39312yes—
252C2×C4C2×C22⋊Q812yes—
253C2×C4C2×C22⋊Q811yescharacteristic
254C2×C4C2×#12412yes—
255C2×C4C2×#12411yescharacteristic
256C2×C4C2×#13712yes—
257C2×C4#57512yes—
258C8—84no—
259C8—82no—
260C22—22no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—82no—
266C22C2×#39311yescharacteristic
267C22#300712yes—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—22no—
274C4—44no—
275C4—82no—
276C4—88no—
277C4—84no—
278C4#307812yes—
279C4#313911yescharacteristic
280C4#314011yescharacteristic
281C2—22no—
282C2—41no—
283C2—41no—
284C2#2345511yescharacteristic
285C2#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. #10367C2semidirect
5. #10363C2semidirect
34. #533Q8semidirect
37. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#353442
#213292
#175842
#103672
#103632
#79622
#78682