Finite Group Databasebeta

← #13301

Subgroups — #13301

Order 256.

202 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 63 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#2568C211yesmaximal, characteristic
2#1801C211yesmaximal, characteristic
3C4×#101C211yesmaximal, characteristic
4#1394C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic
6C2×C4×Q8C2211yescharacteristic
7#423—21no—
8C2×#101C2212yes—
9C2×#101C2212yes—
10C4×M16—21no—
11#400—21no—
12#400—21no—
13#391—21no—
14#391—21no—
15#391C2212yes—
16#382C412yes—
17C4×C4⋊C4C411yescharacteristic
18#343—21no—
19#343—21no—
20C22×Q8—21no—
21C22×Q8C2311yescharacteristic
22C2×M16—22no—
23C2×M16—22no—
24C4⋊Q8—41no—
25C4⋊Q8—41no—
26C4×Q8—41no—
27C4×Q8—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4C2×C412yes—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4D411yescharacteristic
33C2×C4⋊C4D411yescharacteristic
34C2×C42C2×C411yescharacteristic
35C2×C42C2×C412yes—
36C2×C42D411yescharacteristic
37C2×C42D411yescharacteristic
38C4⋊C8—41no—
39C4⋊C8—41no—
40#101—22no—
41#101—24no—
42#101Q814yescomplemented
43#97—41no—
44C4×C8—41no—
45#95—21no—
46#95—22no—
47#95—22no—
48#95—22no—
49#95—22no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×Q8—44no—
53C2×Q8—41no—
54C2×Q8—41no—
55C2×Q8C2×Q812yes—
56C2×Q8#4112yes—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22×C411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C22×C4C2×D411yescharacteristic
66M16—24no—
67M16—42no—
68M16—42no—
69M16—42no—
70C2×C8—42no—
71C2×C8—42no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—21no—
82C42—21no—
83C42—21no—
84C42—42no—
85C23C2×C22⋊C411yescharacteristic
86Q8—42no—
87Q8—42no—
88Q8—82no—
89Q8—82no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—44no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4#9911yescharacteristic
109C2×C4#9911yescharacteristic
110C2×C4C4×Q812yes—
111C2×C4C22⋊Q812yes—
112C2×C4#12411yescharacteristic
113C2×C4C4⋊Q811yescharacteristic
114C8—44no—
115C8—82no—
116C22—21no—
117C22—21no—
118C22#39111yescharacteristic
119C22C2×#9911yescharacteristic
120C22#41011yescharacteristic
121C4—21no—
122C4—22no—
123C4—21no—
124C4—22no—
125C4—21no—
126C4—41no—
127C4—41no—
128C4—41no—
129C4—82no—
130C4—82no—
131C4#35512yes—
132C2—21no—
133C2—21no—
134C2#199611yescharacteristic
135C2C2×#35511yescharacteristic
136C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC23C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
42. #101Q8semidirect

Maximal subgroups

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

subgroupindex
#25682
#18012
C4×#1012
#13942