Finite Group Databasebeta

← #11860

Subgroups — #11860

Order 256.

168 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 59 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#2163C211yesmaximal, characteristic, complemented
3#1834C211yesmaximal, 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#384—21no—
18C4⋊C16—21no—
19C22⋊C16C2212yes—
20#346—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×C4⋊C4—41no—
40C2×C4⋊C4—41no—
41C2×C42D411yescharacteristic
42C2×C16—22no—
43#107C2×C411yescharacteristic
44#107C2×C411yescharacteristic
45#106—21no—
46#103—21no—
47#103—21no—
48#102—21no—
49#102—21no—
50C4×C8C2×C411yescharacteristic
51C4×C8C2×C411yescharacteristic
52C4×C8D411yescharacteristic
53C4×C8D411yescharacteristic
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—
79C4⋊C4—82no—
80C4⋊C4—82no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C42—41no—
87C42C22⋊C411yescharacteristic
88C42C22⋊C411yescharacteristic
89C42SD1611yescharacteristic
90C42SD1611yescharacteristic
91C16—42no—
92C23—41no—
93C23C2×D811yescharacteristic
94Q8—41no—
95Q8—41no—
96D4—41no—
97D4—41no—
98D4—81no—
99D4—81no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—82no—
109C2×C4—82no—
110C2×C4—82no—
111C2×C4—82no—
112C2×C4#10211yescharacteristic
113C2×C4#10211yescharacteristic
114C2×C4#10211yescharacteristic
115C2×C4#10211yescharacteristic
116C2×C4C2×C22⋊C411yescharacteristic
117C2×C4C2×SD1611yescharacteristic
118C8—21no—
119C8—21no—
120C8—21no—
121C8—21no—
122C22—21no—
123C22—21no—
124C22—81no—
125C22—81no—
126C22—81no—
127C22C2×#10211yescharacteristic
128C22#50911yescharacteristic
129C22#51011yescharacteristic
130C4—21no—
131C4—21no—
132C4—21no—
133C4—21no—
134C4—81no—
135C4—81no—
136C4—81no—
137C4—162no—
138C4#36012yes—
139C4#41011yescharacteristic
140C4#42111yescharacteristic
141C2—21no—
142C2—21no—
143C2—161no—
144C2#151711yescharacteristic
145C2#213911yescharacteristic
146C2C2×#36011yescharacteristic
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. #2163C2semidirect
3. #1834C2semidirect
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
#21632
#18342
#13482
#13472