Finite Group Databasebeta

← #27141371

Subgroups — #27141371

Order 12416.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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#20261133C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic
3#20261131C211yesmaximal, characteristic
4#20261114C211yesmaximal, characteristic
5#20261087C211yesmaximal, characteristic
6#20260457C211yesmaximal, characteristic
7#487×C97C211yesmaximal, characteristic
8C2×#13095501—22no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690742—22no—
12#15690741—22no—
13#15690741—22no—
14#15690733—22no—
15C4×#11715895C2211yescharacteristic
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18C4⋊Q8×C97C2211yescharacteristic
19#103×C97—22no—
20C4×C8×C97C2211yescharacteristic
21C2×Dic194—21no—
22C2×Dic194—21no—
23#13095501—44no—
24C2×#11715895D412yes—
25C2×#11715895D412yes—
26C4⋊Dic97—21no—
27C4⋊Dic97—21no—
28C4⋊Dic97—21no—
29C4⋊Dic97—21no—
30C2×Q8×C97—21no—
31C2×Q8×C97—21no—
32C2×C8×C97D412yes—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—21no—
35C4⋊C4×C97—21no—
36C4⋊C4×C97—21no—
37C42×C97C2311yescharacteristic
38Q8×C97—42no—
39Q8×C97—42no—
40C2×C4×C97—41no—
41C2×C4×C97—41no—
42C2×C4×C97C2×D411yescharacteristic
43C2×C4×C97C2×D411yescharacteristic
44C2×C4×C97C2×D411yescharacteristic
45C2×Dic97—41no—
46C2×Dic97—41no—
47Dic194—42no—
48Dic194—42no—
49C8×C97—22no—
50#11715895—22no—
51#11715895SD1614yes—
52C22×C97#12711yescharacteristic
53C4×C97—81no—
54C4×C97—81no—
55C4×C97C2×SD1612yes—
56C4×C97#13712yes—
57C4×C97#13712yes—
58Dic97—81no—
59Dic97—81no—
60C2×C97#49211yescharacteristic
61C2×C97#49711yescharacteristic
62C2×C97#49711yescharacteristic
63#1716—971nomaximal, Sylow
64C97#171611yescharacteristic, complemented, Sylow
65#498—971no—
66#498—971no—
67#498—971no—
68#494—971no—
69#487—971no—
70#487D9711yescharacteristic
71#322—971no—
72C2×Q16—1942no—
73C4⋊Q8—971no—
74C4⋊Q8—971no—
75C4⋊Q8C2×D9711yescharacteristic
76C4⋊Q8C2×D9711yescharacteristic
77#106—1942no—
78#106—1942no—
79#106—1942no—
80#103—22no—
81#103—1942no—
82C4×C8—971no—
83C4×C8—971no—
84C4×C8C2×D9711yescharacteristic
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—1941no—
88C2×Q8—1941no—
89Q16—3884no—
90C2×C8—972no—
91C2×C8—972no—
92C2×C8C22⋊D9712yes—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—1941no—
98C4⋊C4—1941no—
99C4⋊C4—1941no—
100C4⋊C4—1941no—
101C42C22×D9711yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—3882no—
105Q8—3882no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—3881no—
109C2×C4—3881no—
110C2×C4D4×D9711yescharacteristic
111C2×C4D4×D9711yescharacteristic
112C2×C4C2×C22⋊D9711yescharacteristic
113C8—22no—
114C8—974no—
115C8—1942no—
116C22#1569080611yescharacteristic
117C4—81no—
118C4—81no—
119C4—7761no—
120C4—7761no—
121C4SD16×D9712yes—
122C4#1569084012yes—
123C4#1569084212yes—
124C2#2026110611yescharacteristic
125C2#2026112811yescharacteristic
126C2#2026113011yescharacteristic
subgroupG/H
centerC22#15690806
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#487×C97C2
socle, prime radicalC22×C97#127
hypercenter#487D97
hypocenterC97#1716
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
64. C97#1716semidirect

Maximal subgroups

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

subgroupindex
#202611332
#202611312
#202611142
#202610872
#202604572
#487×C972
#171697