Subgroups — C₂×C₃²×#560 #11770574

Order 1152.

4032 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#560	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃²×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#126	C₂	1	2	yes	maximal, complemented
4	C₂×C₃²×#123	C₂	1	3	yes	maximal, complemented
5	C₂×C₃²×#121	C₂	1	6	yes	maximal, complemented
6	C₂×C₃²×#120	C₂	1	3	yes	maximal, complemented
7	C₂×C₃×#560	C₃	1	4	yes	maximal, direct product factor
8	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×D₄	C₂²	1	6	yes	—
10	C₂²×C₃²×D₄	C₂²	1	3	yes	—
11	C₂²×C₃²×D₄	C₂²	1	3	yes	complemented
12	C₂³×C₃²×C₄	C₂²	1	3	yes	complemented
13	C₃²×#127	C₂²	1	8	yes	complemented
14	C₃²×#126	C₂²	1	16	yes	complemented
15	C₃²×#123	C₂²	1	24	yes	complemented
16	C₃²×#121	C₂²	1	48	yes	complemented
17	C₃²×#120	C₂²	1	24	yes	complemented
18	C₂×C₃²×C₄⋊C₄	C₂²	1	6	yes	complemented
19	C₂×C₃²×C₂²⋊C₄	C₂²	1	6	yes	—
20	C₂×C₃²×C₂²⋊C₄	C₂²	1	3	yes	—
21	C₂×C₃²×C₂²⋊C₄	C₂²	1	3	yes	complemented
22	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic, complemented
23	C₃×#560	C₂×C₃	1	64	yes	direct product factor
24	C₂×C₃×#127	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×#126	C₂×C₃	1	8	yes	complemented
26	C₂×C₃×#123	C₂×C₃	1	12	yes	complemented
27	C₂×C₃×#121	C₂×C₃	1	24	yes	complemented
28	C₂×C₃×#120	C₂×C₃	1	12	yes	complemented
29	C₂⁴×C₃²	—	4	6	no	—
30	C₂⁴×C₃²	C₂³	1	3	yes	—
31	C₂⁴×C₃²	C₂³	1	2	yes	—
32	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
33	C₂⁴×C₃²	C₂³	1	4	yes	—
34	C₂×C₃²×D₄	—	4	12	no	—
35	C₂×C₃²×D₄	—	4	6	no	—
36	C₂×C₃²×D₄	—	4	6	no	—
37	C₂×C₃²×D₄	C₂³	1	24	yes	—
38	C₂×C₃²×D₄	C₂³	1	12	yes	—
39	C₂×C₃²×D₄	C₂³	1	12	yes	complemented
40	C₂²×C₃²×C₄	—	4	6	no	—
41	C₂²×C₃²×C₄	C₂³	1	6	yes	—
42	C₂²×C₃²×C₄	C₂³	1	3	yes	—
43	C₂²×C₃²×C₄	C₂³	1	12	yes	complemented
44	C₃²×C₄⋊C₄	C₂³	1	24	yes	complemented
45	C₃²×C₂²⋊C₄	C₂³	1	24	yes	—
46	C₃²×C₂²⋊C₄	C₂³	1	12	yes	—
47	C₃²×C₂²⋊C₄	C₂³	1	12	yes	complemented
48	C₃²×C₄²	C₂³	1	4	yes	complemented
49	C₂×#560	C₃²	1	1	yes	characteristic, direct product factor, Sylow
50	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
51	C₂²×C₃×D₄	C₂²×C₃	1	24	yes	—
52	C₂²×C₃×D₄	C₂²×C₃	1	12	yes	—
53	C₂²×C₃×D₄	C₂²×C₃	1	12	yes	complemented
54	C₂³×C₃×C₄	C₂²×C₃	1	12	yes	complemented
55	C₃×#127	C₂²×C₃	1	32	yes	complemented
56	C₃×#126	C₂²×C₃	1	64	yes	complemented
57	C₃×#123	C₂²×C₃	1	96	yes	complemented
58	C₃×#121	C₂²×C₃	1	192	yes	complemented
59	C₃×#120	C₂²×C₃	1	96	yes	complemented
60	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	24	yes	complemented
61	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	24	yes	—
62	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
63	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
64	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	complemented
65	C₂³×C₃²	—	2	12	no	—
66	C₂³×C₃²	—	2	6	no	—
67	C₂³×C₃²	—	2	6	no	—
68	C₂³×C₃²	—	2	12	no	—
69	C₂³×C₃²	—	4	12	no	—
70	C₂³×C₃²	—	4	16	no	—
71	C₂³×C₃²	C₂⁴	1	4	yes	—
72	C₂³×C₃²	C₂⁴	1	2	yes	—
73	C₂³×C₃²	C₂⁴	1	6	yes	—
74	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
75	C₃²×D₄	—	4	24	no	—
76	C₃²×D₄	—	4	12	no	—
77	C₃²×D₄	—	4	12	no	—
78	C₂×C₃²×C₄	—	2	12	no	—
79	C₂×C₃²×C₄	—	2	6	no	—
80	C₂×C₃²×C₄	—	4	12	no	—
81	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
82	C₂×C₃²×C₄	C₂⁴	1	12	yes	—
83	#560	C₂×C₃²	1	16	yes	direct product factor
84	C₂×#127	C₂×C₃²	1	1	yes	characteristic, complemented
85	C₂×#126	C₂×C₃²	1	2	yes	complemented
86	C₂×#123	C₂×C₃²	1	3	yes	complemented
87	C₂×#121	C₂×C₃²	1	6	yes	complemented
88	C₂×#120	C₂×C₃²	1	3	yes	complemented
89	C₂⁴×C₃	—	4	24	no	—
90	C₂⁴×C₃	C₂³×C₃	1	8	yes	—
91	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
92	C₂⁴×C₃	C₂³×C₃	1	12	yes	—
93	C₂⁴×C₃	C₂³×C₃	1	16	yes	—
94	C₂×C₃×D₄	—	4	48	no	—
95	C₂×C₃×D₄	—	4	24	no	—
96	C₂×C₃×D₄	—	4	24	no	—
97	C₂×C₃×D₄	C₂³×C₃	1	96	yes	—
98	C₂×C₃×D₄	C₂³×C₃	1	48	yes	—
99	C₂×C₃×D₄	C₂³×C₃	1	48	yes	complemented
100	C₂²×C₃×C₄	—	4	24	no	—
101	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
102	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
103	C₂²×C₃×C₄	C₂³×C₃	1	48	yes	complemented
104	C₃×C₄⋊C₄	C₂³×C₃	1	96	yes	complemented
105	C₃×C₂²⋊C₄	C₂³×C₃	1	96	yes	—
106	C₃×C₂²⋊C₄	C₂³×C₃	1	48	yes	—
107	C₃×C₂²⋊C₄	C₂³×C₃	1	48	yes	complemented
108	C₃×C₄²	C₂³×C₃	1	16	yes	complemented
109	C₂²×C₃²	—	2	6	no	—
110	C₂²×C₃²	—	2	12	no	—
111	C₂²×C₃²	—	2	12	no	—
112	C₂²×C₃²	—	2	6	no	—
113	C₂²×C₃²	—	4	16	no	—
114	C₂²×C₃²	—	4	12	no	—
115	C₂²×C₃²	—	4	8	no	—
116	C₂²×C₃²	—	4	4	no	—
117	C₂²×C₃²	H(5,𝔽₂)	1	6	yes	—
118	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
119	C₃²×C₄	—	2	12	no	—
120	C₃²×C₄	—	2	6	no	—
121	C₂⁵	C₂²×C₃²	1	1	yes	characteristic
122	C₂²×D₄	C₂²×C₃²	1	6	yes	—
123	C₂²×D₄	C₂²×C₃²	1	3	yes	—
124	C₂²×D₄	C₂²×C₃²	1	3	yes	complemented
125	C₂³×C₄	C₂²×C₃²	1	3	yes	complemented
126	#127	C₂²×C₃²	1	8	yes	complemented
127	#126	C₂²×C₃²	1	16	yes	complemented
128	#123	C₂²×C₃²	1	24	yes	complemented
129	#121	C₂²×C₃²	1	48	yes	complemented
130	#120	C₂²×C₃²	1	24	yes	complemented
131	C₂×C₄⋊C₄	C₂²×C₃²	1	6	yes	complemented
132	C₂×C₂²⋊C₄	C₂²×C₃²	1	6	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₃²	1	3	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₃²	1	3	yes	complemented
135	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
136	C₂³×C₃	—	2	24	no	—
137	C₂³×C₃	—	2	48	no	—
138	C₂³×C₃	—	2	48	no	—
139	C₂³×C₃	—	2	24	no	—
140	C₂³×C₃	—	4	48	no	—
141	C₂³×C₃	—	4	64	no	—
142	C₂³×C₃	C₂⁴×C₃	1	16	yes	—
143	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
144	C₂³×C₃	C₂⁴×C₃	1	24	yes	—
145	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
146	C₃×D₄	—	4	96	no	—
147	C₃×D₄	—	4	48	no	—
148	C₃×D₄	—	4	48	no	—
149	C₂×C₃×C₄	—	2	48	no	—
150	C₂×C₃×C₄	—	2	24	no	—
151	C₂×C₃×C₄	—	4	48	no	—
152	C₂×C₃×C₄	C₂⁴×C₃	1	24	yes	—
153	C₂×C₃×C₄	C₂⁴×C₃	1	48	yes	—
154	C₂×C₃²	—	4	6	no	—
155	C₂×C₃²	—	4	4	no	—
156	C₂×C₃²	—	4	2	no	—
157	C₂×C₃²	#560	1	4	yes	direct product factor
158	C₂×C₃²	C₂×H(5,𝔽₂)	1	3	yes	—
159	C₂⁴	—	4	6	no	—
160	C₂⁴	C₂³×C₃²	1	2	yes	—
161	C₂⁴	C₂³×C₃²	1	1	yes	characteristic
162	C₂⁴	C₂³×C₃²	1	3	yes	—
163	C₂⁴	C₂³×C₃²	1	4	yes	—
164	C₂×D₄	—	4	12	no	—
165	C₂×D₄	—	4	6	no	—
166	C₂×D₄	—	4	6	no	—
167	C₂×D₄	C₂³×C₃²	1	24	yes	—
168	C₂×D₄	C₂³×C₃²	1	12	yes	—
169	C₂×D₄	C₂³×C₃²	1	12	yes	complemented
170	C₂²×C₄	—	4	6	no	—
171	C₂²×C₄	C₂³×C₃²	1	3	yes	—
172	C₂²×C₄	C₂³×C₃²	1	6	yes	—
173	C₂²×C₄	C₂³×C₃²	1	12	yes	complemented
174	C₄⋊C₄	C₂³×C₃²	1	24	yes	complemented
175	C₂²⋊C₄	C₂³×C₃²	1	24	yes	—
176	C₂²⋊C₄	C₂³×C₃²	1	12	yes	—
177	C₂²⋊C₄	C₂³×C₃²	1	12	yes	complemented
178	C₄²	C₂³×C₃²	1	4	yes	complemented
179	C₂²×C₃	—	2	24	no	—
180	C₂²×C₃	—	2	48	no	—
181	C₂²×C₃	—	2	48	no	—
182	C₂²×C₃	—	2	24	no	—
183	C₂²×C₃	—	4	64	no	—
184	C₂²×C₃	—	4	32	no	—
185	C₂²×C₃	—	4	16	no	—
186	C₂²×C₃	—	4	48	no	—
187	C₂²×C₃	C₃×H(5,𝔽₂)	1	24	yes	—
188	C₂²×C₃	C₂⁵×C₃	1	4	yes	—
189	C₃×C₄	—	2	48	no	—
190	C₃×C₄	—	2	24	no	—
191	C₃²	C₂×#560	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	2	6	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	2	6	no	—
196	C₂³	—	4	12	no	—
197	C₂³	—	4	16	no	—
198	C₂³	C₂⁴×C₃²	1	4	yes	—
199	C₂³	C₂⁴×C₃²	1	2	yes	—
200	C₂³	C₂⁴×C₃²	1	6	yes	—
201	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
202	D₄	—	4	24	no	—
203	D₄	—	4	12	no	—
204	D₄	—	4	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	4	12	no	—
208	C₂×C₄	C₂⁴×C₃²	1	6	yes	—
209	C₂×C₄	C₂⁴×C₃²	1	12	yes	—
210	C₂×C₃	—	4	16	no	—
211	C₂×C₃	—	4	8	no	—
212	C₂×C₃	—	4	24	no	—
213	C₂×C₃	C₃×#560	1	16	yes	direct product factor
214	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	12	yes	—
215	C₂²	—	2	6	no	—
216	C₂²	—	2	12	no	—
217	C₂²	—	2	12	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	4	16	no	—
220	C₂²	—	4	8	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	4	12	no	—
223	C₂²	C₃²×H(5,𝔽₂)	1	6	yes	—
224	C₂²	C₂⁵×C₃²	1	1	yes	characteristic
225	C₄	—	2	12	no	—
226	C₄	—	2	6	no	—
227	C₃	C₂×C₃×#560	1	4	yes	direct product factor
228	C₂	—	4	4	no	—
229	C₂	—	4	2	no	—
230	C₂	—	4	6	no	—
231	C₂	C₃²×#560	1	4	yes	direct product factor
232	C₂	C₂×C₃²×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×#560	C₂	direct (N × H)
2. C₂×C₃²×#127	C₂	semidirect
3. C₂×C₃²×#126	C₂	semidirect
4. C₂×C₃²×#123	C₂	semidirect
5. C₂×C₃²×#121	C₂	semidirect
6. C₂×C₃²×#120	C₂	semidirect
7. C₂×C₃×#560	C₃	direct (N × H)
11. C₂²×C₃²×D₄	C₂²	semidirect
12. C₂³×C₃²×C₄	C₂²	semidirect
13. C₃²×#127	C₂²	semidirect
14. C₃²×#126	C₂²	semidirect
15. C₃²×#123	C₂²	semidirect
16. C₃²×#121	C₂²	semidirect
17. C₃²×#120	C₂²	semidirect
18. C₂×C₃²×C₄⋊C₄	C₂²	semidirect
21. C₂×C₃²×C₂²⋊C₄	C₂²	semidirect
22. C₂×C₃²×C₄²	C₂²	semidirect
23. C₃×#560	C₂×C₃	direct (N × H)
24. C₂×C₃×#127	C₂×C₃	semidirect
25. C₂×C₃×#126	C₂×C₃	semidirect
26. C₂×C₃×#123	C₂×C₃	semidirect
27. C₂×C₃×#121	C₂×C₃	semidirect
28. C₂×C₃×#120	C₂×C₃	semidirect
39. C₂×C₃²×D₄	C₂³	semidirect
43. C₂²×C₃²×C₄	C₂³	semidirect
44. C₃²×C₄⋊C₄	C₂³	semidirect
47. C₃²×C₂²⋊C₄	C₂³	semidirect
48. C₃²×C₄²	C₂³	semidirect
49. C₂×#560	C₃²	direct (N × H)
53. C₂²×C₃×D₄	C₂²×C₃	semidirect
54. C₂³×C₃×C₄	C₂²×C₃	semidirect
55. C₃×#127	C₂²×C₃	semidirect
56. C₃×#126	C₂²×C₃	semidirect
57. C₃×#123	C₂²×C₃	semidirect
58. C₃×#121	C₂²×C₃	semidirect
59. C₃×#120	C₂²×C₃	semidirect
60. C₂×C₃×C₄⋊C₄	C₂²×C₃	semidirect
63. C₂×C₃×C₂²⋊C₄	C₂²×C₃	semidirect
64. C₂×C₃×C₄²	C₂²×C₃	semidirect
83. #560	C₂×C₃²	direct (N × H)
84. C₂×#127	C₂×C₃²	semidirect
85. C₂×#126	C₂×C₃²	semidirect
86. C₂×#123	C₂×C₃²	semidirect
87. C₂×#121	C₂×C₃²	semidirect
88. C₂×#120	C₂×C₃²	semidirect
99. C₂×C₃×D₄	C₂³×C₃	semidirect
103. C₂²×C₃×C₄	C₂³×C₃	semidirect
104. C₃×C₄⋊C₄	C₂³×C₃	semidirect
107. C₃×C₂²⋊C₄	C₂³×C₃	semidirect
108. C₃×C₄²	C₂³×C₃	semidirect
124. C₂²×D₄	C₂²×C₃²	semidirect
125. C₂³×C₄	C₂²×C₃²	semidirect
126. #127	C₂²×C₃²	semidirect
127. #126	C₂²×C₃²	semidirect
128. #123	C₂²×C₃²	semidirect
129. #121	C₂²×C₃²	semidirect
130. #120	C₂²×C₃²	semidirect
131. C₂×C₄⋊C₄	C₂²×C₃²	semidirect
134. C₂×C₂²⋊C₄	C₂²×C₃²	semidirect
135. C₂×C₄²	C₂²×C₃²	semidirect
157. C₂×C₃²	#560	direct (N × H)
169. C₂×D₄	C₂³×C₃²	semidirect
173. C₂²×C₄	C₂³×C₃²	semidirect
174. C₄⋊C₄	C₂³×C₃²	semidirect
177. C₂²⋊C₄	C₂³×C₃²	semidirect
178. C₄²	C₂³×C₃²	semidirect
191. C₃²	C₂×#560	direct (N × H)
213. C₂×C₃	C₃×#560	direct (N × H)
227. C₃	C₂×C₃×#560	direct (N × H)
231. C₂	C₃²×#560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#560	2
C₂×C₃²×#127	2
C₂×C₃²×#126	2
C₂×C₃²×#123	2
C₂×C₃²×#121	2
C₂×C₃²×#120	2
C₂×C₃×#560	3

Subgroups — C2×C32×#560 #11770574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#560 #11770574