Finite Group Databasebeta

← #213843

Subgroups — #213843

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#8748C212yesmaximal
6#8544C212yesmaximal
7C22×#124C2211yescharacteristic
8C2×#518C2211yescharacteristic
9#2483C414yescomplemented
10#2483C2212yes—
11#2483C2212yes—
12#2447—24no—
13#2447—24no—
14#2312C2214yes—
15#2310C412yescomplemented
16#2309C2211yescharacteristic
17#2296C412yescomplemented
18#1819C2214yes—
19#1819C2218yes—
20#1799—24no—
21#1781C2214yes—
22C4×#99—24no—
23#1304C2214yes—
24#1304C2214yes—
25C23×Q8C2×C411yescharacteristic
26C23×D4C2311yescharacteristic
27C2×#124—24no—
28C2×#124—44no—
29#518—44no—
30C2×C4×D4D414yes—
31C2×C4×D4D414yes—
32C2×#117C2×C412yes—
33C2×#117C2×C412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4C2312yes—
37C22×C22⋊C4C2312yes—
38C22×C42C2×C411yescharacteristic
39C2×#99—28no—
40#397—24no—
41#397—24no—
42#397—24no—
43#397—24no—
44#391—24no—
45#391—22no—
46#391—22no—
47#390—24no—
48#389—24no—
49#388—48no—
50#385—24no—
51#385—22no—
52#385—22no—
53#385C2×C414yes—
54#385C2×C414yes—
55#381—24no—
56#381—22no—
57#381—22no—
58#381C2×C414yes—
59#381C2×C414yes—
60#380—24no—
61#379C2×C412yes—
62#379C2318yes—
63#379C2311yescharacteristic
64#379C2311yescharacteristic
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C22⋊C4—24no—
68#342—28no—
69#342—48no—
70C25C22×C411yescharacteristic
71C25C2411yescharacteristic
72C22×Q8—22no—
73C22×Q8—42no—
74C22×Q8—44no—
75C22×D4—24no—
76C22×D4—44no—
77C22×D4C2×D414yes—
78C23×C4C22⋊C414yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82C23×C4C2×D412yes—
83C23×C4C2×D412yes—
84#124—416no—
85C4×D4—48no—
86C4×D4—48no—
87#117—44no—
88#117—44no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4C22⋊C414yes—
94C2×C22⋊C4—28no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—28no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4C22⋊C412yes—
108C2×C22⋊C4C22⋊C412yes—
109C2×C22⋊C4C22×C414yes—
110C2×C22⋊C4C22×C412yes—
111C2×C22⋊C4C22×C412yes—
112C2×C22⋊C4C2×D414yes—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—42no—
116C2×C42—44no—
117C2×C42C22⋊C412yes—
118C2×C42C22⋊C412yes—
119#99—416no—
120#95—24no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—44no—
127#95—44no—
128C24—22no—
129C24—42no—
130C24—42no—
131C24—82no—
132C24—82no—
133C24C2×C22⋊C412yes—
134C24C23×C411yescharacteristic
135C24C22×D412yes—
136C24H(5,𝔽2)14yes—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—44no—
140C2×Q8—88no—
141C2×D4—28no—
142C2×D4—28no—
143C2×D4—44no—
144C2×D4—416no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—84no—
162C22×C4C2×C22⋊C412yes—
163C22×C4C2×C22⋊C414yes—
164C22×C4C2×C22⋊C414yes—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C4⋊C4—164no—
170C22⋊C4—28no—
171C22⋊C4—28no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—48no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—416no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—84no—
183C22⋊C4—88no—
184C22⋊C4—88no—
185C42—42no—
186C42—42no—
187C42—44no—
188C42—44no—
189C42—164no—
190C23—22no—
191C23—24no—
192C23—22no—
193C23—24no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—48no—
198C23—44no—
199C23—44no—
200C23—44no—
201C23—84no—
202C23—164no—
203C23—164no—
204C23C22×C22⋊C411yescharacteristic
205C23#51812yes—
206C23#53414yes—
207Q8—88no—
208D4—416no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4—88no—
229C2×C4—88no—
230C2×C4—88no—
231C2×C4—164no—
232C2×C4#41014yes—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—24no—
237C22—22no—
238C22—24no—
239C22—44no—
240C22—44no—
241C22—42no—
242C22—84no—
243C22—84no—
244C22—164no—
245C22—164no—
246C22#230811yescharacteristic
247C22C2×#41012yes—
248C4—24no—
249C4—44no—
250C4—44no—
251C4—82no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4—168no—
256C2—22no—
257C2—44no—
258C2—42no—
259C2—82no—
260C2#2174711yescharacteristic
261C2#2177712yes—
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
15. #2310C4semidirect
17. #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
#87482
#85442