Finite Group Databasebeta

← #27141042

Subgroups — #27141042

Order 12416.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261130C211yesmaximal, characteristic, complemented
2#20261094C212yesmaximal, complemented
3#20260475C211yesmaximal, characteristic, complemented
4#20260475C212yesmaximal, complemented
5H(3,𝔽4)×C97C211yesmaximal, characteristic
6C2×#13095501—21no—
7C2×#13095499—21no—
8#15690810C2211yescharacteristic, complemented
9#15690734—22no—
10#15690731—22no—
11#15690721C2211yescharacteristic, complemented
12#15690721C2212yescomplemented
13#124×C97—21no—
14#124×C97—22no—
15#124×C97C2211yescharacteristic
16#124×C97C2212yes—
17#120×C97—21no—
18#120×C97—22no—
19C2×Dic194—21no—
20#13095501—42no—
21#13095499—42no—
22C2×#11715895—21no—
23C2×#11715895—22no—
24C4⋊Dic97—22no—
25C24×C97—21no—
26C2×Q8×C97D411yescharacteristic
27C2×Q8×C97D412yes—
28C2×D4×C97—21no—
29C2×D4×C97—22no—
30C2×D4×C97D411yescharacteristic
31C2×D4×C97D412yes—
32C22⋊C4×C97—22no—
33C22⋊C4×C97—21no—
34C22⋊C4×C97—22no—
35C22⋊C4×C97—22no—
36C22⋊C4×C97—22no—
37C42×C97—21no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40C23×C97—22no—
41C23×C97—81no—
42C23×C97—82no—
43Q8×C97—42no—
44Q8×C97—41no—
45D4×C97—42no—
46D4×C97—41no—
47D4×C97—81no—
48D4×C97—82no—
49C2×C4×C97—21no—
50C2×C4×C97—22no—
51C2×C4×C97C2×D412yes—
52C2×C4×C97C2×D411yescharacteristic
53C2×Dic97—41no—
54Dic194—41no—
55Dic194—41no—
56#11715895—22no—
57#11715895—42no—
58C22×C97—41no—
59C22×C97—42no—
60C22×C97—42no—
61C22×C97—42no—
62C22×C97—42no—
63C22×C97—84no—
64C22×C97#12011yescharacteristic
65C4×C97—21no—
66C4×C97—22no—
67C4×C97—41no—
68C4×C97—42no—
69Dic97—81no—
70C2×C97—81no—
71C2×C97—82no—
72C2×C97#45311yescharacteristic
73C2×C97#45512yes—
74#1659—971nomaximal, Sylow
75C97#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)D9711yescharacteristic
77#497—971no—
78#489—972no—
79#329—971no—
80#329—972no—
81C2×Q16—1941no—
82C2×SD16—1941no—
83C4⋊Q8—971no—
84#124—22no—
85#124—21no—
86#124C2×D9712yes—
87#124C2×D9711yescharacteristic
88#120—22no—
89#120—21no—
90#103—1942no—
91#102—1942no—
92#97—972no—
93#97—971no—
94C24—21no—
95C2×Q8—1941no—
96C2×Q8C22⋊D9712yes—
97C2×Q8C22⋊D9711yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4C22⋊D9712yes—
101C2×D4C22⋊D9711yescharacteristic
102Q16—3882no—
103SD16—3882no—
104C2×C8—1942no—
105C2×C8—1941no—
106C4⋊C4—1942no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C22×D9711yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—3881no—
121Q8—3881no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—3881no—
129C2×C4C2×C22⋊D9712yes—
130C2×C4C2×C22⋊D9711yescharacteristic
131C8—3882no—
132C8—1942no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#1569077911yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—7761no—
145C2—81no—
146C2—82no—
147C2#2026090311yescharacteristic
148C2#2026091312yes—
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radicalH(3,𝔽4)×C97C2
socleC22×C97#120
hypercenterH(3,𝔽4)D97
hypocenterC97#1659
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20261130C2semidirect
2. #20261094C2semidirect
3. #20260475C2semidirect
4. #20260475C2semidirect
8. #15690810C22semidirect
11. #15690721C22semidirect
12. #15690721C22semidirect
75. C97#1659semidirect

Maximal subgroups

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

subgroupindex
#202611302
#202610942
#202604752
H(3,𝔽4)×C972
#165997