Finite Group Databasebeta

← #9470

Subgroups — #9470

Order 256.

299 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 77 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#2972C211yesmaximal, characteristic
2#2486C211yesmaximal, characteristic
3#1823C212yesmaximal, complemented
4#1799C211yesmaximal, characteristic
5#1289C212yesmaximal, complemented
6C2×#123—21no—
7C2×C4×D4C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C8×D4—24no—
10#431—21no—
11C2×#99—21no—
12C2×C22⋊C8—21no—
13C2×C22⋊C8C2212yes—
14#398C412yescomplemented
15#397C412yescomplemented
16#389—21no—
17#381—21no—
18#380—21no—
19#380C2212yes—
20#380C2211yescharacteristic
21#342—21no—
22#340—24no—
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28C22×C8D414yes—
29#123—42no—
30#123—42no—
31C4×D4—22no—
32C4×D4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4C2×C412yes—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—22no—
48#99—42no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52C4×C8—22no—
53#95—22no—
54#95—22no—
55#95—22no—
56#95C2×C412yes—
57C24C22⋊C411yescharacteristic
58C24C2×D411yescharacteristic
59C2×D4—22no—
60C2×D4—22no—
61C2×D4D812yes—
62C2×D4SD1612yes—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D412yes—
79C22×C4C2×D411yescharacteristic
80C22×C4#4112yes—
81C2×C8—24no—
82C2×C8—44no—
83C2×C8—82no—
84C4⋊C4—22no—
85C4⋊C4—24no—
86C4⋊C4—42no—
87C4⋊C4—44no—
88C4⋊C4—44no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C22⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—44no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C42—21no—
99C42—21no—
100C23—21no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23#10212yes—
107C23C2×C22⋊C411yescharacteristic
108C23#12112yes—
109C23C2×D811yescharacteristic
110C23C2×SD1611yescharacteristic
111C23#13512yes—
112D4—24no—
113D4—24no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—24no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—81no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4#10212yes—
138C2×C4C4×D412yes—
139C2×C4#12011yescharacteristic
140C2×C4#12311yescharacteristic
141C8—44no—
142C22—21no—
143C22—22no—
144C22—21no—
145C22—22no—
146C22—22no—
147C22—41no—
148C22—41no—
149C22—42no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#38611yescharacteristic
155C22C2×#10211yescharacteristic
156C22#41611yescharacteristic
157C22#46512yes—
158C22#46612yes—
159C4—21no—
160C4—22no—
161C4—22no—
162C4—41no—
163C4—41no—
164C4—82no—
165C4—82no—
166C4—162no—
167C4#45812yes—
168C2—21no—
169C2—22no—
170C2—21no—
171C2—22no—
172C2—42no—
173C2#189011yescharacteristic
174C2#189311yescharacteristic
175C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1823C2semidirect
5. #1289C2semidirect
14. #398C4semidirect
15. #397C4semidirect

Maximal subgroups

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

subgroupindex
#29722
#24862
#18232
#17992
#12892