Finite Group Databasebeta

← #27157125

Subgroups — #27157125

Order 12416.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261757C211yesmaximal, characteristic, complemented
2#20261671C211yesmaximal, characteristic, complemented
3#20261586C214yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5C2×C4×#87604—21no—
6C4⋊#11715897—21no—
7#15690867C2212yes—
8#15690865—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11C2×#13095510—21no—
12#15690810C411yescharacteristic, complemented
13#15690797—21no—
14#15690781—21no—
15#15690781C2212yes—
16D4×Dic97—21no—
17D4×Dic97C412yescomplemented
18C42×D97C2211yescharacteristic
19#127×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4×#87604—24no—
23#13095510—44no—
24C2×Dic194—21no—
25C2×C22⋊D97—22no—
26C22×Dic97—22no—
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97C2311yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C412yes—
32C22⋊Dic97—22no—
33C22⋊Dic97—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C2×D4×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97C2×C412yes—
40C42×C97C2×C411yescharacteristic
41C23×C97—22no—
42C22×D97C2×D411yescharacteristic
43C2×#87604—24no—
44C2×#87604—42no—
45C2×#87604—42no—
46D4×C97—42no—
47D4×C97—42no—
48D4×C97—42no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C22⋊D97—44no—
53C2×Dic97—22no—
54C2×Dic97—44no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C2×D411yescharacteristic
58C4×D97—22no—
59C4×D97—22no—
60C4×D97—41no—
61Dic194—42no—
62#11715897—42no—
63C22×C97—42no—
64C22×C97—42no—
65C22×C97—42no—
66C22×C97C2×C22⋊C411yescharacteristic
67C2×D97—21no—
68C2×D97—21no—
69C2×D97AGL(1,ℤ/8)12yes—
70#87604—44no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97#10412yes—
74Dic97—21no—
75Dic97—21no—
76Dic97—21no—
77Dic97—42no—
78C2×C97—82no—
79C2×C97#41011yescharacteristic
80C2×C97#41811yescharacteristic
81C2×C97C2×#10411yescharacteristic
82D97—22no—
83#1505—971nomaximal, Sylow
84C97#150511yescharacteristic, complemented, Sylow
85#532—971no—
86#432—971no—
87#376—971no—
88#327—974no—
89C2×#41—1941no—
90C4⋊Q8—971no—
91#127#8760411yescharacteristic, complemented
92#124—1941no—
93#121—972no—
94#121—1941no—
95C4×D4—972no—
96C4×D4—1941no—
97C2×C42—971no—
98C2×C42—1941no—
99C4⋊C8—1941no—
100C22⋊C8—972no—
101#97—1941no—
102#95—972no—
103#95—1941no—
104#41—3884no—
105C2×Q8—1941no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—1942no—
109C2×D4C2×#8760412yes—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—1942no—
114C22×C4—1942no—
115C2×C8—1942no—
116C4⋊C4—972no—
117C4⋊C4—1941no—
118C22⋊C4—1942no—
119C22⋊C4—1942no—
120C42—971no—
121C42—971no—
122C42—971no—
123C42—1944no—
124C42C2×#8760411yescharacteristic
125C23—22no—
126C23—971no—
127Q8—3882no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—3884no—
132C2×C4—971no—
133C2×C4—971no—
134C2×C4—971no—
135C2×C4—3881no—
136C2×C4—3882no—
137C2×C4—3882no—
138C2×C4—3884no—
139C2×C4—1942no—
140C2×C4—1944no—
141C2×C4—1942no—
142C2×C4—1942no—
143C2×C4C22⋊#8760411yescharacteristic
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22×#8760411yescharacteristic
146C8—3882no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22—972no—
151C22—1941no—
152C22—1941no—
153C22C2×C22⋊#8760411yescharacteristic
154C4—21no—
155C4—21no—
156C4—3884no—
157C4—3882no—
158C4—1941no—
159C4—1941no—
160C4—1941no—
161C4#1569087412yes—
162C2—82no—
163C2—1942no—
164C2#2026171511yescharacteristic
165C2#2026172711yescharacteristic
166C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC2×C4×D97C23
hypercenter#127#87604
hypocenterC97#1505
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. #20261671C2semidirect
3. #20261586C2semidirect
12. #15690810C4semidirect
17. D4×Dic97C4semidirect
19. #127×C97C4semidirect
84. C97#1505semidirect
91. #127#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616712
#202615862
#202612832
#150597