Finite Group Databasebeta

← #214057

Subgroups — #214057

Order 512.

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

normal subgroups only

This group has 745 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#37387C211yesmaximal, characteristic
2#17410C212yesmaximal
3#9452C214yesmaximal, complemented
4#9450C214yesmaximal, complemented
5#8743C211yesmaximal, characteristic
6#8742C211yesmaximal, characteristic
7#8527C212yesmaximal
8C2×#541C2211yescharacteristic
9C2×#541C2211yescharacteristic
10#2983C2214yes—
11#2857C412yes—
12#2726—24no—
13#2722—24no—
14#2621C412yes—
15#2565C412yes—
16#2565C2212yes—
17#2546C412yes—
18#2546C2212yes—
19#2296C2211yescharacteristic
20#1558C2214yes—
21#1534C2214yes—
22#1473—24no—
23#1470—24no—
24#1467C2214yes—
25#1464C2212yes—
26#1464C2212yes—
27#1285C2214yes—
28#1285C2214yes—
29#541—44no—
30#541—44no—
31C2×C4⋊Q8C2×C412yes—
32C2×C4⋊Q8C2×C412yes—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4⋊Q8C2311yescharacteristic
35C2×#126—22no—
36C2×#126D414yes—
37C2×#125—22no—
38C2×#125C2×C411yescharacteristic
39C2×#125D414yes—
40C2×#125C2311yescharacteristic
41C2×C22⋊Q8C2×C412yes—
42C2×C22⋊Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2312yes—
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#398—24no—
56#398—24no—
57#398—24no—
58#398—24no—
59#397—24no—
60#391C2×C412yes—
61#391C2×C412yes—
62#391C2×C412yes—
63#391C2×C412yes—
64#389—24no—
65#389—24no—
66#389—22no—
67#389—24no—
68#387—24no—
69#387—24no—
70#385—22no—
71#384C2×C412yes—
72#382C2×C412yes—
73#380—24no—
74C4×C4⋊C4C2×C412yes—
75C4×C4⋊C4C2312yes—
76#333—44no—
77#330—44no—
78#324—22no—
79#324—22no—
80#324—28no—
81#323—22no—
82#323—22no—
83C22×Q8C22×C412yes—
84C22×Q8C22×C412yes—
85C23×C4—21no—
86C23×C4C22×C411yescharacteristic
87C4⋊Q8—24no—
88C4⋊Q8—42no—
89C4⋊Q8—82no—
90C4⋊Q8—82no—
91#126—44no—
92#126—44no—
93#126—48no—
94#125—24no—
95#125—42no—
96#125—44no—
97#125—44no—
98#125—48no—
99C22⋊Q8—42no—
100C22⋊Q8—42no—
101C22⋊Q8—44no—
102C4×Q8—24no—
103C4×Q8—24no—
104C4×Q8—44no—
105#117—24no—
106#117—81no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4C22⋊C414yes—
118C2×C4⋊C4C22⋊C414yes—
119C2×C4⋊C4C22⋊C414yes—
120C2×C4⋊C4C22×C412yes—
121C2×C4⋊C4C22×C412yes—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D414yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4C22⋊C414yes—
127C2×C22⋊C4C2×D412yes—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42—81no—
131C2×C42C22×C411yescharacteristic
132C2×C42C22×C412yes—
133C2×C42C22×C412yes—
134C2×C42C2×D414yes—
135C2×C42C2411yescharacteristic
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8—44no—
139C4⋊C8—44no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142#97—24no—
143#97—24no—
144#97—44no—
145C4×C8—44no—
146#95—22no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—22no—
153C24C2×C22⋊C411yescharacteristic
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—24no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—44no—
169C22×C4—81no—
170C22×C4C2×C22⋊C414yes—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C22⋊C414yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4C23×C411yescharacteristic
175C22×C4C22×D412yes—
176C2×C8—44no—
177C2×C8—44no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—48no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—81no—
195C4⋊C4—84no—
196C4⋊C4—84no—
197C22⋊C4—24no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—81no—
201C42—22no—
202C42—24no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42—44no—
207C42—81no—
208C42H(5,𝔽2)14yes—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—42no—
213C23C22×C22⋊C411yescharacteristic
214Q8—82no—
215Q8—82no—
216Q8—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—84no—
243C2×C4#51812yes—
244C2×C4#53414yes—
245C8—84no—
246C8—84no—
247C22—22no—
248C22—42no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22#230811yescharacteristic
253C22#288811yescharacteristic
254C22#288911yescharacteristic
255C4—24no—
256C4—22no—
257C4—44no—
258C4—42no—
259C4—44no—
260C4—41no—
261C4—44no—
262C4—82no—
263C4—82no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—81no—
268C2#2175711yescharacteristic
269C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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
3. #9452C2semidirect
4. #9450C2semidirect

Maximal subgroups

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

subgroupindex
#373872
#174102
#94522
#94502
#87432
#87422
#85272