Finite Group Databasebeta

← #32863

Subgroups — #32863

Order 256.

300 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 109 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#2863C212yesmaximal
2#2088C212yesmaximal
3#2086C211yesmaximal, characteristic
4#2068C212yesmaximal
5#2066C212yesmaximal
6#2066C212yesmaximal
7#2056C212yesmaximal
8#2056C211yesmaximal, characteristic
9#1842C211yesmaximal, characteristic
10C22×Q16C2211yescharacteristic
11C2×#107C2211yescharacteristic
12C2×#106C2212yes—
13C2×C4⋊C8C2211yescharacteristic
14C2×C4⋊C8C2212yes—
15C2×C4⋊C8C2211yescharacteristic
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20#391C2212yes—
21#391C2212yes—
22#391C2212yes—
23#389C2212yes—
24#389C2212yes—
25#389C2212yes—
26#387—22no—
27#387—22no—
28#387—22no—
29#383—22no—
30C43C2211yescharacteristic
31#340C2212yes—
32#340C2211yescharacteristic
33#340C2211yescharacteristic
34#340C2212yes—
35C22×Q8C2312yes—
36C2×Q16—42no—
37C2×Q16—41no—
38C22×C8C2311yescharacteristic
39C22×C8C2312yes—
40C22×C8C2311yescharacteristic
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4C2312yes—
45C2×C4⋊C4C2312yes—
46C2×C4⋊C4C2312yes—
47C2×C42D412yes—
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C2×C42C2312yes—
52#107—41no—
53#106—42no—
54C4⋊C8—41no—
55C4⋊C8—42no—
56C4⋊C8—42no—
57C4⋊C8—41no—
58#103—42no—
59#103—42no—
60#103—42no—
61#95—22no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70C2×Q8—22no—
71C2×Q8—22no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D412yes—
84C22×C4C2411yescharacteristic
85Q16—44no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—21no—
91C2×C8—22no—
92C2×C8—21no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—44no—
100C4⋊C4—44no—
101C4⋊C4—44no—
102C42—22no—
103C42—24no—
104C42—22no—
105C42—21no—
106C42—22no—
107C42—21no—
108C42—41no—
109C23C22×D411yescharacteristic
110Q8—42no—
111Q8—42no—
112Q8—42no—
113Q8—42no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4#13512yes—
136C2×C4#13714yes—
137C2×C4H(5,𝔽2)12yes—
138C2×C4H(5,𝔽2)11yescharacteristic
139C2×C42−1+412yes—
140C2×C42−1+411yescharacteristic
141C8—42no—
142C8—41no—
143C8—41no—
144C22#53712yes—
145C22#53711yescharacteristic
146C22C2×#13511yescharacteristic
147C22C2×#13712yes—
148C22#57511yescharacteristic
149C4—24no—
150C4—22no—
151C4—42no—
152C4—41no—
153C4—82no—
154C4—82no—
155C4—82no—
156C4—82no—
157C4#56011yescharacteristic
158C4#56312yes—
159C4#56311yescharacteristic
160C2#282311yescharacteristic
161C2#324812yes—
162C2#325011yescharacteristic
163C2#325212yes—
164C2#325511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#28632
#20882
#20862
#20682
#20662
#20562
#18422