Finite Group Databasebeta

← #12362

Subgroups — #12362

Order 256.

146 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 41 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#3389C211yesmaximal, characteristic
2#2245C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic
4#1690C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1338C211yesmaximal, characteristic, complemented
7#1333C211yesmaximal, characteristic
8#554—21no—
9C2×SD32—21no—
10#501—21no—
11#499C2211yescharacteristic
12#494C2211yescharacteristic
13#478—21no—
14#478—21no—
15#475—21no—
16#474—21no—
17#446C2211yescharacteristic
18#442—21no—
19C4×D8C2211yescharacteristic
20C4⋊C16C2211yescharacteristic
21#358—21no—
22#358—21no—
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×Q16—21no—
28C2×Q16—21no—
29C2×D8D411yescharacteristic
30C4⋊Q8—21no—
31C4⋊Q8—21no—
32#125—21no—
33#125—21no—
34C22⋊Q8—41no—
35C22⋊Q8—41no—
36C22⋊Q8—41no—
37C4×Q8D411yescharacteristic
38C4×D4—21no—
39C4×D4—21no—
40C4×D4—21no—
41C2×C4⋊C4—41no—
42SD32—42no—
43C2×C16—21no—
44C2×C16—21no—
45#107—21no—
46#107D411yescharacteristic
47#107D411yescharacteristic
48#107D411yescharacteristic
49#106—21no—
50#106—21no—
51#106—41no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8D411yescharacteristic
55#103—41no—
56#102—21no—
57#102—21no—
58#102—21no—
59#97—21no—
60C4×C8C2311yescharacteristic
61C2×Q8—41no—
62C2×Q8—41no—
63C2×Q8D811yescharacteristic
64C2×D4—21no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68Q16—42no—
69Q16—41no—
70Q16—41no—
71D8—21no—
72D8SD1612yes—
73C2×C8—21no—
74C2×C8—41no—
75C2×C8C2×D411yescharacteristic
76C2×C8C2×D411yescharacteristic
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4D811yescharacteristic
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C42—21no—
93C42C2×D411yescharacteristic
94C16—41no—
95C16—41no—
96C23—41no—
97Q8—41no—
98Q8—81no—
99Q8—81no—
100Q8—161no—
101D4—22no—
102D4—41no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—82no—
109C2×C4—81no—
110C2×C4—161no—
111C2×C4—161no—
112C2×C4#12011yescharacteristic
113C2×C4C2×D811yescharacteristic
114C2×C4AGL(1,ℤ/8)11yescharacteristic
115C8—21no—
116C8—41no—
117C8—81no—
118C8C2×SD1611yescharacteristic
119C8#13711yescharacteristic
120C22—42no—
121C22—81no—
122C22—81no—
123C22#44711yescharacteristic
124C4—21no—
125C4—21no—
126C4—41no—
127C4—41no—
128C4—81no—
129C4—81no—
130C4—81no—
131C4—161no—
132C4#44811yescharacteristic
133C4#45411yescharacteristic
134C2—82no—
135C2#162711yescharacteristic
136C2#219111yescharacteristic
137C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radicalC2×D8D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2245C2semidirect
4. #1690C2semidirect
5. #1363C2semidirect
6. #1338C2semidirect

Maximal subgroups

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

subgroupindex
#33892
#22452
#22082
#16902
#13632
#13382
#13332