Finite Group Databasebeta

← #29949

Subgroups — #29949

Order 256.

304 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 111 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#2629C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2084C212yesmaximal, complemented
5#2077C214yesmaximal
6#1989C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1798C211yesmaximal, characteristic
9C2×#125C2211yescharacteristic
10C2×#117C2211yescharacteristic
11C2×#106C2214yes—
12C2×C4⋊C8C2212yes—
13C2×C22⋊C8C2212yes—
14#398—22no—
15#396—22no—
16#389C2212yes—
17#388—22no—
18#387—22no—
19#384C2211yescharacteristic
20#384C2211yescharacteristic
21#382C2211yescharacteristic
22#380C2212yes—
23#380C2212yes—
24C4×C4⋊C4C2212yes—
25#340C2214yes—
26#340C2214yes—
27#340C2214yes—
28#336C2212yes—
29C23×C4C2311yescharacteristic
30C22×C8C2314yes—
31#125—24no—
32#117—24no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4D412yes—
38C2×C4⋊C4C2312yes—
39C2×C4⋊C4C2312yes—
40C2×C4⋊C4C2312yes—
41C2×C4⋊C4C2312yes—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4D412yes—
45C2×C42—22no—
46C2×C42C2311yescharacteristic
47C2×C42C2311yescharacteristic
48#106—44no—
49C4⋊C8—24no—
50C22⋊C8—42no—
51#95—22no—
52#95—22no—
53#95—22no—
54#95—22no—
55#95—22no—
56C24C2×D411yescharacteristic
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D412yes—
67C22×C4C2×D412yes—
68C22×C4C2411yescharacteristic
69C2×C8—24no—
70C2×C8—24no—
71C2×C8—24no—
72C4⋊C4—24no—
73C4⋊C4—24no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—24no—
77C4⋊C4—22no—
78C4⋊C4—22no—
79C4⋊C4—44no—
80C4⋊C4—82no—
81C4⋊C4—82no—
82C4⋊C4—82no—
83C22⋊C4—24no—
84C22⋊C4—82no—
85C22⋊C4—82no—
86C42—22no—
87C42—82no—
88C42#4114yes—
89C23—21no—
90C23—21no—
91C23—21no—
92C23—22no—
93C23—22no—
94C23C22×D411yescharacteristic
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4#12314yes—
118C2×C4C2×#4112yes—
119C2×C4H(5,𝔽2)11yescharacteristic
120C2×C42−1+411yescharacteristic
121C2×C42−1+412yes—
122C8—44no—
123C22—42no—
124C22—41no—
125C22—42no—
126C22—41no—
127C22—41no—
128C22C2×#12311yescharacteristic
129C22#53611yescharacteristic
130C22#53711yescharacteristic
131C22#57712yes—
132C22#57712yes—
133C4—24no—
134C4—42no—
135C4—41no—
136C4—82no—
137C4—82no—
138C4—82no—
139C4—82no—
140C4#54112yes—
141C4#54212yes—
142C2—81no—
143C2#262511yescharacteristic
144C2#309912yes—
145C2#324412yes—
146C2#326011yescharacteristic
147C2#326011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2084C2semidirect
6. #1989C2semidirect
7. #1948C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#23142
#21012
#20842
#20772
#19892
#19482
#17982