Finite Group Databasebeta

← #13008

Subgroups — #13008

Order 256.

258 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 73 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#3054C211yesmaximal, characteristic
2#2911C211yesmaximal, characteristic, complemented
3#2145C212yesmaximal, complemented
4C2×#349C211yesmaximal, characteristic, complemented
5#1384C212yesmaximal, complemented
6C2×#135C2211yescharacteristic
7C22×M16C2211yescharacteristic
8#525—21no—
9C2×#117—21no—
10C2×M32C2212yes—
11#470C412yescomplemented
12#469C412yescomplemented
13#468—22no—
14#428—24no—
15#427—21no—
16#427C2211yescharacteristic
17#427C2212yes—
18#416—21no—
19#360—24no—
20#349—22no—
21C2×#41—21no—
22C23×C4D411yescharacteristic
23#135—22no—
24#135—22no—
25C2×Q16C2×C411yescharacteristic, complemented
26C2×SD16—21no—
27C2×D8C2×C411yescharacteristic, complemented
28C2×M16—22no—
29C2×M16—22no—
30C2×M16C2×C412yes—
31C2×M16D412yes—
32C22×C8D411yescharacteristic
33C22×C8C2311yescharacteristic
34#123—41no—
35C22⋊Q8—22no—
36#121—22no—
37#120—41no—
38C4×D4—42no—
39#117—22no—
40#117—22no—
41#117—21no—
42#117—41no—
43#117—41no—
44C2×C4⋊C4—22no—
45C2×C22⋊C4—41no—
46C2×C42—41no—
47M32—22no—
48M32—22no—
49C2×C16—22no—
50#107—22no—
51#107—22no—
52#107—21no—
53#106—22no—
54#106C2×C412yescomplemented
55#106D414yescomplemented
56#103—22no—
57#102—22no—
58C24D811yescharacteristic
59#41—41no—
60#41—41no—
61#41—41no—
62#41—41no—
63C2×Q8—21no—
64C2×D4—21no—
65C2×D4—42no—
66C2×D4—41no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—42no—
72C22×C4—41no—
73C22×C4C22⋊C412yes—
74C22×C4D811yescharacteristic
75C22×C4SD1612yes—
76C22×C4C2×D411yescharacteristic
77Q16—22no—
78SD16—42no—
79D8—22no—
80M16—24no—
81M16—24no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8C22⋊C412yes—
85C2×C8C22×C411yescharacteristic
86C2×C8C2×D412yes—
87C2×C8C2×D411yescharacteristic
88C2×C8#4112yes—
89C4⋊C4—22no—
90C4⋊C4—24no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—41no—
97C22⋊C4—42no—
98C42—42no—
99C42—41no—
100C42—42no—
101C16—42no—
102C23—21no—
103C23—21no—
104C23—41no—
105C23—81no—
106C23#10212yes—
107C23C2×D811yescharacteristic
108Q8—41no—
109Q8—41no—
110D4—41no—
111D4—41no—
112D4—84no—
113D4—81no—
114D4—81no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—44no—
124C2×C4—84no—
125C2×C4—81no—
126C2×C4—81no—
127C2×C4#10212yes—
128C2×C4C2×C22⋊C411yescharacteristic
129C2×C4#12011yescharacteristic
130C2×C4#12311yescharacteristic
131C2×C4C2×SD1611yescharacteristic
132C2×C4AGL(1,ℤ/8)11yescharacteristic
133C2×C4#13711yescharacteristic
134C8—22no—
135C8C4×D412yes—
136C8#12112yes—
137C22—22no—
138C22—41no—
139C22—42no—
140C22—42no—
141C22—81no—
142C22—82no—
143C22—81no—
144C22C2×#10211yescharacteristic
145C22#44711yescharacteristic
146C22#48011yescharacteristic
147C4—22no—
148C4—84no—
149C4—82no—
150C4—81no—
151C4#38611yescharacteristic
152C4#41711yescharacteristic
153C4#44811yescharacteristic
154C4#48111yescharacteristic
155C2—21no—
156C2—21no—
157C2—21no—
158C2—42no—
159C2—161no—
160C2#187411yescharacteristic
subgroupG/H
centerC4#481
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
socleC2#1874
prime radical#3054C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2911C2semidirect
3. #2145C2semidirect
4. C2×#349C2semidirect
5. #1384C2semidirect
11. #470C4semidirect
12. #469C4semidirect
25. C2×Q16C2×C4semidirect
27. C2×D8C2×C4semidirect
54. #106C2×C4semidirect
55. #106D4semidirect

Maximal subgroups

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

subgroupindex
#30542
#29112
#21452
C2×#3492
#13842