Finite Group Databasebeta

← #12623

Subgroups — #12623

Order 256.

235 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 79 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#2929C211yesmaximal, characteristic
2#2120C212yesmaximal
3#1757C212yesmaximal, complemented
4#1392C211yesmaximal, characteristic, complemented
5#1391C211yesmaximal, characteristic, complemented
6C2×#131C2211yescharacteristic
7C23×C8C411yescharacteristic, complemented
8#525C2211yescharacteristic
9#435—22no—
10C8×D4—22no—
11#424C411yescharacteristic, complemented
12#413C2212yes—
13#410C2212yes—
14#408C2211yescharacteristic
15#407C411yescharacteristic, complemented
16#405—22no—
17#405C411yescharacteristic, complemented
18#352C414yescomplemented
19AΣL(1,𝔽16)C414yescomplemented
20#344—22no—
21C2×#41C2311yescharacteristic
22C23×C4C2×C411yescharacteristic
23#131—41no—
24#131—41no—
25C2×M16—22no—
26C2×M16C2×C411yescharacteristic
27C2×M16C2×C411yescharacteristic
28C2×M16D412yes—
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—41no—
32C22×C8—41no—
33C22×C8C2×C411yescharacteristic
34C22×C8C2×C411yescharacteristic
35C22×C8D412yes—
36#123C2×C412yes—
37C22⋊Q8—21no—
38#121—21no—
39#120C2×C412yes—
40C4×D4—22no—
41#117—22no—
42#117C2×C411yescharacteristic
43C4⋊C8—22no—
44#101—22no—
45#100C2×C414yes—
46#99—22no—
47#99C2×C414yes—
48C22⋊C8—22no—
49C22⋊C8—22no—
50#97—21no—
51#97—42no—
52C4×C8—21no—
53C4×C8—42no—
54C24C4211yescharacteristic
55#41—41no—
56#41—41no—
57C2×Q8#4111yescharacteristic
58C2×D4—22no—
59C2×D4C22×C412yes—
60C2×D4#4111yescharacteristic
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4C4211yescharacteristic
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D412yes—
68M16—24no—
69M16—41no—
70M16—42no—
71M16—42no—
72M16—41no—
73C2×C8—22no—
74C2×C8—21no—
75C2×C8—22no—
76C2×C8—21no—
77C2×C8—21no—
78C2×C8—21no—
79C2×C8—42no—
80C2×C8—42no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8C22⋊C411yescharacteristic
85C2×C8C22⋊C412yes—
86C2×C8C22⋊C411yescharacteristic
87C4⋊C4—21no—
88C4⋊C4—22no—
89C4⋊C4—42no—
90C22⋊C4—22no—
91C22⋊C4—22no—
92C22⋊C4—24no—
93C22⋊C4C4212yes—
94C42—21no—
95C42—42no—
96C23—21no—
97C23—41no—
98C23—42no—
99C23C2×C4211yescharacteristic
100C23C4×D412yes—
101Q8—41no—
102D4—42no—
103D4—41no—
104D4—44no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—21no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—44no—
116C2×C4C2×C22⋊C411yescharacteristic
117C2×C4#11711yescharacteristic
118C2×C4C4×D412yes—
119C8—21no—
120C8—22no—
121C8—44no—
122C8—41no—
123C8—41no—
124C8#9911yescharacteristic
125C8#9911yescharacteristic
126C22—22no—
127C22—22no—
128C22—21no—
129C22—42no—
130C22—41no—
131C22—41no—
132C22—42no—
133C22—42no—
134C22—41no—
135C22C4×C22⋊C411yescharacteristic
136C4—22no—
137C4—21no—
138C4—42no—
139C4—41no—
140C4—84no—
141C4C2×#9911yescharacteristic
142C4#41011yescharacteristic
143C2—21no—
144C2—42no—
145C2—42no—
146C2—41no—
147C2C4×#9911yescharacteristic
subgroupG/H
centerC8#99
commutatorC23C2×C42
Frattini, cosocleC2×#41C23
socleC2C4×#99
prime radical#525C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1757C2semidirect
4. #1392C2semidirect
5. #1391C2semidirect
7. C23×C8C4semidirect
11. #424C4semidirect
15. #407C4semidirect
17. #405C4semidirect
18. #352C4semidirect
19. AΣL(1,𝔽16)C4semidirect

Maximal subgroups

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

subgroupindex
#29292
#21202
#17572
#13922
#13912