Finite Group Databasebeta

← #11564

Subgroups — #11564

Order 256.

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

normal subgroups only

This group has 222 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
1#2547C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic
4#1319C214yesmaximal, complemented
5C2×C4⋊Q8C411yescharacteristic
6C2×#124C412yes—
7C2×C4×D4C2211yescharacteristic
8#407—21no—
9C2×C22⋊C8—21no—
10#397—21no—
11#397—21no—
12#391C2212yes—
13#380—21no—
14#380—21no—
15C4×C4⋊C4C411yescharacteristic
16#324C2212yes—
17#324C2212yes—
18#323—21no—
19#323—21no—
20C22×Q8C2×C412yes—
21C22×D4C2×C411yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C2×M16—41no—
25C22×C8—41no—
26C4⋊Q8—41no—
27C4⋊Q8—41no—
28#124—44no—
29C4×D4—41no—
30C4×D4—41no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2311yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4D411yescharacteristic
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C412yes—
41C22⋊C8—22no—
42C22⋊C8—22no—
43C22⋊C8—41no—
44C22⋊C8—41no—
45#95—21no—
46#95—22no—
47#95—22no—
48C24C22⋊C411yescharacteristic
49C24C22⋊C411yescharacteristic
50C2×Q8—24no—
51C2×Q8—42no—
52C2×Q8—42no—
53C2×D4—22no—
54C2×D4—22no—
55C2×D4—41no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22×C411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69M16—82no—
70C2×C8—42no—
71C2×C8—42no—
72C2×C8—82no—
73C4⋊C4—41no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—44no—
81C22⋊C4—44no—
82C42—21no—
83C42—21no—
84C42—42no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—21no—
90C23—41no—
91C23—41no—
92C23—41no—
93C23—41no—
94C23#10012yes—
95C23C2×C22⋊C411yescharacteristic
96Q8—84no—
97D4—44no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—22no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—81no—
116C2×C4—81no—
117C2×C4#9911yescharacteristic
118C2×C4#9911yescharacteristic
119C2×C4#13112yes—
120C8—82no—
121C8—82no—
122C22—21no—
123C22—21no—
124C22—22no—
125C22—42no—
126C22—42no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—41no—
131C22—41no—
132C22—81no—
133C22—81no—
134C22#40811yescharacteristic
135C22C2×#9911yescharacteristic
136C22C2×#10011yescharacteristic
137C4—21no—
138C4—22no—
139C4—41no—
140C4—41no—
141C4—41no—
142C4—82no—
143C4—82no—
144C4#35412yes—
145C2—21no—
146C2—21no—
147C2—42no—
148C2—81no—
149C2#147111yescharacteristic
150C2C2×#35411yescharacteristic
151C2#213511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#25472
#14652
#14592
#13192