Subgroups — C₃×#3508 #87392

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#559	C₂	1	4	yes	maximal, complemented
2	C₃×#550	C₂	1	4	yes	maximal, complemented
3	C₃×#546	C₂	1	8	yes	maximal, complemented
4	C₃×#540	C₂	1	4	yes	maximal, complemented
5	C₃×#535	C₂	1	2	yes	maximal, complemented
6	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#532	C₂	1	4	yes	maximal, complemented
8	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#518	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
11	#3508	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂×C₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₃×#41	C₂²	1	8	yes	complemented
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×D₄	C₂²	1	4	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₃×#127	C₂²	1	4	yes	—
18	C₃×#127	C₂²	1	2	yes	—
19	C₃×#127	C₂²	1	2	yes	—
20	C₃×#127	C₂²	1	4	yes	complemented
21	C₃×#125	C₂²	1	2	yes	complemented
22	C₃×#124	C₂²	1	4	yes	complemented
23	C₃×#124	C₂²	1	8	yes	complemented
24	C₃×#123	C₂²	1	16	yes	complemented
25	C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₃×#121	C₂²	1	8	yes	complemented
27	C₃×#121	C₂²	1	8	yes	complemented
28	C₃×#121	C₂²	1	16	yes	complemented
29	C₃×#120	C₂²	1	8	yes	—
30	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
31	C₃×C₄×Q₈	C₂²	1	4	yes	complemented
32	C₃×C₄×D₄	C₂²	1	4	yes	—
33	C₃×C₄×D₄	C₂²	1	8	yes	complemented
34	C₃×C₄×D₄	C₂²	1	8	yes	complemented
35	C₃×C₄×D₄	C₂²	1	2	yes	complemented
36	C₃×C₄×D₄	C₂²	1	8	yes	complemented
37	C₃×#117	C₂²	1	1	yes	characteristic
38	C₃×#117	C₂²	1	2	yes	—
39	C₃×#117	C₂²	1	2	yes	complemented
40	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
41	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
43	#559	C₂×C₃	1	4	yes	complemented
44	#550	C₂×C₃	1	4	yes	complemented
45	#546	C₂×C₃	1	8	yes	complemented
46	#540	C₂×C₃	1	4	yes	complemented
47	#535	C₂×C₃	1	2	yes	complemented
48	#533	C₂×C₃	1	1	yes	characteristic, complemented
49	#532	C₂×C₃	1	4	yes	complemented
50	C₂×#127	C₂×C₃	1	1	yes	characteristic, complemented
51	#518	C₂×C₃	1	2	yes	complemented
52	C₄×#41	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂⁴×C₃	C₂³	1	4	yes	—
54	C₃×#41	—	2	4	no	—
55	C₃×#41	—	4	16	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	4	yes	—
58	C₂×C₃×D₄	—	2	8	no	—
59	C₂×C₃×D₄	—	4	8	no	—
60	C₂×C₃×D₄	C₂³	1	2	yes	—
61	C₂×C₃×D₄	C₂³	1	4	yes	—
62	C₂×C₃×D₄	C₂³	1	4	yes	—
63	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
64	C₂×C₃×D₄	C₂³	1	8	yes	—
65	C₂×C₃×D₄	C₂³	1	16	yes	—
66	C₂×C₃×D₄	C₂³	1	8	yes	—
67	C₂²×C₃×C₄	C₂³	1	2	yes	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₂²×C₃×C₄	C₂³	1	4	yes	—
70	C₂²×C₃×C₄	C₂³	1	16	yes	—
71	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₄⋊C₄	C₂³	1	4	yes	—
74	C₃×C₄⋊C₄	C₂³	1	4	yes	—
75	C₃×C₄⋊C₄	C₂³	1	4	yes	—
76	C₃×C₄⋊C₄	C₂³	1	8	yes	—
77	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
78	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
79	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
80	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
81	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
82	C₃×C₄²	C₂³	1	1	yes	characteristic
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₃×C₄²	C₂³	1	2	yes	—
85	C₃×C₄²	C₂³	1	2	yes	—
86	C₃×C₄²	C₂³	1	4	yes	—
87	C₃×C₄²	C₂³	1	4	yes	—
88	C₂×#41	C₂²×C₃	1	1	yes	characteristic
89	C₂×#41	C₂²×C₃	1	8	yes	complemented
90	C₂²×D₄	C₂²×C₃	1	2	yes	—
91	C₂²×D₄	C₂²×C₃	1	4	yes	—
92	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
93	#127	C₂²×C₃	1	2	yes	—
94	#127	C₂²×C₃	1	2	yes	—
95	#127	C₂²×C₃	1	4	yes	—
96	#127	C₂²×C₃	1	4	yes	complemented
97	#125	C₂²×C₃	1	2	yes	complemented
98	#124	C₂²×C₃	1	8	yes	complemented
99	#124	C₂²×C₃	1	4	yes	complemented
100	#123	C₂²×C₃	1	16	yes	complemented
101	C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
102	#121	C₂²×C₃	1	8	yes	complemented
103	#121	C₂²×C₃	1	8	yes	complemented
104	#121	C₂²×C₃	1	16	yes	complemented
105	#120	C₂²×C₃	1	8	yes	—
106	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
107	C₄×Q₈	C₂²×C₃	1	4	yes	complemented
108	C₄×D₄	C₂²×C₃	1	4	yes	—
109	C₄×D₄	C₂²×C₃	1	2	yes	complemented
110	C₄×D₄	C₂²×C₃	1	8	yes	complemented
111	C₄×D₄	C₂²×C₃	1	8	yes	complemented
112	C₄×D₄	C₂²×C₃	1	8	yes	complemented
113	#117	C₂²×C₃	1	1	yes	characteristic
114	#117	C₂²×C₃	1	2	yes	—
115	#117	C₂²×C₃	1	2	yes	complemented
116	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
117	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
119	C₂³×C₃	—	2	8	no	—
120	C₂³×C₃	—	4	8	no	—
121	C₂³×C₃	C₂⁴	1	8	yes	—
122	C₂³×C₃	C₂⁴	1	2	yes	—
123	C₂³×C₃	C₂⁴	1	1	yes	characteristic
124	C₃×Q₈	—	2	2	no	—
125	C₃×Q₈	—	2	8	no	—
126	C₃×D₄	—	2	4	no	—
127	C₃×D₄	—	2	8	no	—
128	C₃×D₄	—	2	8	no	—
129	C₃×D₄	—	2	2	no	—
130	C₃×D₄	—	2	16	no	—
131	C₃×D₄	—	4	8	no	—
132	C₃×D₄	—	4	16	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	4	16	no	—
137	C₂×C₃×C₄	#41	1	4	yes	complemented
138	C₂×C₃×C₄	C₂⁴	1	2	yes	—
139	C₂×C₃×C₄	C₂⁴	1	4	yes	—
140	C₂×C₃×C₄	C₂⁴	1	2	yes	—
141	C₂×C₃×C₄	C₂⁴	1	8	yes	—
142	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂⁴	1	2	yes	—
145	C₂⁴	C₂³×C₃	1	4	yes	—
146	#41	—	2	4	no	—
147	#41	—	4	16	no	—
148	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
149	C₂×Q₈	C₂³×C₃	1	4	yes	—
150	C₂×D₄	—	2	8	no	—
151	C₂×D₄	—	4	8	no	—
152	C₂×D₄	C₂³×C₃	1	2	yes	—
153	C₂×D₄	C₂³×C₃	1	4	yes	—
154	C₂×D₄	C₂³×C₃	1	4	yes	—
155	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
156	C₂×D₄	C₂³×C₃	1	8	yes	—
157	C₂×D₄	C₂³×C₃	1	16	yes	—
158	C₂×D₄	C₂³×C₃	1	8	yes	—
159	C₂²×C₄	C₂³×C₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₃	1	4	yes	—
162	C₂²×C₄	C₂³×C₃	1	16	yes	—
163	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
164	C₄⋊C₄	C₂³×C₃	1	2	yes	—
165	C₄⋊C₄	C₂³×C₃	1	4	yes	—
166	C₄⋊C₄	C₂³×C₃	1	4	yes	—
167	C₄⋊C₄	C₂³×C₃	1	4	yes	—
168	C₄⋊C₄	C₂³×C₃	1	8	yes	—
169	C₂²⋊C₄	C₂³×C₃	1	2	yes	—
170	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
171	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
172	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
173	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
174	C₄²	C₂³×C₃	1	1	yes	characteristic
175	C₄²	C₂³×C₃	1	1	yes	characteristic
176	C₄²	C₂³×C₃	1	2	yes	—
177	C₄²	C₂³×C₃	1	2	yes	—
178	C₄²	C₂³×C₃	1	4	yes	—
179	C₄²	C₂³×C₃	1	4	yes	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	8	no	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	16	no	—
185	C₂²×C₃	—	8	8	no	—
186	C₂²×C₃	C₂×#41	1	2	yes	—
187	C₂²×C₃	C₂⁵	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	8	no	—
192	C₃×C₄	C₂×#41	1	2	yes	—
193	C₃×C₄	C₂×#41	1	2	yes	—
194	C₃×C₄	H(5,𝔽₂)	1	4	yes	—
195	C₂³	—	2	8	no	—
196	C₂³	—	4	8	no	—
197	C₂³	C₂⁴×C₃	1	8	yes	—
198	C₂³	C₂⁴×C₃	1	2	yes	—
199	C₂³	C₂⁴×C₃	1	1	yes	characteristic
200	Q₈	—	2	2	no	—
201	Q₈	—	2	8	no	—
202	D₄	—	2	4	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	8	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	16	no	—
207	D₄	—	4	8	no	—
208	D₄	—	4	16	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	16	no	—
213	C₂×C₄	C₃×#41	1	4	yes	complemented
214	C₂×C₄	C₂⁴×C₃	1	2	yes	—
215	C₂×C₄	C₂⁴×C₃	1	4	yes	—
216	C₂×C₄	C₂⁴×C₃	1	2	yes	—
217	C₂×C₄	C₂⁴×C₃	1	8	yes	—
218	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₃	1	2	yes	—
221	C₂×C₃	—	2	2	no	—
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	—	4	8	no	—
224	C₂×C₃	C₂²×#41	1	1	yes	characteristic
225	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	16	no	—
231	C₂²	—	8	8	no	—
232	C₂²	C₂×C₃×#41	1	2	yes	—
233	C₂²	C₂⁵×C₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	8	no	—
238	C₄	C₂×C₃×#41	1	2	yes	—
239	C₄	C₂×C₃×#41	1	2	yes	—
240	C₄	C₃×H(5,𝔽₂)	1	4	yes	—
241	C₃	#3508	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	4	2	no	—
245	C₂	C₂²×C₃×#41	1	1	yes	characteristic
246	C₂	C₂×C₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#559	C₂	semidirect
2. C₃×#550	C₂	semidirect
3. C₃×#546	C₂	semidirect
4. C₃×#540	C₂	semidirect
5. C₃×#535	C₂	semidirect
6. C₃×#533	C₂	semidirect
7. C₃×#532	C₂	semidirect
8. C₂×C₃×#127	C₂	semidirect
9. C₃×#518	C₂	semidirect
10. C₃×C₄×#41	C₂	semidirect
11. #3508	C₃	direct (N × H)
13. C₂×C₃×#41	C₂²	semidirect
16. C₃×C₄⋊Q₈	C₂²	semidirect
20. C₃×#127	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
22. C₃×#124	C₂²	semidirect
23. C₃×#124	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
26. C₃×#121	C₂²	semidirect
27. C₃×#121	C₂²	semidirect
28. C₃×#121	C₂²	semidirect
30. C₃×C₄×Q₈	C₂²	semidirect
31. C₃×C₄×Q₈	C₂²	semidirect
33. C₃×C₄×D₄	C₂²	semidirect
34. C₃×C₄×D₄	C₂²	semidirect
35. C₃×C₄×D₄	C₂²	semidirect
36. C₃×C₄×D₄	C₂²	semidirect
39. C₃×#117	C₂²	semidirect
42. C₂×C₃×C₄²	C₂²	semidirect
43. #559	C₂×C₃	semidirect
44. #550	C₂×C₃	semidirect
45. #546	C₂×C₃	semidirect
46. #540	C₂×C₃	semidirect
47. #535	C₂×C₃	semidirect
48. #533	C₂×C₃	semidirect
49. #532	C₂×C₃	semidirect
50. C₂×#127	C₂×C₃	semidirect
51. #518	C₂×C₃	semidirect
52. C₄×#41	C₂×C₃	semidirect
89. C₂×#41	C₂²×C₃	semidirect
92. C₄⋊Q₈	C₂²×C₃	semidirect
96. #127	C₂²×C₃	semidirect
97. #125	C₂²×C₃	semidirect
98. #124	C₂²×C₃	semidirect
99. #124	C₂²×C₃	semidirect
100. #123	C₂²×C₃	semidirect
102. #121	C₂²×C₃	semidirect
103. #121	C₂²×C₃	semidirect
104. #121	C₂²×C₃	semidirect
106. C₄×Q₈	C₂²×C₃	semidirect
107. C₄×Q₈	C₂²×C₃	semidirect
109. C₄×D₄	C₂²×C₃	semidirect
110. C₄×D₄	C₂²×C₃	semidirect
111. C₄×D₄	C₂²×C₃	semidirect
112. C₄×D₄	C₂²×C₃	semidirect
115. #117	C₂²×C₃	semidirect
118. C₂×C₄²	C₂²×C₃	semidirect
137. C₂×C₃×C₄	#41	semidirect
213. C₂×C₄	C₃×#41	semidirect
241. C₃	#3508	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#559	2
C₃×#550	2
C₃×#546	2
C₃×#540	2
C₃×#535	2
C₃×#533	2
C₃×#532	2
C₂×C₃×#127	2
C₃×#518	2
C₃×C₄×#41	2
#3508	3

Subgroups — C3×#3508 #87392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3508 #87392