Subgroups — C₂³×D₃×Dic₆ #11919179

Order 1152.

12953 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#634	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#627	C₂	1	2	yes	maximal, complemented
5	C₂³×D₃×Dic₃	C₂	1	2	yes	maximal
6	C₂²×D₃×Dic₆	C₂	1	56	yes	maximal, direct product factor
7	C₂³×D₃×Q₈	—	3	1	no	maximal
8	C₂⁴×Dic₆	—	3	1	no	maximal
9	C₂⁴×C₃×D₃	C₂²	1	1	yes	characteristic
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×#634	C₂²	1	28	yes	complemented
13	C₂³×C₃×Dic₃	C₂²	1	2	yes	—
14	C₂³×C₃×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×D₃	C₂²	1	28	yes	—
16	C₂²×C₃×Dic₆	C₂²	1	28	yes	complemented
17	C₂²×#627	C₂²	1	56	yes	complemented
18	C₂²×D₃×Dic₃	C₂²	1	56	yes	—
19	C₂×D₃×Dic₆	C₂²	1	448	yes	direct product factor
20	C₂³×C₃×Q₈	—	3	1	no	—
21	C₂⁴×C₃×C₄	—	3	1	no	—
22	C₂⁴×Dic₃	—	3	2	no	—
23	C₂²×D₃×Q₈	—	3	56	no	—
24	C₂³×C₄×D₃	—	3	2	no	—
25	C₂³×C₄×D₃	D₃	1	1	yes	characteristic
26	C₂³×Dic₆	—	3	1	no	—
27	C₂³×Dic₆	—	3	1	no	—
28	C₂³×Dic₆	—	3	2	no	—
29	C₂³×Dic₆	—	3	2	no	—
30	C₂³×Dic₆	—	3	56	no	—
31	C₂³×Dic₆	—	6	1	no	—
32	C₂³×Dic₆	D₃	1	1	yes	characteristic, direct product factor
33	C₂⁴×C₃²	C₂³	1	1	yes	characteristic
34	C₂³×C₃×D₃	Q₈	1	16	yes	complemented
35	C₂³×C₃×D₃	C₂³	1	14	yes	—
36	C₂²×C₃²×C₄	C₂³	1	14	yes	—
37	C₂²×C₃⋊Dic₃	C₂³	1	28	yes	—
38	C₂×#634	C₂³	1	112	yes	complemented
39	C₂²×C₃×Dic₃	C₂³	1	28	yes	—
40	C₂²×C₃×Dic₃	C₂³	1	14	yes	—
41	C₂×C₃×C₄×D₃	C₂³	1	112	yes	—
42	C₂×C₃×Dic₆	C₂³	1	112	yes	complemented
43	C₂×#627	C₂³	1	224	yes	complemented
44	C₂×D₃×Dic₃	C₂³	1	224	yes	—
45	D₃×Dic₆	C₂³	1	512	yes	direct product factor
46	C₂⁴×Q₈	—	9	1	no	Sylow
47	C₂⁵×C₃	—	3	1	no	—
48	C₂⁴×D₃	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×Q₈	—	3	28	no	—
50	C₂³×C₃×C₄	—	2	1	no	—
51	C₂³×C₃×C₄	—	3	1	no	—
52	C₂³×C₃×C₄	—	3	2	no	—
53	C₂³×C₃×C₄	—	3	28	no	—
54	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
55	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂³×Dic₃	—	3	2	no	—
57	C₂³×Dic₃	—	3	2	no	—
58	C₂³×Dic₃	—	3	56	no	—
59	C₂³×Dic₃	—	6	2	no	—
60	C₂³×Dic₃	C₂×D₃	1	1	yes	characteristic
61	C₂³×Dic₃	C₂×D₃	1	2	yes	—
62	C₂×D₃×Q₈	—	3	448	no	—
63	C₂²×C₄×D₃	—	3	56	no	—
64	C₂²×C₄×D₃	C₂×D₃	1	28	yes	—
65	C₂²×Dic₆	—	3	28	no	—
66	C₂²×Dic₆	—	3	28	no	—
67	C₂²×Dic₆	—	3	56	no	—
68	C₂²×Dic₆	—	3	56	no	—
69	C₂²×Dic₆	—	3	448	no	—
70	C₂²×Dic₆	—	6	28	no	—
71	C₂²×Dic₆	C₂×D₃	1	28	yes	direct product factor
72	C₂³×C₃²	C₂×Q₈	1	8	yes	complemented
73	C₂³×C₃²	C₂⁴	1	7	yes	—
74	C₂²×C₃×D₃	C₂×Q₈	1	112	yes	complemented
75	C₂²×C₃×D₃	C₂⁴	1	28	yes	—
76	C₂×C₃²×C₄	C₂⁴	1	28	yes	—
77	C₂×C₃⋊Dic₃	C₂⁴	1	56	yes	—
78	#634	C₂⁴	1	64	yes	complemented
79	C₂×C₃×Dic₃	C₂⁴	1	56	yes	—
80	C₂×C₃×Dic₃	C₂⁴	1	28	yes	—
81	C₃×C₄×D₃	C₂⁴	1	64	yes	—
82	C₃×Dic₆	C₂⁴	1	64	yes	complemented
83	#627	C₂⁴	1	128	yes	complemented
84	D₃×Dic₃	C₂⁴	1	128	yes	—
85	C₂³×Q₈	—	3	1	no	—
86	C₂³×Q₈	—	9	2	no	—
87	C₂³×Q₈	—	9	56	no	—
88	C₂³×Q₈	—	9	1	no	—
89	C₂⁴×C₄	—	3	1	no	—
90	C₂⁴×C₄	—	9	2	no	—
91	C₂⁴×C₃	—	2	1	no	—
92	C₂⁴×C₃	—	3	16	no	—
93	C₂⁴×C₃	—	3	14	no	—
94	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
95	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
96	C₂³×D₃	Dic₆	1	16	yes	direct product factor
97	C₂³×D₃	C₂²×D₃	1	14	yes	—
98	C₂×C₃×Q₈	—	3	112	no	—
99	C₂²×C₃×C₄	—	2	14	no	—
100	C₂²×C₃×C₄	—	3	14	no	—
101	C₂²×C₃×C₄	—	3	28	no	—
102	C₂²×C₃×C₄	—	3	112	no	—
103	C₂²×C₃×C₄	C₂²×D₃	1	14	yes	—
104	C₂²×C₃×C₄	C₂²×D₃	1	14	yes	—
105	C₂²×Dic₃	—	3	28	no	—
106	C₂²×Dic₃	—	3	28	no	—
107	C₂²×Dic₃	—	3	224	no	—
108	C₂²×Dic₃	—	6	28	no	—
109	C₂²×Dic₃	C₂²×D₃	1	14	yes	—
110	C₂²×Dic₃	C₂²×D₃	1	28	yes	—
111	D₃×Q₈	—	3	512	no	—
112	C₂×C₄×D₃	—	3	224	no	—
113	C₂×C₄×D₃	C₂²×D₃	1	112	yes	—
114	C₂×Dic₆	—	3	112	no	—
115	C₂×Dic₆	—	3	112	no	—
116	C₂×Dic₆	—	3	224	no	—
117	C₂×Dic₆	—	3	224	no	—
118	C₂×Dic₆	—	3	512	no	—
119	C₂×Dic₆	—	6	112	no	—
120	C₂×Dic₆	C₂²×D₃	1	112	yes	direct product factor
121	C₂²×C₃²	C₂²×Q₈	1	28	yes	complemented
122	C₂²×C₃²	C₂⁵	1	7	yes	—
123	C₂×C₃×D₃	C₂²×Q₈	1	112	yes	complemented
124	C₂×C₃×D₃	C₂⁵	1	8	yes	—
125	C₃²×C₄	C₂⁵	1	8	yes	—
126	C₃⋊Dic₃	C₂⁵	1	16	yes	—
127	C₃×Dic₃	C₂⁵	1	16	yes	—
128	C₃×Dic₃	C₂⁵	1	8	yes	—
129	C₂⁵	—	3	1	no	—
130	C₂²×Q₈	—	3	28	no	—
131	C₂²×Q₈	—	9	56	no	—
132	C₂²×Q₈	—	9	448	no	—
133	C₂²×Q₈	—	9	28	no	—
134	C₂³×C₄	—	3	1	no	—
135	C₂³×C₄	—	3	2	no	—
136	C₂³×C₄	—	3	28	no	—
137	C₂³×C₄	—	9	56	no	—
138	C₂³×C₄	—	9	2	no	—
139	C₂³×C₄	D₃²	1	1	yes	characteristic
140	C₂³×C₃	—	2	7	no	—
141	C₂³×C₃	—	2	8	no	—
142	C₂³×C₃	—	3	112	no	—
143	C₂³×C₃	—	3	28	no	—
144	C₂³×C₃	C₂×Dic₆	1	8	yes	complemented
145	C₂³×C₃	D₃×Q₈	1	8	yes	complemented
146	C₂³×C₃	C₂³×D₃	1	7	yes	—
147	C₂³×C₃	C₂³×D₃	1	7	yes	—
148	C₂²×D₃	C₂×Dic₆	1	112	yes	direct product factor
149	C₂²×D₃	C₂³×D₃	1	28	yes	—
150	C₃×Q₈	—	3	64	no	—
151	C₂×C₃×C₄	—	2	28	no	—
152	C₂×C₃×C₄	—	3	28	no	—
153	C₂×C₃×C₄	—	3	56	no	—
154	C₂×C₃×C₄	—	3	64	no	—
155	C₂×C₃×C₄	C₂³×D₃	1	28	yes	—
156	C₂×C₃×C₄	C₂³×D₃	1	28	yes	—
157	C₂×Dic₃	—	3	56	no	—
158	C₂×Dic₃	—	3	56	no	—
159	C₂×Dic₃	—	3	128	no	—
160	C₂×Dic₃	—	6	56	no	—
161	C₂×Dic₃	C₂³×D₃	1	28	yes	—
162	C₂×Dic₃	C₂³×D₃	1	56	yes	—
163	C₄×D₃	—	3	128	no	—
164	C₄×D₃	C₂³×D₃	1	64	yes	—
165	Dic₆	—	3	64	no	—
166	Dic₆	—	3	64	no	—
167	Dic₆	—	3	128	no	—
168	Dic₆	—	3	128	no	—
169	Dic₆	—	6	64	no	—
170	Dic₆	C₂³×D₃	1	64	yes	direct product factor
171	C₂×C₃²	C₂³×Q₈	1	14	yes	complemented
172	C₂×C₃²	C₂⁶	1	1	yes	characteristic
173	C₃×D₃	C₂³×Q₈	1	16	yes	complemented
174	C₂⁴	—	3	14	no	—
175	C₂⁴	—	3	16	no	—
176	C₂⁴	C₂×D₃²	1	1	yes	characteristic
177	C₂×Q₈	—	3	112	no	—
178	C₂×Q₈	—	9	224	no	—
179	C₂×Q₈	—	9	512	no	—
180	C₂×Q₈	—	9	112	no	—
181	C₂²×C₄	—	3	14	no	—
182	C₂²×C₄	—	3	28	no	—
183	C₂²×C₄	—	3	112	no	—
184	C₂²×C₄	—	9	224	no	—
185	C₂²×C₄	—	9	28	no	—
186	C₂²×C₄	C₂×D₃²	1	14	yes	—
187	C₂²×C₃	—	2	28	no	—
188	C₂²×C₃	—	2	7	no	—
189	C₂²×C₃	—	3	112	no	—
190	C₂²×C₃	—	3	8	no	—
191	C₂²×C₃	C₂²×Dic₆	1	28	yes	complemented
192	C₂²×C₃	C₂×D₃×Q₈	1	28	yes	complemented
193	C₂²×C₃	C₂⁴×D₃	1	7	yes	—
194	C₂²×C₃	C₂⁴×D₃	1	7	yes	—
195	C₂×D₃	C₂²×Dic₆	1	112	yes	direct product factor
196	C₂×D₃	C₂⁴×D₃	1	8	yes	—
197	C₃×C₄	—	2	8	no	—
198	C₃×C₄	—	3	8	no	—
199	C₃×C₄	—	3	16	no	—
200	C₃×C₄	C₂⁴×D₃	1	8	yes	—
201	C₃×C₄	C₂⁴×D₃	1	8	yes	—
202	Dic₃	—	3	16	no	—
203	Dic₃	—	3	16	no	—
204	Dic₃	—	6	16	no	—
205	Dic₃	C₂⁴×D₃	1	8	yes	—
206	Dic₃	C₂⁴×D₃	1	16	yes	—
207	C₃²	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	3	112	no	—
209	C₂³	—	3	28	no	—
210	C₂³	D₃×Dic₆	1	8	yes	direct product factor
211	C₂³	C₂²×D₃²	1	7	yes	—
212	Q₈	—	3	64	no	—
213	Q₈	—	9	128	no	—
214	Q₈	—	9	64	no	—
215	C₂×C₄	—	3	28	no	—
216	C₂×C₄	—	3	56	no	—
217	C₂×C₄	—	3	64	no	—
218	C₂×C₄	—	9	128	no	—
219	C₂×C₄	—	9	56	no	—
220	C₂×C₄	C₂²×D₃²	1	28	yes	—
221	C₂×C₃	—	2	14	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	3	16	no	—
224	C₂×C₃	C₂³×Dic₆	1	14	yes	complemented
225	C₂×C₃	C₂²×D₃×Q₈	1	14	yes	complemented
226	C₂×C₃	C₂⁵×D₃	1	1	yes	characteristic
227	C₂×C₃	C₂⁵×D₃	1	1	yes	characteristic
228	D₃	C₂³×Dic₆	1	16	yes	direct product factor
229	C₂²	—	3	112	no	—
230	C₂²	—	3	8	no	—
231	C₂²	C₂×D₃×Dic₆	1	28	yes	direct product factor
232	C₂²	C₂³×D₃²	1	7	yes	—
233	C₄	—	3	8	no	—
234	C₄	—	3	16	no	—
235	C₄	—	9	16	no	—
236	C₄	C₂³×D₃²	1	8	yes	—
237	C₃	—	2	1	no	—
238	C₃	C₂⁴×Dic₆	1	1	yes	characteristic, complemented
239	C₃	C₂³×D₃×Q₈	1	1	yes	characteristic, complemented
240	C₂	—	3	16	no	—
241	C₂	C₂²×D₃×Dic₆	1	14	yes	direct product factor
242	C₂	C₂⁴×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃²
commutator, cosocle	C₂×C₃²	C₂⁶
Frattini	C₂	C₂⁴×D₃²
Fitting	C₂³×C₃²×C₄	C₂²
socle	C₂⁴×C₃²	C₂³
prime radical	C₂⁴×C₃×D₃	C₂²
hypercenter	C₂³×C₄	D₃²
hypocenter	C₃²	C₂⁴×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#634	C₂	semidirect
3. C₂³×C₃×Dic₆	C₂	semidirect
4. C₂³×#627	C₂	semidirect
6. C₂²×D₃×Dic₆	C₂	direct (N × H)
12. C₂²×#634	C₂²	semidirect
16. C₂²×C₃×Dic₆	C₂²	semidirect
17. C₂²×#627	C₂²	semidirect
19. C₂×D₃×Dic₆	C₂²	direct (N × H)
32. C₂³×Dic₆	D₃	direct (N × H)
34. C₂³×C₃×D₃	Q₈	semidirect
38. C₂×#634	C₂³	semidirect
42. C₂×C₃×Dic₆	C₂³	semidirect
43. C₂×#627	C₂³	semidirect
45. D₃×Dic₆	C₂³	direct (N × H)
71. C₂²×Dic₆	C₂×D₃	direct (N × H)
72. C₂³×C₃²	C₂×Q₈	semidirect
74. C₂²×C₃×D₃	C₂×Q₈	semidirect
78. #634	C₂⁴	semidirect
82. C₃×Dic₆	C₂⁴	semidirect
83. #627	C₂⁴	semidirect
96. C₂³×D₃	Dic₆	direct (N × H)
120. C₂×Dic₆	C₂²×D₃	direct (N × H)
121. C₂²×C₃²	C₂²×Q₈	semidirect
123. C₂×C₃×D₃	C₂²×Q₈	semidirect
144. C₂³×C₃	C₂×Dic₆	semidirect
145. C₂³×C₃	D₃×Q₈	semidirect
148. C₂²×D₃	C₂×Dic₆	direct (N × H)
170. Dic₆	C₂³×D₃	direct (N × H)
171. C₂×C₃²	C₂³×Q₈	semidirect
173. C₃×D₃	C₂³×Q₈	semidirect
191. C₂²×C₃	C₂²×Dic₆	semidirect
192. C₂²×C₃	C₂×D₃×Q₈	semidirect
195. C₂×D₃	C₂²×Dic₆	direct (N × H)
207. C₃²	C₂⁴×Q₈	semidirect
210. C₂³	D₃×Dic₆	direct (N × H)
224. C₂×C₃	C₂³×Dic₆	semidirect
225. C₂×C₃	C₂²×D₃×Q₈	semidirect
228. D₃	C₂³×Dic₆	direct (N × H)
231. C₂²	C₂×D₃×Dic₆	direct (N × H)
238. C₃	C₂⁴×Dic₆	semidirect
239. C₃	C₂³×D₃×Q₈	semidirect
241. C₂	C₂²×D₃×Dic₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#634	2
C₂³×C₃×C₄×D₃	2
C₂³×C₃×Dic₆	2
C₂³×#627	2
C₂³×D₃×Dic₃	2
C₂²×D₃×Dic₆	2
C₂³×D₃×Q₈	3
C₂⁴×Dic₆	3

Subgroups — C23×D3×Dic6 #11919179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₃×Dic₆ #11919179