Finite Group Databasebeta

← #12164

Subgroups — #12164

Order 256.

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

normal subgroups only

This group has 187 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#3259C211yesmaximal, characteristic, complemented
2#2184C212yesmaximal, complemented
3#2045C211yesmaximal, characteristic, complemented
4#1349C212yesmaximal, complemented
5#1323C211yesmaximal, characteristic, complemented
6C22×D8C2211yescharacteristic
7#540—21no—
8C2×D16—22no—
9#478—22no—
10#469—21no—
11#466C2212yes—
12#459—22no—
13#424C2211yescharacteristic
14#418—21no—
15C2×#100—21no—
16#357—22no—
17C22⋊C16C2212yes—
18#341C2211yescharacteristic
19C22×D4—21no—
20C2×D8—21no—
21C2×D8—22no—
22C2×D8—42no—
23C2×D8D412yes—
24C2×M16—21no—
25C2×M16D411yescharacteristic
26C22×C8C2311yescharacteristic
27#127—21no—
28#125—21no—
29#123—42no—
30#121—22no—
31#121—42no—
32C4×D4—22no—
33#117D411yescharacteristic
34D16—44no—
35C2×C16—22no—
36#107D412yes—
37#106—21no—
38C4⋊C8—22no—
39#102—22no—
40#102—22no—
41#102—42no—
42#100—42no—
43C24—41no—
44C2×D4—22no—
45C2×D4—22no—
46C2×D4—41no—
47C2×D4—41no—
48C2×D4—41no—
49C2×D4—42no—
50C2×D4—42no—
51C2×D4—41no—
52C2×D4—41no—
53C22×C4—42no—
54C22×C4C2×D411yescharacteristic
55D8—24no—
56D8—42no—
57D8—42no—
58D8—81no—
59M16—41no—
60M16—41no—
61M16—41no—
62C2×C8—21no—
63C2×C8—41no—
64C2×C8—41no—
65C2×C8C2×D412yes—
66C4⋊C4—22no—
67C4⋊C4—42no—
68C4⋊C4D812yes—
69C22⋊C4—21no—
70C22⋊C4—42no—
71C22⋊C4—42no—
72C42—21no—
73C16—42no—
74C23—42no—
75C23—42no—
76C23—81no—
77C23—81no—
78C23—162no—
79C23AGL(1,ℤ/8)11yescharacteristic
80D4—42no—
81D4—42no—
82D4—42no—
83D4—42no—
84D4—81no—
85D4—81no—
86D4—162no—
87D4—162no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—42no—
92C2×C4—162no—
93C2×C4#12011yescharacteristic
94C2×C4C2×D811yescharacteristic
95C8—22no—
96C8—41no—
97C8—81no—
98C22—21no—
99C22—22no—
100C22—82no—
101C22—82no—
102C22—82no—
103C22—82no—
104C22—162no—
105C22—162no—
106C22—162no—
107C22#44711yescharacteristic
108C4—21no—
109C4—42no—
110C4—82no—
111C4#45411yescharacteristic
112C4B(4,𝔽2)11yescharacteristic
113C2—41no—
114C2—162no—
115C2—162no—
116C2#159811yescharacteristic
117C2#219212yes—
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3259C2semidirect
2. #2184C2semidirect
3. #2045C2semidirect
4. #1349C2semidirect
5. #1323C2semidirect

Maximal subgroups

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

subgroupindex
#32592
#21842
#20452
#13492
#13232