Finite Group Databasebeta

← #32563

Subgroups — #32563

Order 256.

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

normal subgroups only

This group has 325 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#2582C212yesmaximal
2#2099C212yesmaximal
3#2086C211yesmaximal, characteristic
4#2072C212yesmaximal
5#2056C211yesmaximal, characteristic
6#1985C211yesmaximal, characteristic
7#1935C212yesmaximal
8#1878C212yesmaximal
9#1863C212yesmaximal
10C22×Q16C2211yescharacteristic
11C2×C4×Q8C2212yes—
12C2×C4×Q8C2211yescharacteristic
13C2×#107C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#97C2211yescharacteristic
19#398—22no—
20#398—22no—
21#391—22no—
22#391C2212yes—
23#391C2212yes—
24#389—22no—
25#389C2212yes—
26#389C2212yes—
27#384C2212yes—
28#340C2211yescharacteristic
29#340C2211yescharacteristic
30#340C2212yes—
31#340C2212yes—
32#340C2212yes—
33#336C2212yes—
34#336C2212yes—
35C22×Q8D411yescharacteristic
36C22×Q8C2312yes—
37C2×Q16—24no—
38C2×Q16—41no—
39C22×C8C2311yescharacteristic
40C22×C8C2312yes—
41C22×C8C2311yescharacteristic
42C4×Q8—41no—
43C4×Q8—41no—
44C4×Q8—42no—
45C4×Q8—42no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4D412yes—
52C2×C4⋊C4D411yescharacteristic
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C42—22no—
57C2×C42C2312yes—
58C2×C42C2311yescharacteristic
59#107—41no—
60C4⋊C8—41no—
61#103—24no—
62#103—24no—
63#103—42no—
64#97—41no—
65#95—22no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C2×Q8—24no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—41no—
78C2×Q8#4114yes—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D412yes—
86C22×C4C2×D412yes—
87C22×C4C2411yescharacteristic
88Q16—44no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—44no—
105C4⋊C4—44no—
106C4⋊C4—82no—
107C4⋊C4—82no—
108C4⋊C4—82no—
109C42—21no—
110C42—21no—
111C42—22no—
112C42—22no—
113C42—44no—
114C23C22×D411yescharacteristic
115Q8—24no—
116Q8—24no—
117Q8—44no—
118Q8—42no—
119Q8—42no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—24no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4#13714yes—
143C2×C4C2×#4112yes—
144C2×C4H(5,𝔽2)11yescharacteristic
145C2×C4H(5,𝔽2)11yescharacteristic
146C2×C42−1+412yes—
147C8—42no—
148C8—41no—
149C8—41no—
150C22#53411yescharacteristic
151C22#54812yes—
152C22C2×#13712yes—
153C22#57611yescharacteristic
154C22#57811yescharacteristic
155C4—24no—
156C4—41no—
157C4—42no—
158C4—41no—
159C4—44no—
160C4—82no—
161C4—82no—
162C4—82no—
163C4#53912yes—
164C4#55111yescharacteristic
165C4#56311yescharacteristic
166C2#272811yescharacteristic
167C2#319511yescharacteristic
168C2#320311yescharacteristic
169C2#337412yes—
170C2#337512yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ 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
#25822
#20992
#20862
#20722
#20562
#19852
#19352
#18782
#18632