Finite Group Databasebeta

← #391814

Subgroups — #391814

Order 512.

1132 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1132 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#20627C211yesmaximal, characteristic
3#13853C211yesmaximal, characteristic
4#11386C212yesmaximal, complemented
5#11370C212yesmaximal, complemented
6#11209C214yesmaximal, complemented
7#11201C214yesmaximal
8C2×#533C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#423C2212yes—
11C2×#407C2212yes—
12#2629C414yescomplemented
13#2628C414yescomplemented
14#2627—22no—
15#2626—22no—
16#2396—24no—
17#2296C2212yes—
18#2285C2211yescharacteristic
19#2280C2212yes—
20C2×C4×C22⋊C4C2212yes—
21#1798—24no—
22C2×#339C2218yes—
23C2×#339C2218yes—
24#1728C2212yes—
25#1445—22no—
26#1440—22no—
27C4×#95C2212yes—
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⋊C4D412yes—
43C22×C22⋊C4—22no—
44C22×C22⋊C4D412yes—
45C22×C42C2314yes—
46C22×C42C2311yescharacteristic
47C22×C42C2311yescharacteristic
48#423—44no—
49C2×C4⋊C8—24no—
50#407—44no—
51C2×C22⋊C8—24no—
52#400—24no—
53#399—24no—
54#398—28no—
55#396—28no—
56#395—24no—
57#393—24no—
58#389C2×C418yes—
59#385—24no—
60#384C2×C412yes—
61#382C2×C412yes—
62#381—22no—
63#380—22no—
64#380C2×C418yes—
65#379—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—44no—
70C2×#95—24no—
71C2×#95—24no—
72C2×#95—22no—
73#339—48no—
74#339—48no—
75#336—24no—
76C25C2×D411yescharacteristic
77C22×Q8—22no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—24no—
81C23×C4—22no—
82C23×C4—24no—
83C23×C4C22×C412yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C23×C4C2×D412yes—
87C23×C4C2×D412yes—
88C23×C4C2411yescharacteristic
89C2×M16—28no—
90C2×M16—28no—
91C2×M16—28no—
92C22×C8—24no—
93C4⋊Q8—44no—
94#125—44no—
95C22⋊Q8—44no—
96C22⋊Q8—44no—
97C4×Q8—48no—
98#117—28no—
99#117—24no—
100#117—24no—
101#117—44no—
102#117—44no—
103#117—48no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C22⋊C414yes—
111C2×C4⋊C4C22⋊C414yes—
112C2×C4⋊C4C22×C414yes—
113C2×C4⋊C4C22×C414yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—48no—
120C2×C22⋊C4—44no—
121C2×C22⋊C4C22⋊C414yes—
122C2×C22⋊C4C22⋊C414yes—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—24no—
128C2×C42—28no—
129C2×C42—28no—
130C2×C42—24no—
131C2×C42C22×C411yescharacteristic
132C2×C42C22×C411yescharacteristic
133C2×C42C22×C412yes—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C4⋊C8—48no—
137C22⋊C8—48no—
138#95—28no—
139#95—24no—
140#95—28no—
141#95—48no—
142#95—48no—
143#95—44no—
144C24—22no—
145C24—22no—
146C24—22no—
147C24—44no—
148C24C2×C22⋊C412yes—
149C24C22×D411yescharacteristic
150C2×Q8—44no—
151C2×Q8—84no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4C2×C22⋊C412yes—
180C22×C4C2×C22⋊C414yes—
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C412yes—
183C22×C4C23×C411yescharacteristic
184C22×C4C22×D411yescharacteristic
185C22×C4H(5,𝔽2)14yes—
186C22×C4H(5,𝔽2)12yes—
187C22×C42−1+414yes—
188C22×C42−1+412yes—
189M16—416no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—28no—
194C4⋊C4—28no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C4⋊C4—84no—
200C4⋊C4—88no—
201C22⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—416no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—88no—
207C42—24no—
208C42—22no—
209C42—22no—
210C42—24no—
211C42—24no—
212C42—44no—
213C42—42no—
214C42—42no—
215C42—48no—
216C42—48no—
217C23—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—28no—
222C23—44no—
223C23—44no—
224C23—44no—
225C23—84no—
226C23—84no—
227C23C22×C22⋊C411yescharacteristic
228C23#53412yes—
229C23#53612yes—
230C23#53712yes—
231Q8—88no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—28no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4#51812yes—
261C2×C4#51912yes—
262C2×C4#52012yes—
263C2×C4#53412yes—
264C2×C4#53511yescharacteristic
265C2×C4#53612yes—
266C2×C4#53611yescharacteristic
267C2×C4#56218yes—
268C2×C4#56318yes—
269C8—84no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—24no—
274C22—44no—
275C22—42no—
276C22—44no—
277C22—44no—
278C22—88no—
279C22#230811yescharacteristic
280C22#231011yescharacteristic
281C22#231111yescharacteristic
282C22#281414yes—
283C22#290114yes—
284C4—42no—
285C4—44no—
286C4—42no—
287C4—42no—
288C4—88no—
289C4—84no—
290C4—84no—
291C4#237814yes—
292C4#281612yes—
293C4#281712yes—
294C2—24no—
295C2—82no—
296C2#1466011yescharacteristic
297C2#2177512yes—
298C2#2184612yes—
299C2#2185812yes—
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. #11370C2semidirect
6. #11209C2semidirect
12. #2629C4semidirect
13. #2628C4semidirect

Maximal subgroups

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

subgroupindex
#346342
#206272
#138532
#113862
#113702
#112092
#112012