Finite Group Databasebeta

← #29650

Subgroups — #29650

Order 256.

519 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 121 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×#544C211yesmaximal, characteristic, complemented
2#3094C212yesmaximal
3#2501C211yesmaximal, characteristic
4#2060C214yesmaximal, complemented
5#2027C214yesmaximal, complemented
6#1826C212yesmaximal, complemented
7#1821C211yesmaximal, characteristic, complemented
8#544—24no—
9#544—22no—
10#544C2214yes—
11C2×C4⋊Q8—22no—
12C2×#125—22no—
13C2×#123—22no—
14C2×C22⋊Q8—22no—
15C2×#121C2211yescharacteristic
16C2×#117C2212yes—
17C22×C4⋊C4C2211yescharacteristic
18#482—24no—
19#481—22no—
20#480—22no—
21#428C2214yes—
22#418C2214yes—
23#407C2212yes—
24#399—22no—
25#390C2212yes—
26#384C2212yes—
27C4×C22⋊C4C2211yescharacteristic
28#343C2214yes—
29#339C2218yes—
30C22×Q8—22no—
31C22×D4—21no—
32C22×D4C2311yescharacteristic
33C23×C4D412yes—
34C23×C4C2311yescharacteristic
35C2×M16C2314yes—
36C4⋊Q8—28no—
37C4⋊Q8—28no—
38#125—28no—
39#125—28no—
40#123—48no—
41C22⋊Q8—28no—
42C22⋊Q8—28no—
43#121—22no—
44#121—22no—
45#117—22no—
46#117—22no—
47#117D414yes—
48#117C2314yes—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4D412yes—
55C2×C4⋊C4D412yes—
56C2×C4⋊C4D412yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—24no—
63C2×C42—22no—
64C2×C42C2312yes—
65#107—24no—
66#106—24no—
67#102—24no—
68#102—24no—
69C22⋊C8—22no—
70C22⋊C8—22no—
71#95—21no—
72C24—21no—
73C24C2×D411yescharacteristic
74C2×Q8—28no—
75C2×Q8—84no—
76C2×D4—22no—
77C2×D4—24no—
78C2×D4—44no—
79C2×D4—41no—
80C2×D4—41no—
81C22×C4—24no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—44no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D412yes—
91C22×C4C2×D412yes—
92C22×C4C2×D414yes—
93C22×C4C2411yescharacteristic
94M16—28no—
95C2×C8—24no—
96C4⋊C4—28no—
97C4⋊C4—216no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4—48no—
104C4⋊C4C2×D414yes—
105C4⋊C4C2×D414yes—
106C22⋊C4—24no—
107C22⋊C4—24no—
108C22⋊C4—24no—
109C22⋊C4—48no—
110C22⋊C4—42no—
111C42—24no—
112C42—24no—
113C42—24no—
114C23—22no—
115C23—22no—
116C23—41no—
117C23—42no—
118C23—42no—
119C23—41no—
120C23—42no—
121C23#12011yescharacteristic
122C23#12011yescharacteristic
123C23C22×D411yescharacteristic
124Q8—88no—
125D4—44no—
126D4—82no—
127D4—82no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—24no—
134C2×C4—24no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—82no—
144C2×C4#12012yes—
145C2×C4C22×D412yes—
146C2×C42−1+414yes—
147C8—44no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—84no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#45312yes—
161C22C2×#12011yescharacteristic
162C22#53612yes—
163C4—22no—
164C4—44no—
165C4—44no—
166C4—44no—
167C4—44no—
168C4—81no—
169C4#53614yes—
170C2—22no—
171C2—22no—
172C2—41no—
173C2—161no—
174C2#261411yescharacteristic
175C2C2×#45311yescharacteristic
176C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
4. #2060C2semidirect
5. #2027C2semidirect
6. #1826C2semidirect
7. #1821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#30942
#25012
#20602
#20272
#18262
#18212