Finite Group Databasebeta

← #12634

Subgroups — #12634

Order 256.

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

normal subgroups only

This group has 226 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#2932C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic
3#2126C211yesmaximal, characteristic
4#1757C212yesmaximal, complemented
5#1395C212yesmaximal, complemented
6C2×#131C411yescharacteristic, complemented
7C2×#131C2211yescharacteristic
8#532C2211yescharacteristic
9#435—22no—
10#435—22no—
11#423C411yescharacteristic, complemented
12#413C2212yes—
13#410C2212yes—
14#408C411yescharacteristic, complemented
15#408C2211yescharacteristic
16#405—22no—
17C2×#97C411yescharacteristic, complemented
18#356C412yescomplemented
19#355C412yescomplemented
20#354C412yes—
21#353C412yes—
22#344—22no—
23C2×#41C2×C411yescharacteristic
24C2×#41C2311yescharacteristic
25#131—42no—
26#131—42no—
27C2×M16—21no—
28C2×M16—22no—
29C2×M16C2×C411yescharacteristic
30C2×M16C2×C411yescharacteristic
31C2×M16D412yes—
32C22×C8—21no—
33C22×C8C2×C411yescharacteristic
34C22×C8C2×C411yescharacteristic
35C22×C8D412yes—
36C4⋊Q8C2×C411yescharacteristic
37#127C2×C411yescharacteristic
38#124C2×C411yescharacteristic
39#124C2×C411yescharacteristic
40#121—22no—
41C4×D4—22no—
42#117—22no—
43C2×C42C2×C411yescharacteristic
44C4⋊C8—22no—
45#101C2×C414yes—
46#100—22no—
47#99—22no—
48#99C2×C414yes—
49C22⋊C8—22no—
50C22⋊C8—22no—
51#97—21no—
52#97—21no—
53#97—42no—
54C4×C8—42no—
55#41—41no—
56#41—41no—
57#41—41no—
58#41—41no—
59C2×Q8C4211yescharacteristic
60C2×Q8C22×C411yescharacteristic
61C2×D4—21no—
62C2×D4—22no—
63C2×D4C4211yescharacteristic
64C2×D4C22×C411yescharacteristic
65C2×D4#4112yes—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D412yes—
70M16—24no—
71M16—41no—
72M16—42no—
73M16—42no—
74M16—42no—
75M16—41no—
76C2×C8—22no—
77C2×C8—22no—
78C2×C8—22no—
79C2×C8—21no—
80C2×C8—21no—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—41no—
84C2×C8C22⋊C411yescharacteristic
85C2×C8C22⋊C412yes—
86C2×C8C22⋊C411yescharacteristic
87C4⋊C4—22no—
88C4⋊C4—42no—
89C22⋊C4—22no—
90C22⋊C4—22no—
91C22⋊C4—24no—
92C42—21no—
93C42—42no—
94C42C4211yescharacteristic
95C42C4211yescharacteristic
96C23—21no—
97C23#11711yescharacteristic
98C23C4×D412yes—
99Q8—41no—
100Q8—41no—
101D4—41no—
102D4—42no—
103D4—41no—
104D4—42no—
105D4—82no—
106C2×C4—22no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—44no—
116C2×C4—81no—
117C2×C4C2×C4211yescharacteristic
118C2×C4C2×C22⋊C411yescharacteristic
119C2×C4C4×D412yes—
120C8—21no—
121C8—22no—
122C8—44no—
123C8—41no—
124C8—41no—
125C8#9912yes—
126C22—22no—
127C22—21no—
128C22—41no—
129C22—42no—
130C22—42no—
131C22—41no—
132C22C4×C22⋊C411yescharacteristic
133C4—22no—
134C4—21no—
135C4—41no—
136C4—41no—
137C4—41no—
138C4—84no—
139C4C2×#9911yescharacteristic
140C4#41011yescharacteristic
141C2—21no—
142C2—42no—
143C2—41no—
144C2—81no—
145C2C4×#9911yescharacteristic
subgroupG/H
centerC4C2×#99
commutatorC2×C4C2×C42
Frattini, cosocleC2×#41C23
socleC2C4×#99
prime radical#532C22
LCSG ▷ C2×C4 ▷ 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
4. #1757C2semidirect
5. #1395C2semidirect
6. C2×#131C4semidirect
11. #423C4semidirect
14. #408C4semidirect
17. C2×#97C4semidirect
18. #356C4semidirect
19. #355C4semidirect

Maximal subgroups

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

subgroupindex
#29322
#21322
#21262
#17572
#13952