Finite Group Databasebeta

← #2008

Subgroups — #2008

Order 128.

190 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 44 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#575C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#535C211yesmaximal, characteristic, complemented
4#421C211yesmaximal, characteristic, complemented
5#418C211yesmaximal, characteristic, complemented
6#410C211yesmaximal, characteristic, complemented
7#344C211yesmaximal, characteristic, complemented
8C2×#41C2211yescharacteristic
9C2×#41C2211yescharacteristic
10C22×D4C2211yescharacteristic
11#137—21no—
12#137—21no—
13AGL(1,ℤ/8)—22no—
14AGL(1,ℤ/8)—21no—
15AGL(1,ℤ/8)—21no—
16AGL(1,ℤ/8)—22no—
17#135—22no—
18C2×SD16—21no—
19C2×D8—21no—
20C2×M16C2211yescharacteristic
21C2×M16C2211yescharacteristic
22#127—21no—
23#124—21no—
24#121—22no—
25#120—22no—
26#117C2211yescharacteristic
27#117C2211yescharacteristic
28#104—22no—
29#102—22no—
30#99—22no—
31C24—21no—
32#41—21no—
33#41—21no—
34#41—21no—
35#41—21no—
36#41—21no—
37#41—21no—
38#41—21no—
39#41—21no—
40C2×Q8D411yescharacteristic
41C2×Q8D411yescharacteristic, complemented
42C2×D4—21no—
43C2×D4—21no—
44C2×D4—24no—
45C2×D4—21no—
46C2×D4—21no—
47C2×D4—21no—
48C2×D4—21no—
49C2×D4—21no—
50C2×D4D412yes—
51C2×D4D411yescharacteristic
52C2×D4D411yescharacteristic, complemented
53C22×C4—21no—
54C22×C4—21no—
55C22×C4C2311yescharacteristic
56Q16—22no—
57SD16—24no—
58SD16—22no—
59SD16—22no—
60D8—24no—
61D8—22no—
62M16—21no—
63M16—21no—
64M16—21no—
65M16D412yescomplemented
66C2×C8—21no—
67C2×C8D412yescomplemented
68C4⋊C4—21no—
69C4⋊C4—21no—
70C22⋊C4—24no—
71C22⋊C4—21no—
72C22⋊C4D412yes—
73C42—21no—
74C42—21no—
75C23—22no—
76C23—21no—
77C23—21no—
78C23—41no—
79C23—41no—
80C23—82no—
81C23C2×D411yescharacteristic
82Q8—21no—
83Q8—21no—
84Q8—21no—
85Q8—21no—
86D4—21no—
87D4—22no—
88D4—21no—
89D4—21no—
90D4—22no—
91D4—21no—
92D4—41no—
93D4—41no—
94D4—41no—
95D4—41no—
96D4—41no—
97D4—41no—
98D4—84no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4C2×D411yescharacteristic
108C2×C4C2×D411yescharacteristic
109C2×C4C2×D411yescharacteristic
110C2×C4C2×D411yescharacteristic
111C2×C4C2×D411yescharacteristic
112C2×C4#4111yescharacteristic
113C8—22no—
114C8—22no—
115C22—41no—
116C22—42no—
117C22—44no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—81no—
122C22—82no—
123C22—82no—
124C22#12011yescharacteristic
125C22#12111yescharacteristic
126C22#12711yescharacteristic
127C4—42no—
128C4—41no—
129C4—41no—
130C4—81no—
131C4#12011yescharacteristic
132C4#12011yescharacteristic
133C4#12111yescharacteristic
134C4#12111yescharacteristic
135C2—21no—
136C2—21no—
137C2—21no—
138C2—81no—
139C2—82no—
140C2—81no—
141C2#39211yescharacteristic
subgroupG/H
center, socleC2#392
commutator, Frattini, cosocleC22×C4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #575C2semidirect
2. C2×AGL(1,ℤ/8)C2semidirect
3. #535C2semidirect
4. #421C2semidirect
5. #418C2semidirect
6. #410C2semidirect
7. #344C2semidirect
41. C2×Q8D4semidirect
52. C2×D4D4semidirect
65. M16D4semidirect
67. C2×C8D4semidirect

Maximal subgroups

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

subgroupindex
#5752
C2×AGL(1,ℤ/8)2
#5352
#4212
#4182
#4102
#3442