Finite Group Databasebeta

← #375669

Subgroups — #375669

Order 512.

1159 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1159 conjugacy classes of subgroups — too many to draw. Only its 250 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#35672C211yesmaximal, characteristic
2#20399C212yesmaximal, complemented
3#11168C212yesmaximal, complemented
4#10968C212yesmaximal
5#10937C214yesmaximal
6#10916C214yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C2×#519C2211yescharacteristic
9C22×#101C2214yes—
10C2×#407C2214yes—
11#2873C2214yes—
12#2484C414yes—
13#2482C414yes—
14#2423—24no—
15#2421—24no—
16#2317C2212yes—
17#2310C2211yescharacteristic
18#2280C2212yes—
19C4×#101—24no—
20C4×#101—24no—
21C2×#344C2218yes—
22#1742C2214yes—
23C2×#343C2214yes—
24C23×Q8C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16C2318yes—
27C2×C22⋊Q8—24no—
28C2×C22⋊Q8—44no—
29#519—44no—
30C2×C4×Q8D414yes—
31C2×C4×Q8D414yes—
32C2×#117C2×C414yes—
33C2×#117C2312yes—
34C2×#117C2312yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4C2×C412yes—
38C2×#101—28no—
39C2×#101—216no—
40#407—216no—
41C2×C22⋊C8—44no—
42#405—48no—
43C4×M16—28no—
44#400—24no—
45#399—24no—
46#398—24no—
47#397—24no—
48#391—24no—
49#389—24no—
50#388—28no—
51#387—28no—
52#385C2×C418yes—
53#384—22no—
54#382—22no—
55#381C2×C414yes—
56#381C2×C414yes—
57#380—22no—
58#380—22no—
59#379C2×C414yes—
60C4×C22⋊C4—24no—
61#344—216no—
62#343—24no—
63#343—24no—
64#343—44no—
65C25C22×C411yescharacteristic
66C22×Q8—28no—
67C22×Q8—24no—
68C22×Q8C2×D414yes—
69C23×C4—22no—
70C23×C4—28no—
71C23×C4—24no—
72C23×C4C22×C411yescharacteristic
73C23×C4C22×C412yes—
74C23×C4C22×C412yes—
75C23×C4C2411yescharacteristic
76C2×M16—28no—
77C2×M16—28no—
78C2×M16—28no—
79C2×M16—28no—
80C2×M16—28no—
81C2×M16#41116yes—
82C22×C8—28no—
83C22⋊Q8—416no—
84C4×Q8—48no—
85C4×Q8—48no—
86#117—28no—
87#117—28no—
88#117—48no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4C22⋊C414yes—
96C2×C4⋊C4C22⋊C414yes—
97C2×C4⋊C4C2×D412yes—
98C2×C4⋊C4C2×D412yes—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4C22×C414yes—
104C2×C22⋊C4C22×C414yes—
105C2×C42—22no—
106C2×C42C22⋊C418yes—
107C2×C42C2×D414yes—
108#101—232no—
109C22⋊C8—416no—
110#97—48no—
111C4×C8—48no—
112#95—24no—
113#95—24no—
114#95—24no—
115#95—24no—
116#95—28no—
117C24—21no—
118C24—22no—
119C24—24no—
120C24—42no—
121C24—81no—
122C24C23×C411yescharacteristic
123C2×Q8—28no—
124C2×Q8—28no—
125C2×Q8—28no—
126C2×Q8—44no—
127C2×Q8—48no—
128C2×Q8—44no—
129C22×C4—28no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4—82no—
144C22×C4C2×C22⋊C414yes—
145C22×C4C2×C22⋊C414yes—
146C22×C4C2×C22⋊C414yes—
147C22×C4C22×D412yes—
148C22×C4C2×#4118yes—
149C22×C4H(5,𝔽2)14yes—
150M16—216no—
151M16—216no—
152M16—48no—
153M16—48no—
154C2×C8—216no—
155C2×C8—48no—
156C2×C8—48no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—48no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—88no—
165C22⋊C4—28no—
166C22⋊C4—28no—
167C22⋊C4—48no—
168C22⋊C4—48no—
169C22⋊C4—84no—
170C42—28no—
171C42—48no—
172C42—44no—
173C23—22no—
174C23—24no—
175C23—41no—
176C23—41no—
177C23—42no—
178C23—41no—
179C23—42no—
180C23—44no—
181C23—41no—
182C23—44no—
183C23—42no—
184C23—88no—
185C23C22×C22⋊C411yescharacteristic
186C23C4×#4114yes—
187C23#51812yes—
188Q8—416no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—48no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4—84no—
206C2×C4#52514yes—
207C2×C4#53214yes—
208C2×C4#53414yes—
209C2×C4#55114yes—
210C2×C4#55318yes—
211C2×C4#55914yes—
212C8—416no—
213C22—21no—
214C22—22no—
215C22—21no—
216C22—22no—
217C22—44no—
218C22—44no—
219C22—48no—
220C22—42no—
221C22—82no—
222C22—84no—
223C22—81no—
224C22—81no—
225C22#229712yes—
226C22#230811yescharacteristic
227C22#237414yes—
228C4—44no—
229C4—44no—
230C4—44no—
231C4—84no—
232C4—82no—
233C4—81no—
234C4#260218yes—
235C2—22no—
236C2—24no—
237C2—82no—
238C2#1416711yescharacteristic
239C2#2178011yescharacteristic
240C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20399C2semidirect
3. #11168C2semidirect
6. #10916C2semidirect

Maximal subgroups

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

subgroupindex
#356722
#203992
#111682
#109682
#109372
#109162