Finite Group Databasebeta

← C3×#87537 #11919884

Subgroups — C3×#87537 #11919884

Order 1152.

2732 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 2732 conjugacy classes of subgroups — too many to draw. Only its 538 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
1C3×#5945C216yesmaximal, complemented
2C2×C3×2−1+4⋊C3C219yesmaximal, complemented
3C3×2−1+6C311yesmaximal, characteristic, complemented
4#87537C313yesmaximal, direct product factor
5C32×H(5,𝔽2)—41nomaximal
6C3×2−1+4⋊C3C22118yescomplemented
7C3×#994C2212yescomplemented
8C2×C3×#41⋊C3C2219yescomplemented
9C22×C3×SL(2,𝔽3)C2216yescomplemented
10C3×#585—31no—
11C3×#585—39no—
12C3×#585C2×C316yescomplemented
13C2×C3×2−1+4—36no—
14C2×C3×2−1+4C2×C319yescomplemented
15#5945C2×C3118yescomplemented
16C2×2−1+4⋊C3C2×C3127yescomplemented
17C32×#41—46no—
18C2×C32×D4—49no—
19C3×#41⋊C3C2316yes—
20C2×C3×SL(2,𝔽3)C2319yes—
212−1+6C3211yescharacteristic, complemented, Sylow
22C3×2−1+4—336no—
23C3×2−1+4—318no—
24C3×2−1+4—36no—
25C3×2−1+4—36no—
26C3×2−1+4C22×C3118yescomplemented
27C3×H(5,𝔽2)—312no—
28C3×H(5,𝔽2)—39no—
29C3×H(5,𝔽2)—318no—
30C3×H(5,𝔽2)—43no—
31C3×H(5,𝔽2)A411yescharacteristic
32C3×H(5,𝔽2)C22×C312yescomplemented
33C2×C3×#41—318no—
34C2×C3×#41—336no—
35C2×C3×#41—36no—
36C2×C3×#41—39no—
37C2×C3×#41—318no—
38C2×C3×#41C22×C319yescomplemented
39C22×C3×Q8—39no—
40C22×C3×Q8—34no—
41C22×C3×Q8C22×C316yescomplemented
422−1+4⋊C3C22×C3154yescomplemented
43#994C22×C316yescomplemented
44C2×#41⋊C3C22×C3127yescomplemented
45C22×SL(2,𝔽3)C22×C3118yescomplemented
46C23×C32—46no—
47C32×Q8—42no—
48C32×D4—418no—
49C2×C32×C4—49no—
50C3×SL(2,𝔽3)C2411yescharacteristic
51#585—31no—
52#585—39no—
53#585C2×C3216yescomplemented
54C2×2−1+4—36no—
55C2×2−1+4C2×C3219yescomplemented
56C3×#41—372no—
57C3×#41—336no—
58C3×#41—32no—
59C3×#41—336no—
60C3×#41—318no—
61C3×#41—336no—
62C3×#41—318no—
63C3×#41—318no—
64C3×#41—418no—
65C3×#41C2×A416yes—
66C3×#41C23×C316yes—
67C2×C3×Q8—318no—
68C2×C3×Q8—336no—
69C2×C3×Q8—36no—
70C2×C3×Q8—39no—
71C2×C3×Q8—318no—
72C2×C3×Q8C23×C319yes—
73C2×C3×D4—39no—
74C2×C3×D4—336no—
75C2×C3×D4—318no—
76C2×C3×D4—318no—
77C2×C3×D4—36no—
78C2×C3×D4—427no—
79C2×C3×D4C2×A419yes—
80C22×C3×C4—318no—
81C22×C3×C4—312no—
82C22×C3×C4—36no—
83C22×C3×C4—39no—
84#41⋊C3C23×C3118yes—
85C2×SL(2,𝔽3)C23×C3127yes—
86C22×C32—49no—
87C22×C32—166no—
88C32×C4—46no—
892−1+4—336no—
902−1+4—318no—
912−1+4—36no—
922−1+4—36no—
932−1+4C22×C32118yescomplemented
94H(5,𝔽2)—312no—
95H(5,𝔽2)—39no—
96H(5,𝔽2)—318no—
97H(5,𝔽2)C3×A411yescharacteristic
98H(5,𝔽2)C22×C3212yescomplemented
99C2×#41—318no—
100C2×#41—336no—
101C2×#41—36no—
102C2×#41—39no—
103C2×#41—318no—
104C2×#41C22×C3219yescomplemented
105C22×Q8—39no—
106C22×Q8—34no—
107C22×Q8C22×C3216yescomplemented
108C23×C3—34no—
109C23×C3—39no—
110C23×C3—418no—
111C23×C3C22×A416yes—
112C3×Q8—39no—
113C3×Q8—312no—
114C3×Q8—318no—
115C3×Q8—46no—
116C3×Q8C22×A412yes—
117C3×Q8C24×C311yescharacteristic
118C3×D4—36no—
119C3×D4—318no—
120C3×D4—336no—
121C3×D4—36no—
122C3×D4—454no—
123C3×D4C22×A4118yes—
124C2×C3×C4—336no—
125C2×C3×C4—39no—
126C2×C3×C4—318no—
127C2×C3×C4—318no—
128C2×C3×C4—36no—
129C2×C3×C4—427no—
130C2×C3×C4C22×A419yes—
131SL(2,𝔽3)C24×C313yes—
132C2×C32—41no—
133C2×C32—89no—
134#41—372no—
135#41—336no—
136#41—32no—
137#41—318no—
138#41—336no—
139#41—336no—
140#41—318no—
141#41—318no—
142#41C2×C3×A416yes—
143#41C23×C3216yes—
144C2×Q8—318no—
145C2×Q8—336no—
146C2×Q8—36no—
147C2×Q8—39no—
148C2×Q8—318no—
149C2×Q8C23×C3219yes—
150C2×D4—39no—
151C2×D4—336no—
152C2×D4—318no—
153C2×D4—318no—
154C2×D4—36no—
155C2×D4C2×C3×A419yes—
156C22×C4—318no—
157C22×C4—312no—
158C22×C4—36no—
159C22×C4—39no—
160C22×C3—36no—
161C22×C3—46no—
162C22×C3—427no—
163C22×C3—129no—
164C22×C3—124no—
165C22×C3—1618no—
166C22×C3C23×A419yes—
167C3×C4—39no—
168C3×C4—31no—
169C3×C4—418no—
170C3×C4C23×A416yes—
171C32—41noSylow
172C23—34no—
173C23—39no—
174C23C22×C3×A416yes—
175Q8—39no—
176Q8—312no—
177Q8—318no—
178Q8C22×C3×A412yes—
179Q8C24×C3211yescharacteristic
180D4—36no—
181D4—318no—
182D4—336no—
183D4—36no—
184D4C22×C3×A4118yes—
185C2×C4—336no—
186C2×C4—39no—
187C2×C4—318no—
188C2×C4—36no—
189C2×C4—318no—
190C2×C4C22×C3×A419yes—
191C2×C3—29no—
192C2×C3—43no—
193C2×C3—66no—
194C2×C3—827no—
195C2×C3C24×A411yescharacteristic
196C22—36no—
197C22—46no—
198C22—129no—
199C22—124no—
200C22C23×C3×A419yes—
201C4—31no—
202C4—39no—
203C4C23×C3×A416yes—
204C3—43no—
205C3#8753711yescharacteristic, direct product factor
206C2—29no—
207C2—66no—
208C2C24×C3×A411yescharacteristic
subgroupG/H
center, socleC2×C3C24×A4
commutator, cosocle, hypocenterQ8C24×C32
FrattiniC2C24×C3×A4
FittingC3×2−1+6C3
hypercenterC3×H(5,𝔽2)A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C3×#5945C2semidirect
2. C2×C3×2−1+4⋊C3C2semidirect
3. C3×2−1+6C3semidirect
4. #87537C3direct (N × H)
6. C3×2−1+4⋊C3C22semidirect
7. C3×#994C22semidirect
8. C2×C3×#41⋊C3C22semidirect
9. C22×C3×SL(2,𝔽3)C22semidirect
12. C3×#585C2×C3semidirect
14. C2×C3×2−1+4C2×C3semidirect
15. #5945C2×C3semidirect
16. C2×2−1+4⋊C3C2×C3semidirect
21. 2−1+6C32semidirect
26. C3×2−1+4C22×C3semidirect
32. C3×H(5,𝔽2)C22×C3semidirect
38. C2×C3×#41C22×C3semidirect
41. C22×C3×Q8C22×C3semidirect
42. 2−1+4⋊C3C22×C3semidirect
43. #994C22×C3semidirect
44. C2×#41⋊C3C22×C3semidirect
45. C22×SL(2,𝔽3)C22×C3semidirect
53. #585C2×C32semidirect
55. C2×2−1+4C2×C32semidirect
93. 2−1+4C22×C32semidirect
98. H(5,𝔽2)C22×C32semidirect
104. C2×#41C22×C32semidirect
107. C22×Q8C22×C32semidirect
205. C3#87537direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#59452
C2×C3×2−1+4⋊C32
C3×2−1+63
#875373
C32×H(5,𝔽2)4