Finite Group Databasebeta

← #32849

Subgroups — #32849

Order 256.

302 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 117 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#2859C211yesmaximal, characteristic
2#2859C211yesmaximal, characteristic
3#2100C212yesmaximal
4#2099C212yesmaximal
5#2088C214yesmaximal
6#2084C212yesmaximal
7#2084C212yesmaximal
8#1831C211yesmaximal, characteristic
9C2×#125—21no—
10C2×#125—21no—
11C2×#106C2214yes—
12C2×C4⋊C8C2212yes—
13#389—21no—
14#389—21no—
15#389—21no—
16#389—21no—
17#389C2212yes—
18#389C2212yes—
19#389C2212yes—
20#389C2212yes—
21#384C2211yescharacteristic
22#384C2211yescharacteristic
23#384C2211yescharacteristic
24#384C2211yescharacteristic
25#382—21no—
26#382—21no—
27C43C2211yescharacteristic
28#340C2214yes—
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×C8C2314yes—
34#125—44no—
35#125—44no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4C2312yes—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C2×C42D411yescharacteristic
56C2×C42C2311yescharacteristic
57C2×C42C2311yescharacteristic
58C2×C42C2311yescharacteristic
59#106—44no—
60C4⋊C8—42no—
61#95—22no—
62#95—22no—
63#95—22no—
64#95—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2411yescharacteristic
76C2×C8—24no—
77C2×C8—24no—
78C2×C8—24no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—24no—
82C4⋊C4—24no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—44no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—44no—
93C4⋊C4—44no—
94C4⋊C4—42no—
95C4⋊C4—44no—
96C4⋊C4—42no—
97C42—22no—
98C42—22no—
99C42—21no—
100C42—22no—
101C42—22no—
102C42—21no—
103C42—41no—
104C42SD1614yes—
105C23C22×D411yescharacteristic
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
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×C4C2×SD1612yes—
132C2×C4C2×SD1612yes—
133C2×C4C2×SD1612yes—
134C2×C4AGL(1,ℤ/8)12yes—
135C2×C4#13712yes—
136C2×C4H(5,𝔽2)11yescharacteristic
137C2×C42−1+412yes—
138C2×C42−1+412yes—
139C2×C42−1+411yescharacteristic
140C8—44no—
141C22#53611yescharacteristic
142C22#53611yescharacteristic
143C22#53711yescharacteristic
144C22C22×SD1611yescharacteristic
145C22C2×AGL(1,ℤ/8)11yescharacteristic
146C22C2×#13711yescharacteristic
147C22#57511yescharacteristic
148C4—22no—
149C4—22no—
150C4—22no—
151C4—22no—
152C4—22no—
153C4—41no—
154C4—82no—
155C4—82no—
156C4—82no—
157C4—82no—
158C4#56312yes—
159C4#56412yes—
160C2#282211yescharacteristic
161C2#322711yescharacteristic
162C2#322911yescharacteristic
163C2#323411yescharacteristic
164C2#323611yescharacteristic
165C2#324611yescharacteristic
166C2#325411yescharacteristic
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
#28592
#21002
#20992
#20882
#20842
#18312