Finite Group Databasebeta

← #32562

Subgroups — #32562

Order 256.

317 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 107 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#2581C212yesmaximal
2#2101C212yesmaximal
3#2087C211yesmaximal, characteristic
4#2074C212yesmaximal
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#400—22no—
20#395—22no—
21#391C2212yes—
22#391C2212yes—
23#389C2212yes—
24#389C2212yes—
25#387—22no—
26#387—22no—
27#382C2212yes—
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⋊C4D411yescharacteristic
50C2×C4⋊C4D412yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C42—22no—
55C2×C42C2312yes—
56C2×C42C2311yescharacteristic
57#107—41no—
58C4⋊C8—41no—
59#103—24no—
60#103—24no—
61#103—42no—
62#97—41no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C2×Q8—22no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—41no—
77C2×Q8—42no—
78C2×Q8#4114yes—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D412yes—
85C22×C4C2×D412yes—
86C22×C4C2×D411yescharacteristic
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—41no—
103C4⋊C4—42no—
104C4⋊C4—44no—
105C4⋊C4—44no—
106C4⋊C4—44no—
107C42—21no—
108C42—21no—
109C42—22no—
110C42—22no—
111C42—44no—
112C23C22×D411yescharacteristic
113Q8—24no—
114Q8—24no—
115Q8—44no—
116Q8—42no—
117Q8—42no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—24no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
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×C4C2×#4112yes—
142C2×C4H(5,𝔽2)11yescharacteristic
143C2×C4H(5,𝔽2)11yescharacteristic
144C2×C42−1+412yes—
145C8—42no—
146C8—41no—
147C8—41no—
148C22#53411yescharacteristic
149C22#54812yes—
150C22#57512yes—
151C22#57811yescharacteristic
152C22#57811yescharacteristic
153C4—42no—
154C4—41no—
155C4—42no—
156C4—41no—
157C4—44no—
158C4—82no—
159C4—82no—
160C4—82no—
161C4#53912yes—
162C4#55111yescharacteristic
163C4#56311yescharacteristic
164C2#272811yescharacteristic
165C2#319711yescharacteristic
166C2#320211yescharacteristic
167C2#337712yes—
168C2#337712yes—
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
#25812
#21012
#20872
#20742
#20562
#19852
#19352
#18782
#18632