Finite Group Databasebeta

← #12284

Subgroups — #12284

Order 256.

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

normal subgroups only

This group has 169 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#3199C211yesmaximal, characteristic
2#2233C211yesmaximal, characteristic, complemented
3#2232C211yesmaximal, characteristic, complemented
4#2078C211yesmaximal, characteristic, complemented
5#1349C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic, complemented
7#1315C211yesmaximal, characteristic, complemented
8#535—21no—
9#482—21no—
10#482—21no—
11#469—21no—
12#466C2211yescharacteristic
13#466C2211yescharacteristic
14#447—21no—
15#447—21no—
16C2×#107C2211yescharacteristic, complemented
17#417—21no—
18#408C2211yescharacteristic
19#367—21no—
20#366—21no—
21#357—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic, complemented
24C22⋊C16C2211yescharacteristic, complemented
25#343—21no—
26#341C2211yescharacteristic, complemented
27C2×#41D411yescharacteristic
28C22×D4—21no—
29C2×D8—21no—
30C2×D8D411yescharacteristic
31C2×D8D411yescharacteristic
32C2×M16—21no—
33C2×M16D411yescharacteristic
34C22×C8C2311yescharacteristic
35#127—41no—
36#124—41no—
37#121—21no—
38#121—21no—
39#121—21no—
40#121—21no—
41#120—41no—
42#120—41no—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107D411yescharacteristic
50#107D411yescharacteristic
51#106—21no—
52#103—41no—
53#102—21no—
54#102—21no—
55#102—21no—
56#102—21no—
57#102—41no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C24—41no—
61#41—41no—
62#41—41no—
63C2×Q8D811yescharacteristic
64C2×D4—21no—
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—81no—
73C2×D4—81no—
74C2×D4D811yescharacteristic
75C22×C4—21no—
76C22×C4—41no—
77C22×C4C2×D411yescharacteristic
78D8—41no—
79D8—41no—
80D8—81no—
81M16—41no—
82M16—41no—
83M16—41no—
84C2×C8—21no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8C2×D411yescharacteristic
88C2×C8C2×D411yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C42—41no—
99C16—41no—
100C16—41no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—82no—
105C23—161no—
106C23—161no—
107C23AGL(1,ℤ/8)11yescharacteristic
108Q8—41no—
109D4—41no—
110D4—41no—
111D4—41no—
112D4—41no—
113D4—41no—
114D4—81no—
115D4—82no—
116D4—161no—
117D4—161no—
118D4—161no—
119D4—161no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—81no—
128C2×C4—161no—
129C2×C4#12011yescharacteristic
130C2×C4C2×D811yescharacteristic
131C8—21no—
132C8—21no—
133C8—41no—
134C8—81no—
135C22—21no—
136C22—21no—
137C22—21no—
138C22—41no—
139C22—41no—
140C22—41no—
141C22—81no—
142C22—81no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22—81no—
147C22—164no—
148C22#44711yescharacteristic
149C4—21no—
150C4—41no—
151C4—81no—
152C4—81no—
153C4—161no—
154C4#45511yescharacteristic
155C4#45811yescharacteristic
156C2—41no—
157C2—81no—
158C2—161no—
159C2—161no—
160C2#160711yescharacteristic
161C2#219211yescharacteristic
162C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3199C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ 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. #2078C2semidirect
5. #1349C2semidirect
6. #1349C2semidirect
7. #1315C2semidirect
16. C2×#107C22semidirect
23. C22⋊C16C22semidirect
24. C22⋊C16C22semidirect
26. #341C22semidirect

Maximal subgroups

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

subgroupindex
#31992
#22332
#22322
#20782
#13492
#13152