Finite Group Databasebeta

← #27140850

Subgroups — #27140850

Order 12416.

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#20261370C211yesmaximal, characteristic
2#20261022C212yesmaximal, complemented
3#20260479C211yesmaximal, characteristic, complemented
4#20260464C212yesmaximal, complemented
5#492×C97C211yesmaximal, characteristic, complemented
6C2×D4×D97—22no—
7#15690781—22no—
8#15690778—22no—
9C4×D388C2212yes—
10#15690741—22no—
11C4⋊#11715895C2212yescomplemented
12#15690730—22no—
13C2×SD16×C97—22no—
14C4⋊Q8×C97C2211yescharacteristic, complemented
15#127×C97C2211yescharacteristic
16C4×C8×C97C2211yescharacteristic, complemented
17C23×D97—22no—
18C2×D388D412yes—
19C2×C22⋊D97—24no—
20D4×D97—44no—
21D4×D97—44no—
22C2×C4×D97—22no—
23C2×#11715895—22no—
24C4⋊Dic97D412yes—
25C22⋊Dic97—22no—
26#13095488—22no—
27C2×Q8×C97—21no—
28C2×D4×C97—22no—
29C2×D4×C97D412yes—
30SD16×C97—44no—
31C2×C8×C97—21no—
32C4⋊C4×C97—22no—
33C42×C97C2311yescharacteristic
34C23×C97—22no—
35C22×D97—22no—
36C22×D97—48no—
37C22×D97—82no—
38Q8×C97—42no—
39D4×C97—42no—
40D4×C97—42no—
41D4×C97—42no—
42C2×C4×C97—41no—
43C2×C4×C97C2×D412yes—
44C2×C4×C97C2×D411yescharacteristic
45C22⋊D97—48no—
46C2×Dic97—22no—
47D388—22no—
48D388SD1614yescomplemented
49C4×D97—42no—
50C4×D97—42no—
51C8×C97—22no—
52#11715895—42no—
53C22×C97—44no—
54C22×C97—42no—
55C22×C97#12011yescharacteristic
56C2×D97—24no—
57C2×D97—42no—
58C2×D97—42no—
59C2×D97—88no—
60C4×C97—21no—
61C4×C97—81no—
62C4×C97C2×SD1612yes—
63C4×C97AGL(1,ℤ/8)12yes—
64Dic97—42no—
65C2×C97—82no—
66C2×C97#45012yes—
67C2×C97#45311yescharacteristic
68D97—44no—
69#1629—971nomaximal, Sylow
70C97#162911yescharacteristic, complemented, Sylow
71D42—971no—
72#492D9711yescharacteristic, complemented
73#476—972no—
74#331—971no—
75D4⋊C8—972no—
76C22×D4—1942no—
77C2×SD16—22no—
78C4⋊Q8C2×D9711yescharacteristic, complemented
79#127C2×D9711yescharacteristic
80#121—1942no—
81#120—1942no—
82C4×D4—972no—
83#106—1942no—
84C4⋊C8—972no—
85#102—1942no—
86C4×C8C2×D9711yescharacteristic, complemented
87C24—1942no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—972no—
91C2×D4—3884no—
92C2×D4—3884no—
93C2×D4—1944no—
94C2×D4C22⋊D9712yes—
95C22×C4—1942no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—1942no—
99C4⋊C4—22no—
100C4⋊C4—972no—
101C22⋊C4—1942no—
102C22⋊C4—1942no—
103C42C22×D9711yescharacteristic
104C23—22no—
105C23—7762no—
106C23—3888no—
107C23—1942no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—974no—
113D4—3888no—
114D4—1942no—
115C2×C4—41no—
116C2×C4—3882no—
117C2×C4—3882no—
118C2×C4—1942no—
119C2×C4D4×D9712yes—
120C2×C4C2×C22⋊D9711yescharacteristic
121C8—22no—
122C8—3882no—
123C22—44no—
124C22—42no—
125C22—7768no—
126C22—3882no—
127C22—3882no—
128C22—1944no—
129C22#1569077811yescharacteristic
130C4—21no—
131C4—81no—
132C4—3882no—
133C4SD16×D9712yes—
134C4#1569083712yes—
135C2—82no—
136C2—3884no—
137C2#2026088412yes—
138C2#2026090211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#492×C97C2
socleC22×C97#120
prime radical#20261370C2
hypercenter#492D97
hypocenterC97#1629
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20261022C2semidirect
3. #20260479C2semidirect
4. #20260464C2semidirect
5. #492×C97C2semidirect
11. C4⋊#11715895C22semidirect
14. C4⋊Q8×C97C22semidirect
16. C4×C8×C97C22semidirect
48. D388SD16semidirect
70. C97#1629semidirect
72. #492D97semidirect
78. C4⋊Q8C2×D97semidirect
86. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613702
#202610222
#202604792
#202604642
#492×C972
#162997