Subgroups — C₃×#5727 #10595824

Order 576.

594 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#946	C₂	1	2	yes	maximal, complemented
4	C₃×#896	C₂	1	4	yes	maximal, complemented
5	C₃×#893	C₂	1	2	yes	maximal
6	C₃×#888	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×D₄×Q₈	—	3	1	no	maximal
10	#5727	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₃×D₃×Q₈	—	2	8	no	—
13	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₃	C₂²	1	4	yes	—
15	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
16	C₂×C₃×Dic₆	C₂²	1	4	yes	—
17	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄²	C₂²	1	1	yes	characteristic
21	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
26	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄×Dic₃	C₂²	1	2	yes	—
28	C₂²×C₃×Q₈	—	3	2	no	—
29	C₂×D₃×Q₈	C₂×C₃	1	2	yes	complemented
30	C₃×C₄⋊Q₈	—	2	1	no	—
31	C₃×C₄⋊Q₈	—	3	2	no	—
32	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
33	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₂²⋊Q₈	—	3	2	no	—
35	C₃×C₂²⋊Q₈	—	3	4	no	—
36	C₃×C₄×Q₈	—	3	1	no	—
37	C₃×C₄×D₄	—	3	2	no	—
38	C₃×C₄×D₄	—	3	1	no	—
39	#946	C₂×C₃	1	2	yes	complemented
40	#896	C₂×C₃	1	4	yes	complemented
41	#893	C₂×C₃	1	2	yes	—
42	#888	C₂×C₃	1	2	yes	complemented
43	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
44	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂²×C₃×D₃	C₂³	1	2	yes	—
46	C₃²×Q₈	—	2	4	no	—
47	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂³	1	4	yes	—
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃×Dic₃	C₂³	1	2	yes	—
53	C₂×C₃×Dic₃	C₂³	1	2	yes	—
54	C₃×D₁₂	Q₈	1	4	yes	complemented
55	C₃×C₄×D₃	—	2	4	no	—
56	C₃×C₄×D₃	—	2	8	no	—
57	C₃×Dic₆	—	2	8	no	—
58	C₃×Dic₆	D₄	1	4	yes	complemented
59	D₄×Q₈	—	3	1	no	Sylow
60	C₂×C₃×Q₈	—	2	2	no	—
61	C₂×C₃×Q₈	—	3	1	no	—
62	C₂×C₃×Q₈	—	3	4	no	—
63	C₂×C₃×Q₈	—	6	8	no	—
64	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
65	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
66	C₂×C₃×D₄	—	3	1	no	—
67	C₂²×C₃×C₄	—	3	4	no	—
68	C₂²×C₃×C₄	—	3	2	no	—
69	D₃×Q₈	—	2	8	no	—
70	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
72	C₂×C₄×D₃	C₂²×C₃	1	4	yes	—
73	C₂×Dic₆	C₂²×C₃	1	4	yes	—
74	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	3	2	no	—
78	C₃×C₄⋊C₄	—	3	4	no	—
79	C₃×C₄⋊C₄	—	3	1	no	—
80	C₃×C₄⋊C₄	—	3	1	no	—
81	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
82	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
83	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₃×C₂²⋊C₄	—	3	4	no	—
86	C₃×C₂²⋊C₄	—	3	2	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	—	3	2	no	—
89	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
91	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
92	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
93	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
95	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
96	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
97	C₄×Dic₃	C₂²×C₃	1	2	yes	—
98	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
99	C₂×C₃×D₃	—	2	2	no	—
100	C₂×C₃×D₃	—	2	2	no	—
101	C₂×C₃×D₃	C₂×Q₈	1	4	yes	complemented
102	C₃²×C₄	—	2	1	no	—
103	C₃²×C₄	—	2	4	no	—
104	C₃²×C₄	C₂×D₄	1	2	yes	—
105	C₃²×C₄	C₂×Q₈	1	2	yes	complemented
106	C₃×Dic₃	—	2	2	no	—
107	C₃×Dic₃	—	2	2	no	—
108	C₃×Dic₃	C₂×D₄	1	4	yes	—
109	C₂²×Q₈	—	3	2	no	—
110	C₄⋊Q₈	—	3	2	no	—
111	C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
112	C₂²⋊Q₈	—	3	2	no	—
113	C₂²⋊Q₈	—	3	4	no	—
114	C₄×Q₈	—	3	1	no	—
115	C₄×D₄	—	3	2	no	—
116	C₄×D₄	—	3	1	no	—
117	C₂³×C₃	—	3	2	no	—
118	C₂²×D₃	C₂³×C₃	1	2	yes	—
119	C₃×Q₈	—	2	4	no	—
120	C₃×Q₈	—	2	4	no	—
121	C₃×Q₈	—	3	4	no	—
122	C₃×Q₈	—	4	4	no	—
123	C₃×Q₈	—	6	8	no	—
124	C₃×D₄	—	3	4	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	3	2	no	—
130	C₂×C₃×C₄	—	3	2	no	—
131	C₂×C₃×C₄	—	3	2	no	—
132	C₂×C₃×C₄	—	6	8	no	—
133	C₂×C₃×C₄	—	6	4	no	—
134	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
136	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
140	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂×Dic₃	C₂³×C₃	1	2	yes	—
143	C₂×Dic₃	C₂³×C₃	1	2	yes	—
144	C₂×Dic₃	C₂³×C₃	1	2	yes	—
145	D₁₂	C₃×Q₈	1	4	yes	complemented
146	C₄×D₃	—	2	4	no	—
147	C₄×D₃	—	2	8	no	—
148	Dic₆	—	2	8	no	—
149	Dic₆	C₃×D₄	1	4	yes	complemented
150	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
153	C₃×D₃	—	2	4	no	—
154	C₂×Q₈	—	3	1	no	—
155	C₂×Q₈	—	3	4	no	—
156	C₂×Q₈	—	6	8	no	—
157	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
158	C₂×D₄	—	3	1	no	—
159	C₂²×C₄	—	3	4	no	—
160	C₂²×C₄	—	3	2	no	—
161	C₄⋊C₄	—	3	1	no	—
162	C₄⋊C₄	—	3	2	no	—
163	C₄⋊C₄	—	3	1	no	—
164	C₄⋊C₄	—	3	4	no	—
165	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
166	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
167	C₂²⋊C₄	—	3	4	no	—
168	C₂²⋊C₄	—	3	2	no	—
169	C₄²	—	3	2	no	—
170	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	3	4	no	—
173	C₂²×C₃	—	6	2	no	—
174	C₂²×C₃	—	6	2	no	—
175	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
176	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×D₃	—	2	2	no	—
178	C₂×D₃	—	2	2	no	—
179	C₂×D₃	C₂×C₃×Q₈	1	4	yes	complemented
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	3	4	no	—
188	C₃×C₄	—	4	4	no	—
189	C₃×C₄	—	6	2	no	—
190	C₃×C₄	—	6	2	no	—
191	C₃×C₄	D₃×D₄	1	2	yes	—
192	C₃×C₄	D₃×Q₈	1	2	yes	complemented
193	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₃×C₄	C₂×C₃×Q₈	1	2	yes	complemented
195	Dic₃	—	2	2	no	—
196	Dic₃	—	2	2	no	—
197	Dic₃	C₂×C₃×D₄	1	4	yes	—
198	C₃²	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	3	2	no	—
200	Q₈	—	2	4	no	—
201	Q₈	—	3	4	no	—
202	Q₈	—	6	8	no	—
203	D₄	—	3	4	no	—
204	C₂×C₄	—	3	2	no	—
205	C₂×C₄	—	3	2	no	—
206	C₂×C₄	—	3	2	no	—
207	C₂×C₄	—	6	8	no	—
208	C₂×C₄	—	6	4	no	—
209	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₃×D₃	1	4	yes	—
211	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	6	4	no	—
217	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
218	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
219	C₂×C₃	#1007	1	1	yes	characteristic
220	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
221	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
222	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
223	D₃	—	2	4	no	—
224	C₂²	—	3	4	no	—
225	C₂²	—	6	2	no	—
226	C₂²	—	6	2	no	—
227	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	1	no	—
230	C₄	—	3	4	no	—
231	C₄	—	6	2	no	—
232	C₄	—	6	2	no	—
233	C₄	C₃×D₃×D₄	1	2	yes	—
234	C₄	C₃×D₃×Q₈	1	2	yes	complemented
235	C₃	—	2	1	no	—
236	C₃	#5727	1	1	yes	characteristic, direct product factor
237	C₃	C₃×D₄×Q₈	1	1	yes	characteristic, complemented
238	C₂	—	6	4	no	—
239	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
240	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
241	C₂	C₃×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×D₁₂	C₂²
hypercenter	C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₃×Q₈	C₂	semidirect
2. C₃²×C₄⋊Q₈	C₂	semidirect
3. C₃×#946	C₂	semidirect
4. C₃×#896	C₂	semidirect
6. C₃×#888	C₂	semidirect
8. C₃×C₄×Dic₆	C₂	semidirect
10. #5727	C₃	direct (N × H)
29. C₂×D₃×Q₈	C₂×C₃	semidirect
32. C₃×C₄⋊Q₈	D₃	semidirect
33. C₃×C₄⋊Q₈	C₂×C₃	semidirect
39. #946	C₂×C₃	semidirect
40. #896	C₂×C₃	semidirect
42. #888	C₂×C₃	semidirect
44. C₄×Dic₆	C₂×C₃	semidirect
54. C₃×D₁₂	Q₈	semidirect
58. C₃×Dic₆	D₄	semidirect
101. C₂×C₃×D₃	C₂×Q₈	semidirect
105. C₃²×C₄	C₂×Q₈	semidirect
111. C₄⋊Q₈	C₃×D₃	semidirect
145. D₁₂	C₃×Q₈	semidirect
149. Dic₆	C₃×D₄	semidirect
179. C₂×D₃	C₂×C₃×Q₈	semidirect
192. C₃×C₄	D₃×Q₈	semidirect
194. C₃×C₄	C₂×C₃×Q₈	semidirect
198. C₃²	D₄×Q₈	semidirect
234. C₄	C₃×D₃×Q₈	semidirect
236. C₃	#5727	direct (N × H)
237. C₃	C₃×D₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×Q₈	2
C₃²×C₄⋊Q₈	2
C₃×#946	2
C₃×#896	2
C₃×#893	2
C₃×#888	2
C₃×C₄×D₁₂	2
C₃×C₄×Dic₆	2
C₃×D₄×Q₈	3
#5727	3

Subgroups — C3×#5727 #10595824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5727 #10595824