Finite Group Databasebeta

← #12216

Subgroups — #12216

Order 256.

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

normal subgroups only

This group has 194 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#2607C211yesmaximal, characteristic
2#2030C211yesmaximal, characteristic
3#1614C212yesmaximal
4#1352C212yesmaximal
5#1317C211yesmaximal, characteristic
6#543C2211yescharacteristic
7#484—22no—
8#399C2212yes—
9#393—21no—
10#391—22no—
11#387—22no—
12#381—21no—
13#381C2211yescharacteristic
14C4×C22⋊C4—21no—
15#342—21no—
16#342C2211yescharacteristic
17#328—22no—
18#323C2212yes—
19C22×Q8—21no—
20C23×C4D411yescharacteristic
21C22⋊Q8—22no—
22C22⋊Q8—42no—
23C2×C4⋊C4—22no—
24C2×C4⋊C4D412yes—
25C2×C22⋊C4—21no—
26C2×C22⋊C4—22no—
27C2×C22⋊C4—21no—
28C2×C22⋊C4D411yescharacteristic
29C2×C22⋊C4C2311yescharacteristic
30C2×C42—21no—
31C2×C42—21no—
32#107—42no—
33#103—42no—
34C22⋊C8—22no—
35#95—22no—
36#95—21no—
37#95—22no—
38#95—22no—
39#95—42no—
40#95D412yes—
41C24C2×D411yescharacteristic
42C2×Q8—22no—
43C2×Q8—44no—
44C2×Q8—81no—
45C22×C4—21no—
46C22×C4—22no—
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—22no—
50C22×C4—22no—
51C22×C4—41no—
52C22×C4—41no—
53C22×C4—41no—
54C22×C4Q1612yes—
55C22×C4C2×D412yes—
56C2×C8—42no—
57C4⋊C4—22no—
58C4⋊C4—42no—
59C4⋊C4—44no—
60C4⋊C4—42no—
61C22⋊C4—41no—
62C22⋊C4—44no—
63C22⋊C4—41no—
64C22⋊C4—41no—
65C22⋊C4—41no—
66C22⋊C4—41no—
67C22⋊C4—41no—
68C42—81no—
69C42—81no—
70C23—21no—
71C23—21no—
72C23—42no—
73C23#12011yescharacteristic
74C23C2×Q1611yescharacteristic
75C23#13711yescharacteristic
76Q8—84no—
77Q8—82no—
78C2×C4—24no—
79C2×C4—22no—
80C2×C4—22no—
81C2×C4—42no—
82C2×C4—41no—
83C2×C4—42no—
84C2×C4—41no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—81no—
94C2×C4—82no—
95C2×C4—81no—
96C2×C4—82no—
97C8—82no—
98C22—22no—
99C22—21no—
100C22—21no—
101C22—41no—
102C22—41no—
103C22—42no—
104C22—81no—
105C22#45111yescharacteristic
106C22#45511yescharacteristic
107C22B(4,𝔽2)11yescharacteristic
108C4—44no—
109C4—42no—
110C4—42no—
111C4—81no—
112C4—82no—
113C4—82no—
114C4—161no—
115C2—21no—
116C2—21no—
117C2—41no—
118C2—41no—
119C2#160211yescharacteristic
120C2#219812yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#26072
#20302
#16142
#13522
#13172