Finite Group Databasebeta

← C3×#87574 #11919279

Subgroups — C3×#87574 #11919279

Order 1152.

12389 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#585C211yesmaximal, characteristic, complemented
2C3×#5968C2112yesmaximal, complemented
3C3×D3×H(5,𝔽2)C2120yesmaximal, complemented
4C3×#5964C2115yesmaximal, complemented
5C2×C3×#1009C2115yesmaximal, complemented
6C3×H(7,𝔽2)—31nomaximal
7#87574C311yesmaximal, characteristic, direct product factor
8C32×2−1+4C2216yescomplemented
9C32×H(5,𝔽2)C22110yescomplemented
10C2×C32×#41C22115yescomplemented
11C3×#1010C22120yescomplemented
12C3×#1009C22160yescomplemented
13C3×#1009C221120yescomplemented
14C3×D3×#41C221120yescomplemented
15C3×#1007C22130yescomplemented
16C2×C3×#239C22130yescomplemented
17C3×#1004C22190yescomplemented
18C2×C3×D3×D4C22190yescomplemented
19C2×C3×#235C22145yescomplemented
20C22×C3×D12C22115yescomplemented
21C3×#585—21no—
22C3×#585—312no—
23C3×#585—315no—
24C3×#585D311yescharacteristic, complemented
25C3×#585C2×C311yescharacteristic, complemented
26C2×C3×H(5,𝔽2)—320no—
27C2×C3×H(5,𝔽2)—315no—
28#5968C2×C3112yescomplemented
29D3×H(5,𝔽2)C2×C3120yescomplemented
30#5964C2×C3115yescomplemented
31C2×#1009C2×C3115yescomplemented
32C23×C3×D3C23130yescomplemented
33C32×#41C23160yescomplemented
34C32×#41C23120yescomplemented
35C2×C32×Q8C23115yescomplemented
36C2×C32×D4C23145yescomplemented
37C22×C32×C4C23115yescomplemented
38C2×C3×#67C23190yescomplemented
39C3×#239C231120yescomplemented
40C3×D3×Q8C23140yescomplemented
41C3×#237C231120yescomplemented
42C3×D3×D4C231360yescomplemented
43C3×#235C23160yescomplemented
44C3×#235C231180yescomplemented
45C2×C3×D12C23145yescomplemented
46C2×C3×D12C23190yescomplemented
47C2×C3×C4×D3C23190yescomplemented
48C2×C3×Dic6C23115yescomplemented
49H(7,𝔽2)—31noSylow
50C3×2−1+4—26no—
51C3×2−1+4—320no—
52C3×2−1+4—330no—
53C3×2−1+4C2×D316yescomplemented
54C3×2−1+4C22×C316yescomplemented
55C3×H(5,𝔽2)—210no—
56C3×H(5,𝔽2)—3120no—
57C3×H(5,𝔽2)—360no—
58C3×H(5,𝔽2)—390no—
59C3×H(5,𝔽2)C2×D3110yescomplemented
60C3×H(5,𝔽2)C22×C3110yescomplemented
61C2×C3×#41—215no—
62C2×C3×#41—3120no—
63C2×C3×#41—330no—
64C2×C3×#41—345no—
65C2×C3×#41C2×D3115yescomplemented
66C2×C3×#41C22×C3115yescomplemented
67C22×C3×D4—390no—
68C22×C3×D4—315no—
69#1010C22×C3120yescomplemented
70#1009C22×C3160yescomplemented
71#1009C22×C31120yescomplemented
72D3×#41C22×C31120yescomplemented
73#1007C22×C3130yescomplemented
74C2×#239C22×C3130yescomplemented
75#1004C22×C3190yescomplemented
76C2×D3×D4C22×C3190yescomplemented
77C2×#235C22×C3145yescomplemented
78C22×D12C22×C3115yescomplemented
79C23×C32C24115yes—
80C22×C3×D3—830no—
81C22×C3×D3C24190yes—
82C32×Q8C24120yes—
83C32×D4C24160yes—
84C2×C32×C4C24115yes—
85C2×C32×C4C24145yes—
86C3×#67C241120yes—
87C2×C3×Dic3C24130yes—
88C3×D12C24110yes—
89C3×D12C24190yes—
90C3×C4×D3C241120yes—
91C3×Dic6C2416yes—
92C3×Dic6C24130yes—
93#585—312no—
94#585—315no—
95#585C3×D311yescharacteristic, complemented
96C2×H(5,𝔽2)—320no—
97C2×H(5,𝔽2)—315no—
98C24×C3—330no—
99C23×D3C23×C3130yescomplemented
100C3×#41—220no—
101C3×#41—260no—
102C3×#41—3120no—
103C3×#41—3120no—
104C3×#41—3180no—
105C3×#41—360no—
106C3×#41C22×D3160yescomplemented
107C3×#41C22×D3120yescomplemented
108C3×#41C23×C3160yescomplemented
109C3×#41C23×C3120yescomplemented
110C2×C3×Q8—215no—
111C2×C3×Q8—340no—
112C2×C3×Q8—315no—
113C2×C3×Q8C22×D3115yescomplemented
114C2×C3×Q8C23×C3115yescomplemented
115C2×C3×D4—245no—
116C2×C3×D4—390no—
117C2×C3×D4—3360no—
118C2×C3×D4—390no—
119C2×C3×D4—345no—
120C2×C3×D4C22×D3145yescomplemented
121C2×C3×D4C23×C3145yescomplemented
122C22×C3×C4—215no—
123C22×C3×C4—390no—
124C22×C3×C4C22×D3115yescomplemented
125C22×C3×C4C23×C3115yescomplemented
126C2×#67C23×C3190yescomplemented
127#239C23×C31120yescomplemented
128D3×Q8C23×C3140yescomplemented
129#237C23×C31120yescomplemented
130D3×D4C23×C31360yescomplemented
131#235C23×C3160yescomplemented
132#235C23×C31180yescomplemented
133C2×D12C23×C3145yescomplemented
134C2×D12C23×C3190yescomplemented
135C2×C4×D3C23×C3190yescomplemented
136C2×Dic6C23×C3115yescomplemented
137C22×C32—415no—
138C22×C32C25115yes—
139C2×C3×D3—490no—
140C2×C3×D3C25120yes—
141C32×C4C2511yescharacteristic
142C32×C4C25115yes—
143C3×Dic3C25112yes—
1442−1+4—320no—
1452−1+4—330no—
1462−1+4C2×C3×D316yescomplemented
147H(5,𝔽2)—3120no—
148H(5,𝔽2)—360no—
149H(5,𝔽2)—390no—
150H(5,𝔽2)C2×C3×D3110yescomplemented
151C2×#41—3120no—
152C2×#41—330no—
153C2×#41—345no—
154C2×#41C2×C3×D3115yescomplemented
155C22×D4—390no—
156C22×D4—315no—
157C23×C3—215no—
158C23×C3—390no—
159C23×C3—2430no—
160C23×C3C23×D3115yes—
161C23×C3C24×C3115yes—
162C22×D3—830no—
163C22×D3C24×C3190yes—
164C3×Q8—220no—
165C3×Q8—330no—
166C3×Q8—36no—
167C3×Q8C23×D3120yes—
168C3×Q8C24×C3120yes—
169C3×D4—260no—
170C3×D4—3120no—
171C3×D4—390no—
172C3×D4—310no—
173C3×D4C23×D3160yes—
174C3×D4C24×C3160yes—
175C2×C3×C4—215no—
176C2×C3×C4—245no—
177C2×C3×C4—330no—
178C2×C3×C4—3120no—
179C2×C3×C4C23×D3145yes—
180C2×C3×C4C23×D3115yes—
181C2×C3×C4C24×C3115yes—
182C2×C3×C4C24×C3145yes—
183#67C24×C31120yes—
184C2×Dic3C24×C3130yes—
185D12C24×C3110yes—
186D12C24×C3190yes—
187C4×D3C24×C31120yes—
188Dic6C24×C316yes—
189Dic6C24×C3130yes—
190C2×C32—215no—
191C2×C32C2611yescharacteristic
192C3×D3—220no—
193C24—330no—
194#41—3120no—
195#41—3120no—
196#41—3180no—
197#41—360no—
198#41C22×C3×D3160yescomplemented
199#41C22×C3×D3120yescomplemented
200C2×Q8—340no—
201C2×Q8—315no—
202C2×Q8C22×C3×D3115yescomplemented
203C2×D4—390no—
204C2×D4—3360no—
205C2×D4—390no—
206C2×D4—345no—
207C2×D4C22×C3×D3145yescomplemented
208C22×C4—390no—
209C22×C4C22×C3×D3115yescomplemented
210C22×C3—215no—
211C22×C3—320no—
212C22×C3—415no—
213C22×C3—415no—
214C22×C3—815no—
215C22×C3—1290no—
216C22×C3C24×D3115yes—
217C22×C3C25×C3115yes—
218C2×D3—490no—
219C2×D3C25×C3120yes—
220C3×C4—21no—
221C3×C4—215no—
222C3×C4—312no—
223C3×C4C24×D3115yes—
224C3×C4C24×D311yescharacteristic
225C3×C4C25×C311yescharacteristic
226C3×C4C25×C3115yes—
227Dic3C25×C3112yes—
228C32H(7,𝔽2)11yescharacteristic, complemented, Sylow
229C23—390no—
230C23—2430no—
231C23C23×C3×D3115yes—
232Q8—330no—
233Q8—36no—
234Q8C23×C3×D3120yes—
235D4—3120no—
236D4—390no—
237D4—310no—
238D4C23×C3×D3160yes—
239C2×C4—330no—
240C2×C4—3120no—
241C2×C4C23×C3×D3145yes—
242C2×C4C23×C3×D3115yes—
243C2×C3—215no—
244C2×C3—215no—
245C2×C3—21no—
246C2×C3—415no—
247C2×C3—620no—
248C2×C3C25×D311yescharacteristic
249C2×C3C26×C311yescharacteristic
250D3—220no—
251C22—320no—
252C22—415no—
253C22—1290no—
254C22C24×C3×D3115yes—
255C4—312no—
256C4C24×C3×D3115yes—
257C4C24×C3×D311yescharacteristic
258C3—21no—
259C3#8757411yescharacteristic, direct product factor
260C3C3×H(7,𝔽2)11yescharacteristic, complemented
261C2—215no—
262C2—620no—
263C2C25×C3×D311yescharacteristic
subgroupG/H
centerC2×C3C25×D3
commutator, cosocleC2×C3C26×C3
FrattiniC2C25×C3×D3
FittingC32×#585C2
socleC2×C32C26
hypercenterC3×#585D3
hypocenterC3C3×H(7,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C3×#585

Complements

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

NH ≅ G/Ntype
1. C32×#585C2semidirect
2. C3×#5968C2semidirect
3. C3×D3×H(5,𝔽2)C2semidirect
4. C3×#5964C2semidirect
5. C2×C3×#1009C2semidirect
7. #87574C3direct (N × H)
8. C32×2−1+4C22semidirect
9. C32×H(5,𝔽2)C22semidirect
10. C2×C32×#41C22semidirect
11. C3×#1010C22semidirect
12. C3×#1009C22semidirect
13. C3×#1009C22semidirect
14. C3×D3×#41C22semidirect
15. C3×#1007C22semidirect
16. C2×C3×#239C22semidirect
17. C3×#1004C22semidirect
18. C2×C3×D3×D4C22semidirect
19. C2×C3×#235C22semidirect
20. C22×C3×D12C22semidirect
24. C3×#585D3semidirect
25. C3×#585C2×C3semidirect
28. #5968C2×C3semidirect
29. D3×H(5,𝔽2)C2×C3semidirect
30. #5964C2×C3semidirect
31. C2×#1009C2×C3semidirect
32. C23×C3×D3C23semidirect
33. C32×#41C23semidirect
34. C32×#41C23semidirect
35. C2×C32×Q8C23semidirect
36. C2×C32×D4C23semidirect
37. C22×C32×C4C23semidirect
38. C2×C3×#67C23semidirect
39. C3×#239C23semidirect
40. C3×D3×Q8C23semidirect
41. C3×#237C23semidirect
42. C3×D3×D4C23semidirect
43. C3×#235C23semidirect
44. C3×#235C23semidirect
45. C2×C3×D12C23semidirect
46. C2×C3×D12C23semidirect
47. C2×C3×C4×D3C23semidirect
48. C2×C3×Dic6C23semidirect
53. C3×2−1+4C2×D3semidirect
54. C3×2−1+4C22×C3semidirect
59. C3×H(5,𝔽2)C2×D3semidirect
60. C3×H(5,𝔽2)C22×C3semidirect
65. C2×C3×#41C2×D3semidirect
66. C2×C3×#41C22×C3semidirect
69. #1010C22×C3semidirect
70. #1009C22×C3semidirect
71. #1009C22×C3semidirect
72. D3×#41C22×C3semidirect
73. #1007C22×C3semidirect
74. C2×#239C22×C3semidirect
75. #1004C22×C3semidirect
76. C2×D3×D4C22×C3semidirect
77. C2×#235C22×C3semidirect
78. C22×D12C22×C3semidirect
95. #585C3×D3semidirect
99. C23×D3C23×C3semidirect
106. C3×#41C22×D3semidirect
107. C3×#41C22×D3semidirect
108. C3×#41C23×C3semidirect
109. C3×#41C23×C3semidirect
113. C2×C3×Q8C22×D3semidirect
114. C2×C3×Q8C23×C3semidirect
120. C2×C3×D4C22×D3semidirect
121. C2×C3×D4C23×C3semidirect
124. C22×C3×C4C22×D3semidirect
125. C22×C3×C4C23×C3semidirect
126. C2×#67C23×C3semidirect
127. #239C23×C3semidirect
128. D3×Q8C23×C3semidirect
129. #237C23×C3semidirect
130. D3×D4C23×C3semidirect
131. #235C23×C3semidirect
132. #235C23×C3semidirect
133. C2×D12C23×C3semidirect
134. C2×D12C23×C3semidirect
135. C2×C4×D3C23×C3semidirect
136. C2×Dic6C23×C3semidirect
146. 2−1+4C2×C3×D3semidirect
150. H(5,𝔽2)C2×C3×D3semidirect
154. C2×#41C2×C3×D3semidirect
198. #41C22×C3×D3semidirect
199. #41C22×C3×D3semidirect
202. C2×Q8C22×C3×D3semidirect
207. C2×D4C22×C3×D3semidirect
209. C22×C4C22×C3×D3semidirect
228. C32H(7,𝔽2)semidirect
259. C3#87574direct (N × H)
260. C3C3×H(7,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C32×#5852
C3×#59682
C3×D3×H(5,𝔽2)2
C3×#59642
C2×C3×#10092
C3×H(7,𝔽2)3
#875743