Subgroups — C₂×C₃×#577 #87466

Order 384.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#577	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#137	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
6	C₂×C₃×#135	C₂	1	3	yes	maximal, complemented
7	C₂²×C₃×SD₁₆	C₂	1	3	yes	maximal, complemented
8	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#577	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₃×2₋¹⁺⁴	—	2	4	no	—
11	C₃×2₋¹⁺⁴	C₂²	1	8	yes	complemented
12	C₃×H(5,𝔽₂)	—	2	4	no	—
13	C₃×H(5,𝔽₂)	C₂²	1	8	yes	complemented
14	C₂×C₃×#41	—	2	1	no	—
15	C₂×C₃×#41	—	2	3	no	—
16	C₂×C₃×#41	C₂²	1	3	yes	—
17	C₂×C₃×#41	C₂²	1	1	yes	characteristic
18	C₂×C₃×#41	C₂²	1	3	yes	complemented
19	C₂²×C₃×Q₈	—	2	1	no	—
20	C₂²×C₃×Q₈	C₂²	1	3	yes	complemented
21	C₂²×C₃×D₄	—	2	3	no	—
22	C₂²×C₃×D₄	C₂²	1	3	yes	—
23	C₃×#137	C₂²	1	24	yes	complemented
24	C₃×AGL(1,ℤ/8)	C₂²	1	24	yes	complemented
25	C₃×#135	C₂²	1	24	yes	complemented
26	C₂×C₃×Q₁₆	C₂²	1	3	yes	complemented
27	C₂×C₃×SD₁₆	C₂²	1	3	yes	complemented
28	C₂×C₃×SD₁₆	C₂²	1	6	yes	complemented
29	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×SD₁₆	C₂²	1	24	yes	complemented
31	C₂×C₃×D₈	C₂²	1	3	yes	complemented
32	C₃×#131	C₂²	1	8	yes	complemented
33	C₂×C₃×M₁₆	C₂²	1	3	yes	complemented
34	C₂²×C₃×C₈	C₂²	1	3	yes	complemented
35	C₂×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
37	#577	C₂×C₃	1	16	yes	direct product factor
38	C₂×#137	C₂×C₃	1	3	yes	complemented
39	C₂×AGL(1,ℤ/8)	C₂×C₃	1	3	yes	complemented
40	C₂×#135	C₂×C₃	1	3	yes	complemented
41	C₂²×SD₁₆	C₂×C₃	1	3	yes	complemented
42	C₂×#131	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂⁴×C₃	—	2	3	no	—
44	C₃×#41	—	2	8	no	—
45	C₃×#41	—	2	24	no	—
46	C₃×#41	—	2	6	no	—
47	C₃×#41	—	2	6	no	—
48	C₃×#41	D₄	1	4	yes	—
49	C₃×#41	C₂³	1	12	yes	—
50	C₃×#41	C₂³	1	4	yes	—
51	C₃×#41	C₂³	1	12	yes	complemented
52	C₂×C₃×Q₈	—	2	8	no	—
53	C₂×C₃×Q₈	—	2	3	no	—
54	C₂×C₃×Q₈	—	2	6	no	—
55	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
56	C₂×C₃×Q₈	C₂³	1	3	yes	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×Q₈	C₂³	1	12	yes	complemented
59	C₂×C₃×D₄	—	2	3	no	—
60	C₂×C₃×D₄	—	2	3	no	—
61	C₂×C₃×D₄	—	2	24	no	—
62	C₂×C₃×D₄	—	2	3	no	—
63	C₂×C₃×D₄	—	2	6	no	—
64	C₂×C₃×D₄	D₄	1	3	yes	—
65	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
66	C₂×C₃×D₄	C₂³	1	3	yes	—
67	C₂×C₃×D₄	C₂³	1	12	yes	—
68	C₂²×C₃×C₄	—	2	3	no	—
69	C₂²×C₃×C₄	—	2	3	no	—
70	C₂²×C₃×C₄	—	2	3	no	—
71	C₂²×C₃×C₄	C₂³	1	3	yes	—
72	C₃×Q₁₆	C₂³	1	12	yes	complemented
73	C₃×SD₁₆	C₂³	1	12	yes	complemented
74	C₃×SD₁₆	C₂³	1	24	yes	complemented
75	C₃×SD₁₆	C₂³	1	4	yes	complemented
76	C₃×D₈	C₂³	1	12	yes	complemented
77	C₃×M₁₆	C₂³	1	12	yes	complemented
78	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂³	1	3	yes	—
80	C₂×C₃×C₈	C₂³	1	12	yes	complemented
81	2₋¹⁺⁴	—	2	4	no	—
82	2₋¹⁺⁴	C₂²×C₃	1	8	yes	complemented
83	H(5,𝔽₂)	—	2	4	no	—
84	H(5,𝔽₂)	C₂²×C₃	1	8	yes	complemented
85	C₂×#41	—	2	1	no	—
86	C₂×#41	—	2	3	no	—
87	C₂×#41	C₂²×C₃	1	3	yes	—
88	C₂×#41	C₂²×C₃	1	1	yes	characteristic
89	C₂×#41	C₂²×C₃	1	3	yes	complemented
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	C₂²×C₃	1	3	yes	complemented
92	C₂²×D₄	—	2	3	no	—
93	C₂²×D₄	C₂²×C₃	1	3	yes	—
94	#137	C₂²×C₃	1	24	yes	complemented
95	AGL(1,ℤ/8)	C₂²×C₃	1	24	yes	complemented
96	#135	C₂²×C₃	1	24	yes	complemented
97	C₂×Q₁₆	C₂²×C₃	1	3	yes	complemented
98	C₂×SD₁₆	C₂²×C₃	1	3	yes	complemented
99	C₂×SD₁₆	C₂²×C₃	1	6	yes	complemented
100	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₂×SD₁₆	C₂²×C₃	1	24	yes	complemented
102	C₂×D₈	C₂²×C₃	1	3	yes	complemented
103	#131	C₂²×C₃	1	8	yes	complemented
104	C₂×M₁₆	C₂²×C₃	1	3	yes	complemented
105	C₂²×C₈	C₂²×C₃	1	3	yes	complemented
106	C₂³×C₃	—	2	1	no	—
107	C₂³×C₃	—	2	3	no	—
108	C₂³×C₃	—	2	12	no	—
109	C₂³×C₃	—	8	6	no	—
110	C₂³×C₃	C₂×D₄	1	3	yes	—
111	C₃×Q₈	—	2	3	no	—
112	C₃×Q₈	—	2	12	no	—
113	C₃×Q₈	—	2	1	no	—
114	C₃×Q₈	C₂×D₄	1	4	yes	—
115	C₃×Q₈	C₂⁴	1	6	yes	—
116	C₃×Q₈	C₂⁴	1	2	yes	—
117	C₃×D₄	—	2	12	no	—
118	C₃×D₄	—	2	12	no	—
119	C₃×D₄	—	2	12	no	—
120	C₃×D₄	—	2	3	no	—
121	C₃×D₄	—	2	1	no	—
122	C₃×D₄	C₂×D₄	1	12	yes	—
123	C₃×D₄	C₂⁴	1	2	yes	—
124	C₃×D₄	C₂⁴	1	6	yes	—
125	C₂×C₃×C₄	—	2	12	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	3	no	—
128	C₂×C₃×C₄	—	2	12	no	—
129	C₂×C₃×C₄	—	2	12	no	—
130	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
131	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
132	C₂×C₃×C₄	C₂⁴	1	6	yes	—
133	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
134	C₃×C₈	C₂⁴	1	2	yes	—
135	C₃×C₈	C₂⁴	1	6	yes	—
136	C₂⁴	—	2	3	no	—
137	#41	—	2	8	no	—
138	#41	—	2	24	no	—
139	#41	—	2	6	no	—
140	#41	—	2	6	no	—
141	#41	C₃×D₄	1	4	yes	—
142	#41	C₂³×C₃	1	12	yes	—
143	#41	C₂³×C₃	1	4	yes	—
144	#41	C₂³×C₃	1	12	yes	complemented
145	C₂×Q₈	—	2	8	no	—
146	C₂×Q₈	—	2	3	no	—
147	C₂×Q₈	—	2	6	no	—
148	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
149	C₂×Q₈	C₂³×C₃	1	3	yes	—
150	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×Q₈	C₂³×C₃	1	12	yes	complemented
152	C₂×D₄	—	2	3	no	—
153	C₂×D₄	—	2	3	no	—
154	C₂×D₄	—	2	24	no	—
155	C₂×D₄	—	2	3	no	—
156	C₂×D₄	—	2	6	no	—
157	C₂×D₄	C₃×D₄	1	3	yes	—
158	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
159	C₂×D₄	C₂³×C₃	1	3	yes	—
160	C₂×D₄	C₂³×C₃	1	12	yes	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	C₂³×C₃	1	3	yes	—
165	Q₁₆	C₂³×C₃	1	12	yes	complemented
166	SD₁₆	C₂³×C₃	1	12	yes	complemented
167	SD₁₆	C₂³×C₃	1	24	yes	complemented
168	SD₁₆	C₂³×C₃	1	4	yes	complemented
169	D₈	C₂³×C₃	1	12	yes	complemented
170	M₁₆	C₂³×C₃	1	12	yes	complemented
171	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₃	1	3	yes	—
173	C₂×C₈	C₂³×C₃	1	12	yes	complemented
174	C₂²×C₃	—	2	6	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	6	no	—
177	C₂²×C₃	—	4	2	no	—
178	C₂²×C₃	—	4	6	no	—
179	C₂²×C₃	—	8	12	no	—
180	C₂²×C₃	C₂²×D₄	1	6	yes	—
181	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	6	no	—
184	C₃×C₄	C₂²×D₄	1	6	yes	—
185	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
186	C₃×C₄	C₂⁵	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	C₂³	—	2	3	no	—
189	C₂³	—	2	12	no	—
190	C₂³	—	8	6	no	—
191	C₂³	C₂×C₃×D₄	1	3	yes	—
192	Q₈	—	2	3	no	—
193	Q₈	—	2	12	no	—
194	Q₈	—	2	1	no	—
195	Q₈	C₂×C₃×D₄	1	4	yes	—
196	Q₈	C₂⁴×C₃	1	6	yes	—
197	Q₈	C₂⁴×C₃	1	2	yes	—
198	D₄	—	2	12	no	—
199	D₄	—	2	12	no	—
200	D₄	—	2	12	no	—
201	D₄	—	2	3	no	—
202	D₄	—	2	1	no	—
203	D₄	C₂×C₃×D₄	1	12	yes	—
204	D₄	C₂⁴×C₃	1	2	yes	—
205	D₄	C₂⁴×C₃	1	6	yes	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
213	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₃	1	6	yes	—
215	C₈	C₂⁴×C₃	1	2	yes	—
216	C₈	C₂⁴×C₃	1	6	yes	—
217	C₂×C₃	—	2	6	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	—	4	6	no	—
220	C₂×C₃	#577	1	2	yes	direct product factor
221	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
222	C₂²	—	2	6	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	8	12	no	—
228	C₂²	C₂²×C₃×D₄	1	6	yes	—
229	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	6	no	—
232	C₄	C₂²×C₃×D₄	1	6	yes	—
233	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₄	C₂⁵×C₃	1	1	yes	characteristic
235	C₃	C₂×#577	1	1	yes	characteristic, direct product factor, Sylow
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	4	2	no	—
239	C₂	C₃×#577	1	2	yes	direct product factor
240	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ 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×#577	C₂	direct (N × H)
4. C₂×C₃×#137	C₂	semidirect
5. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
6. C₂×C₃×#135	C₂	semidirect
7. C₂²×C₃×SD₁₆	C₂	semidirect
8. C₂×C₃×#131	C₂	semidirect
9. C₂×#577	C₃	direct (N × H)
11. C₃×2₋¹⁺⁴	C₂²	semidirect
13. C₃×H(5,𝔽₂)	C₂²	semidirect
18. C₂×C₃×#41	C₂²	semidirect
20. C₂²×C₃×Q₈	C₂²	semidirect
23. C₃×#137	C₂²	semidirect
24. C₃×AGL(1,ℤ/8)	C₂²	semidirect
25. C₃×#135	C₂²	semidirect
26. C₂×C₃×Q₁₆	C₂²	semidirect
27. C₂×C₃×SD₁₆	C₂²	semidirect
28. C₂×C₃×SD₁₆	C₂²	semidirect
29. C₂×C₃×SD₁₆	C₂²	semidirect
30. C₂×C₃×SD₁₆	C₂²	semidirect
31. C₂×C₃×D₈	C₂²	semidirect
32. C₃×#131	C₂²	semidirect
33. C₂×C₃×M₁₆	C₂²	semidirect
34. C₂²×C₃×C₈	C₂²	semidirect
35. C₂×2₋¹⁺⁴	C₂×C₃	semidirect
36. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
37. #577	C₂×C₃	direct (N × H)
38. C₂×#137	C₂×C₃	semidirect
39. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
40. C₂×#135	C₂×C₃	semidirect
41. C₂²×SD₁₆	C₂×C₃	semidirect
42. C₂×#131	C₂×C₃	semidirect
51. C₃×#41	C₂³	semidirect
58. C₂×C₃×Q₈	C₂³	semidirect
72. C₃×Q₁₆	C₂³	semidirect
73. C₃×SD₁₆	C₂³	semidirect
74. C₃×SD₁₆	C₂³	semidirect
75. C₃×SD₁₆	C₂³	semidirect
76. C₃×D₈	C₂³	semidirect
77. C₃×M₁₆	C₂³	semidirect
80. C₂×C₃×C₈	C₂³	semidirect
82. 2₋¹⁺⁴	C₂²×C₃	semidirect
84. H(5,𝔽₂)	C₂²×C₃	semidirect
89. C₂×#41	C₂²×C₃	semidirect
91. C₂²×Q₈	C₂²×C₃	semidirect
94. #137	C₂²×C₃	semidirect
95. AGL(1,ℤ/8)	C₂²×C₃	semidirect
96. #135	C₂²×C₃	semidirect
97. C₂×Q₁₆	C₂²×C₃	semidirect
98. C₂×SD₁₆	C₂²×C₃	semidirect
99. C₂×SD₁₆	C₂²×C₃	semidirect
100. C₂×SD₁₆	C₂²×C₃	semidirect
101. C₂×SD₁₆	C₂²×C₃	semidirect
102. C₂×D₈	C₂²×C₃	semidirect
103. #131	C₂²×C₃	semidirect
104. C₂×M₁₆	C₂²×C₃	semidirect
105. C₂²×C₈	C₂²×C₃	semidirect
144. #41	C₂³×C₃	semidirect
151. C₂×Q₈	C₂³×C₃	semidirect
165. Q₁₆	C₂³×C₃	semidirect
166. SD₁₆	C₂³×C₃	semidirect
167. SD₁₆	C₂³×C₃	semidirect
168. SD₁₆	C₂³×C₃	semidirect
169. D₈	C₂³×C₃	semidirect
170. M₁₆	C₂³×C₃	semidirect
173. C₂×C₈	C₂³×C₃	semidirect
220. C₂×C₃	#577	direct (N × H)
235. C₃	C₂×#577	direct (N × H)
239. C₂	C₃×#577	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×#577	2
C₂×C₃×#137	2
C₂×C₃×AGL(1,ℤ/8)	2
C₂×C₃×#135	2
C₂²×C₃×SD₁₆	2
C₂×C₃×#131	2
C₂×#577	3

Subgroups — C2×C3×#577 #87466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#577 #87466