Finite Group Databasebeta

← #213901

Subgroups — #213901

Order 512.

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

normal subgroups only

This group has 922 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#35679C211yesmaximal, characteristic
2#20730C212yesmaximal, complemented
3#9512C218yesmaximal, complemented
4#8753C212yesmaximal, complemented
5#8552C212yesmaximal
6C22×#124C2211yescharacteristic
7C2×#519C2211yescharacteristic
8#2921C2214yes—
9#2482C412yes—
10#2482C412yes—
11#2482C2214yes—
12#2448—24no—
13#2448—24no—
14#2310C2211yescharacteristic
15#2309C412yes—
16#2296C412yes—
17#1822C2214yes—
18#1801—24no—
19#1784C2214yes—
20C4×#101—24no—
21#1462C2218yes—
22#1306C2218yes—
23C23×Q8C2311yescharacteristic
24C23×D4C2×C411yescharacteristic
25C2×#124—24no—
26C2×#124—44no—
27#519—44no—
28C2×C4×Q8D414yes—
29C2×C4×Q8D414yes—
30C2×#117C2×C412yes—
31C2×#117C2×C412yes—
32C2×#117C2×C412yes—
33C2×#117C2312yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2312yes—
36C22×C42C2×C411yescharacteristic
37#433—48no—
38#423—24no—
39C2×#101—28no—
40#407C2318yes—
41C4×M16—24no—
42#399—22no—
43#399—22no—
44#399—24no—
45#397—22no—
46#397—22no—
47#397—24no—
48#391—24no—
49#390—24no—
50#390—22no—
51#390—22no—
52#389—24no—
53#388—24no—
54#388—24no—
55#388—24no—
56#388—24no—
57#385C2×C414yes—
58#385C2×C414yes—
59#381C2×C411yescharacteristic
60#381C2×C411yescharacteristic
61#381C2312yes—
62#379C2×C414yes—
63#379C2×C414yes—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66#343—28no—
67#323—48no—
68C25C22×C412yes—
69C22×Q8—24no—
70C22×Q8—44no—
71C22×Q8C2×D414yes—
72C22×D4—22no—
73C22×D4—42no—
74C22×D4—44no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C412yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C2411yescharacteristic
79C2×M16—28no—
80#124—416no—
81C4×Q8—48no—
82C4×Q8—48no—
83#117—28no—
84#117—44no—
85#117—44no—
86#117—44no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4C22⋊C414yes—
91C2×C4⋊C4C22⋊C412yes—
92C2×C4⋊C4C22⋊C412yes—
93C2×C4⋊C4C2×D414yes—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4C22×C414yes—
104C2×C22⋊C4C22×C412yes—
105C2×C22⋊C4C22×C412yes—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—42no—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C412yes—
111C2×C42C22⋊C414yes—
112C2×C42C2×D412yes—
113C2×C42C2×D412yes—
114C4⋊C8—48no—
115#101—416no—
116C22⋊C8—216no—
117#97—44no—
118C4×C8—44no—
119#95—24no—
120#95—24no—
121#95—22no—
122#95—22no—
123#95—24no—
124C24—22no—
125C24—24no—
126C24—44no—
127C24—82no—
128C24—82no—
129C24C23×C411yescharacteristic
130C2×Q8—28no—
131C2×Q8—28no—
132C2×Q8—44no—
133C2×Q8—416no—
134C2×D4—44no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—88no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—82no—
148C22×C4C2×C22⋊C412yes—
149C22×C4C2×C22⋊C412yes—
150C22×C4C2×C22⋊C414yes—
151C22×C4C2×C22⋊C414yes—
152C22×C4C22×D412yes—
153C22×C4H(5,𝔽2)14yes—
154M16—416no—
155C2×C8—48no—
156C4⋊C4—28no—
157C4⋊C4—44no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—84no—
162C4⋊C4—84no—
163C22⋊C4—28no—
164C22⋊C4—44no—
165C22⋊C4—44no—
166C22⋊C4—48no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C22⋊C4—84no—
170C22⋊C4—84no—
171C22⋊C4—84no—
172C22⋊C4—84no—
173C42—24no—
174C42—24no—
175C42—42no—
176C42—42no—
177C42—44no—
178C42—44no—
179C42—44no—
180C23—22no—
181C23—42no—
182C23—44no—
183C23—42no—
184C23—42no—
185C23—48no—
186C23—42no—
187C23—44no—
188C23—84no—
189C23—84no—
190C23—168no—
191C23C22×C22⋊C411yescharacteristic
192C23#51812yes—
193Q8—416no—
194D4—88no—
195C2×C4—28no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—84no—
214C2×C4#41014yes—
215C2×C4#53414yes—
216C8—88no—
217C22—22no—
218C22—24no—
219C22—22no—
220C22—44no—
221C22—84no—
222C22—84no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—168no—
227C22#230811yescharacteristic
228C22C2×#41012yes—
229C4—24no—
230C4—24no—
231C4—44no—
232C4—44no—
233C4—42no—
234C4—82no—
235C4—82no—
236C4—82no—
237C2—22no—
238C2—42no—
239C2—162no—
240C2#2174711yescharacteristic
241C2#2178112yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #9512C2semidirect
4. #8753C2semidirect

Maximal subgroups

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

subgroupindex
#356792
#207302
#95122
#87532
#85522