Finite Group Databasebeta

← #122287

Subgroups — #122287

Order 512.

326 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 76 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#21821C211yesmaximal, characteristic
2#11460C212yesmaximal
3#7061C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C2×#432C411yescharacteristic
6C2×#431C2211yescharacteristic
7#2163—22no—
8#1985C412yes—
9#1985C2212yes—
10#1984—22no—
11#1818—22no—
12#1322C2214yes—
13#1314—22no—
14#533—42no—
15#533—42no—
16C2×C4⋊Q8C811yescharacteristic
17C2×C4⋊Q8C811yescharacteristic
18C2×#125—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C812yes—
21C2×C4×Q8—21no—
22C2×C4×Q8C2×C412yes—
23C2×#117—21no—
24C22×C42C2×C411yescharacteristic
25#432—42no—
26#431—42no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2311yescharacteristic
30C2×C22⋊C8—21no—
31C2×C22⋊C8D412yes—
32C2×#97C2×C412yes—
33C2×C4×C8D412yes—
34C4⋊C16—42no—
35C22⋊C16—24no—
36#346—42no—
37#336—24no—
38#336—24no—
39C22×Q8—21no—
40C22×Q8C2×C812yes—
41C23×C4C22⋊C411yescharacteristic
42C23×C4C2×C811yescharacteristic
43C23×C4M1611yescharacteristic
44C22×C8—22no—
45C22×C8C22⋊C412yes—
46C22×C8C2×D412yes—
47C4⋊Q8—42no—
48C4⋊Q8—42no—
49#125—44no—
50C22⋊Q8—44no—
51C22⋊Q8—44no—
52C4×Q8—42no—
53C4×Q8—42no—
54C4×Q8—42no—
55C4×Q8—42no—
56#117—42no—
57#117—42no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4C2×C812yes—
64C2×C22⋊C4—22no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42C22⋊C411yescharacteristic
69C2×C42C2×C811yescharacteristic
70C2×C42M1611yescharacteristic
71C2×C42C22×C411yescharacteristic
72C2×C16—44no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—42no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78#97—42no—
79C4×C8—22no—
80C4×C8—22no—
81C24C22⋊C811yescharacteristic
82C2×Q8—44no—
83C2×Q8—42no—
84C2×Q8—42no—
85C2×Q8—82no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—81no—
96C22×C4C22⋊C811yescharacteristic
97C22×C4C22⋊C811yescharacteristic
98C22×C4C22⋊C811yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4C22×C811yescharacteristic
101C22×C4C2×M1611yescharacteristic
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—44no—
113C4⋊C4—42no—
114C4⋊C4—44no—
115C22⋊C4—44no—
116C42—22no—
117C42—22no—
118C42—21no—
119C42—21no—
120C42—42no—
121C42—42no—
122C42#9912yes—
123C42#10012yes—
124C16—84no—
125C23—21no—
126C23—22no—
127C23—41no—
128C23—41no—
129C23C2×C22⋊C811yescharacteristic
130Q8—84no—
131Q8—84no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—82no—
151C2×C4#32312yes—
152C2×C4#32312yes—
153C2×C4C2×#9911yescharacteristic
154C2×C4C2×#10011yescharacteristic
155C8—42no—
156C8—42no—
157C8—82no—
158C22—21no—
159C22—21no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—82no—
164C22C2×#32311yescharacteristic
165C22#211712yes—
166C4—22no—
167C4—21no—
168C4—22no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—82no—
173C4—82no—
174C4#212911yescharacteristic
175C4#213511yescharacteristic
176C2—22no—
177C2—81no—
178C2#1142111yescharacteristic
179C2#1143411yescharacteristic
180C2#1143611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7061C2semidirect

Maximal subgroups

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

subgroupindex
#218212
#114602
#70612