Finite Group Databasebeta

← #8540

Subgroups — #8540

Order 256.

359 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 71 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#3491C211yesmaximal, characteristic
2#1822C211yesmaximal, characteristic, complemented
3#1818C211yesmaximal, characteristic, complemented
4#1468C214yesmaximal, complemented
5#552—24no—
6#551—24no—
7#543C411yescharacteristic
8#539—22no—
9#534C411yescharacteristic
10#529—22no—
11C2×#123C2212yes—
12#518C412yes—
13C2×#117C2211yescharacteristic
14#407C2212yes—
15C2×C22⋊C8C2212yes—
16#343—22no—
17#336—22no—
18#324—24no—
19#323—24no—
20C22×D4C2×C412yes—
21C23×C4C2311yescharacteristic
22C2×M16—22no—
23C22×C8—22no—
24#126—24no—
25#126—24no—
26#125—22no—
27#125—24no—
28#124—22no—
29#123—24no—
30#123—22no—
31#123—22no—
32#123—22no—
33#123—24no—
34C22⋊Q8—24no—
35C22⋊Q8—22no—
36C22⋊Q8—24no—
37#121—22no—
38#120—22no—
39C4×Q8—22no—
40C4×D4—22no—
41C4×D4—24no—
42C4×D4—24no—
43#117—24no—
44#117—22no—
45#117—24no—
46C2×C4⋊C4D412yes—
47C2×C4⋊C4D414yes—
48C2×C22⋊C4—22no—
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4C2×C412yes—
52C2×C22⋊C4D414yes—
53C2×C42D412yes—
54C22⋊C8—24no—
55C22⋊C8—24no—
56C24C22⋊C412yes—
57C24C22×C411yescharacteristic
58C2×Q8—22no—
59C2×D4—22no—
60C2×D4—22no—
61C2×D4—22no—
62C2×D4—44no—
63C22×C4—24no—
64C22×C4—42no—
65C22×C4C22⋊C412yes—
66C22×C4C22⋊C412yes—
67C22×C4C22⋊C414yes—
68C22×C4C22⋊C412yes—
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D414yes—
72M16—44no—
73C2×C8—44no—
74C2×C8—42no—
75C2×C8—42no—
76C4⋊C4—24no—
77C4⋊C4—24no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—24no—
81C4⋊C4—24no—
82C4⋊C4—24no—
83C22⋊C4—24no—
84C22⋊C4—24no—
85C22⋊C4—22no—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C22⋊C4—24no—
90C22⋊C4—24no—
91C42—24no—
92C42—22no—
93C42—22no—
94C23—22no—
95C23—22no—
96C23—42no—
97C23—42no—
98C23—44no—
99C23C2×C22⋊C412yes—
100C23C2×C22⋊C411yescharacteristic
101Q8—82no—
102D4—44no—
103D4—44no—
104D4—82no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—24no—
111C2×C4—44no—
112C2×C4—42no—
113C2×C4—44no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—84no—
117C2×C4#12012yes—
118C2×C4#12012yes—
119C8—82no—
120C8—82no—
121C22—22no—
122C22—22no—
123C22—24no—
124C22—42no—
125C22—44no—
126C22—82no—
127C22—162no—
128C22#37911yescharacteristic
129C22#41012yes—
130C4—22no—
131C4—22no—
132C4—42no—
133C4—42no—
134C4—44no—
135C4—42no—
136C2—22no—
137C2—42no—
138C2—82no—
139C2#178211yescharacteristic
140C2#179112yes—
subgroupG/H
center, socleC22#379
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radical#534C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1822C2semidirect
3. #1818C2semidirect
4. #1468C2semidirect

Maximal subgroups

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

subgroupindex
#34912
#18222
#18182
#14682