Subgroups — C₂×3₋¹⁺²×#120 #13154547

Order 1728.

2648 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×3₋¹⁺²	C₂	1	3	yes	maximal, complemented
3	3₋¹⁺²×#120	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₄×3₋¹⁺²	C₂	1	3	yes	maximal, complemented
5	C₂×C₃²×#120	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#120	C₃	1	3	yes	maximal, complemented
7	C₂⁴×3₋¹⁺²	—	2	12	no	—
8	C₂⁴×3₋¹⁺²	C₂²	1	3	yes	—
9	C₂⁴×3₋¹⁺²	C₂²	1	4	yes	complemented
10	C₂⁴×3₋¹⁺²	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄×3₋¹⁺²	—	2	12	no	—
12	C₂×D₄×3₋¹⁺²	C₂²	1	12	yes	complemented
13	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	complemented
14	C₂²⋊C₄×3₋¹⁺²	C₂²	1	12	yes	complemented
15	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃²×D₄	C₂×C₃	1	3	yes	—
17	C₂⁵×C₉	C₂×C₃	1	3	yes	complemented
18	C₃²×#120	C₂×C₃	1	8	yes	—
19	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
20	C₂²×D₄×C₉	C₂×C₃	1	9	yes	complemented
21	C₉×#120	C₂×C₃	1	24	yes	complemented
22	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	9	yes	complemented
23	C₂³×3₋¹⁺²	—	2	32	no	—
24	C₂³×3₋¹⁺²	—	2	12	no	—
25	C₂³×3₋¹⁺²	—	2	24	no	—
26	C₂³×3₋¹⁺²	—	2	6	no	—
27	C₂³×3₋¹⁺²	D₄	1	12	yes	complemented
28	C₂³×3₋¹⁺²	C₂³	1	6	yes	—
29	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
30	C₂³×3₋¹⁺²	C₂³	1	2	yes	complemented
31	D₄×3₋¹⁺²	—	2	24	no	—
32	C₂×C₄×3₋¹⁺²	—	2	6	no	—
33	C₂×C₄×3₋¹⁺²	C₂³	1	6	yes	complemented
34	C₂×C₃×#120	—	3	1	no	—
35	C₂×C₃×#120	C₃²	1	1	yes	characteristic
36	C₂⁴×C₃²	—	2	12	no	—
37	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
38	C₂⁴×C₃²	C₂²×C₃	1	4	yes	—
39	C₂⁴×C₃²	C₂²×C₃	1	3	yes	—
40	C₂×C₃²×D₄	—	2	12	no	—
41	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
43	C₂⁴×C₉	—	2	36	no	—
44	C₂⁴×C₉	C₂²×C₃	1	9	yes	—
45	C₂⁴×C₉	C₂²×C₃	1	12	yes	complemented
46	C₂⁴×C₉	C₂²×C₃	1	3	yes	complemented
47	C₃²×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
48	C₂×D₄×C₉	—	2	36	no	—
49	C₂×D₄×C₉	C₂²×C₃	1	36	yes	complemented
50	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	complemented
51	C₉×C₂²⋊C₄	C₂²×C₃	1	36	yes	complemented
52	C₂²×3₋¹⁺²	—	2	32	no	—
53	C₂²×3₋¹⁺²	—	2	24	no	—
54	C₂²×3₋¹⁺²	—	2	12	no	—
55	C₂²×3₋¹⁺²	—	2	6	no	—
56	C₂²×3₋¹⁺²	—	4	4	no	—
57	C₂²×3₋¹⁺²	C₂×D₄	1	6	yes	—
58	C₂²×3₋¹⁺²	C₂×D₄	1	12	yes	complemented
59	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
60	C₄×3₋¹⁺²	—	2	6	no	—
61	C₂⁵×C₃	—	3	1	no	—
62	C₂⁵×C₃	C₂×C₃²	1	1	yes	characteristic
63	C₂²×C₃×D₄	—	3	3	no	—
64	C₂²×C₃×D₄	C₂×C₃²	1	3	yes	—
65	C₃×#120	—	3	8	no	—
66	C₃×#120	C₂×C₃²	1	8	yes	—
67	C₂×C₃×C₂²⋊C₄	—	3	3	no	—
68	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
69	C₂³×C₃²	—	2	32	no	—
70	C₂³×C₃²	—	2	24	no	—
71	C₂³×C₃²	—	2	12	no	—
72	C₂³×C₃²	—	2	6	no	—
73	C₂³×C₃²	C₃×D₄	1	12	yes	—
74	C₂³×C₃²	C₂³×C₃	1	2	yes	—
75	C₂³×C₃²	C₂³×C₃	1	6	yes	—
76	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
77	C₃²×D₄	—	2	24	no	—
78	C₂×C₃²×C₄	—	2	6	no	—
79	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
80	C₂³×C₉	—	2	96	no	—
81	C₂³×C₉	—	2	36	no	—
82	C₂³×C₉	—	2	72	no	—
83	C₂³×C₉	—	2	18	no	—
84	C₂³×C₉	C₃×D₄	1	36	yes	complemented
85	C₂³×C₉	C₂³×C₃	1	18	yes	—
86	C₂³×C₉	C₂³×C₃	1	3	yes	—
87	C₂³×C₉	C₂³×C₃	1	6	yes	complemented
88	D₄×C₉	—	2	72	no	—
89	C₂×C₄×C₉	—	2	18	no	—
90	C₂×C₄×C₉	C₂³×C₃	1	18	yes	complemented
91	C₂×#120	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×3₋¹⁺²	—	2	12	no	—
93	C₂×3₋¹⁺²	—	4	2	no	—
94	C₂×3₋¹⁺²	#120	1	4	yes	direct product factor
95	C₂×3₋¹⁺²	C₂²×D₄	1	3	yes	—
96	C₂⁴×C₃	—	2	12	no	—
97	C₂⁴×C₃	—	3	3	no	—
98	C₂⁴×C₃	—	3	4	no	—
99	C₂⁴×C₃	—	3	1	no	—
100	C₂⁴×C₃	—	6	12	no	—
101	C₂⁴×C₃	C₂²×C₃²	1	3	yes	—
102	C₂⁴×C₃	C₂²×C₃²	1	4	yes	—
103	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
104	C₂×C₃×D₄	—	2	12	no	—
105	C₂×C₃×D₄	—	3	12	no	—
106	C₂×C₃×D₄	—	6	12	no	—
107	C₂×C₃×D₄	C₂²×C₃²	1	12	yes	—
108	C₂²×C₃×C₄	—	3	3	no	—
109	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
110	C₃×C₂²⋊C₄	—	3	12	no	—
111	C₃×C₂²⋊C₄	C₂²×C₃²	1	12	yes	—
112	C₂²×C₃²	—	2	32	no	—
113	C₂²×C₃²	—	2	24	no	—
114	C₂²×C₃²	—	2	12	no	—
115	C₂²×C₃²	—	2	6	no	—
116	C₂²×C₃²	—	4	4	no	—
117	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
118	C₂²×C₃²	C₂×C₃×D₄	1	6	yes	—
119	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
120	C₃²×C₄	—	2	6	no	—
121	C₂²×C₉	—	2	96	no	—
122	C₂²×C₉	—	2	72	no	—
123	C₂²×C₉	—	2	36	no	—
124	C₂²×C₉	—	2	18	no	—
125	C₂²×C₉	—	4	12	no	—
126	C₂²×C₉	C₂×C₃×D₄	1	18	yes	—
127	C₂²×C₉	C₂×C₃×D₄	1	36	yes	complemented
128	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
129	C₄×C₉	—	2	18	no	—
130	C₂⁵	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
131	C₂²×D₄	C₂×3₋¹⁺²	1	3	yes	complemented
132	#120	C₂×3₋¹⁺²	1	8	yes	direct product factor
133	C₂×C₂²⋊C₄	C₂×3₋¹⁺²	1	3	yes	complemented
134	3₋¹⁺²	C₂×#120	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₃	—	2	32	no	—
136	C₂³×C₃	—	2	12	no	—
137	C₂³×C₃	—	2	24	no	—
138	C₂³×C₃	—	2	6	no	—
139	C₂³×C₃	—	3	12	no	—
140	C₂³×C₃	—	3	1	no	—
141	C₂³×C₃	—	3	6	no	—
142	C₂³×C₃	—	3	2	no	—
143	C₂³×C₃	—	6	12	no	—
144	C₂³×C₃	—	6	24	no	—
145	C₂³×C₃	—	6	6	no	—
146	C₂³×C₃	—	6	32	no	—
147	C₂³×C₃	C₃²×D₄	1	12	yes	—
148	C₂³×C₃	C₂³×C₃²	1	6	yes	—
149	C₂³×C₃	C₂³×C₃²	1	2	yes	—
150	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
151	C₃×D₄	—	2	24	no	—
152	C₃×D₄	—	6	24	no	—
153	C₂×C₃×C₄	—	2	6	no	—
154	C₂×C₃×C₄	—	3	6	no	—
155	C₂×C₃×C₄	—	6	6	no	—
156	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
157	C₂×C₃²	—	2	12	no	—
158	C₂×C₃²	—	4	2	no	—
159	C₂×C₃²	C₃×#120	1	4	yes	—
160	C₂×C₃²	C₂²×C₃×D₄	1	3	yes	—
161	C₂×C₉	—	2	36	no	—
162	C₂×C₉	—	4	6	no	—
163	C₂×C₉	C₃×#120	1	12	yes	complemented
164	C₂×C₉	C₂²×C₃×D₄	1	9	yes	—
165	C₂⁴	—	2	12	no	—
166	C₂⁴	C₂²×3₋¹⁺²	1	3	yes	—
167	C₂⁴	C₂²×3₋¹⁺²	1	4	yes	complemented
168	C₂⁴	C₂²×3₋¹⁺²	1	1	yes	characteristic, complemented
169	C₂×D₄	—	2	12	no	—
170	C₂×D₄	C₂²×3₋¹⁺²	1	12	yes	complemented
171	C₂²×C₄	C₂²×3₋¹⁺²	1	3	yes	complemented
172	C₂²⋊C₄	C₂²×3₋¹⁺²	1	12	yes	complemented
173	C₂²×C₃	—	2	32	no	—
174	C₂²×C₃	—	2	12	no	—
175	C₂²×C₃	—	2	24	no	—
176	C₂²×C₃	—	2	6	no	—
177	C₂²×C₃	—	3	1	no	—
178	C₂²×C₃	—	3	6	no	—
179	C₂²×C₃	—	3	12	no	—
180	C₂²×C₃	—	4	4	no	—
181	C₂²×C₃	—	6	24	no	—
182	C₂²×C₃	—	6	12	no	—
183	C₂²×C₃	—	6	6	no	—
184	C₂²×C₃	—	6	32	no	—
185	C₂²×C₃	—	12	4	no	—
186	C₂²×C₃	C₂×C₃²×D₄	1	6	yes	—
187	C₂²×C₃	C₂×C₃²×D₄	1	12	yes	—
188	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	6	6	no	—
191	C₃²	C₂×C₃×#120	1	1	yes	characteristic
192	C₉	C₂×C₃×#120	1	3	yes	complemented
193	C₂³	—	2	32	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	24	no	—
196	C₂³	—	2	6	no	—
197	C₂³	D₄×3₋¹⁺²	1	12	yes	complemented
198	C₂³	C₂³×3₋¹⁺²	1	6	yes	—
199	C₂³	C₂³×3₋¹⁺²	1	1	yes	characteristic
200	C₂³	C₂³×3₋¹⁺²	1	2	yes	complemented
201	D₄	—	2	24	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	C₂³×3₋¹⁺²	1	6	yes	complemented
204	C₂×C₃	—	2	12	no	—
205	C₂×C₃	—	3	4	no	—
206	C₂×C₃	—	3	3	no	—
207	C₂×C₃	—	4	2	no	—
208	C₂×C₃	—	6	12	no	—
209	C₂×C₃	—	12	2	no	—
210	C₂×C₃	C₃²×#120	1	4	yes	—
211	C₂×C₃	C₂²×C₃²×D₄	1	3	yes	—
212	C₂²	—	2	32	no	—
213	C₂²	—	2	24	no	—
214	C₂²	—	2	12	no	—
215	C₂²	—	2	6	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂×D₄×3₋¹⁺²	1	6	yes	—
218	C₂²	C₂×D₄×3₋¹⁺²	1	12	yes	complemented
219	C₂²	C₂⁴×3₋¹⁺²	1	1	yes	characteristic
220	C₄	—	2	6	no	—
221	C₃	—	3	1	no	—
222	C₃	C₂×C₃²×#120	1	1	yes	characteristic
223	C₂	—	2	12	no	—
224	C₂	—	4	2	no	—
225	C₂	3₋¹⁺²×#120	1	4	yes	direct product factor
226	C₂	C₂²×D₄×3₋¹⁺²	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂³×C₃²
commutator, Frattini, cosocle	C₂²×C₃	C₂⁴×C₃²
prime radical	C₂×C₃²×#120	C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×3₋¹⁺²	C₂	semidirect
2. C₂²×D₄×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²×#120	C₂	direct (N × H)
4. C₂×C₂²⋊C₄×3₋¹⁺²	C₂	semidirect
6. C₂×C₉×#120	C₃	semidirect
9. C₂⁴×3₋¹⁺²	C₂²	semidirect
10. C₂⁴×3₋¹⁺²	C₂²	semidirect
12. C₂×D₄×3₋¹⁺²	C₂²	semidirect
13. C₂²×C₄×3₋¹⁺²	C₂²	semidirect
14. C₂²⋊C₄×3₋¹⁺²	C₂²	semidirect
17. C₂⁵×C₉	C₂×C₃	semidirect
20. C₂²×D₄×C₉	C₂×C₃	semidirect
21. C₉×#120	C₂×C₃	semidirect
22. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
27. C₂³×3₋¹⁺²	D₄	semidirect
30. C₂³×3₋¹⁺²	C₂³	semidirect
33. C₂×C₄×3₋¹⁺²	C₂³	semidirect
45. C₂⁴×C₉	C₂²×C₃	semidirect
46. C₂⁴×C₉	C₂²×C₃	semidirect
49. C₂×D₄×C₉	C₂²×C₃	semidirect
50. C₂²×C₄×C₉	C₂²×C₃	semidirect
51. C₉×C₂²⋊C₄	C₂²×C₃	semidirect
58. C₂²×3₋¹⁺²	C₂×D₄	semidirect
84. C₂³×C₉	C₃×D₄	semidirect
87. C₂³×C₉	C₂³×C₃	semidirect
90. C₂×C₄×C₉	C₂³×C₃	semidirect
91. C₂×#120	3₋¹⁺²	direct (N × H)
94. C₂×3₋¹⁺²	#120	direct (N × H)
127. C₂²×C₉	C₂×C₃×D₄	semidirect
130. C₂⁵	C₂×3₋¹⁺²	semidirect
131. C₂²×D₄	C₂×3₋¹⁺²	semidirect
132. #120	C₂×3₋¹⁺²	direct (N × H)
133. C₂×C₂²⋊C₄	C₂×3₋¹⁺²	semidirect
134. 3₋¹⁺²	C₂×#120	direct (N × H)
163. C₂×C₉	C₃×#120	semidirect
167. C₂⁴	C₂²×3₋¹⁺²	semidirect
168. C₂⁴	C₂²×3₋¹⁺²	semidirect
170. C₂×D₄	C₂²×3₋¹⁺²	semidirect
171. C₂²×C₄	C₂²×3₋¹⁺²	semidirect
172. C₂²⋊C₄	C₂²×3₋¹⁺²	semidirect
192. C₉	C₂×C₃×#120	semidirect
197. C₂³	D₄×3₋¹⁺²	semidirect
200. C₂³	C₂³×3₋¹⁺²	semidirect
203. C₂×C₄	C₂³×3₋¹⁺²	semidirect
218. C₂²	C₂×D₄×3₋¹⁺²	semidirect
225. C₂	3₋¹⁺²×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×3₋¹⁺²	2
C₂²×D₄×3₋¹⁺²	2
3₋¹⁺²×#120	2
C₂×C₂²⋊C₄×3₋¹⁺²	2
C₂×C₃²×#120	3
C₂×C₉×#120	3

Subgroups — C2×3−1+2×#120 #13154547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×#120 #13154547