Finite Group Databasebeta

← #12294

Subgroups — #12294

Order 256.

151 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 37 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#3259C211yesmaximal, characteristic
2#2233C211yesmaximal, characteristic, complemented
3#2232C211yesmaximal, characteristic, complemented
4#2089C211yesmaximal, characteristic, complemented
5#1349C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic, complemented
7#1323C211yesmaximal, characteristic, complemented
8#540—21no—
9#478—21no—
10#478—21no—
11#469—21no—
12#466C2211yescharacteristic
13#466C2211yescharacteristic
14#459—21no—
15#459—21no—
16#428—21no—
17C2×#107C2211yescharacteristic
18#424C2211yescharacteristic
19#367—21no—
20#366—21no—
21#357—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#343—21no—
26#341C2211yescharacteristic
27C2×D8—21no—
28C2×D8D411yescharacteristic
29C2×D8D411yescharacteristic
30C2×M16—21no—
31C2×M16D411yescharacteristic
32C22×C8C2311yescharacteristic
33#127—21no—
34#125—21no—
35#123—41no—
36#123—41no—
37#121—21no—
38#121—21no—
39#121—41no—
40#121—41no—
41C4×D4—21no—
42C4×D4—21no—
43#117D411yescharacteristic
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107—41no—
50#107D411yescharacteristic
51#107D411yescharacteristic
52#106—21no—
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56#102—21no—
57#102—21no—
58#102—21no—
59#102—21no—
60C2×D4—21no—
61C2×D4—21no—
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4C2×D411yescharacteristic
70D8—41no—
71D8—41no—
72D8—81no—
73M16—22no—
74M16—41no—
75M16—41no—
76C2×C8—21no—
77C2×C8—41no—
78C2×C8—41no—
79C2×C8C2×D411yescharacteristic
80C2×C8C2×D411yescharacteristic
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4D811yescharacteristic
88C4⋊C4D811yescharacteristic
89C22⋊C4—21no—
90C22⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42—21no—
95C16—41no—
96C16—41no—
97C23—41no—
98C23—41no—
99C23AGL(1,ℤ/8)11yescharacteristic
100D4—41no—
101D4—41no—
102D4—41no—
103D4—41no—
104D4—161no—
105D4—161no—
106D4—161no—
107D4—161no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—81no—
115C2×C4—161no—
116C2×C4—161no—
117C2×C4—161no—
118C2×C4#12011yescharacteristic
119C2×C4C2×D811yescharacteristic
120C8—21no—
121C8—21no—
122C8—42no—
123C8—41no—
124C22—21no—
125C22—21no—
126C22—21no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22—81no—
132C22—81no—
133C22#44711yescharacteristic
134C4—21no—
135C4—41no—
136C4—41no—
137C4—81no—
138C4—81no—
139C4—161no—
140C4#45611yescharacteristic
141C4#45811yescharacteristic
142C2—41no—
143C2—161no—
144C2—161no—
145C2#161011yescharacteristic
146C2#219211yescharacteristic
147C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3259C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2233C2semidirect
3. #2232C2semidirect
4. #2089C2semidirect
5. #1349C2semidirect
6. #1349C2semidirect
7. #1323C2semidirect

Maximal subgroups

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

subgroupindex
#32592
#22332
#22322
#20892
#13492
#13232