Finite Group Databasebeta

← #12802

Subgroups — #12802

Order 256.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#356C212yesmaximal
3#1801C212yesmaximal
4#1394C212yesmaximal
5C2×C4⋊Q8C2211yescharacteristic
6C2×C4×Q8C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#423—22no—
9C2×#101C2214yes—
10#391C411yescharacteristic
11#391C2211yescharacteristic
12#389—22no—
13#384C411yescharacteristic
14C4×C4⋊C4—22no—
15C43C411yescharacteristic
16#356—24no—
17#343—22no—
18C22×Q8C2×C411yescharacteristic
19C22×Q8C2311yescharacteristic
20C2×M16—24no—
21C4⋊Q8—22no—
22C4⋊Q8D412yes—
23C4⋊Q8Q812yes—
24C4×Q8—41no—
25C4×Q8—41no—
26C4×Q8—41no—
27C4×Q8—41no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4D412yes—
34C2×C42—21no—
35C2×C42—21no—
36C2×C42—21no—
37C2×C42C2×C411yescharacteristic
38C2×C42C2×C411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42D412yes—
41C4⋊C8—42no—
42C4⋊C8—42no—
43#101—24no—
44#101—24no—
45#95—22no—
46#95—22no—
47C2×Q8—21no—
48C2×Q8—41no—
49C2×Q8—41no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×Q8C4⋊C412yes—
53C2×Q8C2×D411yescharacteristic
54C2×Q8C2×Q811yescharacteristic
55C2×Q8#4112yes—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4C22⋊C412yes—
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22×C411yescharacteristic
64C22×C4C2×D412yes—
65M16—44no—
66M16—44no—
67C2×C8—44no—
68C4⋊C4—24no—
69C4⋊C4—24no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—41no—
73C4⋊C4—44no—
74C4⋊C4—42no—
75C42—24no—
76C42—22no—
77C42—22no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—41no—
82C42—42no—
83C42—41no—
84C42—41no—
85C42C4⋊C412yes—
86C23C2×C22⋊C411yescharacteristic
87Q8—41no—
88Q8—41no—
89Q8—42no—
90Q8—42no—
91Q8—42no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—22no—
95C2×C4—21no—
96C2×C4—22no—
97C2×C4—21no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4#9911yescharacteristic
113C2×C4#9911yescharacteristic
114C2×C4C2×C4⋊C411yescharacteristic
115C2×C4#11711yescharacteristic
116C2×C4#12112yes—
117C2×C4C22⋊Q812yes—
118C8—84no—
119C22—21no—
120C22—21no—
121C22#38011yescharacteristic
122C22C2×#9911yescharacteristic
123C22#41011yescharacteristic
124C4—24no—
125C4—22no—
126C4—21no—
127C4—21no—
128C4—22no—
129C4—21no—
130C4—42no—
131C4—42no—
132C4—41no—
133C4—42no—
134C4—81no—
135C4#35612yescomplemented
136C2—21no—
137C2—21no—
138C2#179911yescharacteristic
139C2C2×#35611yescharacteristic
140C2#213211yescharacteristic
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
135. C4#356semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊Q82
C2×#3562
#18012
#13942