Finite Group Databasebeta

← #25907

Subgroups — #25907

Order 256.

558 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×#125C211yesmaximal, characteristic, complemented
2C2×#488C212yesmaximal, complemented
3#2850C211yesmaximal, characteristic
4#2026C212yesmaximal, complemented
5#2025C212yesmaximal, complemented
6#1991C212yesmaximal, complemented
7#1874C212yesmaximal
8#1872C212yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic, complemented
10C2×#125—28no—
11C2×#125—22no—
12C2×#125C2212yes—
13C2×#125C2212yes—
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C22×C4⋊C4—22no—
18C22×C4⋊C4C2212yes—
19C22×C42C2211yescharacteristic
20#488—28no—
21C2×#103C2214yes—
22C2×#102C2214yes—
23C2×C22⋊C8C2212yes—
24C2×C4×C8C2212yes—
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#388—22no—
28#386—22no—
29#382C2211yescharacteristic
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8C2311yescharacteristic
34C22×D4—21no—
35C22×D4C2311yescharacteristic
36C23×C4—22no—
37C23×C4D412yes—
38C23×C4C2311yescharacteristic
39C22×C8C2314yes—
40#125—216no—
41#125—28no—
42#125—24no—
43#125D418yescomplemented
44#124—24no—
45C22⋊Q8—24no—
46#121—24no—
47C2×C4⋊C4—216no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2314yes—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C42—24no—
61C2×C42D412yes—
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#103—28no—
65#102—28no—
66C22⋊C8—24no—
67C4×C8—24no—
68#95—22no—
69#95—22no—
70C24—21no—
71C24C2×D411yescharacteristic
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—42no—
79C2×D4—44no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—28no—
83C22×C4—24no—
84C22×C4—24no—
85C22×C4—24no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—42no—
90C22×C4—44no—
91C22×C4—44no—
92C22×C4C2×D412yes—
93C22×C4C2×D412yes—
94C22×C4C2×D411yescharacteristic
95C22×C4#4114yes—
96C22×C4C2411yescharacteristic
97C2×C8—24no—
98C2×C8—24no—
99C2×C8—24no—
100C4⋊C4—28no—
101C4⋊C4—216no—
102C4⋊C4—216no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4C2×D418yes—
108C22⋊C4—44no—
109C22⋊C4—42no—
110C22⋊C4—44no—
111C22⋊C4—42no—
112C42—24no—
113C42—22no—
114C42C2×D414yes—
115C23—21no—
116C23—21no—
117C23—22no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—42no—
122C23—42no—
123C23#12412yes—
124C23#13514yes—
125C23C22×D411yescharacteristic
126Q8—44no—
127D4—44no—
128D4—84no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—28no—
134C2×C4—22no—
135C2×C4—28no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—48no—
139C2×C4—416no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4#12014yes—
150C2×C4#12412yes—
151C2×C4C22×D412yes—
152C2×C4C2×#4112yes—
153C2×C42−1+412yes—
154C8—44no—
155C22—22no—
156C22—22no—
157C22—24no—
158C22—24no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—82no—
163C22—82no—
164C22#48814yescomplemented
165C22C2×#12011yescharacteristic
166C22C2×#12411yescharacteristic
167C22#53611yescharacteristic
168C22C2×#13512yes—
169C22#57512yes—
170C4—22no—
171C4—24no—
172C4—42no—
173C4—48no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4#54814yes—
178C2—24no—
179C2—161no—
180C2#243611yescharacteristic
181C2#300212yes—
182C2C2×#48811yescharacteristic
183C2#314211yescharacteristic
184C2#323212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#125C2semidirect
2. C2×#488C2semidirect
4. #2026C2semidirect
5. #2025C2semidirect
6. #1991C2semidirect
8. #1872C2semidirect
9. C4×C22⋊C8C2semidirect
43. #125D4semidirect
164. C22#488semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#4882
#28502
#20262
#20252
#19912
#18742
#18722
C4×C22⋊C82