Finite Group Databasebeta

← #27140812

Subgroups — #27140812

Order 12416.

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

normal subgroups only

This group has 228 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#20261465C211yesmaximal, characteristic, complemented
2#20260932C212yesmaximal
3#20260484C211yesmaximal, characteristic, complemented
4#20260467C212yesmaximal, complemented
5#492×C97C211yesmaximal, characteristic
6C2×#13095499—22no—
7#15690809—22no—
8#15690808—22no—
9C4×Dic194C2212yes—
10#15690775—22no—
11C4⋊#11715895C2212yes—
12#15690731—22no—
13C2×SD16×C97—22no—
14C4⋊Q8×C97C2211yescharacteristic
15#127×C97C2211yescharacteristic
16C4×C8×C97C2211yescharacteristic
17C2×Dic194—22no—
18C2×Dic194D412yes—
19#13095499—44no—
20C2×#11715895—22no—
21C4⋊Dic97D412yes—
22#13095490—24no—
23#13095490—22no—
24C4×Dic97—22no—
25C4×Dic97—22no—
26C2×Q8×C97D412yes—
27C2×D4×C97—22no—
28C2×D4×C97—21no—
29SD16×C97—44no—
30C2×C8×C97—21no—
31C4⋊C4×C97—22no—
32C42×C97C2311yescharacteristic
33C23×C97—41no—
34Q8×C97—42no—
35D4×C97—42no—
36D4×C97—42no—
37D4×C97—42no—
38C2×C4×C97—22no—
39C2×C4×C97C2×D412yes—
40C2×C4×C97C2×D411yescharacteristic
41C2×Dic97—22no—
42C2×Dic97—22no—
43Dic194—22no—
44Dic194—82no—
45Dic194SD1614yescomplemented
46C8×C97—22no—
47#11715895—42no—
48C22×C97—82no—
49C22×C97—81no—
50C22×C97#12011yescharacteristic
51C4×C97—21no—
52C4×C97—42no—
53C4×C97C2×SD1612yes—
54C4×C97#13712yes—
55Dic97—24no—
56Dic97—42no—
57C2×C97—161no—
58C2×C97#44812yes—
59C2×C97#45311yescharacteristic
60#1623—971nomaximal, Sylow
61C97#162311yescharacteristic, complemented, Sylow
62Q82—971no—
63#492D9711yescharacteristic
64#460—972no—
65#333—971no—
66Q8⋊C8—972no—
67C2×SD16—22no—
68C2×SD16—1942no—
69C4⋊Q8—1942no—
70C4⋊Q8—1942no—
71C4⋊Q8C2×D9711yescharacteristic
72#127C2×D9711yescharacteristic
73C4×Q8—972no—
74C4×Q8—1942no—
75C4⋊C8—972no—
76#102—1942no—
77C4×C8C2×D9711yescharacteristic
78C2×Q8—972no—
79C2×Q8—1942no—
80C2×Q8C22⋊D9712yes—
81C2×D4—22no—
82C2×D4—21no—
83SD16—44no—
84SD16—3884no—
85C2×C8—21no—
86C2×C8—1942no—
87C4⋊C4—22no—
88C4⋊C4—972no—
89C4⋊C4—1944no—
90C4⋊C4—1942no—
91C42—1942no—
92C42—1942no—
93C42C22×D9711yescharacteristic
94C23—41no—
95Q8—42no—
96Q8—7762no—
97Q8—974no—
98Q8—1942no—
99D4—42no—
100D4—42no—
101D4—42no—
102C2×C4—22no—
103C2×C4—1942no—
104C2×C4—1942no—
105C2×C4D4×D9712yes—
106C2×C4C2×C22⋊D9711yescharacteristic
107C8—22no—
108C8—3882no—
109C22—82no—
110C22—81no—
111C22#1569077811yescharacteristic
112C4—21no—
113C4—42no—
114C4—3882no—
115C4—1944no—
116C4SD16×D9712yes—
117C4#1569083912yes—
118C2—161no—
119C2#2026087212yes—
120C2#2026090211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#492×C97C2
socleC22×C97#120
prime radical#127×C97C22
hypercenter#492D97
hypocenterC97#1623
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20261465C2semidirect
3. #20260484C2semidirect
4. #20260467C2semidirect
45. Dic194SD16semidirect
61. C97#1623semidirect

Maximal subgroups

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

subgroupindex
#202614652
#202609322
#202604842
#202604672
#492×C972
#162397