Finite Group Databasebeta

← #11823

Subgroups — #11823

Order 256.

223 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 51 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#2398C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1465C211yesmaximal, characteristic
4#1351C212yesmaximal, complemented
5#1344C212yesmaximal, complemented
6C2×C4×Q8C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8C2×#99—21no—
9#407—21no—
10#400—21no—
11#397C411yescharacteristic, complemented
12#397C2211yescharacteristic
13#394—21no—
14#393C411yescharacteristic, complemented
15#393C2211yescharacteristic
16#380—21no—
17C4×C22⋊C4—21no—
18C4×C22⋊C4—21no—
19#376C411yescharacteristic, complemented
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×D4D411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—41no—
28C4×Q8—41no—
29C4×Q8—41no—
30C4×D4—41no—
31C4×D4—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4C2×C411yescharacteristic
42C2×C22⋊C4C2311yescharacteristic
43C2×C42—21no—
44C2×C42—21no—
45C2×C42C2×C411yescharacteristic
46#99—42no—
47C22⋊C8—22no—
48C22⋊C8—41no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C2×D411yescharacteristic
59C2×Q8—42no—
60C2×Q8—41no—
61C2×D4—21no—
62C2×D4—21no—
63C2×D4—41no—
64C2×D4—41no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81M16—82no—
82C2×C8—42no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—42no—
86C4⋊C4—82no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C42—21no—
97C42—21no—
98C42—42no—
99C42—41no—
100C42—41no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23C2×C22⋊C411yescharacteristic
111C23AGL(1,ℤ/8)11yescharacteristic
112C23#13711yescharacteristic
113Q8—42no—
114Q8—42no—
115D4—42no—
116D4—42no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—41no—
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—81no—
142C2×C4—81no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4#9911yescharacteristic
147C2×C4#9911yescharacteristic
148C8—82no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—41no—
154C22—41no—
155C22—42no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—81no—
162C22—81no—
163C22C2×#9911yescharacteristic
164C22#41711yescharacteristic
165C22#42111yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4—162no—
176C4#35412yes—
177C2—21no—
178C2—21no—
179C2—41no—
180C2—41no—
181C2—81no—
182C2#151311yescharacteristic
183C2#212011yescharacteristic
184C2C2×#35411yescharacteristic
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. #1351C2semidirect
5. #1344C2semidirect
6. C2×C4×Q8C4semidirect
11. #397C4semidirect
14. #393C4semidirect
19. #376C4semidirect

Maximal subgroups

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

subgroupindex
#23982
#17992
#14652
#13512
#13442