Finite Group Databasebeta

← #9011

Subgroups — #9011

Order 256.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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#2395C211yesmaximal, characteristic
3#1826C212yesmaximal, complemented
4C4×#99C211yesmaximal, characteristic
5#1297C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#435—24no—
9#432—21no—
10C2×#99—21no—
11#407C2212yes—
12C2×C22⋊C8—21no—
13#400C411yescharacteristic, complemented
14#399C411yescharacteristic, complemented
15#398C411yescharacteristic, complemented
16#397—21no—
17#397C411yescharacteristic, complemented
18#394—21no—
19C4×C4⋊C4—21no—
20C4×C22⋊C4—21no—
21C4×C22⋊C4C2212yes—
22C4×C22⋊C4C2211yescharacteristic
23#342—21no—
24#339—24no—
25C22×D4D411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4C2311yescharacteristic
28C2×M16D414yes—
29C22×C8—22no—
30C4×D4—22no—
31C4×D4—22no—
32#117—22no—
33#117—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C42—21no—
47C2×C42—21no—
48C2×C42—22no—
49C2×C42D411yescharacteristic
50C4⋊C8—22no—
51#99—24no—
52C22⋊C8—22no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55#97—22no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—22no—
61#95C2×C411yescharacteristic
62#95C2×C411yescharacteristic
63C24C22⋊C411yescharacteristic
64C24C2×D411yescharacteristic
65C2×D4—22no—
66C2×D4—22no—
67C2×D4—41no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—42no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83C22×C4#4112yes—
84M16—44no—
85C2×C8—24no—
86C2×C8—82no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C22⋊C4—24no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C42—21no—
101C42—21no—
102C42—42no—
103C42—44no—
104C42—41no—
105C42—41no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23#10412yes—
115C23C2×C22⋊C411yescharacteristic
116C23#12112yes—
117D4—44no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—44no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4C4×D412yes—
147C2×C4#12011yescharacteristic
148C2×C4#12311yescharacteristic
149C8—44no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—22no—
154C22—22no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#38611yescharacteristic
163C22C2×#10411yescharacteristic
164C22#42111yescharacteristic
165C4—21no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—84no—
173C4—81no—
174C4—81no—
175C4#45812yes—
176C2—21no—
177C2—21no—
178C2—42no—
179C2—42no—
180C2#187111yescharacteristic
181C2#188411yescharacteristic
182C2#189011yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
socleC22#386
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #1826C2semidirect
5. #1297C2semidirect
13. #400C4semidirect
14. #399C4semidirect
15. #398C4semidirect
17. #397C4semidirect

Maximal subgroups

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

subgroupindex
#29732
#23952
#18262
C4×#992
#12972