Finite Group Databasebeta

← #27140329

Subgroups — #27140329

Order 12416.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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#20261291C211yesmaximal, characteristic
2#20260687C212yesmaximal
3#20260477C211yesmaximal, characteristic, complemented
4#20260476C212yesmaximal, complemented
5#330×C97C211yesmaximal, characteristic, complemented
6C2×#13095494—22no—
7#15690809C412yes—
8#15690802C2211yescharacteristic
9#15690788—22no—
10#15690775—22no—
11#15690768C412yes—
12C42×D97C2211yescharacteristic
13#15690721C2212yes—
14C8×Dic97—22no—
15#125×C97C2211yescharacteristic
16#97×C97C2212yes—
17C2×Dic194—22no—
18C2×C4×D97C2×C411yescharacteristic
19C2×C4×D97D412yes—
20#13095494—28no—
21C2×#11715895—22no—
22C4⋊Dic97C2×C412yes—
23#13095490—22no—
24#13095490—22no—
25#13095488—22no—
26C4×Dic97—22no—
27C4×Dic97C2×C411yescharacteristic
28C4×Dic97D412yes—
29C2×C8×C97—22no—
30C4⋊C4×C97—22no—
31C4⋊C4×C97C2×C412yes—
32C42×C97C2311yescharacteristic
33C22×D97C22⋊C411yescharacteristic
34C2×C4×C97—22no—
35C2×C4×C97C22×C411yescharacteristic
36C2×C4×C97C2×D412yes—
37C2×Dic97—22no—
38C2×Dic97C22⋊C411yescharacteristic
39C2×Dic97C22⋊C412yes—
40C4×D97—22no—
41C4×D97—24no—
42Dic194—44no—
43C8×C97—24no—
44#11715895—24no—
45C22×C97C2×C22⋊C411yescharacteristic
46C2×D97—21no—
47C2×D97—22no—
48C4×C97—21no—
49C4×C97—22no—
50C4×C97—42no—
51Dic97—42no—
52Dic97#10112yes—
53Dic97#10414yes—
54C2×C97C2×#10111yescharacteristic
55C2×C97C2×#10412yes—
56D97—41no—
57#1526—971nomaximal, Sylow
58C97#152611yescharacteristic, complemented, Sylow
59#533—971no—
60C4×M16—972no—
61#330—971no—
62#330—972no—
63#330D9711yescharacteristic, complemented
64C2×M16—1942no—
65C4⋊Q8—972no—
66#125—971no—
67#125C2×D9711yescharacteristic
68C22⋊Q8—1942no—
69C4×Q8—1942no—
70#117—972no—
71C2×C42—971no—
72#97—972no—
73#97C2×D9712yes—
74C4×C8—1942no—
75C2×Q8—1942no—
76C22×C4—971no—
77C22×C4—972no—
78M16—1948no—
79C2×C8—22no—
80C2×C8—1942no—
81C4⋊C4—22no—
82C4⋊C4—972no—
83C4⋊C4—1942no—
84C4⋊C4—1942no—
85C4⋊C4C4×D9712yes—
86C22⋊C4—1942no—
87C42—971no—
88C42—972no—
89C42—1942no—
90C42C22×D9711yescharacteristic
91C23—971no—
92Q8—3884no—
93C2×C4—22no—
94C2×C4—971no—
95C2×C4—972no—
96C2×C4—1944no—
97C2×C4—1942no—
98C2×C4—1942no—
99C2×C4C2×C4×D9711yescharacteristic
100C2×C4D4×D9712yes—
101C8—24no—
102C8—1944no—
103C22—1941no—
104C22—1942no—
105C22C22⋊C4×D9711yescharacteristic
106C4—21no—
107C4—22no—
108C4—42no—
109C4—972no—
110C4—974no—
111C4—3882no—
112C2—3881no—
113C2#101×D9711yescharacteristic
114C2#104×D9712yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#330×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#330D97
hypocenterC97#1526
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #20260477C2semidirect
4. #20260476C2semidirect
5. #330×C97C2semidirect
58. C97#1526semidirect
63. #330D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
#202606872
#202604772
#202604762
#330×C972
#152697