Finite Group Databasebeta

← #27141032

Subgroups — #27141032

Order 12416.

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

normal subgroups only

This group has 206 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#20261485C211yesmaximal, characteristic, complemented
2#20261099C212yesmaximal, complemented
3#20260476C212yesmaximal, complemented
4#20260475C211yesmaximal, characteristic, complemented
5#496×C97C211yesmaximal, characteristic
6#15690808—21no—
7#15690803—22no—
8#15690802C2212yes—
9#15690791—21no—
10#15690787—22no—
11#15690731—22no—
12#15690731—22no—
13#15690721C2212yes—
14C2×SD16×C97—21no—
15C2×D8×C97—21no—
16#127×C97C2211yescharacteristic
17#124×C97C2211yescharacteristic
18#97×C97C2211yescharacteristic
19C2×Dic194—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97D412yes—
23C4⋊Dic97D412yes—
24#13095490—22no—
25#13095490—22no—
26#13095490—22no—
27C22⋊Dic97—22no—
28C22⋊Dic97—21no—
29C4×Dic97—21no—
30C2×Q8×C97D411yescharacteristic
31C2×D4×C97—21no—
32C2×D4×C97—22no—
33C2×D4×C97D411yescharacteristic
34SD16×C97—42no—
35D8×C97—42no—
36C2×C8×C97—21no—
37C22⋊C4×C97—22no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40C23×C97—41no—
41Q8×C97—41no—
42D4×C97—41no—
43D4×C97—41no—
44D4×C97—41no—
45D4×C97—44no—
46C2×C4×C97—21no—
47C2×C4×C97C2×D412yes—
48C2×C4×C97C2×D411yescharacteristic
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97—81no—
52Dic194—81no—
53C8×C97—22no—
54#11715895—42no—
55C22×C97—41no—
56C22×C97—42no—
57C22×C97—81no—
58C22×C97—82no—
59C22×C97#12011yescharacteristic
60C4×C97—21no—
61C4×C97—22no—
62C4×C97—41no—
63Dic97—42no—
64Dic97—42no—
65C2×C97—81no—
66C2×C97—161no—
67C2×C97#45311yescharacteristic
68C2×C97#45412yes—
69#1657—971nomaximal, Sylow
70C97#165711yescharacteristic, complemented, Sylow
71#563—971no—
72#496D9711yescharacteristic
73#490—972no—
74#330—972no—
75#329—971no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—1941no—
79#127C2×D9711yescharacteristic
80#126—1942no—
81#125—972no—
82#124C2×D9711yescharacteristic
83#123—1941no—
84C22⋊Q8—1942no—
85#102—1942no—
86#102—1942no—
87#97—972no—
88#97C2×D9711yescharacteristic
89C2×Q8—1941no—
90C2×Q8C22⋊D9711yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D9711yescharacteristic
94C22×C4—1941no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—1942no—
99C4⋊C4—972no—
100C4⋊C4—972no—
101C4⋊C4—1942no—
102C4⋊C4—1942no—
103C4⋊C4—1942no—
104C22⋊C4—22no—
105C22⋊C4—1942no—
106C22⋊C4—1941no—
107C42—1941no—
108C42C22×D9711yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—7761no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—7761no—
119C2×C4—1942no—
120C2×C4—1942no—
121C2×C4D4×D9712yes—
122C2×C4C2×C22⋊D9711yescharacteristic
123C8—22no—
124C8—3882no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#1569077811yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—3882no—
134C4—3882no—
135C2—81no—
136C2—161no—
137C2#2026090211yescharacteristic
138C2#2026090612yes—
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#496×C97C2
socleC22×C97#120
hypercenter#496D97
hypocenterC97#1657
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #20261485C2semidirect
2. #20261099C2semidirect
3. #20260476C2semidirect
4. #20260475C2semidirect
70. C97#1657semidirect

Maximal subgroups

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

subgroupindex
#202614852
#202610992
#202604762
#202604752
#496×C972
#165797