Finite Group Databasebeta

← #9979

Subgroups — #9979

Order 256.

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

normal subgroups only

This group has 246 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#3189C211yesmaximal, characteristic
2#3095C212yesmaximal
3#1818C211yesmaximal, characteristic
4#1472C212yesmaximal
5#1293C211yesmaximal, characteristic
6C2×C22⋊Q8C2212yes—
7C2×#117C2211yescharacteristic
8#485—24no—
9#484—22no—
10#484—22no—
11#483—22no—
12#467—22no—
13#451—22no—
14#428—22no—
15#419—22no—
16#407C2212yes—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#336—22no—
20#324—24no—
21C22×Q8D412yes—
22C23×C4C2311yescharacteristic
23C2×Q16—22no—
24C2×M16—22no—
25C22×C8—21no—
26C22×C8D412yes—
27C22⋊Q8—22no—
28C22⋊Q8—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31#117—22no—
32#117—22no—
33C2×C4⋊C4—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#103—24no—
44#103—24no—
45#103—24no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48C22⋊C8—22no—
49C22⋊C8D414yes—
50C24#4111yescharacteristic
51C2×Q8—22no—
52C2×Q8—22no—
53C2×Q8—44no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—42no—
59C22×C4C2×D412yes—
60C22×C4C2×D411yescharacteristic
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D412yes—
63Q16—82no—
64M16—44no—
65C2×C8—22no—
66C2×C8—24no—
67C2×C8—42no—
68C2×C8—42no—
69C2×C8—41no—
70C4⋊C4—22no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—42no—
75C4⋊C4—42no—
76C22⋊C4—42no—
77C22⋊C4—42no—
78C22⋊C4—42no—
79C42—41no—
80C42—41no—
81C23—22no—
82C23—41no—
83C23—41no—
84C23#12112yes—
85C23#12111yescharacteristic
86Q8—44no—
87Q8—44no—
88C2×C4—22no—
89C2×C4—24no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—44no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—42no—
99C2×C4—82no—
100C2×C4#12012yes—
101C2×C4#12011yescharacteristic
102C2×C4#12711yescharacteristic
103C8—44no—
104C8—42no—
105C8—81no—
106C22—21no—
107C22—21no—
108C22—22no—
109C22—22no—
110C22—22no—
111C22—82no—
112C22#39211yescharacteristic
113C22#47312yes—
114C4—22no—
115C4—21no—
116C4—81no—
117C4—82no—
118C4—81no—
119C4—82no—
120C4#45812yes—
121C2—22no—
122C2—42no—
123C2#201711yescharacteristic
124C2#202011yescharacteristic
125C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC24#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#31892
#30952
#18182
#14722
#12932