Finite Group Databasebeta

← #149583

Subgroups — #149583

Order 512.

412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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#21322C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3#7358C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C2×C4×M16C411yescharacteristic
6C2×#384C2211yescharacteristic
7#1992C412yes—
8#1992C2212yes—
9#1990—22no—
10#1736—22no—
11C2×#324—22no—
12#1325—22no—
13#1325C2214yes—
14C22×M16—21no—
15#533—22no—
16#533—42no—
17#533C814yescomplemented
18C2×C4⋊Q8—21no—
19C2×#125C2×C412yes—
20C2×C22⋊Q8—41no—
21C2×C4×Q8—41no—
22C2×#117—21no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4D412yes—
25C22×C42C2×C411yescharacteristic
26C2×C22⋊C8—42no—
27C4×M16—42no—
28C2×#97C2×C412yes—
29C2×C4×C8—21no—
30#384—41no—
31#384C2×C411yescharacteristic
32#384D412yes—
33#384C2311yescharacteristic
34#340—24no—
35#340—24no—
36#324—24no—
37#324—42no—
38C22×Q8—41no—
39C23×C4—21no—
40C23×C4—22no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—24no—
43C2×M16—42no—
44C2×M16—42no—
45C22×C8—22no—
46C22×C8—82no—
47C4⋊Q8—24no—
48C4⋊Q8—42no—
49#125—22no—
50#125—42no—
51#125C2×C814yescomplemented
52C22⋊Q8—44no—
53C22⋊Q8—44no—
54C4×Q8—44no—
55C4×Q8—44no—
56#117—24no—
57#117—42no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—84no—
68C2×C4⋊C4C22⋊C412yes—
69C2×C4⋊C4C2×D412yes—
70C2×C22⋊C4—41no—
71C2×C42—21no—
72C2×C42—22no—
73C2×C42—41no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C2×C812yes—
76C2×C42M1612yes—
77C2×C42C22×C411yescharacteristic
78C22⋊C8—44no—
79C22⋊C8—82no—
80#97—42no—
81C4×C8—42no—
82C24#9911yescharacteristic
83C2×Q8—44no—
84C2×Q8—82no—
85C2×Q8—82no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—84no—
100C22×C4C22⋊C812yes—
101C22×C4#9911yescharacteristic
102C22×C4#10012yes—
103C22×C4C2×C22⋊C411yescharacteristic
104M16—48no—
105C2×C8—24no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—84no—
109C2×C8—164no—
110C4⋊C4—24no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—44no—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C4⋊C4—42no—
117C4⋊C4—82no—
118C4⋊C4—84no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C22⋊C4—44no—
122C42—22no—
123C42—22no—
124C42—21no—
125C42—21no—
126C42—44no—
127C42C22⋊C812yes—
128C42C22×C811yescharacteristic
129C42C2×M1611yescharacteristic
130C23—41no—
131C23—41no—
132C23#32312yes—
133C23#35412yes—
134C23#35612yes—
135C23C2×#9911yescharacteristic
136Q8—88no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—44no—
149C2×C4—44no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—84no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4—162no—
161C2×C4#32312yes—
162C2×C4C2×C22⋊C811yescharacteristic
163C2×C4C2×#10011yescharacteristic
164C8—44no—
165C8—164no—
166C22—21no—
167C22—22no—
168C22—21no—
169C22—22no—
170C22—42no—
171C22—42no—
172C22#131914yes—
173C22C2×#32311yescharacteristic
174C22C2×#35411yescharacteristic
175C22C2×#35611yescharacteristic
176C4—24no—
177C4—21no—
178C4—22no—
179C4—21no—
180C4—41no—
181C4—41no—
182C4—84no—
183C4—82no—
184C4—82no—
185C2—21no—
186C2—24no—
187C2—21no—
188C2C2×#131911yescharacteristic
189C2#1359511yescharacteristic
190C2#1359911yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7358C2semidirect
17. #533C8semidirect
51. #125C2×C8semidirect

Maximal subgroups

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

subgroupindex
#213222
C2×#13252
#73582