Finite Group Databasebeta

← #13298

Subgroups — #13298

Order 256.

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

normal subgroups only

This group has 278 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#2542C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3C4×#99C211yesmaximal, characteristic
4#1393C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C2×C22⋊Q8—21no—
8C2×C4×D4C2211yescharacteristic
9C2×#99C2212yes—
10C2×#99C2212yes—
11#399—21no—
12#399—21no—
13#390C2212yes—
14#384C412yescomplemented
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—
26C22⋊Q8—44no—
27C22⋊Q8—44no—
28C4×D4—41no—
29C4×D4—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4C2×C412yes—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42—41no—
44C2×C42C2×C411yescharacteristic
45C2×C42C2×C412yes—
46#99—22no—
47#99—24no—
48#99Q814yescomplemented
49#95—21no—
50#95—41no—
51#95—41no—
52C24C2×D411yescharacteristic
53C24C2×D411yescharacteristic
54C2×Q8—42no—
55C2×Q8—42no—
56C2×Q8—41no—
57C2×Q8—41no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4C2×Q812yes—
61C2×D4#4112yes—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22×C411yescharacteristic
77C4⋊C4—24no—
78C4⋊C4—41no—
79C4⋊C4—42no—
80C4⋊C4—44no—
81C4⋊C4—44no—
82C4⋊C4—42no—
83C4⋊C4—82no—
84C22⋊C4—24no—
85C22⋊C4—41no—
86C22⋊C4—41no—
87C22⋊C4—44no—
88C22⋊C4—44no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C42—21no—
93C42—21no—
94C42—42no—
95C42—82no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23C2×C22⋊C411yescharacteristic
105C23C22⋊Q812yes—
106C23#12411yescharacteristic
107C23C4⋊Q811yescharacteristic
108Q8—82no—
109Q8—82no—
110D4—42no—
111D4—42no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4#9911yescharacteristic
138C2×C4#9911yescharacteristic
139C2×C4C4×Q812yes—
140C22—21no—
141C22—21no—
142C22—22no—
143C22—21no—
144C22—21no—
145C22—42no—
146C22—41no—
147C22—41no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—81no—
153C22—81no—
154C22#39111yescharacteristic
155C22C2×#9911yescharacteristic
156C22#41011yescharacteristic
157C4—21no—
158C4—22no—
159C4—44no—
160C4—41no—
161C4—41no—
162C4—41no—
163C4—84no—
164C4—82no—
165C4—162no—
166C4#35312yes—
167C2—21no—
168C2—21no—
169C2—42no—
170C2—41no—
171C2—41no—
172C2#199611yescharacteristic
173C2C2×#35311yescharacteristic
174C2#212911yescharacteristic
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
14. #384C4semidirect
16. C4×C4⋊C4C4semidirect
48. #99Q8semidirect

Maximal subgroups

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

subgroupindex
#25422
#17992
C4×#992
#13932