Finite Group Databasebeta

← #9889

Subgroups — #9889

Order 256.

299 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 79 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#2857C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic
3#1434C211yesmaximal, characteristic
4#1294C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C22×C42C2211yescharacteristic
8C2×C22⋊C8C2212yes—
9C2×#97—21no—
10#391—21no—
11#389—21no—
12#389C412yescomplemented
13#384C411yescharacteristic, complemented
14#380—21no—
15#380C2212yes—
16C2×#95—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20C22×Q8—21no—
21C23×C4—22no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C4⋊Q8—42no—
27C22⋊Q8—44no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4C2×C412yes—
34C2×C22⋊C4—22no—
35C2×C42—42no—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C22⋊C8—22no—
41C22⋊C8Q814yescomplemented
42#97—42no—
43#95—24no—
44#95—22no—
45#95—22no—
46#95—24no—
47#95—42no—
48C24C2×D411yescharacteristic
49C2×Q8—42no—
50C2×Q8—42no—
51C2×Q8—42no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—24no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22×C411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×Q812yes—
76C22×C4#4112yes—
77C2×C8—24no—
78C2×C8—42no—
79C2×C8—42no—
80C4⋊C4—42no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C4⋊C4—44no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C22⋊C4—44no—
87C42—22no—
88C42—22no—
89C42—41no—
90C42—41no—
91C23—21no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23C2×C22⋊C411yescharacteristic
96C23#12411yescharacteristic
97C23C4⋊Q811yescharacteristic
98C23AGL(1,ℤ/8)12yes—
99C23#13712yes—
100Q8—84no—
101C2×C4—21no—
102C2×C4—21no—
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—21no—
114C2×C4—21no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—44no—
118C2×C4—44no—
119C2×C4—44no—
120C2×C4—44no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4—82no—
132C2×C4#9912yes—
133C2×C4#10412yes—
134C2×C4C4×Q812yes—
135C2×C4C22⋊Q812yes—
136C8—44no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—21no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—22no—
145C22#39111yescharacteristic
146C22C2×#9911yescharacteristic
147C22#41011yescharacteristic
148C22#41811yescharacteristic
149C22#41911yescharacteristic
150C22C2×#10411yescharacteristic
151C22#42111yescharacteristic
152C22#48912yes—
153C22#50112yes—
154C4—22no—
155C4—22no—
156C4—22no—
157C4—22no—
158C4—41no—
159C4—41no—
160C4—84no—
161C4—82no—
162C4—82no—
163C2—22no—
164C2—22no—
165C2#150511yescharacteristic
166C2#150811yescharacteristic
167C2#151011yescharacteristic
168C2#151511yescharacteristic
169C2#198611yescharacteristic
170C2#199311yescharacteristic
171C2#199611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
12. #389C4semidirect
13. #384C4semidirect
41. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#17492
#14342
#12942