Finite Group Databasebeta

← #8464

Subgroups — #8464

Order 256.

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

normal subgroups only

This group has 501 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#3487C211yesmaximal, characteristic
2#1782C211yesmaximal, characteristic, complemented
3#1780C211yesmaximal, characteristic, complemented
4#1516C214yesmaximal, complemented
5C2×2−1+4C411yescharacteristic, complemented
6C22×#41C2211yescharacteristic
7#550—24no—
8#548—24no—
9#535—22no—
10#535C412yescomplemented
11#525—22no—
12#525C2212yes—
13#519C411yescharacteristic, complemented
14C2×#99—22no—
15C2×C22⋊C8—22no—
16C2×C22⋊C8C2212yes—
17#381C2212yes—
18#328—24no—
19#327—24no—
202−1+4—44no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—24no—
24C2×#41—44no—
25C2×#41C2×C412yes—
26C22×Q8—22no—
27C22×Q8C2×C411yescharacteristic
28C22×D4C2×C411yescharacteristic
29C22×D4D412yes—
30C23×C4D412yes—
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33#127—24no—
34#127—22no—
35#124—24no—
36#124—22no—
37#123—22no—
38C22⋊Q8—24no—
39C22⋊Q8D414yes—
40#121—24no—
41#121—28no—
42#121—24no—
43#121D414yes—
44#120—24no—
45#120—22no—
46C4×Q8—24no—
47C4×D4—28no—
48C4×D4—24no—
49#117—22no—
50#117C2×C412yes—
51C2×C22⋊C4—22no—
52#99—44no—
53C22⋊C8—24no—
54C22⋊C8—44no—
55#95—24no—
56C24C22⋊C412yes—
57C24C2×D411yescharacteristic
58#41—44no—
59#41—44no—
60#41—416no—
61#41—48no—
62#41—44no—
63C2×Q8—22no—
64C2×Q8—24no—
65C2×Q8—22no—
66C2×Q8—48no—
67C2×Q8C22⋊C412yes—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—24no—
71C2×D4—28no—
72C2×D4—24no—
73C2×D4—24no—
74C2×D4—22no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4C22⋊C412yes—
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—24no—
92C4⋊C4—24no—
93C4⋊C4C22⋊C414yes—
94C22⋊C4—28no—
95C22⋊C4—22no—
96C22⋊C4—24no—
97C22⋊C4—24no—
98C22⋊C4—44no—
99C42—24no—
100C42—22no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—22no—
105C23—22no—
106C23—42no—
107C23—44no—
108C23C2×C22⋊C411yescharacteristic
109C23#12012yes—
110Q8—28no—
111Q8—42no—
112Q8—44no—
113D4—48no—
114D4—42no—
115D4—44no—
116D4—44no—
117D4—44no—
118D4—84no—
119D4—88no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—44no—
127C2×C4—48no—
128C2×C4—48no—
129C2×C4—44no—
130C2×C4—82no—
131C2×C4—84no—
132C2×C4#9914yes—
133C2×C4C2×C22⋊C412yes—
134C2×C4#12012yes—
135C8—82no—
136C22—22no—
137C22—24no—
138C22—22no—
139C22—44no—
140C22—44no—
141C22—44no—
142C22—42no—
143C22—42no—
144C22—82no—
145C22—82no—
146C22#37911yescharacteristic
147C22C2×#9912yes—
148C4—22no—
149C4—24no—
150C4—22no—
151C4—44no—
152C4—42no—
153C4—162no—
154C2—22no—
155C2—42no—
156C2—82no—
157C2—82no—
158C2#178111yescharacteristic
159C2#179412yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3487C2
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. C2×2−1+4C4semidirect
10. #535C4semidirect
13. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34872
#17822
#17802
#15162