Finite Group Databasebeta

← #12330

Subgroups — #12330

Order 256.

161 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 43 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#3395C211yesmaximal, characteristic, complemented
2#2248C211yesmaximal, characteristic
3#2207C211yesmaximal, characteristic
4#1668C211yesmaximal, characteristic
5#1362C211yesmaximal, characteristic, complemented
6Q8⋊C16C211yesmaximal, characteristic, complemented
7#1332C211yesmaximal, characteristic, complemented
8#557—21no—
9C2×SD32—21no—
10C2×SD32—22no—
11#494—21no—
12#494C2211yescharacteristic
13#493C2211yescharacteristic
14#462—22no—
15#462—21no—
16#460—21no—
17C8×Q8C2211yescharacteristic
18C4×Q16—21no—
19C4×Q16C2211yescharacteristic
20C4⋊C16C2211yescharacteristic
21#357—21no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#331—21no—
25C2×Q16—22no—
26C2×Q16—21no—
27C2×Q16D411yescharacteristic
28C2×Q16D412yes—
29C2×SD16—41no—
30C2×D8—21no—
31C2×D8—21no—
32C4⋊Q8—22no—
33C4⋊Q8—21no—
34#127—21no—
35#125—42no—
36#125—41no—
37C4×Q8—22no—
38C4×Q8—21no—
39C4×Q8—41no—
40C4×Q8D411yescharacteristic
41SD32—44no—
42SD32—42no—
43C2×C16—21no—
44C2×C16—21no—
45#107D411yescharacteristic
46C4⋊C8—21no—
47C4⋊C8—21no—
48C4⋊C8D411yescharacteristic
49#103—22no—
50#103—21no—
51#102—41no—
52C4×C8—21no—
53C4×C8C2311yescharacteristic
54C2×Q8—21no—
55C2×Q8—22no—
56C2×Q8D811yescharacteristic
57C2×D4—41no—
58C2×D4—41no—
59C2×D4—41no—
60Q16—21no—
61Q16—21no—
62Q16—42no—
63Q16—41no—
64Q16—41no—
65Q16—82no—
66SD16—82no—
67D8—42no—
68D8—41no—
69D8—41no—
70C2×C8—21no—
71C2×C8—41no—
72C2×C8C2×D411yescharacteristic
73C2×C8C2×D411yescharacteristic
74C4⋊C4—21no—
75C4⋊C4—21no—
76C4⋊C4—42no—
77C4⋊C4—42no—
78C4⋊C4—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4D811yescharacteristic
83C42—21no—
84C42—42no—
85C42—41no—
86C42C2×D411yescharacteristic
87C16—22no—
88C16—41no—
89C23—81no—
90Q8—21no—
91Q8—41no—
92Q8—42no—
93Q8—42no—
94Q8—41no—
95Q8SD3212yescomplemented
96D4—81no—
97D4—81no—
98D4—81no—
99D4—81no—
100D4—161no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4#12011yescharacteristic
107C2×C4C2×D811yescharacteristic
108C2×C4AGL(1,ℤ/8)11yescharacteristic
109C8—21no—
110C8—41no—
111C8—81no—
112C8#13511yescharacteristic
113C8AGL(1,ℤ/8)11yescharacteristic
114C22—161no—
115C22—161no—
116C22—161no—
117C22#44711yescharacteristic
118C4—22no—
119C4—21no—
120C4—41no—
121C4—81no—
122C4—82no—
123C4—81no—
124C4#44911yescharacteristic
125C4#45311yescharacteristic
126C4C2×SD3211yescharacteristic
127C4#51011yescharacteristic
128C2—321no—
129C2#162111yescharacteristic
130C2#218511yescharacteristic
131C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#493C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3395C2semidirect
5. #1362C2semidirect
6. Q8⋊C16C2semidirect
7. #1332C2semidirect
95. Q8SD32semidirect

Maximal subgroups

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

subgroupindex
#33952
#22482
#22072
#16682
#13622
Q8⋊C162
#13322