Finite Group Databasebeta

← #9977

Subgroups — #9977

Order 256.

272 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 51 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#3094C212yesmaximal, complemented
3#1818C211yesmaximal, characteristic, complemented
4#1471C212yesmaximal, complemented
5#1293C211yesmaximal, characteristic, complemented
6C2×#121C2212yes—
7C2×#117C2211yescharacteristic
8#482—24no—
9#481—22no—
10#480—22no—
11#480—22no—
12#466—22no—
13#447—22no—
14#428—22no—
15#418—22no—
16#407C2212yes—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#336—22no—
20#323—24no—
21C22×D4—22no—
22C22×D4D412yes—
23C23×C4C2311yescharacteristic
24C2×D8—22no—
25C2×M16—22no—
26C22×C8—21no—
27C22×C8D412yes—
28#121—22no—
29#121—22no—
30#121—22no—
31#121—22no—
32#117—22no—
33#117—22no—
34C2×C4⋊C4D411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C4⋊C4D412yes—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C42—21no—
40#107—24no—
41#107—22no—
42#106—24no—
43#102—24no—
44#102—24no—
45#102—24no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8D414yes—
50C24—22no—
51C24#4111yescharacteristic
52C2×D4—22no—
53C2×D4—22no—
54C2×D4—44no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—84no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—42no—
62C22×C4C2×D412yes—
63C22×C4C2×D411yescharacteristic
64C22×C4C2×D411yescharacteristic
65C22×C4C2×D412yes—
66D8—82no—
67M16—44no—
68C2×C8—22no—
69C2×C8—24no—
70C2×C8—42no—
71C2×C8—42no—
72C2×C8—41no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C22⋊C4—42no—
80C42—41no—
81C42—41no—
82C23—22no—
83C23—41no—
84C23—41no—
85C23—44no—
86C23—42no—
87C23—82no—
88C23—82no—
89C23#12112yes—
90C23#12111yescharacteristic
91D4—44no—
92D4—44no—
93D4—84no—
94D4—84no—
95C2×C4—22no—
96C2×C4—24no—
97C2×C4—41no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—82no—
105C2×C4#12012yes—
106C2×C4#12011yescharacteristic
107C2×C4#12711yescharacteristic
108C8—44no—
109C8—42no—
110C8—81no—
111C22—21no—
112C22—21no—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—82no—
117C22—82no—
118C22—82no—
119C22—82no—
120C22—164no—
121C22#39211yescharacteristic
122C22#47212yes—
123C4—22no—
124C4—21no—
125C4—81no—
126C4—81no—
127C4—82no—
128C4#45812yes—
129C2—22no—
130C2—42no—
131C2—162no—
132C2#201711yescharacteristic
133C2#201811yescharacteristic
134C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3094C2semidirect
3. #1818C2semidirect
4. #1471C2semidirect
5. #1293C2semidirect

Maximal subgroups

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

subgroupindex
#31862
#30942
#18182
#14712
#12932