Finite Group Databasebeta

← #29879

Subgroups — #29879

Order 256.

365 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 123 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×#499C211yesmaximal, characteristic, complemented
2C2×#431C211yesmaximal, characteristic
3#2851C212yesmaximal
4#2088C212yesmaximal, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2077C212yesmaximal
8#2076C211yesmaximal, characteristic
9#2075C211yesmaximal, characteristic
10#1991C212yesmaximal, complemented
11#1835C211yesmaximal, characteristic, complemented
12C2×#125C2212yes—
13C22×C42C2211yescharacteristic
14#499—24no—
15#431—24no—
16C2×#107C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#106C2212yes—
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23C2×C4×C8C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#389C2212yes—
26#389C2212yes—
27#387—22no—
28#387—22no—
29#385—22no—
30#385—22no—
31#382C2212yes—
32#380C2212yes—
33#380C2212yes—
34#340C2211yescharacteristic
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47C22×C8C2311yescharacteristic
48#125—24no—
49#125—24no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—22no—
67#107—22no—
68#106—24no—
69C4⋊C8—22no—
70C4⋊C8—22no—
71C22⋊C8—22no—
72C22⋊C8—22no—
73C4×C8—22no—
74C4×C8D414yes—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—42no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8C2×D412yes—
108C2×C8C2×D412yes—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C22⋊C4—44no—
122C22⋊C4—44no—
123C42—22no—
124C42—22no—
125C42—21no—
126C42—21no—
127C42C2×D412yes—
128C42#4112yes—
129C23—21no—
130C23—22no—
131C23—21no—
132C23—22no—
133C23#13512yes—
134C23C22×D411yescharacteristic
135C2×C4—28no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—44no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4#12112yes—
162C2×C4#12112yes—
163C2×C4#13512yes—
164C2×C4C22×D411yescharacteristic
165C2×C4C2×#4111yescharacteristic
166C2×C42−1+412yes—
167C2×C42−1+412yes—
168C8—22no—
169C8—22no—
170C8—41no—
171C8—41no—
172C22—24no—
173C22—22no—
174C22—41no—
175C22—42no—
176C22—41no—
177C22C2×#12111yescharacteristic
178C22#53612yes—
179C22C2×#13511yescharacteristic
180C22C2×#13511yescharacteristic
181C22#57512yes—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—41no—
186C4—41no—
187C4—82no—
188C4—82no—
189C4—82no—
190C4—82no—
191C4#53611yescharacteristic
192C4#53611yescharacteristic
193C4#54112yes—
194C2—42no—
195C2#262111yescharacteristic
196C2#305011yescharacteristic
197C2#305411yescharacteristic
198C2#323212yes—
199C2#323312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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×#499C2semidirect
4. #2088C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
10. #1991C2semidirect
11. #1835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4992
C2×#4312
#28512
#20882
#20872
#20862
#20772
#20762
#20752
#19912
#18352