Finite Group Databasebeta

← #27157165

Subgroups — #27157165

Order 12416.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261753C211yesmaximal, characteristic, complemented
2#20261679C211yesmaximal, characteristic, complemented
3#20261586C214yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6#15690877—21no—
7#15690867C2212yes—
8C4×#11715897—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—42no—
23#13095510—42no—
24#13095510—42no—
25C2×Dic194—21no—
26C2×C22⋊D97—22no—
27C22×Dic97—22no—
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97C2311yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C412yes—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C4×Dic97C2×C411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97D411yescharacteristic
38C2×D4×C97—21no—
39C2×D4×C97—21no—
40C2×D4×C97C2×C412yes—
41C42×C97C2×C411yescharacteristic
42C23×C97—22no—
43C22×D97C2×D411yescharacteristic
44C2×#87604—42no—
45C2×#87604—42no—
46C2×#87604—42no—
47D4×C97—42no—
48D4×C97—42no—
49D4×C97—42no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C22⋊D97—42no—
54C22⋊D97—42no—
55C2×Dic97—22no—
56C2×Dic97—44no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60C4×D97—41no—
61C4×D97—41no—
62C4×D97D812yes—
63C4×D97SD1612yes—
64Dic194—42no—
65#11715897—42no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—21no—
72C2×D97C2×D811yescharacteristic
73C2×D97C2×SD1611yescharacteristic
74#87604—82no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97#10212yes—
78Dic97—21no—
79Dic97—21no—
80Dic97—42no—
81Dic97#10212yes—
82C2×C97—82no—
83C2×C97#41011yescharacteristic
84C2×C97C2×#10211yescharacteristic
85C2×C97#42111yescharacteristic
86D97—21no—
87D97—21no—
88#1517—971nomaximal, Sylow
89C97#151711yescharacteristic, complemented, Sylow
90#532—971no—
91#431—971no—
92#382—971no—
93#327—974no—
94C2×#41—1941no—
95C4⋊Q8—971no—
96#127#8760411yescharacteristic, complemented
97#124—1941no—
98#121—972no—
99#121—1941no—
100C4×D4—972no—
101C4×D4—1941no—
102C2×C4⋊C4—1941no—
103C2×C42—971no—
104C4⋊C8—1941no—
105C22⋊C8—972no—
106C4×C8—1941no—
107#95—972no—
108#95—1941no—
109#41—3882no—
110#41—3882no—
111C2×Q8—1941no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—1942no—
115C2×D4C2×#8760412yes—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—1942no—
120C22×C4—1942no—
121C2×C8—1942no—
122C4⋊C4—972no—
123C4⋊C4—3882no—
124C4⋊C4—1941no—
125C22⋊C4—1942no—
126C22⋊C4—1942no—
127C42—971no—
128C42—971no—
129C42—971no—
130C42C2×#8760411yescharacteristic
131C23—22no—
132C23—971no—
133Q8—3882no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—3882no—
138D4—3882no—
139C2×C4—971no—
140C2×C4—972no—
141C2×C4—972no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—3881no—
145C2×C4—3881no—
146C2×C4—3882no—
147C2×C4—3882no—
148C2×C4—3882no—
149C2×C4—3884no—
150C2×C4—1942no—
151C2×C4C22⋊#8760411yescharacteristic
152C2×C4C22⋊#8760411yescharacteristic
153C2×C4C22×#8760411yescharacteristic
154C8—3882no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—971no—
159C22—971no—
160C22—1941no—
161C22—1941no—
162C22C2×C22⋊#8760411yescharacteristic
163C4—21no—
164C4—21no—
165C4—7762no—
166C4—972no—
167C4—3882no—
168C4—1941no—
169C4—1941no—
170C4#1569087212yes—
171C2—82no—
172C2—1941no—
173C2—1941no—
174C2#2026171511yescharacteristic
175C2C2×#1569087211yescharacteristic
176C2#2026173311yescharacteristic
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#1517
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. #20261753C2semidirect
2. #20261679C2semidirect
3. #20261586C2semidirect
12. #15690810C4semidirect
17. D4×Dic97C4semidirect
19. #127×C97C4semidirect
89. C97#1517semidirect
96. #127#87604semidirect

Maximal subgroups

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

subgroupindex
#202617532
#202616792
#202615862
#202612832
#151797