Finite Group Databasebeta

← #7627

Subgroups — #7627

Order 256.

297 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 75 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#1749C211yesmaximal, characteristic
3#1749C211yesmaximal, characteristic
4#1271C214yesmaximal, complemented
5C22×C4⋊C4—21no—
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C411yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C22⋊C8—21no—
10C2×C22⋊C8—21no—
11C2×#97—21no—
12C2×#97—21no—
13#389—21no—
14#389—21no—
15#389C412yes—
16#389C2212yes—
17C2×#95—21no—
18#336C2212yes—
19#336C2212yes—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C22×C8—22no—
26C22×C8—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—24no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—44no—
33C2×C4⋊C4—42no—
34C2×C4⋊C4C2×C412yes—
35C2×C4⋊C4C2×C412yes—
36C2×C42—42no—
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42C2311yescharacteristic
41C22⋊C8—24no—
42C22⋊C8—24no—
43#97—42no—
44#97—42no—
45#95—22no—
46#95—22no—
47C24C22⋊C411yescharacteristic
48C22×C4—22no—
49C22×C4—22no—
50C22×C4—22no—
51C22×C4—24no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—41no—
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×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C2×C8—24no—
72C2×C8—24no—
73C2×C8—42no—
74C2×C8—42no—
75C2×C8—42no—
76C2×C8—42no—
77C4⋊C4—24no—
78C4⋊C4—22no—
79C4⋊C4—42no—
80C4⋊C4—44no—
81C4⋊C4—42no—
82C4⋊C4—44no—
83C4⋊C4M1614yes—
84C42—22no—
85C42—22no—
86C42—41no—
87C42—41no—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—21no—
92C23#10112yes—
93C23#10412yes—
94C23#10412yes—
95C23C2×C22⋊C411yescharacteristic
96C2×C4—28no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—24no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—44no—
113C2×C4—44no—
114C2×C4—48no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—84no—
121C2×C4#9912yes—
122C2×C4C2×M1612yes—
123C2×C4AGL(1,ℤ/8)12yes—
124C2×C4#13712yes—
125C8—44no—
126C8—44no—
127C22—22no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22#33014yes—
134C22#40711yescharacteristic
135C22C2×#9911yescharacteristic
136C22C2×#10111yescharacteristic
137C22#41811yescharacteristic
138C22#41911yescharacteristic
139C22C2×#10411yescharacteristic
140C22C2×#10411yescharacteristic
141C4—22no—
142C4—22no—
143C4—22no—
144C4—22no—
145C4—44no—
146C4—41no—
147C4—41no—
148C4—82no—
149C2—24no—
150C2#146311yescharacteristic
151C2#149011yescharacteristic
152C2#149111yescharacteristic
153C2#150711yescharacteristic
154C2#150811yescharacteristic
155C2C2×#33011yescharacteristic
156C2#155611yescharacteristic
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

Maximal subgroups

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

subgroupindex
C2×#3892
#17492
#12712