Finite Group Databasebeta

← #8462

Subgroups — #8462

Order 256.

507 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 75 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#3485C211yesmaximal, characteristic
2#1782C211yesmaximal, characteristic, complemented
3#1780C211yesmaximal, characteristic, complemented
4#1516C214yesmaximal, complemented
5C2×H(5,𝔽2)C411yescharacteristic, complemented
6C22×#41C2211yescharacteristic
7#547—24no—
8#546—24no—
9#536C412yescomplemented
10#534—22no—
11#525C2212yes—
12C2×#123—22no—
13#518C411yescharacteristic, complemented
14C2×#99—22no—
15C2×C22⋊C8—22no—
16C2×C22⋊C8C2212yescomplemented
17#381C2212yescomplemented
18#328—24no—
19#327—24no—
20H(5,𝔽2)—44no—
21C2×#41—22no—
22C2×#41—24no—
23C2×#41—44no—
24C2×#41C2×C412yes—
25C22×Q8C2×C411yescharacteristic
26C22×D4—22no—
27C22×D4—22no—
28C22×D4C2×C411yescharacteristic
29C22×D4D412yes—
30C23×C4D412yes—
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33C4⋊Q8—22no—
34#124—22no—
35#123—28no—
36#123—24no—
37#123—24no—
38#123—22no—
39C22⋊Q8—24no—
40C22⋊Q8D414yes—
41#121—28no—
42#121D414yes—
43#120—24no—
44#120—22no—
45C4×D4—24no—
46C4×D4—24no—
47#117C2×C412yes—
48C2×C4⋊C4—24no—
49C2×C22⋊C4—24no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—22no—
52#99—44no—
53C22⋊C8—24no—
54C22⋊C8—44no—
55#95—24no—
56C24—22no—
57C24C22⋊C412yes—
58C24C2×D411yescharacteristic
59#41—44no—
60#41—416no—
61#41—48no—
62C2×Q8—22no—
63C2×Q8—42no—
64C2×Q8C22⋊C412yes—
65C2×D4—24no—
66C2×D4—24no—
67C2×D4—24no—
68C2×D4—22no—
69C2×D4—24no—
70C2×D4—24no—
71C2×D4—22no—
72C2×D4—42no—
73C2×D4—48no—
74C2×D4—44no—
75C2×D4—44no—
76C2×D4C22⋊C412yes—
77C22×C4—22no—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4C22⋊C412yes—
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D414yes—
89C2×C8—44no—
90C2×C8—42no—
91C4⋊C4—28no—
92C4⋊C4—24no—
93C4⋊C4C22⋊C414yes—
94C22⋊C4—24no—
95C22⋊C4—28no—
96C22⋊C4—22no—
97C22⋊C4—24no—
98C22⋊C4—24no—
99C22⋊C4—44no—
100C42—22no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—22no—
105C23—42no—
106C23—44no—
107C23—44no—
108C23—84no—
109C23#9914yes—
110C23C2×C22⋊C411yescharacteristic
111C23#12012yes—
112Q8—42no—
113Q8—44no—
114D4—28no—
115D4—42no—
116D4—48no—
117D4—48no—
118D4—44no—
119D4—84no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—24no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—44no—
128C2×C4—48no—
129C2×C4—44no—
130C2×C4—44no—
131C2×C4—84no—
132C2×C4—82no—
133C2×C4—84no—
134C2×C4C2×C22⋊C412yes—
135C2×C4#12012yes—
136C8—82no—
137C22—22no—
138C22—24no—
139C22—24no—
140C22—22no—
141C22—44no—
142C22—44no—
143C22—44no—
144C22—42no—
145C22—82no—
146C22—82no—
147C22—164no—
148C22#37911yescharacteristic
149C22C2×#9912yes—
150C4—22no—
151C4—22no—
152C4—44no—
153C4—42no—
154C4—42no—
155C4—162no—
156C2—22no—
157C2—42no—
158C2—44no—
159C2—82no—
160C2#178111yescharacteristic
161C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3485C2
LCSG ▷ C22×C4 ▷ 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. #1782C2semidirect
3. #1780C2semidirect
4. #1516C2semidirect
5. C2×H(5,𝔽2)C4semidirect
9. #536C4semidirect
13. #518C4semidirect
16. C2×C22⋊C8C22semidirect
17. #381C22semidirect

Maximal subgroups

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

subgroupindex
#34852
#17822
#17802
#15162