Finite Group Databasebeta

← #13227

Subgroups — #13227

Order 256.

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

normal subgroups only

This group has 236 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#2546C211yesmaximal, characteristic
2C2×#356C214yesmaximal
3#1801C211yesmaximal, characteristic
4C4×#101C211yesmaximal, characteristic
5C2×C4⋊Q8C411yescharacteristic
6C2×C4⋊Q8C2212yes—
7C2×C4×Q8C2211yescharacteristic
8#423—21no—
9C2×#101C2212yes—
10C2×#101C2212yes—
11C4×M16—21no—
12#398—21no—
13#398—21no—
14#391C412yes—
15#389—21no—
16#389—21no—
17C4×C4⋊C4C411yescharacteristic
18#356—28no—
19#343—21no—
20#343—21no—
21C22×Q8C2×C412yes—
22C22×Q8C2311yescharacteristic
23C2×M16—22no—
24C2×M16—22no—
25C4⋊Q8—24no—
26C4⋊Q8—24no—
27C4⋊Q8—41no—
28C4⋊Q8—41no—
29C4×Q8—41no—
30C4×Q8—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C42C2×C412yes—
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C4⋊C8—41no—
44C4⋊C8—41no—
45#101—22no—
46#101—24no—
47#101D414yes—
48#97—41no—
49C4×C8—41no—
50#95—21no—
51#95—22no—
52#95—22no—
53C2×Q8—24no—
54C2×Q8—41no—
55C2×Q8—41no—
56C2×Q8—42no—
57C2×Q8—42no—
58C2×Q8C2×D412yes—
59C2×Q8#4112yes—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4C22⋊C411yescharacteristic
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69M16—24no—
70M16—42no—
71M16—42no—
72M16—42no—
73C2×C8—42no—
74C2×C8—42no—
75C4⋊C4—24no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—42no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—44no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C42—21no—
86C42—21no—
87C42—21no—
88C42—21no—
89C42—21no—
90C42—21no—
91C42—24no—
92C23C2×C22⋊C411yescharacteristic
93Q8—42no—
94Q8—42no—
95Q8—84no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—24no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4#9911yescharacteristic
116C2×C4#9911yescharacteristic
117C2×C4C4×D412yes—
118C2×C4#12112yes—
119C2×C4#12411yescharacteristic
120C2×C4#12711yescharacteristic
121C8—44no—
122C8—82no—
123C22—21no—
124C22—21no—
125C22#39011yescharacteristic
126C22C2×#9911yescharacteristic
127C22#41011yescharacteristic
128C4—21no—
129C4—22no—
130C4—21no—
131C4—22no—
132C4—21no—
133C4—44no—
134C4—41no—
135C4—41no—
136C4—41no—
137C4—82no—
138C4#35612yescomplemented
139C2—21no—
140C2—21no—
141C2#196411yescharacteristic
142C2C2×#35611yescharacteristic
143C2#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
138. C4#356semidirect

Maximal subgroups

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

subgroupindex
#25462
C2×#3562
#18012
C4×#1012