Finite Group Databasebeta

← #12350

Subgroups — #12350

Order 256.

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

normal subgroups only

This group has 149 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#3355C211yesmaximal, characteristic
2#2245C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic, complemented
4#1689C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1333C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8#553—21no—
9C2×SD32—21no—
10#499C2211yescharacteristic
11#494C2211yescharacteristic
12#482—21no—
13#480—21no—
14#478—21no—
15#474—21no—
16#469—21no—
17C4×D8C2211yescharacteristic
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×D8D411yescharacteristic
30C2×M16—21no—
31C4⋊Q8—21no—
32#126—41no—
33#125—21no—
34#123—41no—
35#121—21no—
36#121—21no—
37#121—41no—
38C4×D4—21no—
39C4×D4—41no—
40C4×D4D411yescharacteristic
41#117—21no—
42C2×C4⋊C4—21no—
43SD32—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#102—21no—
56#102—21no—
57#102—21no—
58#102—41no—
59C22⋊C8—21no—
60C4×C8C2311yescharacteristic
61C2×Q8—41no—
62C2×D4—21no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4D811yescharacteristic
66C22×C4—21no—
67C22×C4—41no—
68C22×C4—41no—
69Q16—42no—
70Q16—41no—
71Q16—41no—
72D8—21no—
73D8—21no—
74M16—41no—
75M16—41no—
76C2×C8—21no—
77C2×C8—41no—
78C2×C8C2×D411yescharacteristic
79C2×C8C2×D411yescharacteristic
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—21no—
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—
93C22⋊C4—41no—
94C42—41no—
95C42C2×D411yescharacteristic
96C16—41no—
97C16—41no—
98C23—21no—
99C23—41no—
100Q8—81no—
101Q8—81no—
102D4—41no—
103D4—41no—
104D4—41no—
105D4—82no—
106D4—161no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—81no—
114C2×C4—161no—
115C2×C4—161no—
116C2×C4#12011yescharacteristic
117C2×C4C2×D811yescharacteristic
118C2×C4AGL(1,ℤ/8)11yescharacteristic
119C8—21no—
120C8—41no—
121C8—81no—
122C8#13511yescharacteristic
123C8AGL(1,ℤ/8)11yescharacteristic
124C22—41no—
125C22—41no—
126C22—41no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22#44711yescharacteristic
131C4—21no—
132C4—21no—
133C4—41no—
134C4—81no—
135C4—81no—
136C4—81no—
137C4—161no—
138C4#44911yescharacteristic
139C4#45411yescharacteristic
140C2—81no—
141C2—161no—
142C2#162511yescharacteristic
143C2#219111yescharacteristic
144C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3355C2
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. #2208C2semidirect
4. #1689C2semidirect
5. #1363C2semidirect
6. #1333C2semidirect
7. #1330C2semidirect

Maximal subgroups

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

subgroupindex
#33552
#22452
#22082
#16892
#13632
#13332
#13302