Finite Group Databasebeta

← #11800

Subgroups — #11800

Order 256.

251 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 49 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#2619C211yesmaximal, characteristic
2#1820C211yesmaximal, characteristic, complemented
3#1464C211yesmaximal, characteristic, complemented
4#1343C214yesmaximal, complemented
5C2×#127C411yescharacteristic, complemented
6C2×#121—21no—
7C2×#117C2211yescharacteristic
8#432—21no—
9#394—21no—
10#394C412yescomplemented
11#392—21no—
12#392C2212yes—
13#390—21no—
14#382C411yescharacteristic, complemented
15#381—21no—
16#342—21no—
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×D4—22no—
21C22×D4—21no—
22C22×D4—21no—
23C22×D4C2×C412yes—
24C23×C4D411yescharacteristic
25#127—41no—
26#127—41no—
27#121—44no—
28#117—41no—
29#117—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4D411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4C2311yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42D411yescharacteristic
41C4⋊C8—41no—
42C22⋊C8—22no—
43#97—41no—
44#95—21no—
45#95—41no—
46#95—41no—
47#95C2×C412yes—
48C24—22no—
49C24C2×D411yescharacteristic
50C2×D4—42no—
51C2×D4—42no—
52C2×D4—42no—
53C2×D4—42no—
54C2×D4—44no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—82no—
58C2×D4—82no—
59C2×D4—82no—
60C2×D4—82no—
61C22×C4—22no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—41no—
66C22×C4—42no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C2×C8—42no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4—42no—
77C22⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—44no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—82no—
83C22⋊C4—82no—
84C42—21no—
85C42—21no—
86C42—21no—
87C42—21no—
88C23—21no—
89C23—21no—
90C23—41no—
91C23—41no—
92C23—42no—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23C2×C22⋊C411yescharacteristic
100C23AGL(1,ℤ/8)12yes—
101D4—82no—
102D4—82no—
103D4—84no—
104D4—84no—
105D4—88no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—82no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4#9911yescharacteristic
128C2×C4#9911yescharacteristic
129C8—82no—
130C22—21no—
131C22—21no—
132C22—22no—
133C22—42no—
134C22—41no—
135C22—41no—
136C22—81no—
137C22—82no—
138C22—82no—
139C22—82no—
140C22—82no—
141C22—82no—
142C22—82no—
143C22—82no—
144C22C2×#9911yescharacteristic
145C22#41811yescharacteristic
146C22#42111yescharacteristic
147C4—21no—
148C4—21no—
149C4—41no—
150C4—41no—
151C4—41no—
152C4—41no—
153C4—41no—
154C4—82no—
155C4—162no—
156C2—21no—
157C2—21no—
158C2—42no—
159C2—162no—
160C2#151011yescharacteristic
161C2#212211yescharacteristic
162C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2619C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1820C2semidirect
3. #1464C2semidirect
4. #1343C2semidirect
5. C2×#127C4semidirect
10. #394C4semidirect
14. #382C4semidirect

Maximal subgroups

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

subgroupindex
#26192
#18202
#14642
#13432