Finite Group Databasebeta

← #214055

Subgroups — #214055

Order 512.

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

normal subgroups only

This group has 785 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#37420C211yesmaximal, characteristic
2#17410C212yesmaximal, complemented
3#9458C214yesmaximal, complemented
4#9450C214yesmaximal, complemented
5#8743C211yesmaximal, characteristic, complemented
6#8742C211yesmaximal, characteristic, complemented
7#8526C212yesmaximal
8C2×#541C2211yescharacteristic
9C2×#540C2211yescharacteristic
10#2983C2214yes—
11#2850C412yes—
12#2707—24no—
13#2691—24no—
14#2620C412yes—
15#2566C412yes—
16#2565C2212yes—
17#2548C2212yes—
18#2547C412yes—
19#2296C2211yescharacteristic
20#1555C2214yes—
21#1534C2214yes—
22#1473—24no—
23#1470—24no—
24#1467C2214yes—
25#1464C2212yes—
26#1464C2212yes—
27#1285C2214yes—
28#1285C2214yes—
29#541—44no—
30#540—44no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#127C2311yescharacteristic
33C2×#126D414yes—
34C2×#125C2×C411yescharacteristic
35C2×#125D414yes—
36C2×#125C2311yescharacteristic
37C2×#124C2×C412yes—
38C2×#124C2×C412yes—
39C2×#123—22no—
40C2×C22⋊Q8C2×C412yes—
41C2×#121—22no—
42C2×#121C2×C412yes—
43C2×C4×Q8C2312yes—
44C2×C4×D4C2×C412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2311yescharacteristic
47C22×C42C2×C411yescharacteristic
48#446—44no—
49#446—44no—
50#432—24no—
51#432C2314yes—
52#432C2314yes—
53#431—24no—
54#398—24no—
55#397—24no—
56#396—24no—
57#396—24no—
58#394—24no—
59#394—24no—
60#391C2×C412yes—
61#391C2×C412yes—
62#390—24no—
63#390C2×C412yes—
64#390C2×C412yes—
65#389—22no—
66#388—24no—
67#387—24no—
68#386—24no—
69#385—24no—
70#385—22no—
71#384C2×C412yes—
72#382C2×C412yes—
73#380—24no—
74C4×C4⋊C4C2×C412yes—
75C4×C4⋊C4C2312yes—
76#331—44no—
77#330—44no—
78#324—22no—
79#324—22no—
80#324—28no—
81#323—22no—
82#323—22no—
83C22×Q8C22×C412yes—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4C22×C412yes—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4C22×C411yescharacteristic
90C4⋊Q8—42no—
91#127—24no—
92#126—44no—
93#126—44no—
94#125—24no—
95#125—42no—
96#125—44no—
97#125—44no—
98#124—82no—
99#124—82no—
100#123—48no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103#121—48no—
104#121—44no—
105C4×Q8—24no—
106C4×Q8—24no—
107C4×D4—44no—
108#117—24no—
109#117—81no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4C22⋊C414yes—
116C2×C4⋊C4C22⋊C414yes—
117C2×C4⋊C4C22⋊C414yes—
118C2×C4⋊C4C22×C412yes—
119C2×C4⋊C4C22×C412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D414yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4C22⋊C414yes—
130C2×C22⋊C4C2×D412yes—
131C2×C42—21no—
132C2×C42—81no—
133C2×C42C22×C411yescharacteristic
134C2×C42C22×C412yes—
135C2×C42C22×C412yes—
136C2×C42C2×D414yes—
137C2×C42C2411yescharacteristic
138C4⋊C8—24no—
139C4⋊C8—24no—
140C4⋊C8—44no—
141C4⋊C8—44no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144#97—24no—
145#97—24no—
146#97—44no—
147C4×C8—44no—
148#95—22no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—22no—
153C24—22no—
154C24C2×C22⋊C411yescharacteristic
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—84no—
165C2×D4—84no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—44no—
173C22×C4—81no—
174C22×C4—82no—
175C22×C4—84no—
176C22×C4C2×C22⋊C414yes—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×C22⋊C412yes—
179C22×C4C2×C22⋊C414yes—
180C22×C4C23×C411yescharacteristic
181C22×C4C22×D412yes—
182C2×C8—44no—
183C2×C8—44no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—81no—
195C4⋊C4—82no—
196C4⋊C4—84no—
197C4⋊C4—82no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—44no—
202C22⋊C4—84no—
203C22⋊C4—84no—
204C22⋊C4—81no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—22no—
208C42—24no—
209C42—24no—
210C42—42no—
211C42—42no—
212C42—81no—
213C42H(5,𝔽2)14yes—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—42no—
218C23—44no—
219C23—44no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23C22×C22⋊C411yescharacteristic
224Q8—82no—
225Q8—82no—
226D4—84no—
227D4—164no—
228D4—164no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—164no—
253C2×C4#51812yes—
254C2×C4#53414yes—
255C8—84no—
256C8—84no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—81no—
261C22—81no—
262C22—84no—
263C22—84no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22#230811yescharacteristic
268C22#288811yescharacteristic
269C22#288911yescharacteristic
270C4—24no—
271C4—22no—
272C4—44no—
273C4—42no—
274C4—41no—
275C4—44no—
276C4—82no—
277C4—82no—
278C4—82no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—162no—
283C2#2175711yescharacteristic
284C2#2178512yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17410C2semidirect
3. #9458C2semidirect
4. #9450C2semidirect
5. #8743C2semidirect
6. #8742C2semidirect

Maximal subgroups

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

subgroupindex
#374202
#174102
#94582
#94502
#87432
#87422
#85262