Finite Group Databasebeta

← #11855

Subgroups — #11855

Order 256.

162 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 57 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#3168C211yesmaximal, characteristic
2#2162C211yesmaximal, characteristic, complemented
3#1835C211yesmaximal, characteristic, complemented
4#1348C212yesmaximal, complemented
5#1347C212yesmaximal
6#529—21no—
7#499C411yescharacteristic, complemented
8#495C411yescharacteristic, complemented
9#488—21no—
10#467C2211yescharacteristic
11#466C2211yescharacteristic
12#465—21no—
13C4×Q16C411yescharacteristic, complemented
14C4×SD16—21no—
15C4×D8C411yescharacteristic, complemented
16C2×C4×C8C2211yescharacteristic
17#382—21no—
18C4⋊C16—21no—
19C22⋊C16C2212yes—
20C4×C16—21no—
21#340—21no—
22#340C2212yes—
23C2×Q16C2×C411yescharacteristic
24C2×SD16—21no—
25C2×D8C2×C411yescharacteristic
26C22×C8D411yescharacteristic
27C22×C8C2311yescharacteristic
28#126—41no—
29#125—21no—
30#124—21no—
31#123—41no—
32C22⋊Q8—21no—
33#121—21no—
34C4×Q8—21no—
35C4×D4—21no—
36C4×D4—41no—
37#117—41no—
38C2×C4⋊C4—22no—
39C2×C42D411yescharacteristic
40C2×C16—22no—
41#107C2×C411yescharacteristic
42#107C2×C411yescharacteristic
43#106—21no—
44#103—21no—
45#103—21no—
46#102—21no—
47#102—21no—
48C4×C8C2×C411yescharacteristic
49C4×C8C2×C411yescharacteristic
50C4×C8D411yescharacteristic
51C4×C8D411yescharacteristic
52#95—41no—
53#95—41no—
54C2×Q8—21no—
55C2×D4—21no—
56C2×D4—41no—
57C22×C4—41no—
58C22×C4—42no—
59C22×C4D811yescharacteristic
60C22×C4D811yescharacteristic
61C22×C4C2×D411yescharacteristic
62Q16—41no—
63SD16—42no—
64D8—41no—
65C2×C8—41no—
66C2×C8—41no—
67C2×C8C22⋊C411yescharacteristic
68C2×C8C22⋊C411yescharacteristic
69C2×C8C22×C411yescharacteristic
70C2×C8C2×D411yescharacteristic
71C4⋊C4—21no—
72C4⋊C4—21no—
73C4⋊C4—42no—
74C4⋊C4—42no—
75C4⋊C4—41no—
76C4⋊C4—41no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C22⋊C4—41no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C42—41no—
85C42C22⋊C411yescharacteristic
86C42C22⋊C411yescharacteristic
87C42SD1611yescharacteristic
88C42SD1611yescharacteristic
89C16—42no—
90C23—41no—
91C23C2×D811yescharacteristic
92Q8—41no—
93Q8—41no—
94D4—41no—
95D4—41no—
96D4—82no—
97C2×C4—22no—
98C2×C4—21no—
99C2×C4—21no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—82no—
105C2×C4—82no—
106C2×C4—82no—
107C2×C4—82no—
108C2×C4#10211yescharacteristic
109C2×C4#10211yescharacteristic
110C2×C4#10211yescharacteristic
111C2×C4#10211yescharacteristic
112C2×C4C2×C22⋊C411yescharacteristic
113C2×C4C2×SD1611yescharacteristic
114C8—21no—
115C8—21no—
116C8—21no—
117C8—21no—
118C22—21no—
119C22—21no—
120C22—81no—
121C22—81no—
122C22—81no—
123C22C2×#10211yescharacteristic
124C22#50812yes—
125C4—21no—
126C4—21no—
127C4—21no—
128C4—21no—
129C4—21no—
130C4—81no—
131C4—81no—
132C4—81no—
133C4—162no—
134C4#41011yescharacteristic
135C4#42111yescharacteristic
136C2—22no—
137C2—161no—
138C2#151711yescharacteristic
139C2#213811yescharacteristic
140C2#214511yescharacteristic
subgroupG/H
center, socleC22C2×#102
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radical#466C22
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2162C2semidirect
3. #1835C2semidirect
4. #1348C2semidirect
7. #499C4semidirect
8. #495C4semidirect
13. C4×Q16C4semidirect
15. C4×D8C4semidirect

Maximal subgroups

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

subgroupindex
#31682
#21622
#18352
#13482
#13472