Finite Group Databasebeta

← #391840

Subgroups — #391840

Order 512.

1152 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 242 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#34634C211yesmaximal, characteristic
2#20683C211yesmaximal, characteristic
3#13852C211yesmaximal, characteristic
4#11386C212yesmaximal, complemented
5#11359C212yesmaximal, complemented
6#11226C214yesmaximal, complemented
7#11201C214yesmaximal
8C2×#533C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#407C2212yes—
11#2629C414yescomplemented
12#2628C414yescomplemented
13#2627—22no—
14#2626—22no—
15#2396—24no—
16#2296C2212yes—
17#2285C2211yescharacteristic
18#2280C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21#1818—24no—
22C2×#339C2218yes—
23C2×#339C2218yes—
24#1728C2212yes—
25#1728C2212yes—
26#1441—22no—
27#1439—22no—
28C4×#95C2212yes—
29C24×C4C2311yescharacteristic
30C22×M16C2314yes—
31#533—44no—
32C2×C4⋊Q8C2×C412yes—
33C2×#125C2×C412yes—
34C2×C22⋊Q8—22no—
35C2×C4×Q8—22no—
36C2×#117—44no—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117D414yes—
40C2×#117C2312yes—
41C2×#117C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D412yes—
44C22×C22⋊C4—22no—
45C22×C22⋊C4D412yes—
46C22×C42C2314yes—
47C22×C42C2311yescharacteristic
48C22×C42C2311yescharacteristic
49#407—44no—
50C2×C22⋊C8—24no—
51#400—24no—
52#399—24no—
53#398—28no—
54#396—28no—
55#395—24no—
56#393—24no—
57#389C2×C418yes—
58#385—24no—
59#384C2×C412yes—
60#382C2×C412yes—
61#381—22no—
62#380—22no—
63#380C2×C418yes—
64#379—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—44no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—44no—
70C2×#95—24no—
71C2×#95—22no—
72C2×#95—22no—
73#339—48no—
74#339—48no—
75#336—24no—
76#336—24no—
77C25C2×D411yescharacteristic
78C22×Q8—22no—
79C23×C4—24no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4C22×C412yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2411yescharacteristic
90C2×M16—24no—
91C2×M16—28no—
92C2×M16—28no—
93C2×M16—24no—
94C22×C8—24no—
95C4⋊Q8—44no—
96#125—44no—
97C22⋊Q8—44no—
98C22⋊Q8—44no—
99C4×Q8—48no—
100#117—28no—
101#117—24no—
102#117—24no—
103#117—44no—
104#117—44no—
105#117—48no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4C22⋊C414yes—
115C2×C4⋊C4C22⋊C414yes—
116C2×C4⋊C4C22×C414yes—
117C2×C4⋊C4C22×C414yes—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4C22⋊C414yes—
126C2×C22⋊C4C22⋊C414yes—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—28no—
132C2×C42—28no—
133C2×C42—28no—
134C2×C42C22×C411yescharacteristic
135C2×C42C22×C411yescharacteristic
136C2×C42C22×C412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C22⋊C8—48no—
140#95—28no—
141#95—24no—
142#95—28no—
143#95—48no—
144#95—44no—
145#95—44no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—22no—
150C24—44no—
151C24C2×C22⋊C412yes—
152C24C22×D411yescharacteristic
153C2×Q8—44no—
154C2×Q8—84no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4C2×C22⋊C412yes—
183C22×C4C2×C22⋊C414yes—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C412yes—
186C22×C4C23×C411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4H(5,𝔽2)14yes—
189C22×C4H(5,𝔽2)12yes—
190C22×C42−1+414yes—
191C22×C42−1+412yes—
192M16—48no—
193M16—48no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—44no—
197C4⋊C4—28no—
198C4⋊C4—28no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—416no—
204C4⋊C4—84no—
205C4⋊C4—88no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—416no—
209C22⋊C4—44no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—88no—
213C42—24no—
214C42—22no—
215C42—22no—
216C42—24no—
217C42—24no—
218C42—44no—
219C42—44no—
220C42—416no—
221C23—24no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—28no—
226C23—44no—
227C23—44no—
228C23—42no—
229C23—42no—
230C23—82no—
231C23—82no—
232C23—84no—
233C23C22×C22⋊C411yescharacteristic
234C23#53412yes—
235C23#53511yescharacteristic
236C23#53612yes—
237C23#53611yescharacteristic
238Q8—88no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—28no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4#51812yes—
268C2×C4#51912yes—
269C2×C4#52012yes—
270C2×C4#53412yes—
271C2×C4#53612yes—
272C2×C4#53712yes—
273C2×C4#56218yes—
274C2×C4#56318yes—
275C8—84no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—22no—
281C22—44no—
282C22—42no—
283C22—44no—
284C22—44no—
285C22—84no—
286C22—84no—
287C22#230811yescharacteristic
288C22#231011yescharacteristic
289C22#231111yescharacteristic
290C22#281612yes—
291C22#281712yes—
292C22#290114yes—
293C4—44no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—88no—
298C4—84no—
299C4—84no—
300C4#237814yes—
301C4#281414yes—
302C2—22no—
303C2—22no—
304C2—82no—
305C2#1466011yescharacteristic
306C2#2177512yes—
307C2#2184612yes—
308C2#2190112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11386C2semidirect
5. #11359C2semidirect
6. #11226C2semidirect
11. #2629C4semidirect
12. #2628C4semidirect

Maximal subgroups

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

subgroupindex
#346342
#206832
#138522
#113862
#113592
#112262
#112012