Finite Group Databasebeta

← #7593

Subgroups — #7593

Order 256.

283 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 83 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#1817C211yesmaximal, characteristic
4#1271C214yesmaximal, complemented
5C22×C4⋊C4—21no—
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C411yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C22⋊C8—21no—
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
25C22×C8—22no—
26C22×C8—22no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—42no—
33C2×C4⋊C4—44no—
34C2×C4⋊C4C814yescomplemented
35C2×C4⋊C4C2×C412yes—
36C2×C4⋊C4C2×C412yes—
37C2×C42—42no—
38C2×C42C2×C411yescharacteristic
39C2×C42D411yescharacteristic
40C2×C42D411yescharacteristic
41C2×C42C2311yescharacteristic
42C22⋊C8—42no—
43C22⋊C8—42no—
44#95—22no—
45#95—22no—
46C24C22⋊C411yescharacteristic
47C22×C4—22no—
48C22×C4—22no—
49C22×C4—24no—
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—42no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4C22⋊C411yescharacteristic
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C2×C812yes—
67C22×C4M1612yes—
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C2×C8—42no—
72C2×C8—42no—
73C2×C8—42no—
74C2×C8—42no—
75C2×C8—42no—
76C2×C8—42no—
77C4⋊C4—24no—
78C4⋊C4—22no—
79C4⋊C4—42no—
80C4⋊C4—42no—
81C4⋊C4—44no—
82C4⋊C4—44no—
83C4⋊C4C2×C814yes—
84C42—41no—
85C42—41no—
86C42—41no—
87C42—41no—
88C23—21no—
89C23—21no—
90C23—21no—
91C23—21no—
92C23C22⋊C812yes—
93C23#9912yes—
94C23#10112yes—
95C23C2×C22⋊C411yescharacteristic
96C2×C4—24no—
97C2×C4—24no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—24no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—44no—
114C2×C4—44no—
115C2×C4—48no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—84no—
122C2×C4C22⋊C812yes—
123C2×C4C22×C811yescharacteristic
124C2×C4C2×M1611yescharacteristic
125C2×C4AGL(1,ℤ/8)12yes—
126C2×C4#13712yes—
127C8—82no—
128C8—82no—
129C22—22no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—22no—
134C22—22no—
135C22#32414yes—
136C22C2×C22⋊C811yescharacteristic
137C22C2×#9911yescharacteristic
138C22C2×#10111yescharacteristic
139C22#41811yescharacteristic
140C22#41911yescharacteristic
141C22#42111yescharacteristic
142C22#42111yescharacteristic
143C4—22no—
144C4—21no—
145C4—21no—
146C4—22no—
147C4—22no—
148C4—44no—
149C4—41no—
150C4—41no—
151C4—82no—
152C2—24no—
153C2C2×#32411yescharacteristic
154C2#147811yescharacteristic
155C2#147911yescharacteristic
156C2#151011yescharacteristic
157C2#151111yescharacteristic
158C2#153411yescharacteristic
159C2#155611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C2×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
34. C2×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
#18172
#12712