Finite Group Databasebeta

← #11738

Subgroups — #11738

Order 256.

200 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 51 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#2632C211yesmaximal, characteristic
2C4×#101C211yesmaximal, characteristic, complemented
3#1461C211yesmaximal, characteristic, complemented
4#1354C212yesmaximal, complemented
5#1345C212yesmaximal
6C2×#124C411yescharacteristic
7C2×C4×Q8C2211yescharacteristic
8#431—21no—
9C2×#101—21no—
10C4×M16—21no—
11#399—21no—
12#395C411yescharacteristic
13#395C2211yescharacteristic
14#394C411yescharacteristic
15#394C2211yescharacteristic
16#393—21no—
17#388—21no—
18#388—21no—
19#383C411yescharacteristic
20#343C2212yes—
21#324—21no—
22#324C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×Q8D411yescharacteristic
25C22×D4C2×C411yescharacteristic
26C2×M16—22no—
27#124—41no—
28#124—41no—
29C4×Q8—41no—
30C4×Q8—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4C2311yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C42C2×C411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—41no—
46#101—24no—
47C22⋊C8—22no—
48#97—41no—
49C4×C8—41no—
50C4×C8—41no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24—21no—
58C2×Q8—22no—
59C2×Q8—22no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8—81no—
63C2×D4—41no—
64C2×D4—81no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22×C411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75M16—24no—
76M16—42no—
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—81no—
83C4⋊C4—81no—
84C22⋊C4—42no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—81no—
90C22⋊C4—81no—
91C42—21no—
92C42—21no—
93C42—21no—
94C42—21no—
95C42—21no—
96C42—21no—
97C23—41no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23C2×C22⋊C411yescharacteristic
105Q8—44no—
106Q8—82no—
107D4—82no—
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—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4#9911yescharacteristic
132C2×C4#9911yescharacteristic
133C2×C4#10412yes—
134C2×C4#13512yes—
135C8—44no—
136C8—82no—
137C22—21no—
138C22—21no—
139C22—81no—
140C22—81no—
141C22—81no—
142C22—81no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22C2×#9911yescharacteristic
147C22#41611yescharacteristic
148C22C2×#10411yescharacteristic
149C4—21no—
150C4—21no—
151C4—22no—
152C4—22no—
153C4—41no—
154C4—41no—
155C4—41no—
156C4—41no—
157C4—81no—
158C4—81no—
159C4—81no—
160C2—21no—
161C2—21no—
162C2—161no—
163C2#150111yescharacteristic
164C2#212211yescharacteristic
165C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#101C2semidirect
3. #1461C2semidirect
4. #1354C2semidirect

Maximal subgroups

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

subgroupindex
#26322
C4×#1012
#14612
#13542
#13452