Subgroups — C₃×D₄×Dic₆ #10595634

Order 576.

607 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₆	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×D₄×Dic₃	C₂	1	2	yes	maximal
4	C₃×#924	C₂	1	2	yes	maximal, complemented
5	C₃×#888	C₂	1	2	yes	maximal, complemented
6	C₃×#878	C₂	1	4	yes	maximal, complemented
7	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×D₄×Q₈	—	3	1	no	maximal
10	D₄×Dic₆	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃²×C₄	C₂²	1	2	yes	—
13	C₂²×C₃×Dic₃	C₂²	1	4	yes	—
14	C₂×C₃×Dic₆	—	2	8	no	—
15	C₂×C₃×Dic₆	C₂²	1	4	yes	—
16	C₂×C₃×Dic₆	C₂²	1	2	yes	—
17	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄²	C₂²	1	1	yes	characteristic
21	C₃×#217	C₂²	1	4	yes	—
22	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₃×C₄×Dic₃	C₂²	1	2	yes	—
28	C₂²×C₃×Q₈	—	3	2	no	—
29	C₂²×Dic₆	C₂×C₃	1	2	yes	complemented
30	C₃×C₄⋊Q₈	—	3	2	no	—
31	C₃×C₄⋊Q₈	—	3	1	no	—
32	C₃×C₂²⋊Q₈	—	3	2	no	—
33	C₃×C₂²⋊Q₈	—	3	4	no	—
34	C₃×C₄×Q₈	—	3	1	no	—
35	C₃×C₄×D₄	—	2	1	no	—
36	C₃×C₄×D₄	—	3	2	no	—
37	C₃×C₄×D₄	D₃	1	1	yes	characteristic
38	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
39	D₄×Dic₃	C₂×C₃	1	2	yes	—
40	#924	C₂×C₃	1	2	yes	complemented
41	#888	C₂×C₃	1	2	yes	complemented
42	#878	C₂×C₃	1	4	yes	complemented
43	#869	C₂×C₃	1	1	yes	characteristic, complemented
44	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂³×C₃²	C₂³	1	2	yes	—
46	C₃²×D₄	Q₈	1	4	yes	complemented
47	C₂×C₃²×C₄	—	2	4	no	—
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂³	1	2	yes	—
50	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
52	C₂×C₃×Dic₃	—	2	8	no	—
53	C₂×C₃×Dic₃	C₂³	1	2	yes	—
54	C₂×C₃×Dic₃	C₂³	1	4	yes	—
55	C₂×C₃×Dic₃	C₂³	1	2	yes	—
56	C₃×Dic₆	—	2	4	no	—
57	C₃×Dic₆	—	2	8	no	—
58	C₃×Dic₆	D₄	1	4	yes	direct product factor
59	D₄×Q₈	—	3	1	no	Sylow
60	C₂×C₃×Q₈	—	3	1	no	—
61	C₂×C₃×Q₈	—	3	4	no	—
62	C₂×C₃×Q₈	—	3	2	no	—
63	C₂×C₃×Q₈	—	6	8	no	—
64	C₂×C₃×D₄	—	2	1	no	—
65	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	—	3	4	no	—
69	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
70	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
71	C₂²×Dic₃	C₂²×C₃	1	4	yes	—
72	C₂×Dic₆	—	2	8	no	—
73	C₂×Dic₆	C₂²×C₃	1	2	yes	—
74	C₂×Dic₆	C₂²×C₃	1	4	yes	—
75	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	3	2	no	—
78	C₃×C₄⋊C₄	—	3	4	no	—
79	C₃×C₄⋊C₄	—	3	2	no	—
80	C₃×C₄⋊C₄	—	3	1	no	—
81	C₃×C₄⋊C₄	—	3	2	no	—
82	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	3	4	no	—
86	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
87	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₄²	—	3	2	no	—
90	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
92	#217	C₂²×C₃	1	4	yes	—
93	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
95	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
96	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
97	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
98	C₄×Dic₃	C₂²×C₃	1	2	yes	—
99	C₂²×C₃²	—	2	2	no	—
100	C₂²×C₃²	—	2	2	no	—
101	C₂²×C₃²	C₂×Q₈	1	4	yes	complemented
102	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	—	2	1	no	—
105	C₃²×C₄	C₂×D₄	1	2	yes	—
106	C₃²×C₄	C₂×Q₈	1	2	yes	complemented
107	C₃×Dic₃	—	2	4	no	—
108	C₃×Dic₃	—	2	2	no	—
109	C₃×Dic₃	C₂×D₄	1	4	yes	—
110	C₂²×Q₈	—	3	2	no	—
111	C₄⋊Q₈	—	3	2	no	—
112	C₄⋊Q₈	—	3	1	no	—
113	C₂²⋊Q₈	—	3	2	no	—
114	C₂²⋊Q₈	—	3	4	no	—
115	C₄×Q₈	—	3	1	no	—
116	C₄×D₄	—	3	2	no	—
117	C₄×D₄	C₃×D₃	1	1	yes	characteristic
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	C₂²×D₃	1	2	yes	—
120	C₂³×C₃	C₂³×C₃	1	2	yes	—
121	C₃×Q₈	—	3	4	no	—
122	C₃×Q₈	—	6	8	no	—
123	C₃×Q₈	—	6	4	no	—
124	C₃×D₄	—	2	4	no	—
125	C₃×D₄	Dic₆	1	4	yes	direct product factor
126	C₃×D₄	C₃×Q₈	1	4	yes	complemented
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	3	2	no	—
134	C₂×C₃×C₄	—	3	2	no	—
135	C₂×C₃×C₄	—	3	4	no	—
136	C₂×C₃×C₄	—	4	4	no	—
137	C₂×C₃×C₄	—	6	8	no	—
138	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
140	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
144	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂×Dic₃	—	2	8	no	—
147	C₂×Dic₃	C₂³×C₃	1	2	yes	—
148	C₂×Dic₃	C₂³×C₃	1	4	yes	—
149	C₂×Dic₃	C₂³×C₃	1	2	yes	—
150	Dic₆	—	2	4	no	—
151	Dic₆	—	2	8	no	—
152	Dic₆	C₃×D₄	1	4	yes	direct product factor
153	C₂×C₃²	—	2	4	no	—
154	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×Q₈	—	3	1	no	—
158	C₂×Q₈	—	3	4	no	—
159	C₂×Q₈	—	3	2	no	—
160	C₂×Q₈	—	6	8	no	—
161	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₂²×C₄	—	3	4	no	—
163	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
164	C₄⋊C₄	—	3	2	no	—
165	C₄⋊C₄	—	3	2	no	—
166	C₄⋊C₄	—	3	1	no	—
167	C₄⋊C₄	—	3	4	no	—
168	C₄⋊C₄	—	3	2	no	—
169	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
170	C₂²⋊C₄	—	3	4	no	—
171	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
172	C₄²	—	3	2	no	—
173	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	C₂×Dic₆	1	4	yes	complemented
183	C₂²×C₃	C₂×C₃×Q₈	1	4	yes	complemented
184	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
185	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	3	4	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	6	4	no	—
196	C₃×C₄	—	6	2	no	—
197	C₃×C₄	C₂×Dic₆	1	2	yes	complemented
198	C₃×C₄	D₃×D₄	1	2	yes	—
199	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
200	C₃×C₄	C₂×C₃×Q₈	1	2	yes	complemented
201	Dic₃	—	2	4	no	—
202	Dic₃	—	2	2	no	—
203	Dic₃	C₂×C₃×D₄	1	4	yes	—
204	C₃²	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
205	C₂³	C₂²×C₃×D₃	1	2	yes	—
206	Q₈	—	3	4	no	—
207	Q₈	—	6	8	no	—
208	Q₈	—	6	4	no	—
209	D₄	C₃×Dic₆	1	4	yes	direct product factor
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	3	2	no	—
212	C₂×C₄	—	3	2	no	—
213	C₂×C₄	—	3	4	no	—
214	C₂×C₄	—	6	8	no	—
215	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
217	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	4	4	no	—
225	C₂×C₃	C₂²×Dic₆	1	1	yes	characteristic
226	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
227	C₂×C₃	#1010	1	1	yes	characteristic
228	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
230	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	C₂×C₃×Dic₆	1	4	yes	complemented
234	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	3	4	no	—
238	C₄	—	6	4	no	—
239	C₄	—	6	2	no	—
240	C₄	C₂×C₃×Dic₆	1	2	yes	complemented
241	C₄	C₃×D₃×D₄	1	2	yes	—
242	C₃	—	2	1	no	—
243	C₃	D₄×Dic₆	1	1	yes	characteristic, direct product factor
244	C₃	C₃×D₄×Q₈	1	1	yes	characteristic, complemented
245	C₂	—	2	4	no	—
246	C₂	C₂²×C₃×Dic₆	1	1	yes	characteristic
247	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
248	C₂	C₃×#1010	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×D₄	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃²×D₄	C₂²
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Dic₆	C₂	semidirect
4. C₃×#924	C₂	semidirect
5. C₃×#888	C₂	semidirect
6. C₃×#878	C₂	semidirect
7. C₃×#869	C₂	semidirect
8. C₃×C₄×Dic₆	C₂	semidirect
10. D₄×Dic₆	C₃	direct (N × H)
29. C₂²×Dic₆	C₂×C₃	semidirect
40. #924	C₂×C₃	semidirect
41. #888	C₂×C₃	semidirect
42. #878	C₂×C₃	semidirect
43. #869	C₂×C₃	semidirect
44. C₄×Dic₆	C₂×C₃	semidirect
46. C₃²×D₄	Q₈	semidirect
58. C₃×Dic₆	D₄	direct (N × H)
101. C₂²×C₃²	C₂×Q₈	semidirect
106. C₃²×C₄	C₂×Q₈	semidirect
125. C₃×D₄	Dic₆	direct (N × H)
126. C₃×D₄	C₃×Q₈	semidirect
152. Dic₆	C₃×D₄	direct (N × H)
182. C₂²×C₃	C₂×Dic₆	semidirect
183. C₂²×C₃	C₂×C₃×Q₈	semidirect
197. C₃×C₄	C₂×Dic₆	semidirect
200. C₃×C₄	C₂×C₃×Q₈	semidirect
204. C₃²	D₄×Q₈	semidirect
209. D₄	C₃×Dic₆	direct (N × H)
233. C₂²	C₂×C₃×Dic₆	semidirect
240. C₄	C₂×C₃×Dic₆	semidirect
243. C₃	D₄×Dic₆	direct (N × H)
244. 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
C₂²×C₃×Dic₆	2
C₃²×C₄×D₄	2
C₃×D₄×Dic₃	2
C₃×#924	2
C₃×#888	2
C₃×#878	2
C₃×#869	2
C₃×C₄×Dic₆	2
C₃×D₄×Q₈	3
D₄×Dic₆	3

Subgroups — C3×D4×Dic6 #10595634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×Dic₆ #10595634