Finite Group Databasebeta

← #27170248

Subgroups — #27170248

Order 12928.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265549C211yesmaximal, characteristic, complemented
2#20265467C212yesmaximal, complemented
3#20264921C211yesmaximal, characteristic, complemented
4D4⋊#11717579C212yesmaximal, complemented
5D42×C101C211yesmaximal, characteristic
6C2×#13108062—22no—
7#15692255C2211yescharacteristic, complemented
8#15692186—22no—
9C4⋊#11717579C2212yescomplemented
10#15692176—22no—
11C4×#11717579C2211yescharacteristic, complemented
12C22×D4×C101—22no—
13#127×C101C2211yescharacteristic
14#121×C101—22no—
15#120×C101—22no—
16C4×D4×C101C2212yes—
17C2×Dic202—21no—
18#13108062—44no—
19C2×#11717579—21no—
20C2×#11717579—22no—
21C4⋊Dic101—22no—
22C24×C101—22no—
23C2×D4×C101—24no—
24C2×D4×C101—22no—
25C2×D4×C101—44no—
26C2×D4×C101—44no—
27C2×D4×C101D412yes—
28C2×D4×C101D412yes—
29C22×C4×C101—22no—
30C4⋊C4×C101D412yes—
31C22⋊C4×C101—22no—
32C22⋊C4×C101—22no—
33C42×C101C2311yescharacteristic
34C23×C101—22no—
35C23×C101—22no—
36C23×C101—48no—
37C23×C101—82no—
38D4×C101—22no—
39D4×C101—42no—
40D4×C101—48no—
41D4×C101—42no—
42D4×C101—42no—
43D4×C101SD1614yescomplemented
44C2×C4×C101—22no—
45C2×C4×C101—42no—
46C2×C4×C101—42no—
47C2×C4×C101C2×D411yescharacteristic
48C2×C4×C101C2×D412yes—
49C2×Dic101—41no—
50Dic202—42no—
51#11717579—22no—
52#11717579—42no—
53C22×C101—24no—
54C22×C101—42no—
55C22×C101—42no—
56C22×C101—44no—
57C22×C101—42no—
58C22×C101—88no—
59C22×C101#12011yescharacteristic
60C4×C101—21no—
61C4×C101—42no—
62C4×C101C2×SD1612yes—
63C4×C101AGL(1,ℤ/8)12yes—
64Dic101—81no—
65C2×C101—44no—
66C2×C101—82no—
67C2×C101#45012yes—
68C2×C101#45311yescharacteristic
69#1629—1011nomaximal, Sylow
70C101#162911yescharacteristic, complemented, Sylow
71D42D10111yescharacteristic
72#492—1011no—
73#476—1012no—
74#331—1011no—
75D4⋊C8—1012no—
76C22×D4—22no—
77C2×SD16—2022no—
78C4⋊Q8—1011no—
79#127C2×D10111yescharacteristic
80#121—22no—
81#120—22no—
82C4×D4C2×D10112yes—
83#106—2022no—
84C4⋊C8—1012no—
85#102—2022no—
86C4×C8—1011no—
87C24—22no—
88C2×Q8—2021no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4C22⋊D10112yes—
94C2×D4C22⋊D10112yes—
95C22×C4—22no—
96SD16—4044no—
97C2×C8—2022no—
98C2×C8—2021no—
99C4⋊C4—2022no—
100C4⋊C4C22⋊D10112yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C22×D10111yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—4042no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4#1310806214yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—4041no—
119C2×C4C2×C22⋊D10111yescharacteristic
120C2×C4C2×C22⋊D10112yes—
121C8—4042no—
122C8—2022no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#1569222411yescharacteristic
130C4—21no—
131C4—42no—
132C4—8081no—
133C4C2×#1310806212yes—
134C4#1569228112yes—
135C2—44no—
136C2—82no—
137C2#2026532812yes—
138C2#2026534511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting, prime radicalD42×C101C2
socleC22×C101#120
hypercenterD42D101
hypocenterC101#1629
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20265549C2semidirect
2. #20265467C2semidirect
3. #20264921C2semidirect
4. D4⋊#11717579C2semidirect
7. #15692255C22semidirect
9. C4⋊#11717579C22semidirect
11. C4×#11717579C22semidirect
43. D4×C101SD16semidirect
70. C101#1629semidirect
114. D4#13108062semidirect

Maximal subgroups

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

subgroupindex
#202655492
#202654672
#202649212
D4⋊#117175792
D42×C1012
#1629101