Finite Group Databasebeta

← #12166

Subgroups — #12166

Order 256.

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

normal subgroups only

This group has 181 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#3262C211yesmaximal, characteristic, complemented
2#2187C211yesmaximal, characteristic, complemented
3#2184C211yesmaximal, characteristic
4#2045C211yesmaximal, characteristic, complemented
5#1350C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic, complemented
7#1323C211yesmaximal, characteristic, complemented
8C22×D8C2211yescharacteristic
9#542—21no—
10C2×SD32—21no—
11C2×D16—21no—
12#477—21no—
13#474—21no—
14#468—21no—
15#467C2211yescharacteristic
16#466C2211yescharacteristic
17#464—21no—
18#463—21no—
19#424C2211yescharacteristic
20#418—21no—
21C2×#100—21no—
22#357—21no—
23#357—21no—
24C22⋊C16C2211yescharacteristic
25C22⋊C16C2211yescharacteristic
26#341C2211yescharacteristic
27C22×D4—21no—
28C2×Q16D411yescharacteristic
29C2×SD16—21no—
30C2×D8—21no—
31C2×D8—21no—
32C2×D8—42no—
33C2×D8D411yescharacteristic
34C2×M16—21no—
35C2×M16D411yescharacteristic
36C22×C8C2311yescharacteristic
37C4⋊Q8—21no—
38#126—41no—
39#124—21no—
40#124—41no—
41#123—41no—
42C22⋊Q8—21no—
43C22⋊Q8—41no—
44#121—21no—
45C4×Q8—21no—
46C4×D4—21no—
47#117D411yescharacteristic
48SD32—42no—
49D16—42no—
50C2×C16—21no—
51C2×C16—21no—
52#107—21no—
53#107D411yescharacteristic
54#107D411yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57#103—21no—
58#103—21no—
59#102—21no—
60#102—21no—
61#102—41no—
62#102—41no—
63#100—42no—
64C24—41no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77Q16—41no—
78SD16—81no—
79D8—22no—
80D8—22no—
81D8—41no—
82D8—41no—
83D8—41no—
84M16—41no—
85M16—41no—
86M16—41no—
87C2×C8—21no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8C2×D411yescharacteristic
91C2×C8C2×D411yescharacteristic
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4D811yescharacteristic
98C4⋊C4D811yescharacteristic
99C22⋊C4—21no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C42—21no—
106C42—41no—
107C16—41no—
108C16—41no—
109C23—42no—
110C23—41no—
111C23—81no—
112C23—81no—
113C23—161no—
114C23—161no—
115C23AGL(1,ℤ/8)11yescharacteristic
116Q8—41no—
117Q8—41no—
118Q8—161no—
119D4—42no—
120D4—41no—
121D4—42no—
122D4—41no—
123D4—81no—
124D4—81no—
125D4—161no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—161no—
134C2×C4#12011yescharacteristic
135C2×C4C2×D811yescharacteristic
136C8—21no—
137C8—21no—
138C8—41no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—82no—
144C22—81no—
145C22—81no—
146C22—81no—
147C22—162no—
148C22—162no—
149C22—161no—
150C22—161no—
151C22#44711yescharacteristic
152C4—21no—
153C4—41no—
154C4—41no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4#45411yescharacteristic
159C4B(4,𝔽2)11yescharacteristic
160C2—41no—
161C2—161no—
162C2—162no—
163C2#159811yescharacteristic
164C2#219211yescharacteristic
165C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#2184C2
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
1. #3262C2semidirect
2. #2187C2semidirect
4. #2045C2semidirect
5. #1350C2semidirect
6. #1349C2semidirect
7. #1323C2semidirect

Maximal subgroups

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

subgroupindex
#32622
#21872
#21842
#20452
#13502
#13492
#13232