Finite Group Databasebeta

← #13299

Subgroups — #13299

Order 256.

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

normal subgroups only

This group has 250 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#2563C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3C4×#99C211yesmaximal, characteristic
4#1393C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#99C2212yes—
8C2×#99C2212yes—
9#393—21no—
10#393—21no—
11#390C2212yes—
12#382C412yescomplemented
13#380—21no—
14#380—21no—
15#380—21no—
16C4×C4⋊C4C411yescharacteristic, complemented
17C4×C22⋊C4—21no—
18#342—21no—
19#342—21no—
20C22×Q8—21no—
21C22×D4C2311yescharacteristic
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C4⋊Q8—41no—
25C4⋊Q8—41no—
26C4×D4—41no—
27C4×D4—41no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4C2×C412yes—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4D411yescharacteristic
38C2×C22⋊C4D411yescharacteristic
39C2×C42—41no—
40C2×C42C2×C411yescharacteristic
41C2×C42C2×C412yes—
42#99—22no—
43#99—24no—
44#99Q814yescomplemented
45#95—21no—
46#95—22no—
47#95—22no—
48#95—41no—
49#95—41no—
50C24C2×D411yescharacteristic
51C24C2×D411yescharacteristic
52C2×Q8—42no—
53C2×Q8—82no—
54C2×D4—41no—
55C2×D4—41no—
56C2×D4C2×Q812yes—
57C2×D4#4112yes—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22×C411yescharacteristic
73C4⋊C4—41no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—82no—
79C22⋊C4—24no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C22⋊C4—42no—
85C22⋊C4—82no—
86C22⋊C4—82no—
87C42—21no—
88C42—21no—
89C42—42no—
90C42—82no—
91C23—21no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23—41no—
98C23—41no—
99C23C2×C22⋊C411yescharacteristic
100C23C22⋊Q812yes—
101C23#12411yescharacteristic
102C23C4⋊Q811yescharacteristic
103Q8—84no—
104D4—42no—
105D4—42no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—22no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—44no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4#9911yescharacteristic
132C2×C4#9911yescharacteristic
133C2×C4C4×Q812yes—
134C22—21no—
135C22—21no—
136C22—22no—
137C22—21no—
138C22—21no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22—41no—
146C22—81no—
147C22—81no—
148C22#39111yescharacteristic
149C22C2×#9911yescharacteristic
150C22#41011yescharacteristic
151C4—21no—
152C4—22no—
153C4—41no—
154C4—41no—
155C4—41no—
156C4—82no—
157C4—84no—
158C4—82no—
159C4—162no—
160C4#35412yes—
161C2—21no—
162C2—21no—
163C2—42no—
164C2—41no—
165C2—41no—
166C2#199611yescharacteristic
167C2C2×#35411yescharacteristic
168C2#212811yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
12. #382C4semidirect
16. C4×C4⋊C4C4semidirect
44. #99Q8semidirect

Maximal subgroups

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

subgroupindex
#25632
#17992
C4×#992
#13932