Finite Group Databasebeta

← #11786

Subgroups — #11786

Order 256.

231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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#2487C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic
3#1466C211yesmaximal, characteristic
4#1352C214yesmaximal, complemented
5C2×C4×D4C2211yescharacteristic
6C22×C4⋊C4C411yescharacteristic, complemented
7#432—21no—
8C2×#99—21no—
9#400C412yescomplemented
10#399C2212yes—
11#389—21no—
12#386—21no—
13#385—21no—
14#380—21no—
15#380C411yescharacteristic, complemented
16C4×C22⋊C4—21no—
17#342C2212yes—
18#323—21no—
19#323C2212yes—
20C22×D4D411yescharacteristic
21C23×C4—21no—
22C23×C4C2×C411yescharacteristic
23C23×C4D411yescharacteristic
24C4×D4—41no—
25C4×D4—41no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—41no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4C2×C412yes—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4D411yescharacteristic
37C2×C22⋊C4C2311yescharacteristic
38C2×C42—41no—
39C2×C42D411yescharacteristic
40C4⋊C8—41no—
41#99—24no—
42C22⋊C8—22no—
43#97—41no—
44#95—22no—
45#95—22no—
46#95—41no—
47#95C2×C412yes—
48C24C22⋊C411yescharacteristic
49C24C2×D411yescharacteristic
50C2×D4—22no—
51C2×D4—22no—
52C2×D4—41no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—42no—
60C22×C4—42no—
61C22×C4—42no—
62C22×C4—42no—
63C22×C4—82no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22×C411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C2×C8—42no—
70C4⋊C4—41no—
71C4⋊C4—42no—
72C4⋊C4—42no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—82no—
76C22⋊C4—24no—
77C22⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C22⋊C4—82no—
82C42—21no—
83C42—21no—
84C42—82no—
85C23—21no—
86C23—21no—
87C23—21no—
88C23—21no—
89C23—41no—
90C23—41no—
91C23—41no—
92C23—41no—
93C23#10412yes—
94C23C2×C22⋊C411yescharacteristic
95C23#13712yes—
96D4—44no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—44no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—81no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4—84no—
121C2×C4#9911yescharacteristic
122C2×C4#9911yescharacteristic
123C8—82no—
124C22—21no—
125C22—21no—
126C22—22no—
127C22—22no—
128C22—42no—
129C22—42no—
130C22—42no—
131C22—41no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22C2×#9911yescharacteristic
137C22#41911yescharacteristic
138C22C2×#10411yescharacteristic
139C4—21no—
140C4—41no—
141C4—41no—
142C4—41no—
143C4—41no—
144C4—82no—
145C4—84no—
146C4—82no—
147C4#35212yes—
148C2—21no—
149C2—21no—
150C2—42no—
151C2—42no—
152C2#150811yescharacteristic
153C2C2×#35211yescharacteristic
154C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1352C2semidirect
6. C22×C4⋊C4C4semidirect
9. #400C4semidirect
15. #380C4semidirect

Maximal subgroups

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

subgroupindex
#24872
C4×#992
#14662
#13522