Finite Group Databasebeta

← #2009

Subgroups — #2009

Order 128.

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

normal subgroups only

This group has 174 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
2C2×#137C211yesmaximal, characteristic, complemented
3#536C211yesmaximal, characteristic, complemented
4#421C211yesmaximal, characteristic, complemented
5#419C211yesmaximal, characteristic, complemented
6#410C211yesmaximal, characteristic, complemented
7#344C211yesmaximal, characteristic, complemented
8C2×#41C2211yescharacteristic
9C2×#41C2211yescharacteristic
10C22×Q8C2211yescharacteristic
11#137—22no—
12#137—21no—
13#137—21no—
14#137—22no—
15AGL(1,ℤ/8)—21no—
16AGL(1,ℤ/8)—21no—
17#135—22no—
18C2×Q16—21no—
19C2×SD16—21no—
20C2×M16C2211yescharacteristic
21C2×M16C2211yescharacteristic
22C4⋊Q8—21no—
23#124—21no—
24#123—22no—
25C22⋊Q8—22no—
26#117C2211yescharacteristic
27#117C2211yescharacteristic
28#104—22no—
29#103—22no—
30#99—22no—
31#41—21no—
32#41—21no—
33#41—21no—
34#41—21no—
35#41—21no—
36#41—21no—
37#41—21no—
38#41—21no—
39C2×Q8—21no—
40C2×Q8—21no—
41C2×Q8—21no—
42C2×Q8—21no—
43C2×Q8—21no—
44C2×Q8D412yes—
45C2×Q8D411yescharacteristic
46C2×Q8D411yescharacteristic
47C2×D4—21no—
48C2×D4—21no—
49C2×D4D411yescharacteristic
50C2×D4D411yescharacteristic
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4C2311yescharacteristic
55Q16—24no—
56Q16—22no—
57SD16—24no—
58SD16—22no—
59SD16—22no—
60D8—22no—
61M16—21no—
62M16—21no—
63M16—21no—
64M16D412yes—
65C2×C8—21no—
66C2×C8D412yes—
67C4⋊C4—24no—
68C4⋊C4—21no—
69C4⋊C4—21no—
70C22⋊C4—24no—
71C22⋊C4—21no—
72C22⋊C4D412yes—
73C42—21no—
74C42—21no—
75C23—21no—
76C23—21no—
77C23C2×D411yescharacteristic
78Q8—22no—
79Q8—21no—
80Q8—21no—
81Q8—21no—
82Q8—22no—
83Q8—21no—
84Q8—41no—
85Q8—41no—
86D4—21no—
87D4—21no—
88D4—21no—
89D4—21no—
90D4—41no—
91D4—41no—
92D4—41no—
93D4—41no—
94C2×C4—22no—
95C2×C4—22no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—41no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4C2×D411yescharacteristic
106C2×C4C2×D411yescharacteristic
107C2×C4C2×D411yescharacteristic
108C2×C4C2×D411yescharacteristic
109C2×C4C2×D411yescharacteristic
110C2×C4#4111yescharacteristic
111C8—22no—
112C8—22no—
113C22—41no—
114C22—42no—
115C22—41no—
116C22—41no—
117C22—81no—
118C22#12011yescharacteristic
119C22#12111yescharacteristic
120C22#12711yescharacteristic
121C4—42no—
122C4—41no—
123C4—41no—
124C4—42no—
125C4—81no—
126C4#12011yescharacteristic
127C4#12011yescharacteristic
128C4#12111yescharacteristic
129C4#12111yescharacteristic
130C2—21no—
131C2—21no—
132C2—21no—
133C2—81no—
134C2—81no—
135C2#39211yescharacteristic
subgroupG/H
center, socleC2#392
commutator, Frattini, cosocleC22×C4C23
prime radical#575C2
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
2. C2×#137C2semidirect
3. #536C2semidirect
4. #421C2semidirect
5. #419C2semidirect
6. #410C2semidirect
7. #344C2semidirect

Maximal subgroups

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

subgroupindex
#5752
C2×#1372
#5362
#4212
#4192
#4102
#3442