Finite Group Databasebeta

← #12207

Subgroups — #12207

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#2817C211yesmaximal, characteristic
2#2030C211yesmaximal, characteristic
3#1608C212yesmaximal
4#1344C212yesmaximal
5#1317C211yesmaximal, characteristic
6#543C2211yescharacteristic
7C2×C22⋊Q8—21no—
8#483—22no—
9#400—22no—
10#397—21no—
11#397C2212yes—
12#395—22no—
13#381—21no—
14#381C2211yescharacteristic
15#342—21no—
16#342C2211yescharacteristic
17#328—22no—
18#323C2212yes—
19C22×Q8—21no—
20C23×C4D411yescharacteristic
21C22⋊Q8—22no—
22C22⋊Q8—42no—
23C22⋊Q8—42no—
24C22⋊Q8—42no—
25C2×C4⋊C4—22no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4D412yes—
29C2×C22⋊C4—22no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4D411yescharacteristic
33C2×C22⋊C4C2311yescharacteristic
34#106—42no—
35#103—42no—
36C22⋊C8—22no—
37#95—22no—
38#95—21no—
39#95—22no—
40#95—22no—
41#95—42no—
42#95D412yes—
43C24C2×D411yescharacteristic
44C2×Q8—22no—
45C2×Q8—42no—
46C2×Q8—81no—
47C2×Q8—81no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—41no—
55C22×C4—41no—
56C22×C4—41no—
57C22×C4SD1612yes—
58C22×C4C2×D412yes—
59C2×C8—42no—
60C4⋊C4—22no—
61C4⋊C4—42no—
62C4⋊C4—42no—
63C4⋊C4—44no—
64C4⋊C4—42no—
65C4⋊C4—81no—
66C22⋊C4—41no—
67C22⋊C4—41no—
68C22⋊C4—41no—
69C22⋊C4—42no—
70C22⋊C4—42no—
71C22⋊C4—41no—
72C22⋊C4—41no—
73C22⋊C4—41no—
74C23—21no—
75C23—21no—
76C23—42no—
77C23#12011yescharacteristic
78C23C2×SD1611yescharacteristic
79C23AGL(1,ℤ/8)11yescharacteristic
80Q8—82no—
81Q8—84no—
82C2×C4—22no—
83C2×C4—22no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—44no—
90C2×C4—42no—
91C2×C4—42no—
92C2×C4—41no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—81no—
97C2×C4—82no—
98C2×C4—81no—
99C2×C4—82no—
100C8—82no—
101C22—22no—
102C22—21no—
103C22—21no—
104C22—41no—
105C22—41no—
106C22—42no—
107C22—81no—
108C22#45011yescharacteristic
109C22#45511yescharacteristic
110C22B(4,𝔽2)11yescharacteristic
111C4—42no—
112C4—42no—
113C4—82no—
114C4—81no—
115C4—82no—
116C4—82no—
117C4—161no—
118C2—21no—
119C2—21no—
120C2—41no—
121C2—41no—
122C2#160111yescharacteristic
123C2#220112yes—
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
#28172
#20302
#16082
#13442
#13172