Finite Group Databasebeta

← #112267

Subgroups — #112267

Order 512.

1048 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2621C211yesmaximal, characteristic
2C2×#1558C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8030C212yesmaximal, complemented
5C22×C4⋊Q8C2211yescharacteristic
6C22×C22⋊Q8C412yes—
7C22×#117C2211yescharacteristic
8C2×#432C2214yes—
9#2621—24no—
10#2621—24no—
11C2×#398—22no—
12C2×#397—22no—
13C2×#391C412yes—
14C2×#382C2211yescharacteristic
15#1558—28no—
16C2×#333—24no—
17#1464—28no—
18C2×#324—22no—
19C2×#323—22no—
20#1295—24no—
21C23×Q8C2×C412yes—
22C24×C4C2×C411yescharacteristic
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—24no—
25C2×C4⋊Q8—24no—
26C2×C4⋊Q8D414yes—
27C2×C22⋊Q8—24no—
28C2×C22⋊Q8—24no—
29C2×C22⋊Q8—44no—
30C2×C22⋊Q8—44no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—24no—
34C2×#117—24no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4D412yes—
39C22×C22⋊C4—22no—
40C22×C22⋊C4D412yes—
41C22×C42C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#432—24no—
44#432—24no—
45#432—24no—
46#432D418yes—
47C2×C4⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×#97—24no—
50#398—216no—
51#397—216no—
52#391—24no—
53#391—24no—
54#382—22no—
55#382D414yes—
56C2×#95—24no—
57C2×#95—22no—
58#333—216no—
59#324—28no—
60#323—28no—
61C25C22⋊C411yescharacteristic
62C22×Q8—22no—
63C22×Q8—28no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—44no—
67C22×Q8C22⋊C414yes—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—42no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C22×C8—44no—
81C4⋊Q8—24no—
82C4⋊Q8—28no—
83C4⋊Q8—24no—
84C4⋊Q8—44no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C22⋊Q8—416no—
88C22⋊Q8—416no—
89#117—42no—
90#117—44no—
91#117—44no—
92#117—42no—
93#117—42no—
94#117—42no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C22⋊C414yes—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—44no—
116C2×C22⋊C4—44no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—24no—
120C2×C42C2×D412yes—
121C2×C42C2×D414yes—
122C2×C42C2×D412yes—
123C2×C42#4114yes—
124C4⋊C8—28no—
125C4⋊C8—44no—
126C22⋊C8—28no—
127C22⋊C8—44no—
128#97—28no—
129#97—44no—
130#95—216no—
131#95—28no—
132C24—21no—
133C24—21no—
134C24—22no—
135C24—22no—
136C24—44no—
137C24#10012yes—
138C24C2×C22⋊C411yescharacteristic
139C2×Q8—28no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—88no—
146C2×Q8—88no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—48no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4#10012yes—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C4×D414yes—
180C22×C4#12014yes—
181C22×C4#12012yes—
182C22×C4#12114yes—
183C22×C4#12312yes—
184C2×C8—48no—
185C2×C8—88no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—416no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—48no—
193C22⋊C4—416no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C42—42no—
197C42—42no—
198C42#12012yes—
199C42#12114yes—
200C42#12114yes—
201C42#12312yes—
202C42#12412yes—
203C42#12712yes—
204C23—24no—
205C23—22no—
206C23—24no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23—84no—
214C23—82no—
215C23—82no—
216C23#37911yescharacteristic
217C23#38612yes—
218C23C2×#10011yescharacteristic
219C23C2×#10011yescharacteristic
220C23#41312yes—
221Q8—88no—
222Q8—88no—
223C2×C4—28no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—48no—
236C2×C4—48no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—84no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#38612yes—
254C2×C4#39012yes—
255C2×C4#39214yes—
256C2×C4#39414yes—
257C8—88no—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—44no—
264C22—42no—
265C22—84no—
266C22—84no—
267C22—82no—
268C22—82no—
269C22#143811yescharacteristic
270C22#178311yescharacteristic
271C22#178511yescharacteristic
272C22#179614yes—
273C22#190012yes—
274C22#190012yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—24no—
279C4—42no—
280C4—84no—
281C4—84no—
282C4—82no—
283C2—22no—
284C2—22no—
285C2—82no—
286C2#852412yes—
287C2#852712yes—
288C2#1096711yescharacteristic
289C2#1098512yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1558C2semidirect
4. #8030C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#15582
C2×#14642
#80302