Finite Group Databasebeta

← #11798

Subgroups — #11798

Order 256.

243 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 49 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#2392C211yesmaximal, characteristic
2#1820C211yesmaximal, characteristic, complemented
3#1464C211yesmaximal, characteristic, complemented
4#1344C212yesmaximal, complemented
5#1343C212yesmaximal
6C2×C4×D4—21no—
7C2×C4×D4C411yescharacteristic, complemented
8C2×#117C2211yescharacteristic
9#432—21no—
10#398C411yescharacteristic, complemented
11#397C2211yescharacteristic
12#396—21no—
13#394—21no—
14#394C411yescharacteristic, complemented
15#392C2211yescharacteristic
16#381—21no—
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19#342—21no—
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4D411yescharacteristic
27C4×D4—41no—
28C4×D4—41no—
29C4×D4—44no—
30#117—41no—
31#117—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4D411yescharacteristic
44C2×C22⋊C4C2311yescharacteristic
45C2×C42—21no—
46C2×C42C2×C411yescharacteristic
47C2×C42D411yescharacteristic
48C4⋊C8—41no—
49C22⋊C8—22no—
50#97—41no—
51#95—21no—
52#95—21no—
53#95—41no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24—21no—
58C24C2×D411yescharacteristic
59C2×D4—41no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C2×D4—42no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—41no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—42no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C2×C8—42no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C42—21no—
98C42—21no—
99C42—21no—
100C42—21no—
101C42—42no—
102C23—21no—
103C23—21no—
104C23—22no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23C2×C22⋊C411yescharacteristic
114C23AGL(1,ℤ/8)12yes—
115D4—42no—
116D4—48no—
117D4—42no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4#9911yescharacteristic
148C2×C4#9911yescharacteristic
149C8—82no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22—41no—
157C22—41no—
158C22—42no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22C2×#9911yescharacteristic
165C22#41811yescharacteristic
166C22#42111yescharacteristic
167C4—21no—
168C4—21no—
169C4—42no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—41no—
175C4—81no—
176C4—81no—
177C4—162no—
178C2—21no—
179C2—21no—
180C2—42no—
181C2—82no—
182C2#151011yescharacteristic
183C2#212011yescharacteristic
184C2#212611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1820C2semidirect
3. #1464C2semidirect
4. #1344C2semidirect
7. C2×C4×D4C4semidirect
10. #398C4semidirect
14. #394C4semidirect
17. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#23922
#18202
#14642
#13442
#13432