Finite Group Databasebeta

← #11813

Subgroups — #11813

Order 256.

191 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 49 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#2628C211yesmaximal, characteristic
2#1820C211yesmaximal, characteristic
3#1464C211yesmaximal, characteristic
4#1352C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×#117C2211yescharacteristic
7#432—21no—
8#400C412yescomplemented
9#399C2212yes—
10#395—21no—
11#393—21no—
12#389—21no—
13#382C411yescharacteristic, complemented
14#381—21no—
15#380—21no—
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×Q8—21no—
21C23×C4D411yescharacteristic
22C4⋊Q8—41no—
23C4⋊Q8—41no—
24#117—41no—
25#117—41no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4C2×C412yes—
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4D411yescharacteristic
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4D411yescharacteristic
35C2×C22⋊C4C2311yescharacteristic
36C2×C42C2×C411yescharacteristic
37C2×C42D411yescharacteristic
38C4⋊C8—41no—
39C22⋊C8—22no—
40#97—41no—
41#95—22no—
42#95—21no—
43#95—22no—
44#95—41no—
45#95—41no—
46#95C2×C412yes—
47C24C2×D411yescharacteristic
48C2×Q8—42no—
49C2×Q8—82no—
50C22×C4—22no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—41no—
56C22×C4—42no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22×C411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C2×C8—42no—
64C4⋊C4—41no—
65C4⋊C4—41no—
66C4⋊C4—42no—
67C4⋊C4—42no—
68C4⋊C4—41no—
69C4⋊C4—41no—
70C4⋊C4—82no—
71C22⋊C4—41no—
72C22⋊C4—41no—
73C22⋊C4—42no—
74C22⋊C4—42no—
75C22⋊C4—82no—
76C42—21no—
77C42—21no—
78C42—21no—
79C42—21no—
80C23—21no—
81C23—21no—
82C23—41no—
83C23—41no—
84C23C2×C22⋊C411yescharacteristic
85C23#13712yes—
86Q8—84no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—82no—
109C2×C4—82no—
110C2×C4—82no—
111C2×C4#9911yescharacteristic
112C2×C4#9911yescharacteristic
113C8—82no—
114C22—21no—
115C22—21no—
116C22—22no—
117C22—42no—
118C22—41no—
119C22—41no—
120C22—81no—
121C22C2×#9911yescharacteristic
122C22#41911yescharacteristic
123C22#42111yescharacteristic
124C4—21no—
125C4—21no—
126C4—41no—
127C4—41no—
128C4—41no—
129C4—41no—
130C4—41no—
131C4—82no—
132C4—82no—
133C4—162no—
134C2—21no—
135C2—21no—
136C2—42no—
137C2#151111yescharacteristic
138C2#212311yescharacteristic
139C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
8. #400C4semidirect
13. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26282
#18202
#14642
#13522