Finite Group Databasebeta

← #213917

Subgroups — #213917

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#8754C212yesmaximal, complemented
5#8552C212yesmaximal
6C22×#124C2211yescharacteristic
7C2×#519C2211yescharacteristic
8#2921C2214yes—
9#2482C414yes—
10#2482C2214yes—
11#2448—24no—
12#2448—24no—
13#2310C2211yescharacteristic
14#2309C412yes—
15#2296C412yes—
16#1822C2214yes—
17#1801—24no—
18#1784C2214yes—
19C4×#101—24no—
20#1462C2218yes—
21#1306C2218yes—
22C23×Q8C2311yescharacteristic
23C23×D4C2×C411yescharacteristic
24C2×#124—24no—
25C2×#124—44no—
26#519—44no—
27C2×C4×Q8D414yes—
28C2×C4×Q8D414yes—
29C2×#117C2×C412yes—
30C2×#117C2×C412yes—
31C2×#117C2×C412yes—
32C2×#117C2312yes—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2312yes—
35C22×C42C2×C411yescharacteristic
36#433—48no—
37#423—24no—
38C2×#101—28no—
39#407C2318yes—
40C4×M16—24no—
41#399—24no—
42#399—24no—
43#397—24no—
44#397—24no—
45#391—24no—
46#390—22no—
47#390—22no—
48#390—24no—
49#389—22no—
50#389—22no—
51#388—24no—
52#388—24no—
53#388—24no—
54#388—24no—
55#385C2×C414yes—
56#385C2×C414yes—
57#381C2×C412yes—
58#381C2312yes—
59#379C2×C414yes—
60#379C2×C414yes—
61C4×C4⋊C4—24no—
62#343—28no—
63#323—48no—
64C25C22×C412yes—
65C22×Q8—24no—
66C22×Q8—44no—
67C22×Q8C2×D414yes—
68C22×D4—22no—
69C22×D4—42no—
70C22×D4—44no—
71C23×C4C22×C411yescharacteristic
72C23×C4C22×C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2411yescharacteristic
75C2×M16—28no—
76#124—416no—
77C4×Q8—48no—
78C4×Q8—48no—
79#117—28no—
80#117—44no—
81#117—44no—
82#117—44no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4C22⋊C412yes—
87C2×C4⋊C4C22⋊C412yes—
88C2×C4⋊C4C22⋊C414yes—
89C2×C4⋊C4C2×D414yes—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—28no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—44no—
98C2×C22⋊C4C22×C414yes—
99C2×C22⋊C4C22×C414yes—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42C22⋊C414yes—
104C2×C42C22⋊C412yes—
105C2×C42C22⋊C412yes—
106C2×C42C2×D412yes—
107C2×C42C2×D412yes—
108C4⋊C8—48no—
109#101—416no—
110C22⋊C8—216no—
111#97—44no—
112C4×C8—44no—
113#95—24no—
114#95—24no—
115#95—24no—
116#95—24no—
117C24—22no—
118C24—24no—
119C24—44no—
120C24—82no—
121C24—82no—
122C24C23×C411yescharacteristic
123C2×Q8—28no—
124C2×Q8—28no—
125C2×Q8—44no—
126C2×Q8—416no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—88no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—82no—
140C22×C4C2×C22⋊C412yes—
141C22×C4C2×C22⋊C412yes—
142C22×C4C2×C22⋊C414yes—
143C22×C4C2×C22⋊C414yes—
144C22×C4C22×D412yes—
145C22×C4H(5,𝔽2)14yes—
146M16—416no—
147C2×C8—48no—
148C4⋊C4—28no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—44no—
152C4⋊C4—44no—
153C4⋊C4—84no—
154C4⋊C4—84no—
155C22⋊C4—28no—
156C22⋊C4—48no—
157C22⋊C4—48no—
158C22⋊C4—44no—
159C22⋊C4—84no—
160C22⋊C4—84no—
161C22⋊C4—84no—
162C22⋊C4—84no—
163C42—24no—
164C42—24no—
165C42—44no—
166C42—42no—
167C42—42no—
168C42—44no—
169C42—44no—
170C23—22no—
171C23—42no—
172C23—44no—
173C23—44no—
174C23—48no—
175C23—42no—
176C23—44no—
177C23—84no—
178C23—84no—
179C23—168no—
180C23C22×C22⋊C411yescharacteristic
181C23#51812yes—
182Q8—416no—
183D4—88no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—84no—
203C2×C4#41014yes—
204C2×C4#53414yes—
205C8—88no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—44no—
210C22—88no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—168no—
215C22#230811yescharacteristic
216C22C2×#41012yes—
217C4—24no—
218C4—24no—
219C4—44no—
220C4—44no—
221C4—42no—
222C4—84no—
223C4—82no—
224C2—22no—
225C2—42no—
226C2—162no—
227C2#2174711yescharacteristic
228C2#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. #8754C2semidirect

Maximal subgroups

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

subgroupindex
#356792
#207302
#95122
#87542
#85522