Subgroups — C₃²×#546 #10596347

Order 576.

1002 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#123	C₂	1	2	yes	maximal, complemented
5	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#121	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#121	C₂	1	2	yes	maximal, complemented
8	C₃²×#120	C₂	1	2	yes	maximal, complemented
9	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₃×#546	C₃	1	4	yes	maximal, direct product factor
13	C₂⁴×C₃²	C₂²	1	2	yes	—
14	C₃²×#41	—	2	4	no	—
15	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃²×D₄	—	2	4	no	—
17	C₂×C₃²×D₄	C₂²	1	2	yes	—
18	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₃²×D₄	C₂²	1	2	yes	complemented
22	C₂×C₃²×D₄	C₂²	1	2	yes	complemented
23	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂²	1	2	yes	complemented
25	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₃²×C₄	C₂²	1	2	yes	complemented
27	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	C₃²×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
33	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
34	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
35	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
36	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
37	C₃²×C₄²	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃×#41	C₂×C₃	1	4	yes	complemented
39	C₂²×C₃×D₄	C₂×C₃	1	4	yes	complemented
40	C₃×#124	C₂×C₃	1	4	yes	complemented
41	C₃×#123	C₂×C₃	1	8	yes	complemented
42	C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
43	C₃×#121	C₂×C₃	1	4	yes	complemented
44	C₃×#121	C₂×C₃	1	8	yes	complemented
45	C₃×#120	C₂×C₃	1	8	yes	complemented
46	C₃×C₄×D₄	C₂×C₃	1	4	yes	complemented
47	C₃×C₄×D₄	C₂×C₃	1	4	yes	complemented
48	C₂×C₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
49	C₂³×C₃²	—	2	4	no	—
50	C₂³×C₃²	—	2	4	no	—
51	C₂³×C₃²	—	4	2	no	—
52	C₂³×C₃²	C₂³	1	2	yes	—
53	C₂³×C₃²	C₂³	1	1	yes	characteristic
54	C₂³×C₃²	C₂³	1	2	yes	—
55	C₂³×C₃²	C₂³	1	1	yes	characteristic
56	C₃²×Q₈	—	2	2	no	—
57	C₃²×D₄	—	2	4	no	—
58	C₃²×D₄	—	2	4	no	—
59	C₃²×D₄	—	2	2	no	—
60	C₃²×D₄	—	2	2	no	—
61	C₃²×D₄	—	4	2	no	—
62	C₃²×D₄	D₄	1	4	yes	complemented
63	C₂×C₃²×C₄	—	2	4	no	—
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	4	2	no	—
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂³	1	2	yes	—
70	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂³	1	2	yes	—
72	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
74	#546	C₃²	1	1	yes	characteristic, direct product factor, Sylow
75	C₂⁴×C₃	C₂²×C₃	1	8	yes	—
76	C₃×#41	—	2	16	no	—
77	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
78	C₂×C₃×D₄	—	2	16	no	—
79	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
80	C₂×C₃×D₄	C₂²×C₃	1	8	yes	—
81	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
82	C₂×C₃×D₄	C₂²×C₃	1	4	yes	—
83	C₂×C₃×D₄	C₂²×C₃	1	8	yes	complemented
84	C₂×C₃×D₄	C₂²×C₃	1	8	yes	complemented
85	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	complemented
87	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	complemented
88	C₂²×C₃×C₄	C₂²×C₃	1	8	yes	complemented
89	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
90	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
91	C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	complemented
92	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
93	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
94	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
95	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
96	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
97	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
98	C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
99	C₃×C₄²	C₂²×C₃	1	4	yes	complemented
100	C₂²×C₃²	—	2	1	no	—
101	C₂²×C₃²	—	2	2	no	—
102	C₂²×C₃²	—	2	2	no	—
103	C₂²×C₃²	—	2	1	no	—
104	C₂²×C₃²	—	2	1	no	—
105	C₂²×C₃²	—	2	2	no	—
106	C₂²×C₃²	—	2	1	no	—
107	C₂²×C₃²	—	2	2	no	—
108	C₂²×C₃²	—	2	1	no	—
109	C₂²×C₃²	—	2	2	no	—
110	C₂²×C₃²	—	4	8	no	—
111	C₂²×C₃²	C₂×D₄	1	4	yes	—
112	C₂²×C₃²	#41	1	2	yes	complemented
113	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	—	2	2	no	—
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	—	2	1	no	—
119	C₃²×C₄	—	2	2	no	—
120	C₃²×C₄	C₂×D₄	1	2	yes	—
121	C₂×#41	C₂×C₃²	1	1	yes	characteristic, complemented
122	C₂²×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
123	#124	C₂×C₃²	1	1	yes	characteristic, complemented
124	#123	C₂×C₃²	1	2	yes	complemented
125	C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
126	#121	C₂×C₃²	1	1	yes	characteristic, complemented
127	#121	C₂×C₃²	1	2	yes	complemented
128	#120	C₂×C₃²	1	2	yes	complemented
129	C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
130	C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
131	C₂×C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
132	C₂³×C₃	—	2	16	no	—
133	C₂³×C₃	—	2	16	no	—
134	C₂³×C₃	—	4	8	no	—
135	C₂³×C₃	C₂³×C₃	1	8	yes	—
136	C₂³×C₃	C₂³×C₃	1	4	yes	—
137	C₂³×C₃	C₂³×C₃	1	8	yes	—
138	C₂³×C₃	C₂³×C₃	1	4	yes	—
139	C₃×Q₈	—	2	8	no	—
140	C₃×D₄	—	2	16	no	—
141	C₃×D₄	—	2	16	no	—
142	C₃×D₄	—	2	8	no	—
143	C₃×D₄	—	2	8	no	—
144	C₃×D₄	—	4	8	no	—
145	C₃×D₄	C₃×D₄	1	16	yes	complemented
146	C₂×C₃×C₄	—	2	16	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	2	8	no	—
149	C₂×C₃×C₄	—	4	8	no	—
150	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
151	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
152	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
153	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
154	C₂×C₃×C₄	C₂³×C₃	1	8	yes	—
155	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
156	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
157	C₂×C₃²	—	2	2	no	—
158	C₂×C₃²	—	2	4	no	—
159	C₂×C₃²	—	4	1	no	—
160	C₂×C₃²	—	4	2	no	—
161	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃²	C₂×#41	1	1	yes	characteristic
163	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂⁴	C₂²×C₃²	1	2	yes	—
165	#41	—	2	4	no	—
166	C₂×Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
169	C₂×D₄	C₂²×C₃²	1	2	yes	—
170	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
171	C₂×D₄	C₂²×C₃²	1	1	yes	characteristic
172	C₂×D₄	C₂²×C₃²	1	2	yes	complemented
173	C₂×D₄	C₂²×C₃²	1	2	yes	complemented
174	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₃²	1	2	yes	complemented
176	C₂²×C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
177	C₂²×C₄	C₂²×C₃²	1	2	yes	complemented
178	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
179	C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
180	C₄⋊C₄	C₂²×C₃²	1	2	yes	complemented
181	C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic
182	C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
183	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
184	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
185	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
186	C₂²⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
187	C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
188	C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	—	2	8	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	2	4	no	—
194	C₂²×C₃	—	2	8	no	—
195	C₂²×C₃	—	2	4	no	—
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	2	4	no	—
198	C₂²×C₃	—	2	8	no	—
199	C₂²×C₃	—	4	32	no	—
200	C₂²×C₃	C₂×C₃×D₄	1	16	yes	—
201	C₂²×C₃	C₃×#41	1	8	yes	complemented
202	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	4	no	—
205	C₃×C₄	—	2	8	no	—
206	C₃×C₄	—	2	4	no	—
207	C₃×C₄	—	2	4	no	—
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	C₂×C₃×D₄	1	8	yes	—
210	C₃²	#546	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	—	2	4	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂³×C₃²	1	2	yes	—
215	C₂³	C₂³×C₃²	1	1	yes	characteristic
216	C₂³	C₂³×C₃²	1	2	yes	—
217	C₂³	C₂³×C₃²	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	4	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	4	2	no	—
224	D₄	C₃²×D₄	1	4	yes	complemented
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
230	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
231	C₂×C₄	C₂³×C₃²	1	2	yes	—
232	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₃²	1	2	yes	—
234	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₃²	1	1	yes	characteristic
236	C₂×C₃	—	2	8	no	—
237	C₂×C₃	—	2	16	no	—
238	C₂×C₃	—	4	8	no	—
239	C₂×C₃	—	4	4	no	—
240	C₂×C₃	C₂²×C₃×D₄	1	4	yes	—
241	C₂×C₃	C₂×C₃×#41	1	4	yes	—
242	C₂×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	8	no	—
254	C₂²	C₂×C₃²×D₄	1	4	yes	—
255	C₂²	C₃²×#41	1	2	yes	complemented
256	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	C₂×C₃²×D₄	1	2	yes	—
264	C₃	C₃×#546	1	4	yes	direct product factor
265	C₂	—	2	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	4	2	no	—
268	C₂	—	4	1	no	—
269	C₂	C₂²×C₃²×D₄	1	1	yes	characteristic
270	C₂	C₂×C₃²×#41	1	1	yes	characteristic
271	C₂	C₃²×H(5,𝔽₂)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#41	C₂	semidirect
2. C₂²×C₃²×D₄	C₂	semidirect
3. C₃²×#124	C₂	semidirect
4. C₃²×#123	C₂	semidirect
5. C₃²×C₂²⋊Q₈	C₂	semidirect
6. C₃²×#121	C₂	semidirect
7. C₃²×#121	C₂	semidirect
8. C₃²×#120	C₂	semidirect
9. C₃²×C₄×D₄	C₂	semidirect
10. C₃²×C₄×D₄	C₂	semidirect
11. C₂×C₃²×C₂²⋊C₄	C₂	semidirect
12. C₃×#546	C₃	direct (N × H)
15. C₂×C₃²×Q₈	C₂²	semidirect
21. C₂×C₃²×D₄	C₂²	semidirect
22. C₂×C₃²×D₄	C₂²	semidirect
24. C₂²×C₃²×C₄	C₂²	semidirect
25. C₂²×C₃²×C₄	C₂²	semidirect
26. C₂²×C₃²×C₄	C₂²	semidirect
27. C₃²×C₄⋊C₄	C₂²	semidirect
28. C₃²×C₄⋊C₄	C₂²	semidirect
29. C₃²×C₄⋊C₄	C₂²	semidirect
32. C₃²×C₂²⋊C₄	C₂²	semidirect
33. C₃²×C₂²⋊C₄	C₂²	semidirect
34. C₃²×C₂²⋊C₄	C₂²	semidirect
35. C₃²×C₂²⋊C₄	C₂²	semidirect
36. C₃²×C₂²⋊C₄	C₂²	semidirect
37. C₃²×C₄²	C₂²	semidirect
38. C₂×C₃×#41	C₂×C₃	semidirect
39. C₂²×C₃×D₄	C₂×C₃	semidirect
40. C₃×#124	C₂×C₃	semidirect
41. C₃×#123	C₂×C₃	semidirect
42. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
43. C₃×#121	C₂×C₃	semidirect
44. C₃×#121	C₂×C₃	semidirect
45. C₃×#120	C₂×C₃	semidirect
46. C₃×C₄×D₄	C₂×C₃	semidirect
47. C₃×C₄×D₄	C₂×C₃	semidirect
48. C₂×C₃×C₂²⋊C₄	C₂×C₃	semidirect
62. C₃²×D₄	D₄	semidirect
74. #546	C₃²	direct (N × H)
77. C₂×C₃×Q₈	C₂²×C₃	semidirect
83. C₂×C₃×D₄	C₂²×C₃	semidirect
84. C₂×C₃×D₄	C₂²×C₃	semidirect
86. C₂²×C₃×C₄	C₂²×C₃	semidirect
87. C₂²×C₃×C₄	C₂²×C₃	semidirect
88. C₂²×C₃×C₄	C₂²×C₃	semidirect
89. C₃×C₄⋊C₄	C₂²×C₃	semidirect
90. C₃×C₄⋊C₄	C₂²×C₃	semidirect
91. C₃×C₄⋊C₄	C₂²×C₃	semidirect
94. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
95. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
96. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
97. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
98. C₃×C₂²⋊C₄	C₂²×C₃	semidirect
99. C₃×C₄²	C₂²×C₃	semidirect
112. C₂²×C₃²	#41	semidirect
121. C₂×#41	C₂×C₃²	semidirect
122. C₂²×D₄	C₂×C₃²	semidirect
123. #124	C₂×C₃²	semidirect
124. #123	C₂×C₃²	semidirect
125. C₂²⋊Q₈	C₂×C₃²	semidirect
126. #121	C₂×C₃²	semidirect
127. #121	C₂×C₃²	semidirect
128. #120	C₂×C₃²	semidirect
129. C₄×D₄	C₂×C₃²	semidirect
130. C₄×D₄	C₂×C₃²	semidirect
131. C₂×C₂²⋊C₄	C₂×C₃²	semidirect
145. C₃×D₄	C₃×D₄	semidirect
166. C₂×Q₈	C₂²×C₃²	semidirect
172. C₂×D₄	C₂²×C₃²	semidirect
173. C₂×D₄	C₂²×C₃²	semidirect
175. C₂²×C₄	C₂²×C₃²	semidirect
176. C₂²×C₄	C₂²×C₃²	semidirect
177. C₂²×C₄	C₂²×C₃²	semidirect
178. C₄⋊C₄	C₂²×C₃²	semidirect
179. C₄⋊C₄	C₂²×C₃²	semidirect
180. C₄⋊C₄	C₂²×C₃²	semidirect
183. C₂²⋊C₄	C₂²×C₃²	semidirect
184. C₂²⋊C₄	C₂²×C₃²	semidirect
185. C₂²⋊C₄	C₂²×C₃²	semidirect
186. C₂²⋊C₄	C₂²×C₃²	semidirect
187. C₂²⋊C₄	C₂²×C₃²	semidirect
188. C₄²	C₂²×C₃²	semidirect
201. C₂²×C₃	C₃×#41	semidirect
210. C₃²	#546	direct (N × H)
224. D₄	C₃²×D₄	semidirect
255. C₂²	C₃²×#41	semidirect
264. C₃	C₃×#546	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×C₃²×#41	2
C₂²×C₃²×D₄	2
C₃²×#124	2
C₃²×#123	2
C₃²×C₂²⋊Q₈	2
C₃²×#121	2
C₃²×#120	2
C₃²×C₄×D₄	2
C₂×C₃²×C₂²⋊C₄	2
C₃×#546	3

Subgroups — C32×#546 #10596347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#546 #10596347