Finite Group Databasebeta

← #27140881

Subgroups — #27140881

Order 12416.

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

normal subgroups only

This group has 218 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#20261114C211yesmaximal, characteristic, complemented
2#20261025C212yesmaximal, complemented
3#20260484C211yesmaximal, characteristic, complemented
4D4⋊#11715895C212yesmaximal, complemented
5#555×C97C211yesmaximal, characteristic
6C2×#13095501—22no—
7#15690810C2211yescharacteristic
8#15690741—22no—
9C4⋊#11715895C2212yes—
10#15690731—22no—
11C4×#11715895C2211yescharacteristic
12C4⋊Q8×C97C2211yescharacteristic
13#124×C97—22no—
14#121×C97—22no—
15C4×D4×C97C2212yes—
16#117×C97—22no—
17C2×Dic194—21no—
18#13095501—44no—
19C2×#11715895—21no—
20C2×#11715895—22no—
21C4⋊Dic97—22no—
22C2×Q8×C97D412yes—
23C2×D4×C97—22no—
24C2×D4×C97D412yes—
25C22×C4×C97—22no—
26C4⋊C4×C97—22no—
27C4⋊C4×C97D412yes—
28C22⋊C4×C97—24no—
29C22⋊C4×C97—22no—
30C42×C97—22no—
31C42×C97C2311yescharacteristic
32C23×C97—22no—
33Q8×C97—42no—
34D4×C97—22no—
35D4×C97—22no—
36D4×C97—82no—
37C2×C4×C97—22no—
38C2×C4×C97—22no—
39C2×C4×C97—42no—
40C2×C4×C97—42no—
41C2×C4×C97C2×D411yescharacteristic
42C2×C4×C97C2×D412yes—
43C2×Dic97—41no—
44Dic194—42no—
45#11715895—22no—
46#11715895—42no—
47C22×C97—42no—
48C22×C97—42no—
49C22×C97—42no—
50C22×C97#12011yescharacteristic
51C4×C97—21no—
52C4×C97—42no—
53C4×C97—42no—
54C4×C97#13512yes—
55C4×C97#13712yes—
56Dic97—81no—
57C2×C97—82no—
58C2×C97#45212yes—
59C2×C97#45311yescharacteristic
60#1634—971nomaximal, Sylow
61C97#163411yescharacteristic, complemented, Sylow
62#555D9711yescharacteristic
63#494—971no—
64#476—972no—
65#333—971no—
66D4⋊C8—972no—
67C2×Q16—1942no—
68C4⋊Q8—971no—
69C4⋊Q8C2×D9711yescharacteristic
70#124—22no—
71#121—22no—
72C4×D4C2×D9712yes—
73#117—22no—
74#106—1942no—
75C4⋊C8—972no—
76#102—1942no—
77C4×C8—971no—
78C2×Q8—1941no—
79C2×Q8C22⋊D9712yes—
80C2×D4—22no—
81C2×D4C22⋊D9712yes—
82C22×C4—22no—
83Q16—3884no—
84C2×C8—1942no—
85C2×C8—1941no—
86C4⋊C4—22no—
87C4⋊C4—1942no—
88C4⋊C4C22⋊D9712yes—
89C22⋊C4—24no—
90C22⋊C4—22no—
91C42—22no—
92C42C22×D9711yescharacteristic
93C23—22no—
94Q8—42no—
95Q8—3882no—
96D4—22no—
97D4—22no—
98D4—82no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—3881no—
104C2×C4C2×C22⋊D9711yescharacteristic
105C2×C4C2×C22⋊D9712yes—
106C8—3882no—
107C8—1942no—
108C22—42no—
109C22—42no—
110C22—42no—
111C22#1569077911yescharacteristic
112C4—21no—
113C4—42no—
114C4—42no—
115C4—7761no—
116C4#1569083012yes—
117C4#1569084012yes—
118C2—82no—
119C2#2026089712yes—
120C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#555×C97C2
socleC22×C97#120
hypercenter#555D97
hypocenterC97#1634
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20261114C2semidirect
2. #20261025C2semidirect
3. #20260484C2semidirect
4. D4⋊#11715895C2semidirect
61. C97#1634semidirect

Maximal subgroups

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

subgroupindex
#202611142
#202610252
#202604842
D4⋊#117158952
#555×C972
#163497