Finite Group Databasebeta

← #13429

Subgroups — #13429

Order 256.

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

normal subgroups only

This group has 217 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#3199C211yesmaximal, characteristic
2#2134C212yesmaximal, complemented
3#1914C212yesmaximal, complemented
4#1858C211yesmaximal, characteristic, complemented
5#1392C211yesmaximal, characteristic, complemented
6#568C2211yescharacteristic
7#535C2212yes—
8#482—22no—
9#469—21no—
10#466—21no—
11#447—22no—
12#430—21no—
13#413C2212yes—
14#413C2211yescharacteristic
15C2×#100—22no—
16#408C2211yescharacteristic
17#355—24no—
18#344—22no—
19#341—21no—
20C2×#41C2311yescharacteristic
21C22×D4D412yes—
22C2×D8—21no—
23C2×D8—21no—
24#131—42no—
25C2×M16—21no—
26C2×M16—22no—
27C2×M16—21no—
28C2×M16—22no—
29C2×M16D411yescharacteristic
30C22×C8D411yescharacteristic
31#127—22no—
32#124—22no—
33#121—22no—
34#121—22no—
35#120—22no—
36#120—22no—
37#117D412yes—
38#108—42no—
39#107—21no—
40#106—21no—
41#102—22no—
42#102—22no—
43#101—22no—
44#101D412yes—
45#100—21no—
46#100—22no—
47#100—44no—
48C22⋊C8—22no—
49C24—22no—
50#41—41no—
51#41—41no—
52C2×Q8C2×D411yescharacteristic
53C2×D4—22no—
54C2×D4—22no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4#4111yescharacteristic
60C2×D4#4112yes—
61C22×C4C2×D412yes—
62C22×C4C2×D411yescharacteristic
63D8—81no—
64D8—81no—
65M16—24no—
66M16—22no—
67M16—22no—
68M16—41no—
69M16—41no—
70M16—41no—
71M16—42no—
72M16—42no—
73C2×C8—21no—
74C2×C8—21no—
75C2×C8—41no—
76C2×C8—41no—
77C2×C8—42no—
78C2×C8—41no—
79C2×C8—42no—
80C4⋊C4—22no—
81C22⋊C4—22no—
82C22⋊C4—42no—
83C22⋊C4—42no—
84C42—22no—
85C23—44no—
86C23—42no—
87C23—162no—
88C23#12312yes—
89C23#12411yescharacteristic
90Q8—41no—
91D4—41no—
92D4—42no—
93D4—42no—
94D4—42no—
95D4—82no—
96D4—162no—
97D4—162no—
98C2×C4—21no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—22no—
102C2×C4—42no—
103C2×C4#12011yescharacteristic
104C2×C4#12112yes—
105C2×C4#12111yescharacteristic
106C8—42no—
107C8—41no—
108C8—42no—
109C8—41no—
110C8—82no—
111C22—22no—
112C22—21no—
113C22—41no—
114C22—42no—
115C22—82no—
116C22—82no—
117C22—82no—
118C22—164no—
119C22#39411yescharacteristic
120C4—21no—
121C4—22no—
122C4—82no—
123C4B(4,𝔽2)11yescharacteristic
124C4#45811yescharacteristic
125C2—21no—
126C2—41no—
127C2—42no—
128C2—162no—
129C2#204411yescharacteristic
subgroupG/H
center, socleC2#2044
commutator, Frattini, cosocleC2×#41C23
prime radical#3199C2
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2134C2semidirect
3. #1914C2semidirect
4. #1858C2semidirect
5. #1392C2semidirect

Maximal subgroups

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

subgroupindex
#31992
#21342
#19142
#18582
#13922