Finite Group Databasebeta

← #11803

Subgroups — #11803

Order 256.

239 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×#398C211yesmaximal, characteristic
2#1821C211yesmaximal, characteristic
3#1463C211yesmaximal, characteristic
4#1346C214yesmaximal, complemented
5C22×C4⋊C4—21no—
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8#407—21no—
9#407—21no—
10#398—21no—
11#398—21no—
12#398C412yes—
13#398C2212yes—
14C2×#95—21no—
15C2×#95C411yescharacteristic
16#343—21no—
17#343C2212yes—
18#324—21no—
19#324C2212yes—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—22no—
26C2×M16—41no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4—41no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4—44no—
35C2×C4⋊C4C2×C412yes—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4C2311yescharacteristic
40C22⋊C8—22no—
41C22⋊C8—41no—
42C22⋊C8—41no—
43C22⋊C8—41no—
44#95—22no—
45#95—21no—
46#95—22no—
47#95C2×C412yes—
48C24C22⋊C411yescharacteristic
49C22×C4—24no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—41no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22×C411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74M16—42no—
75M16—42no—
76M16—82no—
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—44no—
86C4⋊C4—44no—
87C23—22no—
88C23—22no—
89C23—21no—
90C23—21no—
91C23#9911yescharacteristic
92C23#9911yescharacteristic
93C23C2×C22⋊C411yescharacteristic
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—24no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—48no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—81no—
120C2×C4—82no—
121C2×C4—82no—
122C2×C4AGL(1,ℤ/8)12yes—
123C8—82no—
124C8—82no—
125C22—24no—
126C22—24no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22#35212yes—
134C22#35612yes—
135C22C2×#9911yescharacteristic
136C22#41811yescharacteristic
137C22#42111yescharacteristic
138C4—22no—
139C4—44no—
140C4—41no—
141C4—41no—
142C4—41no—
143C4—41no—
144C4—41no—
145C4—82no—
146C2—21no—
147C2—22no—
148C2—22no—
149C2—21no—
150C2#151011yescharacteristic
151C2C2×#35211yescharacteristic
152C2C2×#35611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1346C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
#18212
#14632
#13462