Finite Group Databasebeta

← #7610

Subgroups — #7610

Order 256.

306 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 81 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×#389C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1728C211yesmaximal, characteristic
4#1271C214yesmaximal, complemented
5C22×M16—21no—
6C22×C4⋊C4—21no—
7C22×C4⋊C4C411yescharacteristic
8C22×C4⋊C4C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C22⋊C8—21no—
11#389—21no—
12#389—21no—
13#389C412yes—
14#389C2212yes—
15C2×#95—21no—
16#336—21no—
17#336—21no—
18#336C2212yes—
19#336C2212yes—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C2×M16—24no—
26C2×M16—42no—
27C2×M16—42no—
28C22×C8—22no—
29C22×C8—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—24no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—42no—
36C2×C4⋊C4—42no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4C2×C412yes—
39C2×C4⋊C4C2×C412yes—
40C2×C42—42no—
41C2×C42C2×C411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42C2311yescharacteristic
45C22⋊C8—42no—
46#95—22no—
47#95—22no—
48C24C22⋊C411yescharacteristic
49C22×C4—22no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—24no—
55C22×C4—22no—
56C22×C4—22no—
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×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4D812yes—
69C22×C4SD1612yes—
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73M16—48no—
74C2×C8—24no—
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C2×C8—42no—
80C4⋊C4—24no—
81C4⋊C4—22no—
82C4⋊C4—42no—
83C4⋊C4—42no—
84C4⋊C4—44no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4M1614yescomplemented
88C42—41no—
89C42—41no—
90C42—41no—
91C42—41no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23#9912yes—
97C23#10212yes—
98C23#10412yes—
99C23C2×C22⋊C411yescharacteristic
100C2×C4—28no—
101C2×C4—24no—
102C2×C4—24no—
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—24no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—44no—
118C2×C4—48no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—84no—
125C2×C4#10112yes—
126C2×C4#10212yes—
127C2×C4C2×M1612yes—
128C2×C4C2×D811yescharacteristic
129C2×C4C2×SD1611yescharacteristic
130C2×C4#13712yes—
131C8—44no—
132C8—82no—
133C22—22no—
134C22—22no—
135C22—22no—
136C22—22no—
137C22—22no—
138C22—22no—
139C22#32714yes—
140C22#40711yescharacteristic
141C22C2×#9911yescharacteristic
142C22C2×#10111yescharacteristic
143C22C2×#10211yescharacteristic
144C22#41911yescharacteristic
145C22C2×#10411yescharacteristic
146C22#42111yescharacteristic
147C4—22no—
148C4—22no—
149C4—21no—
150C4—22no—
151C4—21no—
152C4—22no—
153C4—44no—
154C4—41no—
155C4—82no—
156C2—24no—
157C2#146311yescharacteristic
158C2#148611yescharacteristic
159C2#149111yescharacteristic
160C2C2×#32711yescharacteristic
161C2#151111yescharacteristic
162C2#152811yescharacteristic
163C2#154611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1271C2semidirect
87. C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
#18172
#17282
#12712