Finite Group Databasebeta

← #27141043

Subgroups — #27141043

Order 12416.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#20261477C211yesmaximal, characteristic
2#20261130C211yesmaximal, characteristic, complemented
3#20261095C212yesmaximal, complemented
4#20260474C212yesmaximal, complemented
5#329×C97C211yesmaximal, characteristic, complemented
6C2×#13095501—21no—
7C2×#13095499—21no—
8#15690810C2211yescharacteristic, complemented
9#15690799—21no—
10#15690798C2212yes—
11#15690796—22no—
12#15690780—22no—
13#15690778—21no—
14#15690735—22no—
15#15690732—22no—
16#15690721C2211yescharacteristic, complemented
17#124×C97C2211yescharacteristic
18#97×C97C2212yescomplemented
19C23×D97—21no—
20C2×Dic194—21no—
21C2×Dic194D412yes—
22C2×D388—21no—
23C2×D388D412yes—
24C2×C22⋊D97—22no—
25#13095501—42no—
26#13095499—42no—
27C2×#11715895—21no—
28C4⋊Dic97—22no—
29C22⋊Dic97—21no—
30#13095488—22no—
31#13095488—22no—
32#13095488—22no—
33C4×Dic97—21no—
34C2×Q8×C97D411yescharacteristic
35C2×D4×C97D411yescharacteristic
36C2×C8×C97—22no—
37C22⋊C4×C97—22no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40C22×D97—22no—
41C22×D97—81no—
42C22×D97—82no—
43Q8×C97—41no—
44D4×C97—41no—
45C2×C4×C97—21no—
46C2×C4×C97C2×D412yes—
47C2×C4×C97C2×D411yescharacteristic
48C22⋊D97—82no—
49C2×Dic97—22no—
50C2×Dic97—41no—
51D388—42no—
52D388—81no—
53Dic194—42no—
54Dic194—41no—
55Dic194—41no—
56C8×C97—42no—
57#11715895—22no—
58C22×C97—41no—
59C22×C97—42no—
60C22×C97#12011yescharacteristic
61C2×D97—42no—
62C2×D97—42no—
63C2×D97—42no—
64C2×D97—84no—
65C4×C97—21no—
66C4×C97—22no—
67C4×C97—41no—
68Dic97—42no—
69Dic97—81no—
70C2×C97—81no—
71C2×C97#45311yescharacteristic
72C2×C97#45512yes—
73D97—82no—
74#1659—971nomaximal, Sylow
75C97#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—971no—
77#497—971no—
78#489—972no—
79#329—972no—
80#329D9711yescharacteristic, complemented
81C2×Q16—1941no—
82C2×SD16—1941no—
83C4⋊Q8—971no—
84#124—972no—
85#124—1942no—
86#124—1941no—
87#124C2×D9711yescharacteristic
88#120—1942no—
89#120—1941no—
90#103—1942no—
91#102—1942no—
92#97—971no—
93#97C2×D9712yescomplemented
94C24—1941no—
95C2×Q8—972no—
96C2×Q8—1941no—
97C2×Q8D38811yescharacteristic
98C2×D4—972no—
99C2×D4—1941no—
100C2×D4—1942no—
101C2×D4D38811yescharacteristic
102Q16—3882no—
103SD16—3882no—
104C2×C8—22no—
105C2×C8—1941no—
106C4⋊C4—1942no—
107C22⋊C4—22no—
108C22⋊C4—1942no—
109C22⋊C4—1941no—
110C22⋊C4—1942no—
111C22⋊C4—1942no—
112C42—1941no—
113C42C22×D9711yescharacteristic
114C23—21no—
115C23—7761no—
116C23—7762no—
117C23—1942no—
118Q8—41no—
119Q8—3881no—
120Q8—3881no—
121Q8—3882no—
122D4—41no—
123D4—7761no—
124D4—7762no—
125D4—3882no—
126C2×C4—21no—
127C2×C4—3881no—
128C2×C4—1942no—
129C2×C4D4×D9712yes—
130C2×C4C2×D38811yescharacteristic
131C8—42no—
132C8—1942no—
133C22—41no—
134C22—42no—
135C22—7764no—
136C22—3882no—
137C22—3882no—
138C22—3882no—
139C22#1569078011yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—7761no—
144C4—3882no—
145C2—81no—
146C2—7762no—
147C2#2026090511yescharacteristic
148C2#2026091412yes—
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97#120
prime radical#20261477C2
hypercenter#329D97
hypocenterC97#1659
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20261130C2semidirect
3. #20261095C2semidirect
4. #20260474C2semidirect
5. #329×C97C2semidirect
8. #15690810C22semidirect
16. #15690721C22semidirect
18. #97×C97C22semidirect
75. C97#1659semidirect
80. #329D97semidirect
93. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614772
#202611302
#202610952
#202604742
#329×C972
#165997