Finite Group Databasebeta

← #213866

Subgroups — #213866

Order 512.

994 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 190 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#35626C211yesmaximal, characteristic
2#20724C212yesmaximal
3#9502C214yesmaximal, complemented
4#9500C214yesmaximal, complemented
5#8749C212yesmaximal
6#8544C212yesmaximal
7C22×#124C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2483C412yescomplemented
10#2483C412yescomplemented
11#2483C2212yes—
12#2483C2212yes—
13#2447—24no—
14#2447—24no—
15#2312C2214yes—
16#2310C412yescomplemented
17#2309C2211yescharacteristic
18#2296C412yescomplemented
19#1819C2214yes—
20#1819C2214yes—
21#1819C2214yes—
22#1799—24no—
23#1781C2214yes—
24C4×#99—24no—
25#1304C2218yes—
26C23×Q8C2×C411yescharacteristic
27C23×D4C2311yescharacteristic
28C2×#124—24no—
29C2×#124—44no—
30#518—44no—
31C2×C4×D4D414yes—
32C2×C4×D4D414yes—
33C2×#117C2×C412yes—
34C2×#117C2×C412yes—
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2311yescharacteristic
39C22×C22⋊C4C2311yescharacteristic
40C22×C22⋊C4C2312yes—
41C22×C42C2×C411yescharacteristic
42C2×#99—28no—
43#397—24no—
44#397—24no—
45#397—24no—
46#397—24no—
47#391—22no—
48#391—22no—
49#391—24no—
50#390—24no—
51#389—22no—
52#389—22no—
53#388—48no—
54#385—22no—
55#385—22no—
56#385—22no—
57#385—22no—
58#385C2×C414yes—
59#385C2×C414yes—
60#381—22no—
61#381—22no—
62#381—22no—
63#381—22no—
64#381C2×C414yes—
65#381C2×C414yes—
66#380—24no—
67#379C2×C412yes—
68#379C2318yes—
69#379C2312yes—
70C4×C4⋊C4—24no—
71C4×C22⋊C4—24no—
72#342—28no—
73#342—44no—
74#342—44no—
75C25C22×C411yescharacteristic
76C25C2411yescharacteristic
77C22×Q8—22no—
78C22×Q8—42no—
79C22×Q8—44no—
80C22×D4—24no—
81C22×D4—44no—
82C22×D4C2×D414yes—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92#124—416no—
93C4×D4—48no—
94C4×D4—48no—
95#117—44no—
96#117—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4C22⋊C412yes—
102C2×C4⋊C4C22⋊C412yes—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4C22⋊C414yes—
121C2×C22⋊C4C22×C414yes—
122C2×C22⋊C4C22×C414yes—
123C2×C22⋊C4C2×D414yes—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—42no—
127C2×C42—44no—
128C2×C42C22⋊C414yes—
129#99—416no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—44no—
137#95—44no—
138C24—22no—
139C24—42no—
140C24—42no—
141C24—81no—
142C24—81no—
143C24—81no—
144C24—81no—
145C24C2×C22⋊C412yes—
146C24C23×C411yescharacteristic
147C24C22×D412yes—
148C24H(5,𝔽2)14yes—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—88no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—44no—
156C2×D4—416no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—48no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4C2×C22⋊C412yes—
182C22×C4C2×C22⋊C414yes—
183C22×C4C2×C22⋊C414yes—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—84no—
188C4⋊C4—84no—
189C4⋊C4—164no—
190C22⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—416no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—88no—
204C22⋊C4—88no—
205C42—44no—
206C42—44no—
207C42—44no—
208C42—164no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—44no—
218C23—44no—
219C23—44no—
220C23—44no—
221C23—42no—
222C23—42no—
223C23—82no—
224C23—82no—
225C23—164no—
226C23—164no—
227C23C22×C22⋊C411yescharacteristic
228C23#51812yes—
229C23#53414yes—
230Q8—88no—
231D4—416no—
232C2×C4—28no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—88no—
257C2×C4—88no—
258C2×C4—88no—
259C2×C4—162no—
260C2×C4—162no—
261C2×C4#41014yes—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—24no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—84no—
275C22—84no—
276C22—164no—
277C22—164no—
278C22#230811yescharacteristic
279C22C2×#41012yes—
280C4—24no—
281C4—44no—
282C4—44no—
283C4—84no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—168no—
289C2—22no—
290C2—44no—
291C2—42no—
292C2—82no—
293C2#2174711yescharacteristic
294C2#2177711yescharacteristic
295C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9502C2semidirect
4. #9500C2semidirect
9. #2483C4semidirect
10. #2483C4semidirect
16. #2310C4semidirect
18. #2296C4semidirect

Maximal subgroups

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

subgroupindex
#356262
#207242
#95022
#95002
#87492
#85442