Subgroups — C₃×#3500 #87384

Order 384.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#548	C₂	1	4	yes	maximal, complemented
4	C₃×#546	C₂	1	12	yes	maximal, complemented
5	C₃×#536	C₂	1	3	yes	maximal, complemented
6	C₃×#535	C₂	1	3	yes	maximal, complemented
7	C₃×#532	C₂	1	3	yes	maximal, complemented
8	C₂×C₃×#124	C₂	1	3	yes	maximal, complemented
9	C₃×C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
10	#3500	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×2₋¹⁺⁴	—	2	8	no	—
12	C₃×H(5,𝔽₂)	—	2	8	no	—
13	C₂×C₃×#41	C₂²	1	3	yes	—
14	C₂×C₃×#41	C₂²	1	2	yes	—
15	C₂×C₃×#41	C₂²	1	1	yes	characteristic
16	C₂×C₃×#41	C₂²	1	3	yes	complemented
17	C₂×C₃×#41	C₂²	1	6	yes	complemented
18	C₂²×C₃×Q₈	C₂²	1	3	yes	complemented
19	C₂²×C₃×Q₈	C₂²	1	2	yes	complemented
20	C₂²×C₃×D₄	C₂²	1	3	yes	—
21	C₂²×C₃×D₄	C₂²	1	6	yes	—
22	C₃×C₄⋊Q₈	C₂²	1	3	yes	complemented
23	C₃×#127	C₂²	1	3	yes	complemented
24	C₃×#124	C₂²	1	3	yes	complemented
25	C₃×#124	C₂²	1	6	yes	complemented
26	C₃×#124	C₂²	1	12	yes	complemented
27	C₃×#124	C₂²	1	1	yes	characteristic, complemented
28	C₃×#123	C₂²	1	12	yes	complemented
29	C₃×C₂²⋊Q₈	C₂²	1	12	yes	complemented
30	C₃×#121	C₂²	1	12	yes	complemented
31	C₃×#121	C₂²	1	12	yes	complemented
32	C₃×#120	C₂²	1	12	yes	complemented
33	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
34	C₃×C₄×D₄	C₂²	1	6	yes	complemented
35	C₃×C₄×D₄	C₂²	1	12	yes	complemented
36	C₃×#117	C₂²	1	3	yes	complemented
37	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
38	C₂×C₃×C₄²	C₂²	1	3	yes	complemented
39	C₂×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
41	#548	C₂×C₃	1	4	yes	complemented
42	#546	C₂×C₃	1	12	yes	complemented
43	#536	C₂×C₃	1	3	yes	complemented
44	#535	C₂×C₃	1	3	yes	complemented
45	#532	C₂×C₃	1	3	yes	complemented
46	C₂×#124	C₂×C₃	1	3	yes	complemented
47	C₄×#41	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂⁴×C₃	C₂³	1	6	yes	—
49	C₃×#41	—	2	8	no	—
50	C₃×#41	—	2	24	no	—
51	C₃×#41	—	2	12	no	—
52	C₃×#41	—	2	12	no	—
53	C₃×#41	D₄	1	8	yes	complemented
54	C₂×C₃×Q₈	—	2	8	no	—
55	C₂×C₃×Q₈	—	2	12	no	—
56	C₂×C₃×Q₈	C₂³	1	6	yes	—
57	C₂×C₃×Q₈	C₂³	1	3	yes	—
58	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	24	no	—
61	C₂×C₃×D₄	—	2	12	no	—
62	C₂×C₃×D₄	C₂³	1	6	yes	—
63	C₂×C₃×D₄	C₂³	1	6	yes	—
64	C₂×C₃×D₄	C₂³	1	6	yes	—
65	C₂×C₃×D₄	C₂³	1	3	yes	—
66	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×D₄	C₂³	1	12	yes	—
68	C₂×C₃×D₄	C₂³	1	3	yes	—
69	C₂²×C₃×C₄	C₂³	1	6	yes	—
70	C₂²×C₃×C₄	C₂³	1	6	yes	—
71	C₂²×C₃×C₄	C₂³	1	6	yes	—
72	C₂²×C₃×C₄	C₂³	1	6	yes	—
73	C₂²×C₃×C₄	C₂³	1	3	yes	—
74	C₃×C₄⋊C₄	C₂³	1	6	yes	—
75	C₃×C₄⋊C₄	C₂³	1	12	yes	—
76	C₃×C₂²⋊C₄	C₂³	1	12	yes	—
77	C₃×C₂²⋊C₄	C₂³	1	24	yes	—
78	C₃×C₂²⋊C₄	C₂³	1	6	yes	—
79	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₃×C₄²	C₂³	1	3	yes	—
82	C₃×C₄²	C₂³	1	6	yes	—
83	2₋¹⁺⁴	—	2	8	no	—
84	H(5,𝔽₂)	—	2	8	no	—
85	C₂×#41	C₂²×C₃	1	3	yes	—
86	C₂×#41	C₂²×C₃	1	2	yes	—
87	C₂×#41	C₂²×C₃	1	1	yes	characteristic
88	C₂×#41	C₂²×C₃	1	3	yes	complemented
89	C₂×#41	C₂²×C₃	1	6	yes	complemented
90	C₂²×Q₈	C₂²×C₃	1	3	yes	complemented
91	C₂²×Q₈	C₂²×C₃	1	2	yes	complemented
92	C₂²×D₄	C₂²×C₃	1	3	yes	—
93	C₂²×D₄	C₂²×C₃	1	6	yes	—
94	C₄⋊Q₈	C₂²×C₃	1	3	yes	complemented
95	#127	C₂²×C₃	1	3	yes	complemented
96	#124	C₂²×C₃	1	3	yes	complemented
97	#124	C₂²×C₃	1	6	yes	complemented
98	#124	C₂²×C₃	1	12	yes	complemented
99	#124	C₂²×C₃	1	1	yes	characteristic, complemented
100	#123	C₂²×C₃	1	12	yes	complemented
101	C₂²⋊Q₈	C₂²×C₃	1	12	yes	complemented
102	#121	C₂²×C₃	1	12	yes	complemented
103	#121	C₂²×C₃	1	12	yes	complemented
104	#120	C₂²×C₃	1	12	yes	complemented
105	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
106	C₄×D₄	C₂²×C₃	1	6	yes	complemented
107	C₄×D₄	C₂²×C₃	1	12	yes	complemented
108	#117	C₂²×C₃	1	3	yes	complemented
109	C₂×C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
110	C₂×C₄²	C₂²×C₃	1	3	yes	complemented
111	C₂³×C₃	—	2	12	no	—
112	C₂³×C₃	—	4	12	no	—
113	C₂³×C₃	C₂⁴	1	6	yes	—
114	C₂³×C₃	C₂⁴	1	2	yes	—
115	C₂³×C₃	C₂⁴	1	3	yes	—
116	C₃×Q₈	—	2	12	no	—
117	C₃×Q₈	—	2	6	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	C₂×D₄	1	4	yes	—
120	C₃×D₄	—	2	12	no	—
121	C₃×D₄	—	2	12	no	—
122	C₃×D₄	—	2	12	no	—
123	C₃×D₄	—	2	6	no	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	4	12	no	—
126	C₃×D₄	C₂×D₄	1	12	yes	—
127	C₂×C₃×C₄	—	2	12	no	—
128	C₂×C₃×C₄	—	2	12	no	—
129	C₂×C₃×C₄	—	2	12	no	—
130	C₂×C₃×C₄	—	4	6	no	—
131	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
132	C₂×C₃×C₄	C₂⁴	1	2	yes	—
133	C₂×C₃×C₄	C₂⁴	1	6	yes	—
134	C₂×C₃×C₄	C₂⁴	1	6	yes	—
135	C₂×C₃×C₄	C₂⁴	1	2	yes	—
136	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂⁴	1	3	yes	—
138	C₂⁴	C₂³×C₃	1	6	yes	—
139	#41	—	2	8	no	—
140	#41	—	2	24	no	—
141	#41	—	2	12	no	—
142	#41	—	2	12	no	—
143	#41	C₃×D₄	1	8	yes	complemented
144	C₂×Q₈	—	2	8	no	—
145	C₂×Q₈	—	2	12	no	—
146	C₂×Q₈	C₂³×C₃	1	6	yes	—
147	C₂×Q₈	C₂³×C₃	1	3	yes	—
148	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
149	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
150	C₂×D₄	—	2	24	no	—
151	C₂×D₄	—	2	12	no	—
152	C₂×D₄	C₂³×C₃	1	6	yes	—
153	C₂×D₄	C₂³×C₃	1	6	yes	—
154	C₂×D₄	C₂³×C₃	1	6	yes	—
155	C₂×D₄	C₂³×C₃	1	3	yes	—
156	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
157	C₂×D₄	C₂³×C₃	1	12	yes	—
158	C₂×D₄	C₂³×C₃	1	3	yes	—
159	C₂²×C₄	C₂³×C₃	1	6	yes	—
160	C₂²×C₄	C₂³×C₃	1	6	yes	—
161	C₂²×C₄	C₂³×C₃	1	6	yes	—
162	C₂²×C₄	C₂³×C₃	1	6	yes	—
163	C₂²×C₄	C₂³×C₃	1	3	yes	—
164	C₄⋊C₄	C₂³×C₃	1	6	yes	—
165	C₄⋊C₄	C₂³×C₃	1	12	yes	—
166	C₂²⋊C₄	C₂³×C₃	1	12	yes	—
167	C₂²⋊C₄	C₂³×C₃	1	24	yes	—
168	C₂²⋊C₄	C₂³×C₃	1	6	yes	—
169	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
170	C₄²	C₂³×C₃	1	1	yes	characteristic
171	C₄²	C₂³×C₃	1	3	yes	—
172	C₄²	C₂³×C₃	1	6	yes	—
173	C₂²×C₃	—	2	6	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	6	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	12	no	—
178	C₂²×C₃	—	8	12	no	—
179	C₂²×C₃	C₂²×D₄	1	6	yes	—
180	C₂²×C₃	C₂⁵	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	6	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	6	no	—
185	C₃×C₄	C₂²×D₄	1	6	yes	—
186	C₃×C₄	C₂²×D₄	1	2	yes	—
187	C₂³	—	2	12	no	—
188	C₂³	—	4	12	no	—
189	C₂³	C₂⁴×C₃	1	6	yes	—
190	C₂³	C₂⁴×C₃	1	2	yes	—
191	C₂³	C₂⁴×C₃	1	3	yes	—
192	Q₈	—	2	12	no	—
193	Q₈	—	2	6	no	—
194	Q₈	—	2	2	no	—
195	Q₈	C₂×C₃×D₄	1	4	yes	—
196	D₄	—	2	12	no	—
197	D₄	—	2	12	no	—
198	D₄	—	2	12	no	—
199	D₄	—	2	6	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	12	no	—
202	D₄	C₂×C₃×D₄	1	12	yes	—
203	C₂×C₄	—	2	12	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	4	6	no	—
207	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
208	C₂×C₄	C₂⁴×C₃	1	2	yes	—
209	C₂×C₄	C₂⁴×C₃	1	6	yes	—
210	C₂×C₄	C₂⁴×C₃	1	6	yes	—
211	C₂×C₄	C₂⁴×C₃	1	2	yes	—
212	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
213	C₂×C₄	C₂⁴×C₃	1	3	yes	—
214	C₂×C₃	—	2	6	no	—
215	C₂×C₃	—	4	2	no	—
216	C₂×C₃	—	4	6	no	—
217	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
218	C₂×C₃	#585	1	2	yes	—
219	C₂²	—	2	6	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	6	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	12	no	—
224	C₂²	—	8	12	no	—
225	C₂²	C₂²×C₃×D₄	1	6	yes	—
226	C₂²	C₂⁵×C₃	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	6	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	6	no	—
231	C₄	C₂²×C₃×D₄	1	6	yes	—
232	C₄	C₂²×C₃×D₄	1	2	yes	—
233	C₃	#3500	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	2	6	no	—
235	C₂	—	4	6	no	—
236	C₂	—	4	2	no	—
237	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
238	C₂	C₃×#585	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. C₂×C₃×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₃×H(5,𝔽₂)	C₂	semidirect
3. C₃×#548	C₂	semidirect
4. C₃×#546	C₂	semidirect
5. C₃×#536	C₂	semidirect
6. C₃×#535	C₂	semidirect
7. C₃×#532	C₂	semidirect
8. C₂×C₃×#124	C₂	semidirect
9. C₃×C₄×#41	C₂	semidirect
10. #3500	C₃	direct (N × H)
16. C₂×C₃×#41	C₂²	semidirect
17. C₂×C₃×#41	C₂²	semidirect
18. C₂²×C₃×Q₈	C₂²	semidirect
19. C₂²×C₃×Q₈	C₂²	semidirect
22. C₃×C₄⋊Q₈	C₂²	semidirect
23. C₃×#127	C₂²	semidirect
24. C₃×#124	C₂²	semidirect
25. C₃×#124	C₂²	semidirect
26. C₃×#124	C₂²	semidirect
27. C₃×#124	C₂²	semidirect
28. C₃×#123	C₂²	semidirect
29. C₃×C₂²⋊Q₈	C₂²	semidirect
30. C₃×#121	C₂²	semidirect
31. C₃×#121	C₂²	semidirect
32. C₃×#120	C₂²	semidirect
33. C₃×C₄×Q₈	C₂²	semidirect
34. C₃×C₄×D₄	C₂²	semidirect
35. C₃×C₄×D₄	C₂²	semidirect
36. C₃×#117	C₂²	semidirect
37. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
38. C₂×C₃×C₄²	C₂²	semidirect
39. C₂×2₋¹⁺⁴	C₂×C₃	semidirect
40. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
41. #548	C₂×C₃	semidirect
42. #546	C₂×C₃	semidirect
43. #536	C₂×C₃	semidirect
44. #535	C₂×C₃	semidirect
45. #532	C₂×C₃	semidirect
46. C₂×#124	C₂×C₃	semidirect
47. C₄×#41	C₂×C₃	semidirect
53. C₃×#41	D₄	semidirect
88. C₂×#41	C₂²×C₃	semidirect
89. C₂×#41	C₂²×C₃	semidirect
90. C₂²×Q₈	C₂²×C₃	semidirect
91. C₂²×Q₈	C₂²×C₃	semidirect
94. C₄⋊Q₈	C₂²×C₃	semidirect
95. #127	C₂²×C₃	semidirect
96. #124	C₂²×C₃	semidirect
97. #124	C₂²×C₃	semidirect
98. #124	C₂²×C₃	semidirect
99. #124	C₂²×C₃	semidirect
100. #123	C₂²×C₃	semidirect
101. C₂²⋊Q₈	C₂²×C₃	semidirect
102. #121	C₂²×C₃	semidirect
103. #121	C₂²×C₃	semidirect
104. #120	C₂²×C₃	semidirect
105. C₄×Q₈	C₂²×C₃	semidirect
106. C₄×D₄	C₂²×C₃	semidirect
107. C₄×D₄	C₂²×C₃	semidirect
108. #117	C₂²×C₃	semidirect
109. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
110. C₂×C₄²	C₂²×C₃	semidirect
143. #41	C₃×D₄	semidirect
233. C₃	#3500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×2₋¹⁺⁴	2
C₂×C₃×H(5,𝔽₂)	2
C₃×#548	2
C₃×#546	2
C₃×#536	2
C₃×#535	2
C₃×#532	2
C₂×C₃×#124	2
C₃×C₄×#41	2
#3500	3

Subgroups — C3×#3500 #87384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3500 #87384