Finite Group Databasebeta

← #27158013

Subgroups — #27158013

Order 12416.

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#15690869C212yesmaximal, complemented
2#20261642C212yesmaximal, complemented
3#20261640C212yesmaximal, complemented
4#20261304C211yesmaximal, characteristic
5C2×C22⋊#87604—22no—
6#15690869—22no—
7#15690869C2214yes—
8C2×#13095511C411yescharacteristic, complemented
9C22×D388C2211yescharacteristic
10#15690807C411yescharacteristic, complemented
11#15690807C2211yescharacteristic
12#15690798C411yescharacteristic, complemented
13#15690798C2211yescharacteristic
14#15690785—22no—
15#15690780—22no—
16#117×C97C411yescharacteristic, complemented
17C22×#87604—42no—
18C22⋊#87604—24no—
19C22⋊#87604—42no—
20C23×D97D412yes—
21#13095511—42no—
22C2×Dic194C2×C411yescharacteristic, complemented
23C2×D388—22no—
24C2×D388—22no—
25C2×D388—42no—
26C2×D388C2×C411yescharacteristic
27C2×D388C2×C411yescharacteristic, complemented
28C2×D388D412yes—
29C2×D388C2311yescharacteristic
30C2×C22⋊D97—21no—
31C2×C4×D97—21no—
32#13095488—22no—
33#13095488—22no—
34C22×C4×C97C2×C411yescharacteristic
35C4⋊C4×C97—21no—
36C22⋊C4×C97—21no—
37C42×C97C2×C411yescharacteristic, complemented
38C42×C97C2×C411yescharacteristic, complemented
39C23×C97C22⋊C411yescharacteristic
40C22×D97—22no—
41C22×D97—22no—
42C22×D97—21no—
43C22×D97—82no—
44C22×D97C22⋊C412yes—
45C22×D97C2×D412yes—
46C2×#87604—44no—
47C2×#87604—84no—
48C2×C4×C97—22no—
49C2×C4×C97—21no—
50C2×C4×C97—21no—
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C22⋊D97—42no—
54C2×Dic97—21no—
55D388—41no—
56D388—42no—
57D388—41no—
58D388—41no—
59D388—82no—
60D388—82no—
61C4×D97—42no—
62Dic194—41no—
63C22×C97—41no—
64C22×C97#9912yes—
65C22×C97C2×C22⋊C411yescharacteristic
66C2×D97—22no—
67C2×D97—22no—
68C2×D97—44no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—42no—
72C2×D97—41no—
73C2×D97—82no—
74#87604—84no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79Dic97—41no—
80C2×C97—21no—
81C2×C97—22no—
82C2×C97C2×#9911yescharacteristic
83D97—42no—
84D97—42no—
85D97—81no—
86#2127—971nomaximal, Sylow
87C97#212711yescharacteristic, complemented, Sylow
88#535—971no—
89C2×#99—972no—
90#354—972no—
91#353—972no—
92C2×#41—971no—
93C22×D4—971no—
94#127—971no—
95#127—971no—
96#124—971no—
97#124—971no—
98#121—1942no—
99#120—1942no—
100#117#8760411yescharacteristic, complemented
101C2×C22⋊C4—1942no—
102#99—974no—
103#99—1942no—
104C24—972no—
105#41—3882no—
106C2×Q8—971no—
107C2×D4—972no—
108C2×D4—971no—
109C2×D4—971no—
110C2×D4—971no—
111C2×D4—3882no—
112C2×D4—1942no—
113C2×D4—1942no—
114C2×D4—1941no—
115C22×C4—3882no—
116C22×C4—1941no—
117C22×C4C2×#8760411yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—3882no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C22⋊C4—1944no—
124C42C2×#8760411yescharacteristic, complemented
125C42C2×#8760411yescharacteristic, complemented
126C23—7762no—
127C23—972no—
128C23—972no—
129C23—1942no—
130C23—1941no—
131C23—1942no—
132C23C22⋊#8760411yescharacteristic
133Q8—3881no—
134D4—7762no—
135D4—7762no—
136D4—3881no—
137D4—3882no—
138D4—3881no—
139D4—3881no—
140D4—3882no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—7764no—
145C2×C4—3882no—
146C2×C4—3884no—
147C2×C4—1941no—
148C2×C4C22⋊#8760411yescharacteristic
149C2×C4C22×#8760411yescharacteristic
150C22—41no—
151C22—7762no—
152C22—3882no—
153C22—3882no—
154C22—3882no—
155C22—3884no—
156C22—3881no—
157C22—1942no—
158C22—1942no—
159C22#1569086912yes—
160C22C2×C22⋊#8760411yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—7764no—
166C4—3881no—
167C2—21no—
168C2—22no—
169C2—7761no—
170C2—3882no—
171C2—3882no—
172C2C2×#1569086911yescharacteristic
subgroupG/H
centerC2C2×#15690869
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#99
prime radical#20261304C2
cosocleC2×D388C23
hypercenter#117#87604
hypocenterC97#2127
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#15690869C2semidirect
2. #20261642C2semidirect
3. #20261640C2semidirect
8. C2×#13095511C4semidirect
10. #15690807C4semidirect
12. #15690798C4semidirect
16. #117×C97C4semidirect
22. C2×Dic194C2×C4semidirect
27. C2×D388C2×C4semidirect
37. C42×C97C2×C4semidirect
38. C42×C97C2×C4semidirect
87. C97#2127semidirect
100. #117#87604semidirect
124. C42C2×#87604semidirect
125. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908692
#202616422
#202616402
#202613042
#212797