Finite Group Databasebeta

← #27170246

Subgroups — #27170246

Order 12928.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265812C211yesmaximal, characteristic
2#20265464C212yesmaximal, complemented
3#20264921C211yesmaximal, characteristic, complemented
4#20264906C212yesmaximal, complemented
5#492×C101C211yesmaximal, characteristic, complemented
6C2×D4×D101—22no—
7#15692226—22no—
8#15692223—22no—
9C4×D404C2212yes—
10#15692186—22no—
11C4⋊#11717579C2212yescomplemented
12#15692175—22no—
13C2×SD16×C101—22no—
14C4⋊Q8×C101C2211yescharacteristic, complemented
15#127×C101C2211yescharacteristic
16C4×C8×C101C2211yescharacteristic, complemented
17C23×D101—22no—
18C2×D404D412yes—
19C2×C22⋊D101—24no—
20D4×D101—44no—
21D4×D101—44no—
22C2×C4×D101—22no—
23C2×#11717579—22no—
24C4⋊Dic101D412yes—
25C22⋊Dic101—22no—
26#13108051—22no—
27C2×Q8×C101—21no—
28C2×D4×C101—22no—
29C2×D4×C101D412yes—
30SD16×C101—44no—
31C2×C8×C101—21no—
32C4⋊C4×C101—22no—
33C42×C101C2311yescharacteristic
34C23×C101—22no—
35C22×D101—22no—
36C22×D101—48no—
37C22×D101—82no—
38Q8×C101—42no—
39D4×C101—42no—
40D4×C101—42no—
41D4×C101—42no—
42C2×C4×C101—41no—
43C2×C4×C101C2×D412yes—
44C2×C4×C101C2×D411yescharacteristic
45C22⋊D101—48no—
46C2×Dic101—22no—
47D404—22no—
48D404SD1614yescomplemented
49C4×D101—42no—
50C4×D101—42no—
51C8×C101—22no—
52#11717579—42no—
53C22×C101—44no—
54C22×C101—42no—
55C22×C101#12011yescharacteristic
56C2×D101—24no—
57C2×D101—42no—
58C2×D101—42no—
59C2×D101—88no—
60C4×C101—21no—
61C4×C101—81no—
62C4×C101C2×SD1612yes—
63C4×C101AGL(1,ℤ/8)12yes—
64Dic101—42no—
65C2×C101—82no—
66C2×C101#45012yes—
67C2×C101#45311yescharacteristic
68D101—44no—
69#1629—1011nomaximal, Sylow
70C101#162911yescharacteristic, complemented, Sylow
71D42—1011no—
72#492D10111yescharacteristic, complemented
73#476—1012no—
74#331—1011no—
75D4⋊C8—1012no—
76C22×D4—2022no—
77C2×SD16—22no—
78C4⋊Q8C2×D10111yescharacteristic, complemented
79#127C2×D10111yescharacteristic
80#121—2022no—
81#120—2022no—
82C4×D4—1012no—
83#106—2022no—
84C4⋊C8—1012no—
85#102—2022no—
86C4×C8C2×D10111yescharacteristic, complemented
87C24—2022no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—1012no—
91C2×D4—4044no—
92C2×D4—4044no—
93C2×D4—2024no—
94C2×D4C22⋊D10112yes—
95C22×C4—2022no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—2022no—
99C4⋊C4—22no—
100C4⋊C4—1012no—
101C22⋊C4—2022no—
102C22⋊C4—2022no—
103C42C22×D10111yescharacteristic
104C23—22no—
105C23—8082no—
106C23—4048no—
107C23—2022no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—1014no—
113D4—4048no—
114D4—2022no—
115C2×C4—41no—
116C2×C4—4042no—
117C2×C4—4042no—
118C2×C4—2022no—
119C2×C4D4×D10112yes—
120C2×C4C2×C22⋊D10111yescharacteristic
121C8—22no—
122C8—4042no—
123C22—44no—
124C22—42no—
125C22—8088no—
126C22—4042no—
127C22—4042no—
128C22—2024no—
129C22#1569222311yescharacteristic
130C4—21no—
131C4—81no—
132C4—4042no—
133C4SD16×D10112yes—
134C4#1569228212yes—
135C2—82no—
136C2—4044no—
137C2#2026532612yes—
138C2#2026534411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#492×C101C2
socleC22×C101#120
prime radical#20265812C2
hypercenter#492D101
hypocenterC101#1629
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20265464C2semidirect
3. #20264921C2semidirect
4. #20264906C2semidirect
5. #492×C101C2semidirect
11. C4⋊#11717579C22semidirect
14. C4⋊Q8×C101C22semidirect
16. C4×C8×C101C22semidirect
48. D404SD16semidirect
70. C101#1629semidirect
72. #492D101semidirect
78. C4⋊Q8C2×D101semidirect
86. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658122
#202654642
#202649212
#202649062
#492×C1012
#1629101