Finite Group Databasebeta

← #12206

Subgroups — #12206

Order 256.

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

normal subgroups only

This group has 250 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#2815C211yesmaximal, characteristic
2#2030C211yesmaximal, characteristic, complemented
3#1596C212yesmaximal, complemented
4#1343C212yesmaximal, complemented
5#1317C211yesmaximal, characteristic, complemented
6#543C2211yescharacteristic
7C2×#121—21no—
8#450—22no—
9#397—22no—
10#392—22no—
11#392—21no—
12#392C2212yes—
13#381—21no—
14#381C2211yescharacteristic
15#342—21no—
16#342C2211yescharacteristic
17#328—22no—
18#323C2212yes—
19C22×D4—21no—
20C22×D4—22no—
21C22×D4—21no—
22C22×D4D412yes—
23C23×C4D411yescharacteristic
24C2×SD16—42no—
25C22⋊Q8—22no—
26C22⋊Q8—42no—
27#121—42no—
28#121—42no—
29C2×C4⋊C4—21no—
30C2×C22⋊C4—22no—
31C2×C22⋊C4—21no—
32C2×C22⋊C4—21no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4D411yescharacteristic
36C2×C22⋊C4C2311yescharacteristic
37#102—42no—
38C22⋊C8—22no—
39#95—21no—
40#95—22no—
41#95—42no—
42#95D412yes—
43C24—22no—
44C24C2×D411yescharacteristic
45C2×Q8—22no—
46C2×D4—42no—
47C2×D4—42no—
48C2×D4—42no—
49C2×D4—42no—
50C2×D4—82no—
51C2×D4—81no—
52C2×D4—81no—
53C2×D4—82no—
54C2×D4—82no—
55C2×D4—82no—
56C2×D4—81no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4SD1612yescomplemented
66C22×C4C2×D412yes—
67SD16—84no—
68C2×C8—42no—
69C4⋊C4—22no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C22⋊C4—41no—
73C22⋊C4—44no—
74C22⋊C4—41no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—41no—
78C22⋊C4—41no—
79C22⋊C4—82no—
80C22⋊C4—82no—
81C23—21no—
82C23—21no—
83C23—42no—
84C23—42no—
85C23—42no—
86C23—42no—
87C23—42no—
88C23—42no—
89C23—42no—
90C23—42no—
91C23#12011yescharacteristic
92C23C2×SD1611yescharacteristic
93C23AGL(1,ℤ/8)11yescharacteristic
94Q8—82no—
95D4—44no—
96D4—82no—
97D4—84no—
98D4—84no—
99D4—84no—
100D4—84no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—81no—
114C2×C4—82no—
115C2×C4—81no—
116C2×C4—82no—
117C8—82no—
118C22—22no—
119C22—21no—
120C22—21no—
121C22—41no—
122C22—41no—
123C22—42no—
124C22—81no—
125C22—82no—
126C22—82no—
127C22—82no—
128C22—82no—
129C22—82no—
130C22—82no—
131C22—82no—
132C22#45011yescharacteristic
133C22#45511yescharacteristic
134C22B(4,𝔽2)11yescharacteristic
135C4—42no—
136C4—42no—
137C4—82no—
138C4—81no—
139C4—82no—
140C4—161no—
141C2—21no—
142C2—21no—
143C2—41no—
144C2—41no—
145C2—162no—
146C2#160111yescharacteristic
147C2#220012yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#2815C2
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
2. #2030C2semidirect
3. #1596C2semidirect
4. #1343C2semidirect
5. #1317C2semidirect
65. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28152
#20302
#15962
#13432
#13172