Finite Group Databasebeta

← #1600

Subgroups — #1600

Order 128.

160 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 32 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#538C211yesmaximal, characteristic, complemented
2#450C211yesmaximal, characteristic, complemented
3#447C211yesmaximal, characteristic, complemented
4#392C211yesmaximal, characteristic, complemented
5#328C211yesmaximal, characteristic, complemented
6#327C211yesmaximal, characteristic, complemented
7#323C211yesmaximal, characteristic, complemented
8C22×D4—21no—
9C22×D4C2211yescharacteristic
10C2×SD16—21no—
11C2×D8—21no—
12#126—21no—
13#124—21no—
14#123—21no—
15C22⋊Q8C2211yescharacteristic
16#121—21no—
17#121C2211yescharacteristic
18#120—21no—
19C4×D4—21no—
20C2×C22⋊C4—21no—
21C2×C22⋊C4C2211yescharacteristic
22#102—21no—
23#102—21no—
24C22⋊C8C2211yescharacteristic
25C22⋊C8C2211yescharacteristic
26#95C2211yescharacteristic
27C24—21no—
28C24D411yescharacteristic
29C2×Q8D411yescharacteristic, complemented
30C2×D4—21no—
31C2×D4—21no—
32C2×D4—21no—
33C2×D4—21no—
34C2×D4—21no—
35C2×D4—42no—
36C2×D4—42no—
37C2×D4—42no—
38C2×D4—42no—
39C2×D4D411yescharacteristic, complemented
40C22×C4—21no—
41C22×C4—21no—
42C22×C4D411yescharacteristic
43C22×C4C2311yescharacteristic
44SD16—42no—
45D8—42no—
46C2×C8—21no—
47C2×C8—21no—
48C4⋊C4—21no—
49C4⋊C4—21no—
50C4⋊C4D411yescharacteristic, complemented
51C4⋊C4D411yescharacteristic, complemented
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—21no—
57C22⋊C4—21no—
58C22⋊C4—21no—
59C22⋊C4—42no—
60C42—21no—
61C23—22no—
62C23—21no—
63C23—21no—
64C23—21no—
65C23—21no—
66C23—41no—
67C23—41no—
68C23—41no—
69C23—41no—
70C23—41no—
71C23—41no—
72C23—41no—
73C23—41no—
74C23C2×D411yescharacteristic
75Q8—41no—
76D4—22no—
77D4—22no—
78D4—41no—
79D4—41no—
80D4—41no—
81D4—48no—
82D4—41no—
83D4—41no—
84D4—81no—
85D4—81no—
86C2×C4—22no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—81no—
96C2×C4C2×D411yescharacteristic
97C2×C4C2×D411yescharacteristic
98C8—41no—
99C8—41no—
100C22—21no—
101C22—21no—
102C22—42no—
103C22—42no—
104C22—42no—
105C22—41no—
106C22—41no—
107C22—41no—
108C22—41no—
109C22—42no—
110C22—41no—
111C22—41no—
112C22—81no—
113C22—81no—
114C22—81no—
115C22—81no—
116C22—81no—
117C22#12011yescharacteristic
118C22#13511yescharacteristic
119C22AGL(1,ℤ/8)11yescharacteristic
120C4—21no—
121C4—21no—
122C4—42no—
123C4—41no—
124C4—41no—
125C4—41no—
126C4—41no—
127C2—21no—
128C2—21no—
129C2—81no—
130C2—81no—
131C2—82no—
132C2#44911yescharacteristic
133C2#45311yescharacteristic
134C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC22×C4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #538C2semidirect
2. #450C2semidirect
3. #447C2semidirect
4. #392C2semidirect
5. #328C2semidirect
6. #327C2semidirect
7. #323C2semidirect
29. C2×Q8D4semidirect
39. C2×D4D4semidirect
50. C4⋊C4D4semidirect
51. C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#5382
#4502
#4472
#3922
#3282
#3272
#3232