Finite Group Databasebeta

← #149785

Subgroups — #149785

Order 512.

400 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 62 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
2#11512C212yesmaximal
3#7358C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C2×C4×M16C411yescharacteristic
6C2×#384C2211yescharacteristic
7#1992C412yes—
8#1992C2212yes—
9#1990—22no—
10#1736—22no—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C22×M16—21no—
15#533—24no—
16#533—42no—
17C2×C4⋊Q8—21no—
18C2×#125C2×C412yes—
19C2×C22⋊Q8—41no—
20C2×C4×Q8—41no—
21C2×#117—21no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4D412yes—
24C22×C42C2×C411yescharacteristic
25#407—42no—
26C4×M16—42no—
27C2×#97C2×C412yes—
28C2×C4×C8—21no—
29#384—41no—
30#384C2×C411yescharacteristic
31#384D412yes—
32#384C2311yescharacteristic
33#340—24no—
34#340—24no—
35#324—24no—
36#324—42no—
37C22×Q8—41no—
38C23×C4—21no—
39C23×C4—22no—
40C23×C4C22⋊C411yescharacteristic
41C2×M16—24no—
42C2×M16—42no—
43C2×M16—42no—
44C2×M16—82no—
45C22×C8—22no—
46C4⋊Q8—24no—
47C4⋊Q8—42no—
48#125—22no—
49#125—42no—
50#125M1614yescomplemented
51C22⋊Q8—48no—
52C4×Q8—48no—
53#117—24no—
54#117—42no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—84no—
65C2×C4⋊C4C22⋊C412yes—
66C2×C4⋊C4C2×D412yes—
67C2×C22⋊C4—41no—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—41no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22×C411yescharacteristic
74C22⋊C8—44no—
75C22⋊C8—82no—
76#97—42no—
77C4×C8—42no—
78C24#9911yescharacteristic
79C2×Q8—44no—
80C2×Q8—82no—
81C2×Q8—82no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—24no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—84no—
97C22×C4#9911yescharacteristic
98C22×C4C2×C22⋊C411yescharacteristic
99M16—48no—
100M16—164no—
101C2×C8—24no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—84no—
105C4⋊C4—24no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4—44no—
110C4⋊C4—44no—
111C4⋊C4—42no—
112C4⋊C4—82no—
113C4⋊C4—84no—
114C4⋊C4—82no—
115C4⋊C4—82no—
116C22⋊C4—44no—
117C42—24no—
118C42—21no—
119C42—21no—
120C42—44no—
121C42#10012yes—
122C42C2×M1612yes—
123C23—21no—
124C23—21no—
125C23—41no—
126C23—41no—
127C23#35412yes—
128C23C2×#9911yescharacteristic
129Q8—88no—
130C2×C4—22no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—48no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—84no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—162no—
152C2×C4#35612yes—
153C2×C4#40711yescharacteristic
154C2×C4C2×#10011yescharacteristic
155C8—44no—
156C8—164no—
157C22—21no—
158C22—21no—
159C22—22no—
160C22—22no—
161C22—22no—
162C22—42no—
163C22—42no—
164C22#146211yescharacteristic
165C22C2×#35411yescharacteristic
166C22C2×#35611yescharacteristic
167C4—24no—
168C4—21no—
169C4—22no—
170C4—21no—
171C4—42no—
172C4—84no—
173C4—82no—
174C4—82no—
175C2—21no—
176C2—21no—
177C2—42no—
178C2#1150611yescharacteristic
179C2#1359511yescharacteristic
180C2#1359911yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×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
50. #125M16semidirect

Maximal subgroups

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

subgroupindex
#213222
#115122
#73582