Subgroups — C₃×#5816 #10595913

Order 576.

618 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#235	C₂	1	1	yes	maximal, characteristic
3	C₃×#947	C₂	1	2	yes	maximal, complemented
4	C₃×#946	C₂	1	4	yes	maximal, complemented
5	C₃×#944	C₂	1	2	yes	maximal, complemented
6	C₃×#929	C₂	1	4	yes	maximal, complemented
7	C₃×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#536	—	3	1	no	maximal
9	#5816	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×Q₈	—	2	4	no	—
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₂×C₃²×Q₈	C₂²	1	2	yes	—
13	C₂²×C₃²×C₄	C₂²	1	2	yes	—
14	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×#67	C₂²	1	2	yes	—
16	C₃×#235	—	2	4	no	—
17	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
19	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
20	C₃×#217	C₂²	1	2	yes	—
21	C₃×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
22	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₃×Dic₃⋊C₄	C₂²	1	8	yes	—
24	C₃×C₄×Dic₃	C₂²	1	2	yes	—
25	C₂×C₃×#41	—	3	1	no	—
26	C₂²×C₃×Q₈	—	2	1	no	—
27	C₂²×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
28	C₂²×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#235	C₂×C₃	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	—	3	2	no	—
31	C₃×#124	—	3	2	no	—
32	C₃×#123	—	3	4	no	—
33	C₃×C₂²⋊Q₈	—	3	4	no	—
34	C₃×#117	—	3	1	no	—
35	#947	C₂×C₃	1	2	yes	complemented
36	#946	C₂×C₃	1	4	yes	complemented
37	#944	C₂×C₃	1	2	yes	complemented
38	#929	C₂×C₃	1	4	yes	complemented
39	#926	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃²	C₂³	1	1	yes	characteristic
41	C₂²×C₃×D₃	C₂³	1	2	yes	—
42	C₃²×Q₈	—	2	4	no	—
43	C₃²×Q₈	—	2	4	no	—
44	C₂×C₃²×C₄	—	2	4	no	—
45	C₂×C₃²×C₄	D₄	1	4	yes	—
46	C₂×C₃²×C₄	C₂³	1	4	yes	—
47	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₃×#67	—	2	4	no	—
50	C₂×C₃×Dic₃	C₂³	1	4	yes	—
51	C₂×C₃×Dic₃	C₂³	1	2	yes	—
52	C₃×D₁₂	—	2	2	no	—
53	C₃×C₄×D₃	—	2	4	no	—
54	C₃×Dic₆	—	2	2	no	—
55	#536	—	3	1	no	Sylow
56	C₃×#41	—	6	4	no	—
57	C₂×C₃×Q₈	—	2	8	no	—
58	C₂×C₃×Q₈	—	2	4	no	—
59	C₂×C₃×Q₈	—	2	4	no	—
60	C₂×C₃×Q₈	—	2	2	no	—
61	C₂×C₃×Q₈	—	2	2	no	—
62	C₂×C₃×Q₈	—	3	1	no	—
63	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
64	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
65	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
66	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
67	C₂×C₃×D₄	—	3	2	no	—
68	C₂×C₃×D₄	—	3	1	no	—
69	C₂²×C₃×C₄	—	2	2	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	3	2	no	—
72	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
73	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
75	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×#67	C₂²×C₃	1	2	yes	—
77	#235	—	2	4	no	—
78	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
80	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
81	C₃×C₄⋊C₄	—	3	8	no	—
82	C₃×C₄⋊C₄	—	3	2	no	—
83	C₃×C₂²⋊C₄	—	3	2	no	—
84	C₃×C₂²⋊C₄	—	3	8	no	—
85	C₃×C₄²	—	3	2	no	—
86	#217	C₂²×C₃	1	2	yes	—
87	(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
88	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
89	Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
90	C₄×Dic₃	C₂²×C₃	1	2	yes	—
91	C₂²×C₃²	—	2	2	no	—
92	C₂²×C₃²	C₂×D₄	1	2	yes	—
93	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
94	C₂×C₃×D₃	—	2	2	no	—
95	C₂×C₃×D₃	—	2	4	no	—
96	C₃²×C₄	—	2	4	no	—
97	C₃²×C₄	C₂×D₄	1	2	yes	—
98	C₃²×C₄	C₂×D₄	1	2	yes	—
99	C₃×Dic₃	—	2	2	no	—
100	C₃×Dic₃	—	2	4	no	—
101	C₂×#41	—	3	1	no	—
102	C₂²×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
103	C₄⋊Q₈	—	3	2	no	—
104	#124	—	3	2	no	—
105	#123	—	3	4	no	—
106	C₂²⋊Q₈	—	3	4	no	—
107	#117	—	3	1	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	—	3	2	no	—
110	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
111	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
112	C₂²×D₃	C₂³×C₃	1	2	yes	—
113	C₃×Q₈	—	2	8	no	—
114	C₃×Q₈	—	2	8	no	—
115	C₃×Q₈	—	2	4	no	—
116	C₃×Q₈	—	2	4	no	—
117	C₃×Q₈	—	2	4	no	—
118	C₃×Q₈	—	2	4	no	—
119	C₃×Q₈	—	6	2	no	—
120	C₃×D₄	—	6	4	no	—
121	C₃×D₄	—	6	2	no	—
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	3	4	no	—
130	C₂×C₃×C₄	—	3	2	no	—
131	C₂×C₃×C₄	—	6	4	no	—
132	C₂×C₃×C₄	#67	1	4	yes	—
133	C₂×C₃×C₄	C₃×D₄	1	4	yes	—
134	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
135	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
138	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
140	#67	—	2	4	no	—
141	C₂×Dic₃	C₂³×C₃	1	4	yes	—
142	C₂×Dic₃	C₂³×C₃	1	2	yes	—
143	D₁₂	—	2	2	no	—
144	C₄×D₃	—	2	4	no	—
145	Dic₆	—	2	2	no	—
146	C₂×C₃²	—	2	2	no	—
147	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
149	C₃×D₃	—	4	2	no	—
150	#41	—	6	4	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	3	1	no	—
153	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
154	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
155	C₂×D₄	—	3	2	no	—
156	C₂×D₄	—	3	1	no	—
157	C₂²×C₄	—	3	2	no	—
158	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
159	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
160	C₄⋊C₄	—	3	2	no	—
161	C₄⋊C₄	—	3	8	no	—
162	C₂²⋊C₄	—	3	2	no	—
163	C₂²⋊C₄	—	3	8	no	—
164	C₄²	—	3	2	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	6	4	no	—
171	C₂²×C₃	—	6	2	no	—
172	C₂²×C₃	C₂×#67	1	2	yes	—
173	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
174	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
175	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
176	C₂×D₃	—	2	2	no	—
177	C₂×D₃	—	2	4	no	—
178	C₃×C₄	—	2	8	no	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	6	4	no	—
184	C₃×C₄	—	6	2	no	—
185	C₃×C₄	C₂×#67	1	2	yes	—
186	C₃×C₄	C₂×#67	1	2	yes	—
187	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
188	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
189	Dic₃	—	2	2	no	—
190	Dic₃	—	2	4	no	—
191	C₃²	#536	1	1	yes	characteristic, complemented, Sylow
192	C₂³	—	3	2	no	—
193	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
194	Q₈	—	2	4	no	—
195	Q₈	—	2	4	no	—
196	Q₈	—	6	2	no	—
197	D₄	—	6	4	no	—
198	D₄	—	6	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	3	4	no	—
201	C₂×C₄	—	3	2	no	—
202	C₂×C₄	—	6	4	no	—
203	C₂×C₄	C₃×#67	1	4	yes	—
204	C₂×C₄	C₂²×C₃×D₃	1	4	yes	—
205	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	2	2	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	12	2	no	—
213	C₂×C₃	#1007	1	2	yes	—
214	C₂×C₃	C₂²×#67	1	1	yes	characteristic
215	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
216	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
217	D₃	—	4	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	6	4	no	—
220	C₂²	—	6	2	no	—
221	C₂²	C₂×C₃×#67	1	2	yes	—
222	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	6	4	no	—
225	C₄	—	6	2	no	—
226	C₄	C₂×C₃×#67	1	2	yes	—
227	C₄	C₂×C₃×#67	1	2	yes	—
228	C₃	—	2	1	no	—
229	C₃	#5816	1	1	yes	characteristic, direct product factor
230	C₃	C₃×#536	1	1	yes	characteristic, complemented
231	C₂	—	2	2	no	—
232	C₂	—	12	2	no	—
233	C₂	C₃×#1007	1	2	yes	—
234	C₂	C₂²×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₂²×C₃²×Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×#235	C₂
hypercenter	C₂²×C₃×Q₈	D₃
hypocenter	C₃	C₃×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×Q₈	C₂	semidirect
3. C₃×#947	C₂	semidirect
4. C₃×#946	C₂	semidirect
5. C₃×#944	C₂	semidirect
6. C₃×#929	C₂	semidirect
7. C₃×#926	C₂	semidirect
9. #5816	C₃	direct (N × H)
27. C₂²×C₃×Q₈	D₃	semidirect
28. C₂²×C₃×Q₈	C₂×C₃	semidirect
35. #947	C₂×C₃	semidirect
36. #946	C₂×C₃	semidirect
37. #944	C₂×C₃	semidirect
38. #929	C₂×C₃	semidirect
39. #926	C₂×C₃	semidirect
102. C₂²×Q₈	C₃×D₃	semidirect
191. C₃²	#536	semidirect
229. C₃	#5816	direct (N × H)
230. C₃	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×Q₈	2
C₂×C₃×#235	2
C₃×#947	2
C₃×#946	2
C₃×#944	2
C₃×#929	2
C₃×#926	2
C₃×#536	3
#5816	3

Subgroups — C3×#5816 #10595913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5816 #10595913