Finite Group Databasebeta

← #25906

Subgroups — #25906

Order 256.

552 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#533C211yesmaximal, characteristic, complemented
2C2×#488C212yesmaximal, complemented
3#2850C211yesmaximal, characteristic
4#2024C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1991C212yesmaximal, complemented
7#1875C212yesmaximal
8#1873C212yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic, complemented
10#533—28no—
11C2×C4⋊Q8C2212yes—
12C2×#125C2212yes—
13C2×#124C2211yescharacteristic
14C2×C22⋊Q8—22no—
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C2×C4×Q8—22no—
18C2×#117C2212yes—
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×Q8—22no—
34C22×Q8C2311yescharacteristic
35C22×D4—21no—
36C22×D4C2311yescharacteristic
37C23×C4D412yes—
38C23×C4C2311yescharacteristic
39C22×C8C2314yes—
40C4⋊Q8—28no—
41#125—24no—
42#125D418yescomplemented
43#124—24no—
44C22⋊Q8—216no—
45C22⋊Q8—24no—
46#121—24no—
47C4×Q8—216no—
48#117—28no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2311yescharacteristic
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2314yes—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C42—22no—
61C2×C42—24no—
62C2×C42D412yes—
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65#103—28no—
66#102—28no—
67C22⋊C8—24no—
68C4×C8—24no—
69#95—22no—
70#95—22no—
71C24—21no—
72C24C2×D411yescharacteristic
73C2×Q8—28no—
74C2×Q8—22no—
75C2×Q8—22no—
76C2×Q8—22no—
77C2×Q8—44no—
78C2×Q8—44no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—44no—
83C2×D4—42no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—24no—
88C22×C4—24no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4C2×D412yes—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4#4114yes—
97C22×C4C2411yescharacteristic
98C2×C8—24no—
99C2×C8—24no—
100C2×C8—24no—
101C4⋊C4—28no—
102C4⋊C4—216no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4C2×D418yes—
108C22⋊C4—28no—
109C22⋊C4—42no—
110C22⋊C4—44no—
111C22⋊C4—42no—
112C22⋊C4—44no—
113C42—28no—
114C42—24no—
115C42—22no—
116C42C2×D414yes—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—22no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—42no—
125C23—42no—
126C23#12412yes—
127C23C22×D411yescharacteristic
128Q8—416no—
129Q8—44no—
130D4—44no—
131D4—84no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—28no—
139C2×C4—28no—
140C2×C4—48no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4#12014yes—
151C2×C4#12412yes—
152C2×C4#13514yes—
153C2×C4C22×D412yes—
154C2×C4C2×#4112yes—
155C2×C42−1+412yes—
156C8—44no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—82no—
166C22—82no—
167C22C2×#12011yescharacteristic
168C22C2×#12411yescharacteristic
169C22#53611yescharacteristic
170C22C2×#13512yes—
171C22#57512yes—
172C4—22no—
173C4—24no—
174C4—24no—
175C4—48no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4#54814yes—
180C2—42no—
181C2—161no—
182C2#243611yescharacteristic
183C2#300212yes—
184C2#313111yescharacteristic
185C2#314011yescharacteristic
186C2#323312yes—
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. C2×#533C2semidirect
2. C2×#488C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #1991C2semidirect
8. #1873C2semidirect
9. C4×C22⋊C8C2semidirect
42. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4882
#28502
#20242
#20232
#19912
#18752
#18732
C4×C22⋊C82