Finite Group Databasebeta

← #12351

Subgroups — #12351

Order 256.

143 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 39 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#3352C211yesmaximal, characteristic
2#2245C211yesmaximal, characteristic, complemented
3#2209C211yesmaximal, characteristic, complemented
4#1689C211yesmaximal, characteristic
5#1363C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic
8#552—21no—
9C2×Q32—21no—
10#499C2211yescharacteristic
11#494C2211yescharacteristic
12#485—21no—
13#484—21no—
14#479—21no—
15#474—21no—
16#470—21no—
17C4×Q16C2211yescharacteristic
18#436C2211yescharacteristic
19#428—21no—
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×Q16D411yescharacteristic
30C2×M16—21no—
31C4⋊Q8—21no—
32#126—41no—
33#125—21no—
34#125—41no—
35#123—41no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C4×Q8—21no—
39C4×D4D411yescharacteristic
40#117—21no—
41#117—41no—
42C2×C4⋊C4—21no—
43Q32—42no—
44C2×C16—21no—
45C2×C16—21no—
46#107—21no—
47#107—41no—
48#107D411yescharacteristic
49#107D411yescharacteristic
50#107D411yescharacteristic
51#106—21no—
52#106—21no—
53C4⋊C8—21no—
54C4⋊C8D411yescharacteristic
55#103—21no—
56#103—21no—
57#103—21no—
58#102—41no—
59C22⋊C8—21no—
60C4×C8C2311yescharacteristic
61C2×Q8—21no—
62C2×Q8—41no—
63C2×D4D811yescharacteristic
64C22×C4—21no—
65C22×C4—41no—
66Q16—21no—
67Q16—21no—
68Q16—42no—
69Q16—41no—
70Q16—41no—
71M16—41no—
72M16—41no—
73C2×C8—21no—
74C2×C8—41no—
75C2×C8C2×D411yescharacteristic
76C2×C8C2×D411yescharacteristic
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—41no—
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—21no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C42—41no—
94C42—41no—
95C42C2×D411yescharacteristic
96C16—41no—
97C16—41no—
98C23—21no—
99Q8—41no—
100Q8—41no—
101Q8—81no—
102Q8—81no—
103D4—41no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—82no—
112C2×C4—81no—
113C2×C4#12011yescharacteristic
114C2×C4C2×D811yescharacteristic
115C2×C4AGL(1,ℤ/8)11yescharacteristic
116C8—21no—
117C8—41no—
118C8—81no—
119C8#13511yescharacteristic
120C8AGL(1,ℤ/8)11yescharacteristic
121C22—41no—
122C22—41no—
123C22—41no—
124C22#44711yescharacteristic
125C4—21no—
126C4—21no—
127C4—41no—
128C4—81no—
129C4—81no—
130C4—81no—
131C4—81no—
132C4—161no—
133C4#44911yescharacteristic
134C4#45411yescharacteristic
135C2—81no—
136C2#162511yescharacteristic
137C2#219111yescharacteristic
138C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radicalC2×D4D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2245C2semidirect
3. #2209C2semidirect
5. #1363C2semidirect
6. #1334C2semidirect

Maximal subgroups

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

subgroupindex
#33522
#22452
#22092
#16892
#13632
#13342
#13302