Subgroups — C₂×C₃×#942 #10595905

Order 576.

594 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×C₃×#216	C₂	1	2	yes	maximal, complemented
4	C₃×#942	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#215	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#137	—	3	1	no	maximal
8	C₂×#942	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃²×Q₈	—	2	1	no	—
10	C₂×C₃²×Q₈	—	2	2	no	—
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₂×C₃²×Q₈	C₂²	1	4	yes	complemented
13	C₂²×C₃²×C₄	—	2	1	no	—
14	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×#67	—	2	1	no	—
16	C₃×#235	—	2	2	no	—
17	C₃×#235	C₂²	1	4	yes	—
18	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	—	2	1	no	—
20	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
21	C₃×#216	C₂²	1	8	yes	complemented
22	C₃×#215	C₂²	1	8	yes	complemented
23	C₃×#208	C₂²	1	4	yes	complemented
24	C₂×C₃×C₃⋊C₈	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×#137	—	3	8	no	—
31	C₂×C₃×Q₁₆	—	3	2	no	—
32	C₂×C₃×SD₁₆	—	3	2	no	—
33	C₂×C₃×M₁₆	—	3	1	no	—
34	C₂×#216	C₂×C₃	1	2	yes	complemented
35	#942	C₂×C₃	1	8	yes	direct product factor
36	C₂×#215	C₂×C₃	1	2	yes	complemented
37	C₂×#208	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃²	D₄	1	1	yes	characteristic
39	C₂²×C₃×D₃	—	2	1	no	—
40	C₃²×Q₈	—	2	2	no	—
41	C₃²×Q₈	—	2	4	no	—
42	C₃²×Q₈	C₂³	1	4	yes	—
43	C₂×C₃²×C₄	—	2	2	no	—
44	C₂×C₃²×C₄	—	2	4	no	—
45	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	D₄	1	2	yes	—
47	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂³	1	2	yes	—
49	C₃×#67	—	2	4	no	—
50	C₂×C₃×Dic₃	—	2	1	no	—
51	C₃×D₁₂	—	2	1	no	—
52	C₃×D₁₂	C₂³	1	2	yes	—
53	C₃×C₄×D₃	—	2	4	no	—
54	C₃×Dic₆	—	2	1	no	—
55	C₃×Dic₆	C₂³	1	2	yes	—
56	C₃×C₃⋊C₈	C₂³	1	4	yes	—
57	C₂×#137	—	3	1	no	Sylow
58	C₃×#41	—	3	4	no	—
59	C₃×#41	—	6	2	no	—
60	C₂×C₃×Q₈	—	2	2	no	—
61	C₂×C₃×Q₈	—	2	4	no	—
62	C₂×C₃×Q₈	—	2	2	no	—
63	C₂×C₃×Q₈	—	2	1	no	—
64	C₂×C₃×Q₈	—	2	1	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂×C₃×Q₈	—	2	4	no	—
68	C₂×C₃×Q₈	—	3	1	no	—
69	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
70	C₂×C₃×Q₈	C₂×D₃	1	4	yes	complemented
71	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
72	C₂×C₃×Q₈	C₂²×C₃	1	4	yes	complemented
73	C₂×C₃×D₄	—	3	1	no	—
74	C₂×C₃×D₄	—	6	1	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	6	1	no	—
80	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×#67	—	2	1	no	—
83	#235	—	2	2	no	—
84	#235	C₂²×C₃	1	4	yes	—
85	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄×D₃	—	2	1	no	—
87	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
88	C₃×Q₁₆	—	3	8	no	—
89	C₃×SD₁₆	—	3	8	no	—
90	C₃×M₁₆	—	3	4	no	—
91	C₂×C₃×C₈	—	3	2	no	—
92	#216	C₂²×C₃	1	8	yes	complemented
93	#215	C₂²×C₃	1	8	yes	complemented
94	#208	C₂²×C₃	1	4	yes	complemented
95	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	—
96	C₂²×C₃²	—	2	2	no	—
97	C₂²×C₃²	C₂×D₄	1	2	yes	—
98	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×D₃	—	2	2	no	—
100	C₂×C₃×D₃	—	4	2	no	—
101	C₃²×C₄	—	2	4	no	—
102	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₃²×C₄	C₂×D₄	1	2	yes	—
104	C₃²×C₄	C₂⁴	1	1	yes	characteristic
105	C₃×Dic₃	—	2	2	no	—
106	C₂×#41	—	3	1	no	—
107	C₂²×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
108	#137	—	3	8	no	—
109	C₂×Q₁₆	—	3	2	no	—
110	C₂×SD₁₆	—	3	2	no	—
111	C₂×M₁₆	—	3	1	no	—
112	C₂³×C₃	—	2	1	no	—
113	C₂³×C₃	—	6	1	no	—
114	C₂³×C₃	#67	1	1	yes	characteristic
115	C₂³×C₃	C₃×D₄	1	1	yes	characteristic
116	C₂²×D₃	—	2	1	no	—
117	C₃×Q₈	—	2	4	no	—
118	C₃×Q₈	—	2	8	no	—
119	C₃×Q₈	—	2	2	no	—
120	C₃×Q₈	—	2	4	no	—
121	C₃×Q₈	—	2	2	no	—
122	C₃×Q₈	—	2	4	no	—
123	C₃×Q₈	—	2	4	no	—
124	C₃×Q₈	—	3	2	no	—
125	C₃×Q₈	—	6	1	no	—
126	C₃×Q₈	C₂²×D₃	1	4	yes	—
127	C₃×Q₈	C₂³×C₃	1	4	yes	—
128	C₃×D₄	—	3	2	no	—
129	C₃×D₄	—	6	4	no	—
130	C₃×D₄	—	6	1	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	8	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	6	1	no	—
142	C₂×C₃×C₄	—	6	4	no	—
143	C₂×C₃×C₄	#67	1	2	yes	—
144	C₂×C₃×C₄	#67	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₃×D₄	1	2	yes	—
147	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
151	#67	—	2	4	no	—
152	C₂×Dic₃	—	2	1	no	—
153	D₁₂	—	2	1	no	—
154	D₁₂	C₂³×C₃	1	2	yes	—
155	C₄×D₃	—	2	4	no	—
156	Dic₆	—	2	1	no	—
157	Dic₆	C₂³×C₃	1	2	yes	—
158	C₃×C₈	—	3	4	no	—
159	C₃⋊C₈	C₂³×C₃	1	4	yes	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	#137	1	2	yes	complemented
162	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
163	C₃×D₃	—	4	2	no	—
164	#41	—	3	4	no	—
165	#41	—	6	2	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	3	1	no	—
169	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
170	C₂×Q₈	C₂×C₃×D₃	1	4	yes	complemented
171	C₂×D₄	—	3	1	no	—
172	C₂×D₄	—	6	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	6	1	no	—
175	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
176	Q₁₆	—	3	8	no	—
177	SD₁₆	—	3	8	no	—
178	M₁₆	—	3	4	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂²×C₃	—	2	4	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	6	2	no	—
186	C₂²×C₃	—	12	2	no	—
187	C₂²×C₃	C₂×#67	1	2	yes	—
188	C₂²×C₃	C₂×#67	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
190	C₂²×C₃	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×D₃	—	2	2	no	—
192	C₂×D₃	—	4	2	no	—
193	C₃×C₄	—	2	8	no	—
194	C₃×C₄	—	2	4	no	—
195	C₃×C₄	—	2	4	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	6	2	no	—
200	C₃×C₄	C₂×#67	1	2	yes	—
201	C₃×C₄	C₂×#67	1	1	yes	characteristic
202	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
204	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
205	C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	Dic₃	—	2	2	no	—
207	C₃²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	6	1	no	—
209	C₂³	C₃×#67	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	Q₈	—	2	4	no	—
212	Q₈	—	3	2	no	—
213	Q₈	—	6	1	no	—
214	Q₈	C₂²×C₃×D₃	1	4	yes	—
215	D₄	—	3	2	no	—
216	D₄	—	6	4	no	—
217	D₄	—	6	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	6	1	no	—
221	C₂×C₄	—	6	4	no	—
222	C₂×C₄	C₃×#67	1	1	yes	characteristic
223	C₂×C₄	C₃×#67	1	2	yes	—
224	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
226	C₈	—	3	4	no	—
227	C₂×C₃	—	2	4	no	—
228	C₂×C₃	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	2	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	12	2	no	—
233	C₂×C₃	#942	1	2	yes	direct product factor
234	C₂×C₃	C₃×#137	1	2	yes	complemented
235	C₂×C₃	C₂²×#67	1	1	yes	characteristic
236	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
237	D₃	—	4	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	6	2	no	—
240	C₂²	—	12	2	no	—
241	C₂²	C₂×C₃×#67	1	2	yes	—
242	C₂²	C₂×C₃×#67	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	6	2	no	—
245	C₄	C₂×C₃×#67	1	2	yes	—
246	C₄	C₂×C₃×#67	1	1	yes	characteristic
247	C₄	C₂³×C₃×D₃	1	1	yes	characteristic
248	C₃	—	2	1	no	—
249	C₃	C₂×#942	1	1	yes	characteristic, direct product factor
250	C₃	C₂×C₃×#137	1	1	yes	characteristic, complemented
251	C₂	—	2	2	no	—
252	C₂	—	12	2	no	—
253	C₂	C₃×#942	1	2	yes	direct product factor
254	C₂	C₂²×C₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₃
Fitting	C₂²×C₃²×Q₈	C₂
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂×C₃×#235	C₂
hypercenter	C₂²×C₃×Q₈	D₃
hypocenter	C₃	C₂×C₃×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ 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. C₂×C₃×#216	C₂	semidirect
4. C₃×#942	C₂	direct (N × H)
5. C₂×C₃×#215	C₂	semidirect
6. C₂×C₃×#208	C₂	semidirect
8. C₂×#942	C₃	direct (N × H)
12. C₂×C₃²×Q₈	C₂²	semidirect
21. C₃×#216	C₂²	semidirect
22. C₃×#215	C₂²	semidirect
23. C₃×#208	C₂²	semidirect
27. C₂²×C₃×Q₈	D₃	semidirect
28. C₂²×C₃×Q₈	C₂×C₃	semidirect
34. C₂×#216	C₂×C₃	semidirect
35. #942	C₂×C₃	direct (N × H)
36. C₂×#215	C₂×C₃	semidirect
37. C₂×#208	C₂×C₃	semidirect
70. C₂×C₃×Q₈	C₂×D₃	semidirect
72. C₂×C₃×Q₈	C₂²×C₃	semidirect
92. #216	C₂²×C₃	semidirect
93. #215	C₂²×C₃	semidirect
94. #208	C₂²×C₃	semidirect
107. C₂²×Q₈	C₃×D₃	semidirect
161. C₂×C₃²	#137	semidirect
170. C₂×Q₈	C₂×C₃×D₃	semidirect
207. C₃²	C₂×#137	semidirect
233. C₂×C₃	#942	direct (N × H)
234. C₂×C₃	C₃×#137	semidirect
249. C₃	C₂×#942	direct (N × H)
250. C₃	C₂×C₃×#137	semidirect
253. C₂	C₃×#942	direct (N × H)

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
C₂×C₃×#216	2
C₃×#942	2
C₂×C₃×#215	2
C₂×C₃×#208	2
C₂×C₃×#137	3
C₂×#942	3

Subgroups — C2×C3×#942 #10595905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#942 #10595905