Finite Group Databasebeta

← #8852

Subgroups — #8852

Order 256.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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#2973C211yesmaximal, characteristic
2#1869C211yesmaximal, characteristic
3#1850C212yesmaximal, complemented
4C4×#99C211yesmaximal, characteristic
5#1282C212yesmaximal, complemented
6C23×C8C2211yescharacteristic
7C2×C4×D4C2211yescharacteristic
8#435—22no—
9#435—22no—
10#432—21no—
11C2×C4⋊C8—22no—
12C2×#99C412yes—
13#407—21no—
14C2×C22⋊C8—21no—
15C2×C22⋊C8C2211yescharacteristic
16C2×C22⋊C8C2212yes—
17C4×C22⋊C4C2212yes—
18#342C411yescharacteristic
19#342C411yescharacteristic
20#336—22no—
21#336—21no—
22C22×D4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C2×M16—21no—
26C2×M16—21no—
27C22×C8—21no—
28C22×C8—22no—
29C22×C8—21no—
30C22×C8—42no—
31C22×C8D412yes—
32C22×C8D412yes—
33C4×D4—22no—
34C4×D4—22no—
35C2×C4⋊C4Q811yescharacteristic
36C2×C22⋊C4C2×C412yes—
37C2×C22⋊C4C2×C411yescharacteristic
38C2×C22⋊C4C2×C412yes—
39C2×C22⋊C4Q811yescharacteristic
40C2×C42—22no—
41C2×C42D411yescharacteristic
42C4⋊C8—21no—
43C4⋊C8—21no—
44C4⋊C8—44no—
45#99—24no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8—22no—
50C22⋊C8—42no—
51#97—22no—
52#95—22no—
53C24C4⋊C411yescharacteristic
54C24C22×C411yescharacteristic
55C2×D4—22no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—21no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4C4⋊C411yescharacteristic
68C22×C4C4⋊C411yescharacteristic
69C22×C4C4⋊C411yescharacteristic
70C22×C4C2×D412yes—
71C22×C4C2×D411yescharacteristic
72C22×C4C2×Q811yescharacteristic
73C22×C4#4112yes—
74M16—81no—
75M16—81no—
76C2×C8—22no—
77C2×C8—24no—
78C2×C8—22no—
79C2×C8—22no—
80C2×C8—42no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—22no—
89C22⋊C4—22no—
90C22⋊C4—22no—
91C22⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4M1614yescomplemented
95C42—21no—
96C42—21no—
97C42—42no—
98C42—42no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C2×C4⋊C411yescharacteristic
107C23C4×D412yes—
108C23C2×M1612yes—
109C23#13111yescharacteristic
110C23#13111yescharacteristic
111D4—42no—
112D4—42no—
113C2×C4—22no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—24no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—81no—
135C2×C4#12011yescharacteristic
136C2×C4C22⋊Q812yes—
137C2×C4#12311yescharacteristic
138C8—42no—
139C8—42no—
140C8—42no—
141C8—81no—
142C22—22no—
143C22—21no—
144C22—22no—
145C22—21no—
146C22—42no—
147C22—41no—
148C22—41no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#38511yescharacteristic
155C22#42311yescharacteristic
156C22#42411yescharacteristic
157C22#43512yes—
158C22#43612yes—
159C4—21no—
160C4—22no—
161C4—44no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—82no—
166C4B(4,𝔽2)11yescharacteristic
167C4#45811yescharacteristic
168C2—22no—
169C2—22no—
170C2—21no—
171C2—21no—
172C2—81no—
173C2#185111yescharacteristic
174C2#186711yescharacteristic
175C2#186811yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
socleC22#385
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1850C2semidirect
5. #1282C2semidirect
94. C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
#29732
#18692
#18502
C4×#992
#12822