Finite Group Databasebeta

← #12208

Subgroups — #12208

Order 256.

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

normal subgroups only

This group has 332 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#2679C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic, complemented
3#1599C212yesmaximal, complemented
4#1345C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic, complemented
6#544C2211yescharacteristic, complemented
7C2×#120—22no—
8C2×#120—21no—
9#450—22no—
10#418—21no—
11#394—22no—
12#394C2212yes—
13#390—21no—
14#390C2211yescharacteristic
15#343C2211yescharacteristic, complemented
16#339—21no—
17#328—22no—
18#324C2212yescomplemented
19C25—21no—
20C22×D4—21no—
21C22×D4—22no—
22C22×D4D411yescharacteristic
23C22×D4D412yes—
24C2×SD16—42no—
25C2×M16—21no—
26C4⋊Q8—41no—
27#125—41no—
28C22⋊Q8—22no—
29#120—48no—
30#120—42no—
31#120—42no—
32#117—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4C2311yescharacteristic
35C2×C22⋊C4—22no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C42D411yescharacteristic
41#102—41no—
42#102—41no—
43#102—42no—
44C22⋊C8—22no—
45#95D412yes—
46C24—22no—
47C24—21no—
48C24—44no—
49C24—42no—
50C24—82no—
51C24—81no—
52C24—81no—
53C2×Q8—22no—
54C2×D4—24no—
55C2×D4—21no—
56C2×D4—41no—
57C2×D4—42no—
58C2×D4—41no—
59C2×D4—44no—
60C2×D4—44no—
61C2×D4—44no—
62C2×D4—81no—
63C2×D4—81no—
64C2×D4—82no—
65C2×D4SD1612yescomplemented
66C22×C4—21no—
67C22×C4—22no—
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D412yes—
70SD16—84no—
71M16—41no—
72M16—41no—
73C2×C8—42no—
74C2×C8—41no—
75C4⋊C4—21no—
76C4⋊C4—21no—
77C4⋊C4—22no—
78C4⋊C4—41no—
79C4⋊C4—42no—
80C4⋊C4—81no—
81C22⋊C4—44no—
82C22⋊C4—44no—
83C22⋊C4—44no—
84C22⋊C4—42no—
85C22⋊C4—41no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C42—41no—
89C42—41no—
90C23—24no—
91C23—22no—
92C23—42no—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—42no—
97C23—42no—
98C23—41no—
99C23—42no—
100C23—41no—
101C23—42no—
102C23—88no—
103C23—84no—
104C23—88no—
105C23—84no—
106C23—84no—
107C23#12011yescharacteristic
108Q8—82no—
109D4—42no—
110D4—44no—
111D4—81no—
112D4—84no—
113D4—82no—
114D4—88no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4C2×SD1611yescharacteristic
128C2×C4AGL(1,ℤ/8)11yescharacteristic
129C8—81no—
130C8—82no—
131C22—21no—
132C22—21no—
133C22—44no—
134C22—44no—
135C22—44no—
136C22—42no—
137C22—44no—
138C22—82no—
139C22—82no—
140C22—82no—
141C22—816no—
142C22—82no—
143C22—81no—
144C22—81no—
145C22—82no—
146C22#45011yescharacteristic
147C22#45511yescharacteristic
148C22B(4,𝔽2)11yescharacteristic
149C4—21no—
150C4—21no—
151C4—42no—
152C4—42no—
153C4—82no—
154C4—81no—
155C4—81no—
156C2—21no—
157C2—21no—
158C2—82no—
159C2—84no—
160C2#160111yescharacteristic
161C2#219611yescharacteristic
162C2#220211yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#2679C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2027C2semidirect
3. #1599C2semidirect
4. #1345C2semidirect
5. #1318C2semidirect
6. #544C22semidirect
15. #343C22semidirect
18. #324C22semidirect
65. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#26792
#20272
#15992
#13452
#13182