Finite Group Databasebeta

← #12286

Subgroups — #12286

Order 256.

153 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 37 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#3202C211yesmaximal, characteristic
2#2236C211yesmaximal, characteristic, complemented
3#2235C211yesmaximal, characteristic, complemented
4#2078C211yesmaximal, characteristic
5#1350C211yesmaximal, characteristic, complemented
6#1350C211yesmaximal, characteristic, complemented
7#1315C211yesmaximal, characteristic
8#536—21no—
9#485—21no—
10#485—21no—
11#470—21no—
12#467C2211yescharacteristic
13#467C2211yescharacteristic
14#451—21no—
15#451—21no—
16C2×#107C2211yescharacteristic
17#417—21no—
18#408C2211yescharacteristic
19#367—21no—
20#366—21no—
21#358—21no—
22#358—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#343—21no—
26#341C2211yescharacteristic
27C2×#41D411yescharacteristic
28C22×Q8—21no—
29C2×Q16—21no—
30C2×Q16D411yescharacteristic
31C2×Q16D411yescharacteristic
32C2×M16—21no—
33C2×M16D411yescharacteristic
34C22×C8C2311yescharacteristic
35C4⋊Q8—41no—
36#124—41no—
37#123—41no—
38#123—41no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107D411yescharacteristic
50#107D411yescharacteristic
51#106—21no—
52#103—21no—
53#103—21no—
54#103—21no—
55#103—21no—
56#103—41no—
57#102—41no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#41—41no—
61#41—41no—
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8—81no—
66C2×Q8—81no—
67C2×Q8D811yescharacteristic
68C2×D4—21no—
69C2×D4D811yescharacteristic
70C22×C4—21no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74Q16—41no—
75Q16—41no—
76Q16—81no—
77M16—41no—
78M16—41no—
79M16—41no—
80C2×C8—21no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8C2×D411yescharacteristic
84C2×C8C2×D411yescharacteristic
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C42—41no—
99C16—41no—
100C16—41no—
101C23—21no—
102C23AGL(1,ℤ/8)11yescharacteristic
103Q8—41no—
104Q8—41no—
105Q8—41no—
106Q8—41no—
107Q8—41no—
108Q8—82no—
109D4—41no—
110D4—81no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—82no—
121C2×C4—81no—
122C2×C4—161no—
123C2×C4#12011yescharacteristic
124C2×C4C2×D811yescharacteristic
125C8—21no—
126C8—21no—
127C8—41no—
128C8—81no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22#44711yescharacteristic
136C4—21no—
137C4—41no—
138C4—81no—
139C4—81no—
140C4—81no—
141C4—81no—
142C4—161no—
143C4#45511yescharacteristic
144C4#45811yescharacteristic
145C2—41no—
146C2—81no—
147C2#160711yescharacteristic
148C2#219311yescharacteristic
149C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radicalC2×#41D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2236C2semidirect
3. #2235C2semidirect
5. #1350C2semidirect
6. #1350C2semidirect

Maximal subgroups

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

subgroupindex
#32022
#22362
#22352
#20782
#13502
#13152