Finite Group Databasebeta

← #7598

Subgroups — #7598

Order 256.

257 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 83 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#2296C211yesmaximal, characteristic
2#1797C212yesmaximal
3#1271C214yesmaximal, complemented
4C2×#117C411yescharacteristic
5C2×#117C411yescharacteristic
6C22×C42C2211yescharacteristic
7C2×C4⋊C8—22no—
8C2×C22⋊C8—22no—
9#389C412yes—
10#389C2212yes—
11#385—22no—
12C4×C4⋊C4—22no—
13#336C2214yes—
14C23×C4C2×C411yescharacteristic
15C23×C4D412yes—
16C22×C8—24no—
17#117—22no—
18#117—42no—
19#117C814yescomplemented
20C2×C4⋊C4—22no—
21C2×C4⋊C4—22no—
22C2×C4⋊C4C2×C412yes—
23C2×C4⋊C4C2×C412yes—
24C2×C22⋊C4—21no—
25C2×C22⋊C4—21no—
26C2×C42—21no—
27C2×C42—21no—
28C2×C42—42no—
29C2×C42C2×C411yescharacteristic
30C2×C42D412yes—
31C2×C42C2311yescharacteristic
32C4⋊C8—44no—
33C22⋊C8—44no—
34#95—22no—
35#95—22no—
36C24C22⋊C411yescharacteristic
37C22×C4—22no—
38C22×C4—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—22no—
42C22×C4—22no—
43C22×C4—42no—
44C22×C4—42no—
45C22×C4C22⋊C412yes—
46C22×C4C22⋊C411yescharacteristic
47C22×C4C2×C812yes—
48C22×C4M1612yes—
49C22×C4C22×C411yescharacteristic
50C22×C4C2×D412yes—
51C2×C8—44no—
52C2×C8—44no—
53C2×C8—44no—
54C4⋊C4—22no—
55C4⋊C4—42no—
56C4⋊C4—44no—
57C4⋊C4—42no—
58C4⋊C4—42no—
59C4⋊C4C2×C814yes—
60C22⋊C4—24no—
61C22⋊C4—42no—
62C42—24no—
63C42—24no—
64C42—41no—
65C42—41no—
66C42—42no—
67C23—21no—
68C23—21no—
69C23—22no—
70C23—22no—
71C23C22⋊C812yes—
72C23C2×C22⋊C411yescharacteristic
73C2×C4—22no—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—22no—
77C2×C4—24no—
78C2×C4—24no—
79C2×C4—24no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—44no—
84C2×C4—44no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4C22⋊C812yes—
91C2×C4#9912yes—
92C2×C4#10112yes—
93C2×C4C22×C811yescharacteristic
94C2×C4C2×M1611yescharacteristic
95C2×C4AGL(1,ℤ/8)12yes—
96C2×C4#13712yes—
97C8—84no—
98C22—22no—
99C22—24no—
100C22—41no—
101C22—41no—
102C22—42no—
103C22C2×C22⋊C811yescharacteristic
104C22C2×#9911yescharacteristic
105C22C2×#10111yescharacteristic
106C22#41712yes—
107C22#42112yes—
108C4—21no—
109C4—21no—
110C4—22no—
111C4—22no—
112C4—44no—
113C4—42no—
114C4—82no—
115C4#32414yes—
116C2—42no—
117C2C2×#32411yescharacteristic
118C2#147712yes—
119C2#151312yes—
120C2#153711yescharacteristic
121C2#156111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1271C2semidirect
19. #117C8semidirect

Maximal subgroups

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

subgroupindex
#22962
#17972
#12712