Finite Group Databasebeta

← #12348

Subgroups — #12348

Order 256.

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

normal subgroups only

This group has 157 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, complemented
2#2244C211yesmaximal, characteristic, complemented
3#2206C211yesmaximal, characteristic, complemented
4#1687C211yesmaximal, characteristic, complemented
5#1363C211yesmaximal, characteristic, complemented
6#1333C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8#553—21no—
9C2×D16—21no—
10#499C2211yescharacteristic
11#493C2211yescharacteristic
12#482—21no—
13#480—21no—
14#478—21no—
15#469—21no—
16#459—21no—
17C4×D8C2211yescharacteristic
18#436C2211yescharacteristic
19#428—21no—
20C4⋊C16C2211yescharacteristic
21#357—21no—
22#357—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#331—21no—
27C2×D8—21no—
28C2×D8—21no—
29C2×D8—41no—
30C2×D8D411yescharacteristic
31C2×M16—21no—
32#127—21no—
33#126—41no—
34#125—21no—
35#123—41no—
36#121—21no—
37#121—21no—
38#121—41no—
39C4×D4—21no—
40C4×D4—41no—
41C4×D4D411yescharacteristic
42#117—21no—
43C2×C4⋊C4—21no—
44D16—42no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
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×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4D811yescharacteristic
68C22×C4—21no—
69C22×C4—41no—
70C22×C4—41no—
71D8—21no—
72D8—21no—
73D8—42no—
74D8—41no—
75D8—41no—
76D8—82no—
77M16—41no—
78M16—41no—
79C2×C8—21no—
80C2×C8—41no—
81C2×C8C2×D411yescharacteristic
82C2×C8C2×D411yescharacteristic
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4D811yescharacteristic
90C22⋊C4—21no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C42—41no—
96C42C2×D411yescharacteristic
97C16—41no—
98C16—41no—
99C23—21no—
100C23—41no—
101C23—81no—
102D4—41no—
103D4—41no—
104D4—41no—
105D4—81no—
106D4—81no—
107D4—82no—
108D4—81no—
109D4—81no—
110D4—161no—
111D4—161no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—161no—
119C2×C4—161no—
120C2×C4#12011yescharacteristic
121C2×C4C2×D811yescharacteristic
122C2×C4AGL(1,ℤ/8)11yescharacteristic
123C8—21no—
124C8—41no—
125C8—81no—
126C8#13511yescharacteristic
127C8AGL(1,ℤ/8)11yescharacteristic
128C22—41no—
129C22—41no—
130C22—41no—
131C22—81no—
132C22—81no—
133C22—81no—
134C22—161no—
135C22—161no—
136C22—161no—
137C22#44711yescharacteristic
138C4—21no—
139C4—21no—
140C4—41no—
141C4—81no—
142C4—81no—
143C4—81no—
144C4#44911yescharacteristic
145C4#45411yescharacteristic
146C2—81no—
147C2—161no—
148C2—321no—
149C2#162511yescharacteristic
150C2#219011yescharacteristic
151C2#219211yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
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
1. #3355C2semidirect
2. #2244C2semidirect
3. #2206C2semidirect
4. #1687C2semidirect
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
#22442
#22062
#16872
#13632
#13332
#13302