Finite Group Databasebeta

← #223155

Subgroups — #223155

Order 512.

783 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic
2#21270C212yesmaximal
3#9891C212yesmaximal, complemented
4#9125C214yesmaximal, complemented
5#8722C212yesmaximal, complemented
6#8580C211yesmaximal, characteristic, complemented
7C4×#330C211yesmaximal, characteristic, complemented
8#8031C212yesmaximal, complemented
9C2×#432C2214yes—
10#2859C2211yescharacteristic
11#2851C2212yes—
12#2552C412yes—
13#2522—24no—
14#2454—22no—
15#2441C414yes—
16#2370—22no—
17#2354C412yes—
18C4×#125C2211yescharacteristic
19#2285C2212yes—
20C42×D4C2211yescharacteristic
21#1954—22no—
22#1939—22no—
23#1924—24no—
24#1767C2212yes—
25#1749C2214yes—
26C4×#97C2212yes—
27C2×#330C2212yes—
28C2×#330C2212yes—
29#1295C2211yescharacteristic
30#1295C2212yes—
31#1295C2214yes—
32#1295C2211yescharacteristic
33#1295C2214yes—
34C2×#125—22no—
35C2×#125C2×C412yes—
36C2×#125C2311yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D412yes—
39C2×#117C2×C414yes—
40C22×C42C2312yes—
41#432—28no—
42#432—44no—
43C2×C4⋊C8—24no—
44C2×C4⋊C8—22no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—24no—
47C2×#97—22no—
48C2×#97C2314yes—
49C2×#97C2312yes—
50C2×#97C2312yes—
51C2×C4×C8—22no—
52#399—24no—
53#398—24no—
54#396—24no—
55#390—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389—24no—
60#389C2×C412yes—
61#389C2×C414yes—
62#388—24no—
63#387—22no—
64#387—24no—
65#387—24no—
66#385—24no—
67#384C2×C412yes—
68#384C2×C414yes—
69#382C2311yescharacteristic
70#382C2312yes—
71#380—24no—
72#380C2×C414yes—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4C2×C412yes—
77C4×C4⋊C4C2×C411yescharacteristic
78C4×C4⋊C4D412yes—
79C4×C22⋊C4C2×C412yes—
80C4×C22⋊C4D414yes—
81C43C2311yescharacteristic
82#336—22no—
83#336—24no—
84#330—24no—
85#330—42no—
86C22×D4C22⋊C411yescharacteristic
87C23×C4C22×C412yes—
88C23×C4C2×D414yes—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—24no—
92#125—22no—
93#125—21no—
94#125—21no—
95#125—48no—
96#125—82no—
97C4×D4—24no—
98C4×D4—44no—
99#117—84no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22×C412yes—
111C2×C4⋊C4C22×C412yes—
112C2×C4⋊C4C22×C414yes—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4C22⋊C412yes—
116C2×C22⋊C4C22⋊C414yes—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—24no—
122C2×C42C22×C411yescharacteristic
123C2×C42C22×C412yes—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2×D414yes—
127C2×C42C2×D412yes—
128C2×C42C2411yescharacteristic
129C4⋊C8—48no—
130C4⋊C8—44no—
131C22⋊C8—48no—
132C22⋊C8—48no—
133#97—22no—
134#97—24no—
135#97—22no—
136#97—24no—
137#97—22no—
138#97—44no—
139#97#4114yes—
140C4×C8—44no—
141#95—24no—
142#95—22no—
143#95—22no—
144#95—24no—
145#95—22no—
146#95—24no—
147#95C22⋊C414yes—
148#95C22⋊C412yes—
149C24C2×C22⋊C412yes—
150C2×D4—22no—
151C2×D4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×C22⋊C412yes—
165C22×C4C2×C22⋊C414yes—
166C22×C4C2×C22⋊C412yes—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C23×C411yescharacteristic
169C22×C4C22×D412yes—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—44no—
178C2×C8—44no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C4⋊C4—48no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4#10114yes—
194C22⋊C4—24no—
195C22⋊C4—44no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C42—22no—
199C42—22no—
200C42—24no—
201C42—24no—
202C42—22no—
203C42—24no—
204C42—42no—
205C42—41no—
206C42—44no—
207C42—84no—
208C42C2×#4112yes—
209C42H(5,𝔽2)11yescharacteristic
210C422−1+411yescharacteristic
211C23—22no—
212C23—22no—
213C23—22no—
214C23—24no—
215C23—24no—
216C23C22×C22⋊C411yescharacteristic
217D4—44no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4C2×#10114yes—
246C2×C4C2×#10112yes—
247C2×C4#42114yes—
248C2×C4C4×#4111yescharacteristic
249C2×C4#52011yescharacteristic
250C2×C4#54612yes—
251C2×C4#54712yes—
252C8—44no—
253C8—44no—
254C8—84no—
255C22—44no—
256C22—42no—
257C22—42no—
258C22—44no—
259C22—42no—
260C22#233911yescharacteristic
261C22C22×#10111yescharacteristic
262C22#288811yescharacteristic
263C22C2×#42112yes—
264C22#290112yes—
265C4—21no—
266C4—21no—
267C4—24no—
268C4—22no—
269C4—22no—
270C4—42no—
271C4—41no—
272C4—44no—
273C4—82no—
274C4—84no—
275C4—82no—
276C4#153714yes—
277C2—82no—
278C2C2×#153711yescharacteristic
279C2#1768412yes—
280C2#1769211yescharacteristic
281C2#2248111yescharacteristic
282C2#2250712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9891C2semidirect
4. #9125C2semidirect
5. #8722C2semidirect
6. #8580C2semidirect
7. C4×#330C2semidirect
8. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#212702
#98912
#91252
#87222
#85802
C4×#3302
#80312