Finite Group Databasebeta

← #27157117

Subgroups — #27157117

Order 12416.

284 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261754C211yesmaximal, characteristic, complemented
2C4×C4⋊#87604C211yesmaximal, characteristic, complemented
3#20261587C214yesmaximal, complemented
4#20261286C211yesmaximal, characteristic
5C2×C4⋊#87604—21no—
6C2×C4×#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9C4×#11715897—21no—
10(C2×#87604)⋊C4C2212yes—
11C2×#13095512—21no—
12#15690807C411yescharacteristic, complemented
13#15690799—21no—
14#15690785—21no—
15#15690785C2212yes—
16#15690772—21no—
17#15690772C412yescomplemented
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4⋊#87604—24no—
23C4×#87604—42no—
24#13095512—42no—
25#13095512—42no—
26C2×D388—22no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388C2×C412yes—
30C2×C4×D97—22no—
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97C2311yescharacteristic
34#13095488—22no—
35#13095488—22no—
36C4×Dic97C2×C411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97D411yescharacteristic
39C2×Q8×C97—21no—
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C42×C97C2×C411yescharacteristic
43C22×D97—22no—
44C22×D97C2×D411yescharacteristic
45C2×#87604—24no—
46C2×#87604—42no—
47C2×#87604—42no—
48Q8×C97—42no—
49C2×C4×C97—22no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C2×D411yescharacteristic
56D388—42no—
57D388—42no—
58D388—42no—
59D388—42no—
60D388—42no—
61C4×D97—22no—
62C4×D97—41no—
63C4×D97—44no—
64C4×D97D812yes—
65C4×D97SD1612yes—
66#11715897—42no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—21no—
69C2×D97—21no—
70C2×D97—42no—
71C2×D97—42no—
72C2×D97—42no—
73C2×D97C2×D811yescharacteristic
74C2×D97C2×SD1611yescharacteristic
75#87604—44no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97#10212yes—
79C4×C97#10412yes—
80Dic97—21no—
81Dic97—21no—
82Dic97#10212yes—
83C2×C97#41011yescharacteristic
84C2×C97C2×#10211yescharacteristic
85C2×C97C2×#10411yescharacteristic
86D97—21no—
87D97—21no—
88D97—82no—
89#1502—971nomaximal, Sylow
90C97#150211yescharacteristic, complemented, Sylow
91#532—971no—
92#431—971no—
93C4×C4⋊C4—971no—
94#327—974no—
95C2×#41—1941no—
96C4⋊Q8#8760411yescharacteristic, complemented
97#127—971no—
98#124—1941no—
99#121—972no—
100#121—1941no—
101C4×D4—972no—
102C4×D4—1941no—
103C2×C4⋊C4—1941no—
104C2×C42—971no—
105C2×C42—1941no—
106C4⋊C8—1941no—
107C22⋊C8—972no—
108C4×C8—1941no—
109#95—972no—
110#41—3882no—
111#41—3882no—
112C2×Q8—21no—
113C2×D4—972no—
114C2×D4—1942no—
115C2×D4—1941no—
116C2×D4—1941no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4—1942no—
121C22×C4—1942no—
122C2×C8—1942no—
123C4⋊C4—21no—
124C4⋊C4—1944no—
125C4⋊C4C2×#8760412yes—
126C22⋊C4—1942no—
127C22⋊C4—1942no—
128C42—971no—
129C42—971no—
130C42—971no—
131C42—3882no—
132C42C2×#8760411yescharacteristic
133C23—971no—
134C23—1942no—
135Q8—42no—
136D4—3882no—
137D4—3882no—
138D4—3882no—
139D4—3882no—
140D4—3882no—
141C2×C4—22no—
142C2×C4—971no—
143C2×C4—972no—
144C2×C4—972no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—3881no—
148C2×C4—3884no—
149C2×C4—3882no—
150C2×C4—3882no—
151C2×C4—1944no—
152C2×C4—1942no—
153C2×C4C22⋊#8760411yescharacteristic
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22×#8760411yescharacteristic
156C8—3882no—
157C22—971no—
158C22—971no—
159C22—3882no—
160C22—3882no—
161C22—3882no—
162C22—1941no—
163C22—1941no—
164C22C2×C22⋊#8760411yescharacteristic
165C4—21no—
166C4—42no—
167C4—972no—
168C4—3884no—
169C4—1941no—
170C4—1941no—
171C4#1569087212yes—
172C4#1569087512yes—
173C2—7762no—
174C2—1941no—
175C2—1941no—
176C2#2026171611yescharacteristic
177C2C2×#1569087211yescharacteristic
178C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1502
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261754C2semidirect
2. C4×C4⋊#87604C2semidirect
3. #20261587C2semidirect
12. #15690807C4semidirect
17. #15690772C4semidirect
19. C4⋊Q8×C97C4semidirect
90. C97#1502semidirect
96. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617542
C4×C4⋊#876042
#202615872
#202612862
#150297