Subgroups — C₂²×C₃²×#239 #13181718

Order 1728.

5964 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#239	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃²×D₁₂	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃²×C₄×D₃	C₂	1	3	yes	maximal, complemented
5	C₂²×C₃²×#41	—	3	1	no	maximal
6	C₂²×C₃×#239	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²×D₃	C₂²	1	3	yes	—
8	C₂×C₃³×Q₈	C₂²	1	12	yes	complemented
9	C₂²×C₃³×C₄	C₂²	1	3	yes	—
10	C₂²×C₃²×Dic₃	C₂²	1	1	yes	characteristic
11	C₃²×#239	C₂²	1	64	yes	direct product factor
12	C₂×C₃²×D₁₂	C₂²	1	36	yes	complemented
13	C₂×C₃²×C₄×D₃	C₂²	1	36	yes	complemented
14	C₂×C₃²×#41	—	3	24	no	—
15	C₂²×C₃²×Q₈	—	2	4	no	—
16	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
17	C₂²×C₃²×Q₈	C₂×C₃	1	4	yes	complemented
18	C₂²×C₃²×D₄	—	3	3	no	—
19	C₂³×C₃²×C₄	—	3	3	no	—
20	C₂×C₃×#239	C₂×C₃	1	96	yes	direct product factor
21	C₂²×C₃×D₁₂	C₂×C₃	1	12	yes	complemented
22	C₂²×C₃×C₄×D₃	C₂×C₃	1	12	yes	complemented
23	C₂³×C₃³	C₂³	1	1	yes	characteristic
24	C₂²×C₃²×D₃	—	2	12	no	—
25	C₂²×C₃²×D₃	C₂³	1	18	yes	—
26	C₃³×Q₈	C₂³	1	16	yes	complemented
27	C₂×C₃³×C₄	C₂³	1	18	yes	—
28	C₂×C₃²×Dic₃	C₂³	1	6	yes	—
29	C₃²×D₁₂	C₂³	1	48	yes	complemented
30	C₃²×C₄×D₃	C₂³	1	48	yes	complemented
31	C₂²×C₃×#41	—	3	4	no	—
32	C₂²×#239	C₃²	1	1	yes	characteristic, direct product factor
33	C₂⁴×C₃²	—	3	3	no	—
34	C₂³×C₃×D₃	C₂²×C₃	1	12	yes	—
35	C₃²×#41	—	3	64	no	—
36	C₂×C₃²×Q₈	—	2	48	no	—
37	C₂×C₃²×Q₈	C₂×D₃	1	12	yes	complemented
38	C₂×C₃²×Q₈	C₂²×C₃	1	48	yes	complemented
39	C₂×C₃²×D₄	—	3	36	no	—
40	C₂²×C₃²×C₄	—	2	12	no	—
41	C₂²×C₃²×C₄	—	3	1	no	—
42	C₂²×C₃²×C₄	—	3	36	no	—
43	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	12	yes	—
45	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
46	C₃×#239	C₂²×C₃	1	256	yes	direct product factor
47	C₂×C₃×D₁₂	C₂²×C₃	1	144	yes	complemented
48	C₂×C₃×C₄×D₃	C₂²×C₃	1	144	yes	complemented
49	C₂²×C₃³	#41	1	4	yes	complemented
50	C₂²×C₃³	C₂⁴	1	3	yes	—
51	C₂×C₃²×D₃	—	2	36	no	—
52	C₂×C₃²×D₃	C₂⁴	1	12	yes	—
53	C₃³×C₄	C₂⁴	1	12	yes	—
54	C₃²×Dic₃	C₂⁴	1	4	yes	—
55	C₂×C₃×#41	—	3	96	no	—
56	C₂²×C₃×Q₈	—	2	4	no	—
57	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
58	C₂²×C₃×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
59	C₂²×C₃×D₄	—	3	12	no	—
60	C₂³×C₃×C₄	—	3	12	no	—
61	C₂×#239	C₂×C₃²	1	24	yes	direct product factor
62	C₂²×D₁₂	C₂×C₃²	1	3	yes	complemented
63	C₂²×C₄×D₃	C₂×C₃²	1	3	yes	complemented
64	C₂³×C₃²	—	2	4	no	—
65	C₂³×C₃²	—	3	18	no	—
66	C₂³×C₃²	—	6	12	no	—
67	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
68	C₂³×C₃²	C₂³×C₃	1	4	yes	—
69	C₂²×C₃×D₃	—	2	48	no	—
70	C₂²×C₃×D₃	C₂³×C₃	1	72	yes	—
71	C₃²×Q₈	—	2	64	no	—
72	C₃²×Q₈	C₂²×D₃	1	16	yes	complemented
73	C₃²×Q₈	C₂³×C₃	1	64	yes	complemented
74	C₃²×D₄	—	3	48	no	—
75	C₂×C₃²×C₄	—	2	72	no	—
76	C₂×C₃²×C₄	—	3	6	no	—
77	C₂×C₃²×C₄	—	3	48	no	—
78	C₂×C₃²×C₄	C₂²×D₃	1	18	yes	—
79	C₂×C₃²×C₄	C₂³×C₃	1	72	yes	—
80	C₂×C₃×Dic₃	C₂³×C₃	1	24	yes	—
81	C₃×D₁₂	C₂³×C₃	1	192	yes	complemented
82	C₃×C₄×D₃	C₂³×C₃	1	192	yes	complemented
83	C₂²×#41	—	3	1	no	Sylow
84	C₂×C₃³	C₂×#41	1	6	yes	complemented
85	C₂×C₃³	C₂⁵	1	1	yes	characteristic
86	C₃²×D₃	—	2	12	no	—
87	C₂⁴×C₃	—	3	12	no	—
88	C₂³×D₃	C₂²×C₃²	1	3	yes	—
89	C₃×#41	—	3	256	no	—
90	C₂×C₃×Q₈	—	2	48	no	—
91	C₂×C₃×Q₈	C₂×C₃×D₃	1	48	yes	complemented
92	C₂×C₃×Q₈	C₂²×C₃²	1	12	yes	complemented
93	C₂×C₃×D₄	—	3	144	no	—
94	C₂²×C₃×C₄	—	2	12	no	—
95	C₂²×C₃×C₄	—	3	4	no	—
96	C₂²×C₃×C₄	—	3	144	no	—
97	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
98	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
99	C₂²×Dic₃	C₂²×C₃²	1	1	yes	characteristic
100	#239	C₂²×C₃²	1	64	yes	direct product factor
101	C₂×D₁₂	C₂²×C₃²	1	36	yes	complemented
102	C₂×C₄×D₃	C₂²×C₃²	1	36	yes	complemented
103	C₂²×C₃²	—	2	12	no	—
104	C₂²×C₃²	—	2	16	no	—
105	C₂²×C₃²	—	3	12	no	—
106	C₂²×C₃²	—	6	36	no	—
107	C₂²×C₃²	#239	1	4	yes	direct product factor
108	C₂²×C₃²	C₃×#41	1	16	yes	complemented
109	C₂²×C₃²	C₂³×D₃	1	3	yes	—
110	C₂²×C₃²	C₂⁴×C₃	1	12	yes	—
111	C₂×C₃×D₃	—	2	144	no	—
112	C₂×C₃×D₃	C₂⁴×C₃	1	48	yes	—
113	C₃²×C₄	—	2	48	no	—
114	C₃²×C₄	—	3	4	no	—
115	C₃²×C₄	C₂³×D₃	1	12	yes	—
116	C₃²×C₄	C₂⁴×C₃	1	48	yes	—
117	C₃×Dic₃	C₂⁴×C₃	1	16	yes	—
118	C₂×#41	—	3	24	no	—
119	C₂²×Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
120	C₂²×D₄	—	3	3	no	—
121	C₂³×C₄	—	3	3	no	—
122	C₃³	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	3	72	no	—
125	C₂³×C₃	—	6	48	no	—
126	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
127	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
128	C₂²×D₃	—	2	12	no	—
129	C₂²×D₃	C₂³×C₃²	1	18	yes	—
130	C₃×Q₈	—	2	64	no	—
131	C₃×Q₈	C₂²×C₃×D₃	1	64	yes	complemented
132	C₃×Q₈	C₂³×C₃²	1	16	yes	complemented
133	C₃×D₄	—	3	192	no	—
134	C₂×C₃×C₄	—	2	72	no	—
135	C₂×C₃×C₄	—	3	24	no	—
136	C₂×C₃×C₄	—	3	192	no	—
137	C₂×C₃×C₄	C₂²×C₃×D₃	1	72	yes	—
138	C₂×C₃×C₄	C₂³×C₃²	1	18	yes	—
139	C₂×Dic₃	C₂³×C₃²	1	6	yes	—
140	D₁₂	C₂³×C₃²	1	48	yes	complemented
141	C₄×D₃	C₂³×C₃²	1	48	yes	complemented
142	C₂×C₃²	—	2	24	no	—
143	C₂×C₃²	—	2	4	no	—
144	C₂×C₃²	—	6	12	no	—
145	C₂×C₃²	C₂×#239	1	6	yes	direct product factor
146	C₂×C₃²	C₂×C₃×#41	1	24	yes	complemented
147	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
148	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
149	C₃×D₃	—	2	48	no	—
150	C₂⁴	—	3	3	no	—
151	#41	—	3	64	no	—
152	C₂×Q₈	C₂×C₃²×D₃	1	12	yes	complemented
153	C₂×D₄	—	3	36	no	—
154	C₂²×C₄	—	3	1	no	—
155	C₂²×C₄	—	3	36	no	—
156	C₂²×C₄	C₂×C₃²×D₃	1	3	yes	—
157	C₂²×C₃	—	2	12	no	—
158	C₂²×C₃	—	2	16	no	—
159	C₂²×C₃	—	3	48	no	—
160	C₂²×C₃	—	6	144	no	—
161	C₂²×C₃	C₃×#239	1	16	yes	direct product factor
162	C₂²×C₃	C₃²×#41	1	4	yes	complemented
163	C₂²×C₃	C₂³×C₃×D₃	1	12	yes	—
164	C₂²×C₃	C₂⁴×C₃²	1	3	yes	—
165	C₂×D₃	—	2	36	no	—
166	C₂×D₃	C₂⁴×C₃²	1	12	yes	—
167	C₃×C₄	—	2	48	no	—
168	C₃×C₄	—	3	16	no	—
169	C₃×C₄	C₂³×C₃×D₃	1	48	yes	—
170	C₃×C₄	C₂⁴×C₃²	1	12	yes	—
171	Dic₃	C₂⁴×C₃²	1	4	yes	—
172	C₃²	—	2	4	no	—
173	C₃²	C₂²×#239	1	1	yes	characteristic, direct product factor
174	C₃²	C₂²×C₃×#41	1	4	yes	complemented
175	C₂³	—	3	18	no	—
176	C₂³	—	6	12	no	—
177	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
178	Q₈	C₂²×C₃²×D₃	1	16	yes	complemented
179	D₄	—	3	48	no	—
180	C₂×C₄	—	3	6	no	—
181	C₂×C₄	—	3	48	no	—
182	C₂×C₄	C₂²×C₃²×D₃	1	18	yes	—
183	C₂×C₃	—	2	24	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	6	48	no	—
186	C₂×C₃	C₂×C₃×#239	1	24	yes	direct product factor
187	C₂×C₃	C₂×C₃²×#41	1	6	yes	complemented
188	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
189	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
190	D₃	—	2	12	no	—
191	C₂²	—	3	12	no	—
192	C₂²	—	6	36	no	—
193	C₂²	C₃²×#239	1	4	yes	direct product factor
194	C₂²	C₂³×C₃²×D₃	1	3	yes	—
195	C₄	—	3	4	no	—
196	C₄	C₂³×C₃²×D₃	1	12	yes	—
197	C₃	—	2	4	no	—
198	C₃	C₂²×C₃×#239	1	4	yes	direct product factor
199	C₃	C₂²×C₃²×#41	1	1	yes	characteristic, complemented
200	C₂	—	6	12	no	—
201	C₂	C₂×C₃²×#239	1	6	yes	direct product factor
202	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle	C₂³×C₃³	C₂³
hypercenter	C₂²×C₃²×Q₈	D₃
hypocenter	C₃	C₂²×C₃²×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
2. C₂×C₃²×#239	C₂	direct (N × H)
3. C₂²×C₃²×D₁₂	C₂	semidirect
4. C₂²×C₃²×C₄×D₃	C₂	semidirect
6. C₂²×C₃×#239	C₃	direct (N × H)
8. C₂×C₃³×Q₈	C₂²	semidirect
11. C₃²×#239	C₂²	direct (N × H)
12. C₂×C₃²×D₁₂	C₂²	semidirect
13. C₂×C₃²×C₄×D₃	C₂²	semidirect
16. C₂²×C₃²×Q₈	D₃	semidirect
17. C₂²×C₃²×Q₈	C₂×C₃	semidirect
20. C₂×C₃×#239	C₂×C₃	direct (N × H)
21. C₂²×C₃×D₁₂	C₂×C₃	semidirect
22. C₂²×C₃×C₄×D₃	C₂×C₃	semidirect
26. C₃³×Q₈	C₂³	semidirect
29. C₃²×D₁₂	C₂³	semidirect
30. C₃²×C₄×D₃	C₂³	semidirect
32. C₂²×#239	C₃²	direct (N × H)
37. C₂×C₃²×Q₈	C₂×D₃	semidirect
38. C₂×C₃²×Q₈	C₂²×C₃	semidirect
46. C₃×#239	C₂²×C₃	direct (N × H)
47. C₂×C₃×D₁₂	C₂²×C₃	semidirect
48. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
49. C₂²×C₃³	#41	semidirect
57. C₂²×C₃×Q₈	C₃×D₃	semidirect
58. C₂²×C₃×Q₈	C₂×C₃²	semidirect
61. C₂×#239	C₂×C₃²	direct (N × H)
62. C₂²×D₁₂	C₂×C₃²	semidirect
63. C₂²×C₄×D₃	C₂×C₃²	semidirect
72. C₃²×Q₈	C₂²×D₃	semidirect
73. C₃²×Q₈	C₂³×C₃	semidirect
81. C₃×D₁₂	C₂³×C₃	semidirect
82. C₃×C₄×D₃	C₂³×C₃	semidirect
84. C₂×C₃³	C₂×#41	semidirect
91. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
92. C₂×C₃×Q₈	C₂²×C₃²	semidirect
100. #239	C₂²×C₃²	direct (N × H)
101. C₂×D₁₂	C₂²×C₃²	semidirect
102. C₂×C₄×D₃	C₂²×C₃²	semidirect
107. C₂²×C₃²	#239	direct (N × H)
108. C₂²×C₃²	C₃×#41	semidirect
119. C₂²×Q₈	C₃²×D₃	semidirect
122. C₃³	C₂²×#41	semidirect
131. C₃×Q₈	C₂²×C₃×D₃	semidirect
132. C₃×Q₈	C₂³×C₃²	semidirect
140. D₁₂	C₂³×C₃²	semidirect
141. C₄×D₃	C₂³×C₃²	semidirect
145. C₂×C₃²	C₂×#239	direct (N × H)
146. C₂×C₃²	C₂×C₃×#41	semidirect
152. C₂×Q₈	C₂×C₃²×D₃	semidirect
161. C₂²×C₃	C₃×#239	direct (N × H)
162. C₂²×C₃	C₃²×#41	semidirect
173. C₃²	C₂²×#239	direct (N × H)
174. C₃²	C₂²×C₃×#41	semidirect
178. Q₈	C₂²×C₃²×D₃	semidirect
186. C₂×C₃	C₂×C₃×#239	direct (N × H)
187. C₂×C₃	C₂×C₃²×#41	semidirect
193. C₂²	C₃²×#239	direct (N × H)
198. C₃	C₂²×C₃×#239	direct (N × H)
199. C₃	C₂²×C₃²×#41	semidirect
201. C₂	C₂×C₃²×#239	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×C₃³×Q₈	2
C₂×C₃²×#239	2
C₂²×C₃²×D₁₂	2
C₂²×C₃²×C₄×D₃	2
C₂²×C₃²×#41	3
C₂²×C₃×#239	3

Subgroups — C22×C32×#239 #13181718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×#239 #13181718