Finite Group Databasebeta

← #12217

Subgroups — #12217

Order 256.

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

normal subgroups only

This group has 242 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#2840C211yesmaximal, characteristic
2#2029C211yesmaximal, characteristic
3#1602C212yesmaximal, complemented
4#1351C212yesmaximal, complemented
5#1316C211yesmaximal, characteristic
6#543C2211yescharacteristic
7#451—22no—
8#393—22no—
9#393—21no—
10#393—22no—
11#393C2212yes—
12#379—21no—
13#379—21no—
14#379C2211yescharacteristic
15#342—21no—
16#342C2211yescharacteristic
17#328—22no—
18#323C2212yes—
19C25D411yescharacteristic
20C22×Q8—21no—
21C22×Q8D412yes—
22C2×Q16—42no—
23C22⋊Q8—22no—
24C22⋊Q8—42no—
25C2×C22⋊C4—21no—
26C2×C22⋊C4—21no—
27C2×C22⋊C4—21no—
28C2×C22⋊C4—22no—
29C2×C22⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—42no—
33C2×C22⋊C4D411yescharacteristic
34C2×C22⋊C4C2311yescharacteristic
35#103—42no—
36C22⋊C8—22no—
37#95—21no—
38#95—22no—
39#95—22no—
40#95D412yes—
41C24—21no—
42C24—21no—
43C24—41no—
44C24—41no—
45C24—82no—
46C24Q1612yescomplemented
47C24C2×D411yescharacteristic
48C2×Q8—22no—
49C2×Q8—42no—
50C2×Q8—82no—
51C2×Q8—82no—
52C2×Q8—81no—
53C22×C4—22no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—41no—
58C22×C4C2×D412yes—
59Q16—84no—
60C2×C8—42no—
61C4⋊C4—22no—
62C4⋊C4—42no—
63C22⋊C4—41no—
64C22⋊C4—41no—
65C22⋊C4—41no—
66C22⋊C4—41no—
67C22⋊C4—41no—
68C22⋊C4—41no—
69C22⋊C4—82no—
70C22⋊C4—82no—
71C22⋊C4—82no—
72C22⋊C4—81no—
73C22⋊C4—81no—
74C22⋊C4—84no—
75C23—21no—
76C23—21no—
77C23—22no—
78C23—42no—
79C23—42no—
80C23—42no—
81C23—42no—
82C23—44no—
83C23—41no—
84C23—41no—
85C23—81no—
86C23—82no—
87C23—82no—
88C23—82no—
89C23—84no—
90C23#12011yescharacteristic
91C23C2×Q1611yescharacteristic
92C23#13711yescharacteristic
93Q8—44no—
94Q8—82no—
95Q8—82no—
96Q8—84no—
97C2×C4—22no—
98C2×C4—42no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—42no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—82no—
109C2×C4—81no—
110C2×C4—82no—
111C8—82no—
112C22—22no—
113C22—21no—
114C22—21no—
115C22—41no—
116C22—41no—
117C22—42no—
118C22—42no—
119C22—42no—
120C22—44no—
121C22—42no—
122C22—81no—
123C22—82no—
124C22—82no—
125C22—84no—
126C22—82no—
127C22#45111yescharacteristic
128C22#45511yescharacteristic
129C22B(4,𝔽2)11yescharacteristic
130C4—42no—
131C4—81no—
132C4—82no—
133C4—82no—
134C4—82no—
135C4—161no—
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2—82no—
141C2#160211yescharacteristic
142C2#219912yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC25D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1602C2semidirect
4. #1351C2semidirect
46. C24Q16semidirect

Maximal subgroups

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

subgroupindex
#28402
#20292
#16022
#13512
#13162