Finite Group Databasebeta

← #12281

Subgroups — #12281

Order 256.

198 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 41 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#3186C211yesmaximal, characteristic
2#2233C212yesmaximal, complemented
3#2076C211yesmaximal, characteristic, complemented
4#1347C212yesmaximal, complemented
5#1314C211yesmaximal, characteristic, complemented
6C2×#121—21no—
7#480—22no—
8#466—21no—
9#466C2212yes—
10#447—22no—
11C2×#107C2211yescharacteristic, complemented
12C2×C22⋊C8C2211yescharacteristic
13#380—21no—
14#367—22no—
15#357—22no—
16C22⋊C16C2212yescomplemented
17#340—21no—
18#340C2211yescharacteristic, complemented
19C22×D4—21no—
20C22×D4—41no—
21C23×C4D411yescharacteristic
22C2×D8—21no—
23C2×D8D412yes—
24C22×C8D411yescharacteristic
25C22×C8C2311yescharacteristic
26#121—22no—
27#121—22no—
28#121—41no—
29#121—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C22⋊C4—41no—
34C2×C22⋊C4—41no—
35C2×C16—22no—
36#107—21no—
37#107—22no—
38#107D412yes—
39#102—22no—
40#102—22no—
41C22⋊C8—22no—
42#95—41no—
43C24—41no—
44C24D811yescharacteristic
45C2×D4—22no—
46C2×D4—41no—
47C2×D4—42no—
48C2×D4—42no—
49C2×D4—81no—
50C2×D4—81no—
51C2×D4—82no—
52C2×D4—82no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4D811yescharacteristic
61C22×C4C2×D411yescharacteristic
62D8—42no—
63D8—81no—
64C2×C8—21no—
65C2×C8—41no—
66C2×C8—41no—
67C2×C8C2×D412yes—
68C4⋊C4—22no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—41no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C22⋊C4—42no—
75C22⋊C4—42no—
76C22⋊C4—82no—
77C16—42no—
78C23—21no—
79C23—21no—
80C23—41no—
81C23—41no—
82C23—42no—
83C23—82no—
84C23—82no—
85C23—82no—
86C23SD3212yes—
87C23C2×D811yescharacteristic
88D4—42no—
89D4—42no—
90D4—82no—
91D4—84no—
92D4—82no—
93D4—82no—
94C2×C4—21no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—81no—
102C2×C4—82no—
103C2×C4—81no—
104C2×C4—81no—
105C2×C4—81no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4—81no—
109C2×C4#12011yescharacteristic
110C2×C4AGL(1,ℤ/8)11yescharacteristic
111C8—22no—
112C8—41no—
113C22—21no—
114C22—21no—
115C22—22no—
116C22—42no—
117C22—41no—
118C22—41no—
119C22—81no—
120C22—82no—
121C22—82no—
122C22—82no—
123C22—164no—
124C22#44711yescharacteristic
125C22C2×SD3211yescharacteristic
126C22#50911yescharacteristic
127C4—21no—
128C4—41no—
129C4—82no—
130C4—161no—
131C4—161no—
132C4#45511yescharacteristic
133C4#45811yescharacteristic
134C2—21no—
135C2—21no—
136C2—42no—
137C2—162no—
138C2#160711yescharacteristic
139C2#218711yescharacteristic
140C2#219011yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3186C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2233C2semidirect
3. #2076C2semidirect
4. #1347C2semidirect
5. #1314C2semidirect
11. C2×#107C22semidirect
16. C22⋊C16C22semidirect
18. #340C22semidirect

Maximal subgroups

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

subgroupindex
#31862
#22332
#20762
#13472
#13142