Finite Group Databasebeta

← #9909

Subgroups — #9909

Order 256.

342 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 93 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#1797C211yesmaximal, characteristic
3C4×#95C211yesmaximal, characteristic
4#1294C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C22×C42—21no—
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8C2212yes—
11#391—21no—
12#389—21no—
13#389C412yescomplemented
14#384C411yescharacteristic, complemented
15#380—21no—
16#380C2212yes—
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—42no—
28C22⋊Q8—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C412yes—
34C2×C4⋊C4C2×C412yes—
35C2×C22⋊C4—22no—
36C2×C42—24no—
37C2×C42—42no—
38C2×C42—42no—
39C2×C42—42no—
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C4⋊C8—42no—
45C22⋊C8—24no—
46#95—24no—
47#95—22no—
48#95—22no—
49#95—22no—
50#95Q814yescomplemented
51C24C2×D411yescharacteristic
52C2×Q8—42no—
53C2×Q8—42no—
54C2×Q8—42no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—24no—
58C22×C4—24no—
59C22×C4—24no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4D812yes—
76C22×C4SD1612yes—
77C22×C4C22×C411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2×Q812yes—
80C22×C4#4112yes—
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—42no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—44no—
89C4⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C42—22no—
93C42—22no—
94C42—41no—
95C42—41no—
96C42—48no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23C2×C22⋊C411yescharacteristic
102C23C22⋊Q812yes—
103C23C2×D811yescharacteristic
104C23C2×SD1611yescharacteristic
105C23#13712yes—
106Q8—84no—
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—24no—
116C2×C4—24no—
117C2×C4—28no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—44no—
124C2×C4—44no—
125C2×C4—44no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4#9912yes—
137C2×C4#10212yes—
138C2×C4#10212yes—
139C2×C4#10412yes—
140C2×C4#10412yes—
141C2×C4C4×Q812yes—
142C2×C4#12411yescharacteristic
143C2×C4C4⋊Q811yescharacteristic
144C8—82no—
145C22—22no—
146C22—22no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—22no—
152C22—22no—
153C22#39111yescharacteristic
154C22C2×#9911yescharacteristic
155C22#41011yescharacteristic
156C22C2×#10211yescharacteristic
157C22#41911yescharacteristic
158C22C2×#10411yescharacteristic
159C22C2×#10411yescharacteristic
160C22#47412yes—
161C22#47512yes—
162C4—21no—
163C4—22no—
164C4—21no—
165C4—22no—
166C4—22no—
167C4—48no—
168C4—41no—
169C4—82no—
170C4—82no—
171C4#32714yes—
172C2—22no—
173C2—22no—
174C2C2×#32711yescharacteristic
175C2#150211yescharacteristic
176C2#150611yescharacteristic
177C2#150811yescharacteristic
178C2#198811yescharacteristic
179C2#199411yescharacteristic
180C2#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
13. #389C4semidirect
14. #384C4semidirect
50. #95Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#17972
C4×#952
#12942