Finite Group Databasebeta

← #9101

Subgroups — #9101

Order 256.

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

normal subgroups only

This group has 315 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
1#2972C211yesmaximal, characteristic, complemented
2#2328C211yesmaximal, characteristic
3#1478C212yesmaximal, complemented
4#1460C211yesmaximal, characteristic, complemented
5#1274C212yesmaximal, complemented
6C2×#127C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C8×D4—24no—
10#431C2212yes—
11#431C2212yes—
12C2×C22⋊C8—22no—
13#390C411yescharacteristic
14#390C411yescharacteristic
15#386—22no—
16#386—22no—
17#382C411yescharacteristic
18#376C2211yescharacteristic
19D4⋊C8—24no—
20#323—22no—
21C22×D4—22no—
22C22×D4C2×C411yescharacteristic
23C22×D4C2×C411yescharacteristic
24C23×C4—21no—
25C23×C4D412yes—
26C22×C8—24no—
27#127—22no—
28#127C814yescomplemented
29C4×D4—22no—
30C4×D4—22no—
31C4×D4—22no—
32C4×D4—22no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C22⋊C4—21no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4D412yes—
39C2×C42C2×C411yescharacteristic
40C2×C42C2×C411yescharacteristic
41C2×C42C2311yescharacteristic
42C4⋊C8—22no—
43C4⋊C8—22no—
44C22⋊C8—24no—
45C22⋊C8—22no—
46C22⋊C8—22no—
47C4×C8—22no—
48C4×C8D414yescomplemented
49#95—22no—
50#95—21no—
51#95—21no—
52C24—21no—
53C24C22⋊C412yes—
54C2×D4—28no—
55C2×D4—22no—
56C2×D4—22no—
57C2×D4—21no—
58C2×D4—22no—
59C2×D4—22no—
60C2×D4—82no—
61C2×D4—82no—
62C2×D4C2×C812yes—
63C2×D4C2×C812yescomplemented
64C2×D4M1612yes—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—42no—
71C22×C4—82no—
72C22×C4—82no—
73C22×C4C22⋊C412yes—
74C22×C4C22×C411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C2×C8—24no—
78C2×C8—42no—
79C2×C8—84no—
80C4⋊C4—22no—
81C4⋊C4—22no—
82C22⋊C4—24no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—82no—
86C22⋊C4—82no—
87C42—21no—
88C42—22no—
89C42C2×C812yescomplemented
90C42C2×D412yes—
91C42#4112yes—
92C23—22no—
93C23—22no—
94C23—22no—
95C23—42no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—41no—
100C23—41no—
101C23C22⋊C814yes—
102C23C2×C22⋊C411yescharacteristic
103D4—24no—
104D4—24no—
105D4—24no—
106D4—42no—
107D4—88no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—44no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—82no—
122C2×C4—84no—
123C2×C4C4×D412yes—
124C2×C4#12011yescharacteristic
125C2×C4#12112yes—
126C2×C4#12311yescharacteristic
127C2×C4C22×C811yescharacteristic
128C2×C4C2×M1611yescharacteristic
129C2×C4#13112yes—
130C8—44no—
131C8—82no—
132C22—21no—
133C22—21no—
134C22—24no—
135C22—44no—
136C22—44no—
137C22—42no—
138C22—42no—
139C22—42no—
140C22—82no—
141C22—82no—
142C22—81no—
143C22—81no—
144C22#38611yescharacteristic
145C22C2×C22⋊C811yescharacteristic
146C22#40811yescharacteristic
147C4—22no—
148C4—21no—
149C4—21no—
150C4—42no—
151C4—42no—
152C4—81no—
153C4—81no—
154C4C8×D412yes—
155C4#43512yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2—82no—
160C2#186911yescharacteristic
161C2#191011yescharacteristic
162C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC2×#127C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2972C2semidirect
3. #1478C2semidirect
4. #1460C2semidirect
5. #1274C2semidirect
28. #127C8semidirect
48. C4×C8D4semidirect
63. C2×D4C2×C8semidirect
89. C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#29722
#23282
#14782
#14602
#12742