Finite Group Databasebeta

← #12974

Subgroups — #12974

Order 256.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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#2980C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic
3#1884C211yesmaximal, characteristic
4#1858C212yesmaximal, complemented
5#1395C212yesmaximal, complemented
6#568C2211yescharacteristic
7#532C2211yescharacteristic
8#435—22no—
9#435—22no—
10#432—21no—
11#430—22no—
12#421—21no—
13#413C2212yes—
14#410C2211yescharacteristic
15#408—21no—
16#408C2212yes—
17#356C412yescomplemented
18#355C412yescomplemented
19#341—22no—
20#339—21no—
21C2×#41D411yescharacteristic
22C2×#41C2311yescharacteristic
23#131—42no—
24C2×M16—21no—
25C2×M16—21no—
26C2×M16—22no—
27C2×M16—22no—
28C2×M16D412yes—
29C22×C8—21no—
30C22×C8D412yes—
31C4⋊Q8C2×C411yescharacteristic
32#127Q811yescharacteristic
33#124C2×C411yescharacteristic
34#124Q811yescharacteristic
35#121—22no—
36C4×D4—22no—
37#117—21no—
38C2×C42D411yescharacteristic
39#108—44no—
40C4⋊C8—21no—
41C4⋊C8—21no—
42#104—42no—
43#101C2×C414yes—
44#100—22no—
45#99—22no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49#97—22no—
50#41—41no—
51#41—41no—
52#41—41no—
53#41—41no—
54C2×Q8C4⋊C411yescharacteristic
55C2×Q8C22×C411yescharacteristic
56C2×D4—22no—
57C2×D4—21no—
58C2×D4C4⋊C411yescharacteristic
59C2×D4C2×Q811yescharacteristic
60C2×D4#4112yes—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D412yes—
65M16—22no—
66M16—24no—
67M16—24no—
68M16—42no—
69M16—41no—
70M16—42no—
71M16—81no—
72C2×C8—22no—
73C2×C8—22no—
74C2×C8—42no—
75C2×C8—41no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—81no—
79C4⋊C4—22no—
80C4⋊C4—41no—
81C22⋊C4—22no—
82C22⋊C4—22no—
83C22⋊C4—41no—
84C42—21no—
85C42—41no—
86C42C4⋊C411yescharacteristic
87C42C4⋊C411yescharacteristic
88C23—21no—
89C23C22⋊Q812yes—
90C23#12311yescharacteristic
91Q8—41no—
92Q8—41no—
93D4—41no—
94D4—42no—
95D4—41no—
96D4—81no—
97D4—82no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—81no—
108C2×C4—81no—
109C2×C4C2×C4⋊C411yescharacteristic
110C2×C4C4×D412yes—
111C2×C4#12011yescharacteristic
112C8—44no—
113C8—42no—
114C8—42no—
115C8—42no—
116C22—21no—
117C22—22no—
118C22—41no—
119C22—42no—
120C22—42no—
121C22—41no—
122C22#38511yescharacteristic
123C4—22no—
124C4—21no—
125C4—41no—
126C4—41no—
127C4—41no—
128C4—161no—
129C4B(4,𝔽2)11yescharacteristic
130C4#45811yescharacteristic
131C2—21no—
132C2—41no—
133C2—42no—
134C2—81no—
135C2#186711yescharacteristic
subgroupG/H
centerC4B(4,𝔽2)
commutatorC2×Q8C22×C4
Frattini, cosocleC2×#41C23
socleC2#1867
prime radical#532C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1858C2semidirect
5. #1395C2semidirect
17. #356C4semidirect
18. #355C4semidirect

Maximal subgroups

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

subgroupindex
#29802
#21322
#18842
#18582
#13952