Finite Group Databasebeta

← #7692

Subgroups — #7692

Order 256.

222 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 63 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#2860C211yesmaximal, characteristic
2#1745C211yesmaximal, characteristic, complemented
3#1744C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal
6C2×#126—21no—
7C2×#97—21no—
8C2×C4×C8—21no—
9#390C411yescharacteristic
10#390C411yescharacteristic
11#390C2211yescharacteristic
12#389C411yescharacteristic
13#389C411yescharacteristic
14#389C2211yescharacteristic
15#388—21no—
16#388—21no—
17#388—21no—
18C43C2211yescharacteristic
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4C2×C411yescharacteristic
24C22×C8—22no—
25C22×C8—22no—
26#126—44no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4C2×C411yescharacteristic
31C2×C4⋊C4C2×C411yescharacteristic
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C22⋊C4—21no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42C2311yescharacteristic
46#97—24no—
47C4×C8—24no—
48#95—21no—
49#95—21no—
50#95—21no—
51C24—21no—
52C2×D4—22no—
53C2×D4—41no—
54C2×D4—41no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C22⋊C411yescharacteristic
62C22×C4C22×C411yescharacteristic
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C2×C8—24no—
66C2×C8—24no—
67C2×C8—42no—
68C2×C8—42no—
69C2×C8—42no—
70C2×C8—42no—
71C4⋊C4—22no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C22⋊C4—42no—
82C22⋊C4—42no—
83C42—22no—
84C42—22no—
85C42—22no—
86C42—22no—
87C42—22no—
88C42—41no—
89C42—41no—
90C23—41no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23—41no—
95C23—41no—
96C23—41no—
97C23C2×C22⋊C411yescharacteristic
98D4—82no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
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—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4#10412yes—
125C2×C4#10412yes—
126C2×C4#13112yes—
127C2×C4#13512yes—
128C2×C4AGL(1,ℤ/8)12yes—
129C8—44no—
130C8—44no—
131C22—81no—
132C22—81no—
133C22—81no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22#40811yescharacteristic
139C22#41011yescharacteristic
140C22#41311yescharacteristic
141C22#41611yescharacteristic
142C22#41811yescharacteristic
143C22C2×#10411yescharacteristic
144C22C2×#10411yescharacteristic
145C4—22no—
146C4—22no—
147C4—22no—
148C4—22no—
149C4—22no—
150C4—41no—
151C4—41no—
152C4—81no—
153C4—81no—
154C4—81no—
155C2—161no—
156C2#147311yescharacteristic
157C2#148411yescharacteristic
158C2#149411yescharacteristic
159C2#150011yescharacteristic
160C2#150511yescharacteristic
161C2#152411yescharacteristic
162C2#155311yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1745C2semidirect
3. #1744C2semidirect
4. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28602
#17452
#17442
#12712
#12702