Finite Group Databasebeta

← #12624

Subgroups — #12624

Order 256.

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

normal subgroups only

This group has 228 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#2930C211yesmaximal, characteristic
2#2120C212yesmaximal
3#1757C212yesmaximal, complemented
4#1392C211yesmaximal, characteristic, complemented
5#1391C211yesmaximal, characteristic, complemented
6C2×#131C2211yescharacteristic
7C22×M16C411yescharacteristic, complemented
8#525C2211yescharacteristic
9#436—22no—
10#435—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—42no—
24C2×M16—21no—
25C2×M16—21no—
26C2×M16—22no—
27C2×M16—42no—
28C2×M16C2×C411yescharacteristic
29C2×M16C2×C411yescharacteristic
30C2×M16D412yes—
31C22×C8C2×C411yescharacteristic
32C22×C8C2×C411yescharacteristic
33C22×C8D412yes—
34#123C2×C412yes—
35C22⋊Q8—21no—
36#121—21no—
37#120C2×C412yes—
38C4×D4—22no—
39#117—22no—
40#117C2×C411yescharacteristic
41C4⋊C8—22no—
42#101—22no—
43#100C2×C414yes—
44#99—22no—
45#99C2×C414yes—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#97—21no—
49#97—42no—
50C4×C8—21no—
51C4×C8—42no—
52C24C4211yescharacteristic
53#41—41no—
54#41—41no—
55C2×Q8#4111yescharacteristic
56C2×D4—22no—
57C2×D4C22×C412yes—
58C2×D4#4111yescharacteristic
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4C4211yescharacteristic
64C22×C4C22×C411yescharacteristic
65C22×C4C2×D412yes—
66M16—24no—
67M16—41no—
68M16—42no—
69M16—42no—
70M16—42no—
71M16—42no—
72M16—41no—
73C2×C8—22no—
74C2×C8—22no—
75C2×C8—22no—
76C2×C8—21no—
77C2×C8—21no—
78C2×C8—42no—
79C2×C8—41no—
80C2×C8C22⋊C411yescharacteristic
81C2×C8C22⋊C412yes—
82C2×C8C22⋊C411yescharacteristic
83C4⋊C4—21no—
84C4⋊C4—22no—
85C4⋊C4—42no—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—24no—
89C22⋊C4C4212yes—
90C42—21no—
91C42—42no—
92C23—21no—
93C23—41no—
94C23—81no—
95C23C2×C4211yescharacteristic
96C23C4×D412yes—
97Q8—41no—
98D4—42no—
99D4—41no—
100D4—82no—
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—22no—
104C2×C4—21no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4C2×C22⋊C411yescharacteristic
113C2×C4#11711yescharacteristic
114C2×C4C4×D412yes—
115C8—22no—
116C8—21no—
117C8—44no—
118C8—41no—
119C8—41no—
120C8#9912yes—
121C22—22no—
122C22—21no—
123C22—42no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—41no—
128C22—82no—
129C22C4×C22⋊C411yescharacteristic
130C4—22no—
131C4—21no—
132C4—42no—
133C4—41no—
134C4—84no—
135C4C2×#9911yescharacteristic
136C4#41011yescharacteristic
137C2—21no—
138C2—42no—
139C2—41no—
140C2—81no—
141C2C4×#9911yescharacteristic
subgroupG/H
centerC4C2×#99
commutatorC23C2×C42
Frattini, cosocleC2×#41C23
socleC2C4×#99
prime radical#525C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ 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. C22×M16C4semidirect
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
#29302
#21202
#17572
#13922
#13912