Subgroups — C₂×C₃×#535 #87330

Order 384.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#41	C₂	semidirect
2. C₂³×C₃×D₄	C₂	semidirect
3. C₃×#535	C₂	direct (N × H)
4. C₂×C₃×#127	C₂	semidirect
5. C₂×C₃×#124	C₂	semidirect
6. C₂×C₃×#121	C₂	semidirect
7. C₂×C₃×#120	C₂	semidirect
8. C₂×C₃×#117	C₂	semidirect
9. C₂×#535	C₃	direct (N × H)
12. C₂×C₃×#41	C₂²	semidirect
13. C₂²×C₃×Q₈	C₂²	semidirect
18. C₂²×C₃×D₄	C₂²	semidirect
19. C₂²×C₃×D₄	C₂²	semidirect
20. C₂²×C₃×D₄	C₂²	semidirect
22. C₂³×C₃×C₄	C₂²	semidirect
23. C₃×#127	C₂²	semidirect
24. C₃×#124	C₂²	semidirect
25. C₃×#121	C₂²	semidirect
26. C₃×#120	C₂²	semidirect
27. C₃×#117	C₂²	semidirect
28. C₂×C₃×C₄⋊C₄	C₂²	semidirect
30. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
31. C₂×C₃×C₄²	C₂²	semidirect
32. C₂²×#41	C₂×C₃	semidirect
33. C₂³×D₄	C₂×C₃	semidirect
34. #535	C₂×C₃	direct (N × H)
35. C₂×#127	C₂×C₃	semidirect
36. C₂×#124	C₂×C₃	semidirect
37. C₂×#121	C₂×C₃	semidirect
38. C₂×#120	C₂×C₃	semidirect
39. C₂×#117	C₂×C₃	semidirect
48. C₂×C₃×Q₈	C₂³	semidirect
58. C₂×C₃×D₄	C₂³	semidirect
59. C₂×C₃×D₄	C₂³	semidirect
61. C₂²×C₃×C₄	D₄	semidirect
67. C₂²×C₃×C₄	C₂³	semidirect
68. C₃×C₄⋊C₄	C₂³	semidirect
70. C₃×C₂²⋊C₄	C₂³	semidirect
71. C₃×C₄²	C₂³	semidirect
74. C₂×#41	C₂²×C₃	semidirect
75. C₂²×Q₈	C₂²×C₃	semidirect
80. C₂²×D₄	C₂²×C₃	semidirect
81. C₂²×D₄	C₂²×C₃	semidirect
82. C₂²×D₄	C₂²×C₃	semidirect
84. C₂³×C₄	C₂²×C₃	semidirect
85. #127	C₂²×C₃	semidirect
86. #124	C₂²×C₃	semidirect
87. #121	C₂²×C₃	semidirect
88. #120	C₂²×C₃	semidirect
89. #117	C₂²×C₃	semidirect
90. C₂×C₄⋊C₄	C₂²×C₃	semidirect
92. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
93. C₂×C₄²	C₂²×C₃	semidirect
118. C₂×C₃×C₄	C₂×D₄	semidirect
131. C₂×Q₈	C₂³×C₃	semidirect
141. C₂×D₄	C₂³×C₃	semidirect
142. C₂×D₄	C₂³×C₃	semidirect
144. C₂²×C₄	C₃×D₄	semidirect
150. C₂²×C₄	C₂³×C₃	semidirect
151. C₄⋊C₄	C₂³×C₃	semidirect
153. C₂²⋊C₄	C₂³×C₃	semidirect
154. C₄²	C₂³×C₃	semidirect
196. C₂×C₄	C₂×C₃×D₄	semidirect
204. C₂×C₃	#535	direct (N × H)
224. C₃	C₂×#535	direct (N × H)
228. C₂	C₃×#535	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#41	2
C₂³×C₃×D₄	2
C₃×#535	2
C₂×C₃×#127	2
C₂×C₃×#124	2
C₂×C₃×#121	2
C₂×C₃×#120	2
C₂×C₃×#117	2
C₂×#535	3

Subgroups — C2×C3×#535 #87330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#535 #87330