Finite Group Databasebeta

← #12633

Subgroups — #12633

Order 256.

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

normal subgroups only

This group has 230 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#2931C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic
3#2126C211yesmaximal, characteristic
4#1757C212yesmaximal, complemented
5#1395C212yesmaximal, complemented
6C2×#131C411yescharacteristic, complemented
7C2×#131C2211yescharacteristic
8#532C2211yescharacteristic
9#436—22no—
10C8×D4—22no—
11#423C411yescharacteristic, complemented
12#413C2212yes—
13#410C2212yes—
14#408C411yescharacteristic, complemented
15#408C2211yescharacteristic
16#405—22no—
17C2×C4×C8C411yescharacteristic, complemented
18#356C412yescomplemented
19#355C412yescomplemented
20#354C412yes—
21#353C412yes—
22#344—22no—
23C2×#41C2×C411yescharacteristic
24C2×#41C2311yescharacteristic
25#131—41no—
26#131—41no—
27#131—41no—
28#131—41no—
29C2×M16—21no—
30C2×M16—22no—
31C2×M16C2×C411yescharacteristic
32C2×M16C2×C411yescharacteristic
33C2×M16D412yes—
34C22×C8—21no—
35C22×C8C2×C411yescharacteristic
36C22×C8C2×C411yescharacteristic
37C22×C8D412yes—
38C4⋊Q8C2×C411yescharacteristic
39#127C2×C411yescharacteristic
40#124C2×C411yescharacteristic
41#124C2×C411yescharacteristic
42#121—22no—
43C4×D4—22no—
44#117—22no—
45C2×C42C2×C411yescharacteristic
46C4⋊C8—22no—
47#101C2×C414yes—
48#100—22no—
49#99—22no—
50#99C2×C414yes—
51C22⋊C8—22no—
52C22⋊C8—22no—
53#97—42no—
54C4×C8—21no—
55C4×C8—21no—
56C4×C8—42no—
57#41—41no—
58#41—41no—
59#41—41no—
60#41—41no—
61C2×Q8C4211yescharacteristic
62C2×Q8C22×C411yescharacteristic
63C2×D4—21no—
64C2×D4—22no—
65C2×D4C4211yescharacteristic
66C2×D4C22×C411yescharacteristic
67C2×D4#4112yes—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D412yes—
72M16—24no—
73M16—41no—
74M16—42no—
75M16—42no—
76M16—42no—
77M16—41no—
78C2×C8—22no—
79C2×C8—21no—
80C2×C8—22no—
81C2×C8—21no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—42no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8C22⋊C411yescharacteristic
89C2×C8C22⋊C412yes—
90C2×C8C22⋊C411yescharacteristic
91C4⋊C4—22no—
92C4⋊C4—42no—
93C22⋊C4—22no—
94C22⋊C4—22no—
95C22⋊C4—24no—
96C42—21no—
97C42—42no—
98C42C4211yescharacteristic
99C42C4211yescharacteristic
100C23—21no—
101C23#11711yescharacteristic
102C23C4×D412yes—
103Q8—41no—
104Q8—41no—
105D4—41no—
106D4—42no—
107D4—41no—
108D4—44no—
109D4—42no—
110C2×C4—21no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—44no—
121C2×C4C2×C4211yescharacteristic
122C2×C4C2×C22⋊C411yescharacteristic
123C2×C4C4×D412yes—
124C8—22no—
125C8—21no—
126C8—44no—
127C8—41no—
128C8—41no—
129C8#9911yescharacteristic
130C8#9911yescharacteristic
131C22—22no—
132C22—21no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—41no—
137C22C4×C22⋊C411yescharacteristic
138C4—22no—
139C4—22no—
140C4—21no—
141C4—41no—
142C4—41no—
143C4—84no—
144C4C2×#9911yescharacteristic
145C4#41011yescharacteristic
146C2—21no—
147C2—42no—
148C2—41no—
149C2—81no—
150C2C4×#9911yescharacteristic
subgroupG/H
centerC8#99
commutatorC2×C4C2×C42
Frattini, cosocleC2×#41C23
socleC2C4×#99
prime radical#532C22
LCSG ▷ C2×C4 ▷ 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
4. #1757C2semidirect
5. #1395C2semidirect
6. C2×#131C4semidirect
11. #423C4semidirect
14. #408C4semidirect
17. C2×C4×C8C4semidirect
18. #356C4semidirect
19. #355C4semidirect

Maximal subgroups

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

subgroupindex
#29312
#21322
#21262
#17572
#13952