Finite Group Databasebeta

← #190091

Subgroups — #190091

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34048C211yesmaximal, characteristic
2#17357C212yesmaximal
3#8745C214yesmaximal, complemented
4#8744C214yesmaximal, complemented
5#7953C214yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2920C2214yes—
8#2314—22no—
9#2314—22no—
10#2313—22no—
11#2312—22no—
12#2298C412yes—
13#2298C412yes—
14#2298C2212yes—
15#2298C2212yes—
16#2285C2212yes—
17#2280C412yes—
18C4×#117C411yescharacteristic
19C4×#117C411yescharacteristic
20#1464C2214yes—
21#1464C2214yes—
22#1463—22no—
23#1462—22no—
24#1458—22no—
25#1458—22no—
26#1285C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C2×#117—24no—
29C2×#117—24no—
30C2×#117C2×C412yes—
31C2×#117C2×C412yes—
32C2×#117C2×C411yescharacteristic
33C2×#117C2×C411yescharacteristic
34C2×#117D412yes—
35C2×#117D412yes—
36C2×#117C2311yescharacteristic
37C2×#117C2311yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C22⋊C4D412yes—
40C22×C42C2×C411yescharacteristic
41C22×C42C2311yescharacteristic
42#432—24no—
43#432C2318yes—
44#431—24no—
45#431—24no—
46#423—24no—
47#407—24no—
48C4×M16—24no—
49#388—24no—
50#388—24no—
51#388—24no—
52#388—24no—
53#387—24no—
54#387—24no—
55#387—24no—
56#387—24no—
57#384—22no—
58#382—22no—
59#381—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—22no—
64#379—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2×C414yes—
68C4×C4⋊C4C2314yes—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—22no—
71C4×C22⋊C4C2×C414yes—
72C4×C22⋊C4C2×C414yes—
73#376C2×C412yes—
74#376C2×C412yes—
75C43C2×C412yes—
76C43C2×C412yes—
77#324—24no—
78#324—24no—
79#323—24no—
80#323—24no—
81C25C22⋊C411yescharacteristic
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—44no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D412yes—
93C2×M16—44no—
94#117—24no—
95#117—24no—
96#117—24no—
97#117—24no—
98#117—24no—
99#117—24no—
100#117—24no—
101#117—24no—
102#117—44no—
103#117—44no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4C22⋊C412yes—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4C22⋊C412yes—
121C2×C22⋊C4C22⋊C412yes—
122C2×C22⋊C4C2×D412yes—
123C2×C22⋊C4C2×D412yes—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—24no—
127C2×C42—24no—
128C2×C42C22⋊C412yes—
129C2×C42C22×C411yescharacteristic
130C2×C42C22×C411yescharacteristic
131C2×C42C22×C414yes—
132C2×C42C22×C414yes—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D412yes—
135C2×C42#4114yes—
136C2×C42#4114yes—
137C2×C42C2411yescharacteristic
138C4⋊C8—28no—
139C4⋊C8—44no—
140C22⋊C8—28no—
141C22⋊C8—44no—
142#97—28no—
143C4×C8—44no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150C24—21no—
151C24—21no—
152C24—42no—
153C24—82no—
154C24C2×C22⋊C411yescharacteristic
155C24C2×C22⋊C411yescharacteristic
156C24C2×C22⋊C411yescharacteristic
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C22⋊C412yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4#11712yes—
184C22×C4#11712yes—
185C22×C4#12314yes—
186C22×C4#12314yes—
187C22×C4C23×C411yescharacteristic
188C22×C4C22×D412yes—
189M16—88no—
190C2×C8—48no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—22no—
211C42—44no—
212C42—44no—
213C42—84no—
214C42—84no—
215C42#11714yes—
216C42#12314yes—
217C42#12314yes—
218C42C2×#4114yes—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—82no—
230C23—82no—
231C23—88no—
232C23#38111yescharacteristic
233C23#38111yescharacteristic
234C23C22×C22⋊C411yescharacteristic
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4#38114yes—
265C2×C4#38112yes—
266C2×C4#41314yes—
267C2×C4C2×#11712yes—
268C2×C4C2×#12314yes—
269C8—88no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—44no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22—82no—
282C22—84no—
283C22—84no—
284C22C2×#38111yescharacteristic
285C22C2×#41312yes—
286C4—22no—
287C4—24no—
288C4—24no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—84no—
294C4—84no—
295C4#182214yes—
296C2—22no—
297C2—41no—
298C2—41no—
299C2—81no—
300C2—81no—
301C2C2×#182211yescharacteristic
302C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ 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
3. #8745C2semidirect
4. #8744C2semidirect
5. #7953C2semidirect
26. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#340482
#173572
#87452
#87442
#79532