Finite Group Databasebeta

← #32791

Subgroups — #32791

Order 256.

314 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 105 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#2851C212yesmaximal
2#2101C212yesmaximal, complemented
3#2101C212yesmaximal, complemented
4#2077C212yesmaximal
5#2076C211yesmaximal, characteristic
6#2075C211yesmaximal, characteristic
7#2059C212yesmaximal, complemented
8#2059C212yesmaximal, complemented
9#1817C211yesmaximal, characteristic
10C2×#125C2212yes—
11C22×C42C2211yescharacteristic
12C2×#107C2211yescharacteristic
13C2×#107C2211yescharacteristic
14C2×#106C2212yes—
15C2×C22⋊C8C2211yescharacteristic
16C2×C22⋊C8C2211yescharacteristic
17#389C2212yes—
18#389C2212yes—
19#387—22no—
20#387—22no—
21#385—22no—
22#385—22no—
23#382C2212yes—
24#380C2212yes—
25#380C2212yes—
26#340C2212yes—
27#340C2212yes—
28#340C2212yes—
29#340C2212yes—
30#340C2211yescharacteristic
31#340C2211yescharacteristic
32#340C2212yes—
33#336C2212yes—
34#336C2212yes—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8C2312yes—
39C22×C8C2311yescharacteristic
40C22×C8C2311yescharacteristic
41#125—42no—
42#125—42no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4C2312yes—
48C2×C4⋊C4C2312yes—
49C2×C4⋊C4C2312yes—
50C2×C4⋊C4C2312yes—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C42—41no—
54C2×C42—41no—
55C2×C42D412yes—
56C2×C42C2312yes—
57#107—41no—
58#107—41no—
59#106—42no—
60C22⋊C8—41no—
61C22⋊C8—41no—
62#95—22no—
63#95—22no—
64#95—22no—
65#95—22no—
66#95—22no—
67#95—22no—
68C24C2×D411yescharacteristic
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—42no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D412yes—
84C22×C4C2×D412yes—
85C22×C4C2411yescharacteristic
86C2×C8—22no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C4⋊C4—22no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—44no—
102C4⋊C4—44no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C22⋊C4—44no—
106C22⋊C4—44no—
107C42—22no—
108C42—22no—
109C42—42no—
110C23—21no—
111C23—22no—
112C23—22no—
113C23—21no—
114C23#13512yes—
115C23C22×D411yescharacteristic
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—42no—
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×C4H(5,𝔽2)11yescharacteristic
142C2×C4H(5,𝔽2)11yescharacteristic
143C2×C42−1+412yes—
144C2×C42−1+412yes—
145C8—42no—
146C8—41no—
147C8—41no—
148C22—22no—
149C22—24no—
150C22—41no—
151C22—41no—
152C22—42no—
153C22#53411yescharacteristic
154C22#53612yes—
155C22C2×#13511yescharacteristic
156C22#57511yescharacteristic
157C22#57512yes—
158C4—41no—
159C4—42no—
160C4—42no—
161C4—82no—
162C4—82no—
163C4—82no—
164C4—82no—
165C4#56312yes—
166C4#56311yescharacteristic
167C4#56311yescharacteristic
168C2—42no—
169C2#281711yescharacteristic
170C2#319111yescharacteristic
171C2#319711yescharacteristic
172C2#323212yes—
173C2#324012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2101C2semidirect
3. #2101C2semidirect
7. #2059C2semidirect
8. #2059C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#21012
#20772
#20762
#20752
#20592
#18172