Finite Group Databasebeta

← #13603

Subgroups — #13603

Order 256.

167 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 45 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#3140C211yesmaximal, characteristic
2C2×#355C211yesmaximal, characteristic
3C2×#354C211yesmaximal, characteristic
4#1414C214yesmaximal, complemented
5#533C411yescharacteristic, complemented
6C2×#124C2211yescharacteristic
7#489—21no—
8#489—21no—
9#488C412yescomplemented
10#488C2212yes—
11#419—21no—
12#418—21no—
13C2×#101—21no—
14C2×#99—21no—
15C4×M16C411yescharacteristic, complemented
16#355—21no—
17#355C2212yes—
18#354—21no—
19#354C2212yes—
20C22×Q8D411yescharacteristic
21C22×D4D411yescharacteristic
22C2×M16—21no—
23C2×M16—41no—
24C4⋊Q8—21no—
25#125C2×C412yescomplemented
26#124—41no—
27#124C2×C411yescharacteristic
28#124D411yescharacteristic
29#124D411yescharacteristic
30#124C2311yescharacteristic
31C22⋊Q8—41no—
32C4×Q8—41no—
33#117—21no—
34C2×C22⋊C4—41no—
35C2×C22⋊C4—41no—
36C2×C42C2×C411yescharacteristic
37#103—22no—
38#103—22no—
39#102—22no—
40#102—22no—
41#101—22no—
42#101—41no—
43#99—22no—
44#99—41no—
45#97—21no—
46C4×C8C2×C412yescomplemented
47C24—21no—
48C2×Q8—21no—
49C2×Q8—41no—
50C2×Q8—41no—
51C2×Q8—41no—
52C2×Q8C22⋊C411yescharacteristic
53C2×Q8C2×D411yescharacteristic
54C2×D4—21no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4C22⋊C411yescharacteristic
58C2×D4C2×D411yescharacteristic
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—81no—
62C22×C4C22⋊C411yescharacteristic
63M16—41no—
64M16—41no—
65M16—42no—
66M16—81no—
67C2×C8—22no—
68C2×C8—81no—
69C4⋊C4—22no—
70C4⋊C4—42no—
71C4⋊C4—41no—
72C22⋊C4—41no—
73C22⋊C4—41no—
74C22⋊C4—41no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C22⋊C4—42no—
78C22⋊C4—81no—
79C42—21no—
80C42—41no—
81C42C22⋊C411yescharacteristic
82C42C22×C411yescharacteristic
83C23—21no—
84C23—21no—
85C23—41no—
86C23—41no—
87C23—82no—
88C23#9911yescharacteristic
89Q8—41no—
90Q8—41no—
91Q8—81no—
92Q8—82no—
93D4—41no—
94D4—41no—
95D4—81no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—81no—
106C2×C4—82no—
107C2×C4—162no—
108C2×C4#9911yescharacteristic
109C2×C4C2×C22⋊C411yescharacteristic
110C8—42no—
111C8—82no—
112C22—41no—
113C22—41no—
114C22—41no—
115C22—81no—
116C22—81no—
117C22—82no—
118C22—161no—
119C22#35411yescharacteristic
120C22#35411yescharacteristic
121C22C2×#9911yescharacteristic
122C4—21no—
123C4—21no—
124C4—41no—
125C4—41no—
126C4—41no—
127C4—41no—
128C4—82no—
129C4—162no—
130C2—21no—
131C2—21no—
132C2—21no—
133C2—81no—
134C2—81no—
135C2C2×#35411yescharacteristic
subgroupG/H
center, socleC2C2×#354
commutatorC42C22×C4
Frattini, cosocle#124C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1414C2semidirect
5. #533C4semidirect
9. #488C4semidirect
15. C4×M16C4semidirect
25. #125C2×C4semidirect
46. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31402
C2×#3552
C2×#3542
#14142