Subgroups — C₃²×Q₈×D₁₂ #13178952

Order 1728.

1960 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×Q₈	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#897	C₂	1	6	yes	maximal, complemented
4	C₃²×C₄×D₁₂	C₂	1	3	yes	maximal
5	C₃²×#869	C₂	1	3	yes	maximal, complemented
6	C₃²×D₄×Q₈	—	3	1	no	maximal
7	C₃×Q₈×D₁₂	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
9	C₃²×D₃×Q₈	—	2	8	no	—
10	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×D₃	C₂²	1	6	yes	—
12	C₂×C₃²×Dic₆	C₂²	1	6	yes	—
13	C₃³×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃³×C₄²	C₂²	1	3	yes	—
15	C₃²×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
16	C₃²×C₄⋊Dic₃	C₂²	1	3	yes	—
17	C₃²×C₄⋊Dic₃	C₂²	1	6	yes	—
18	C₂²×C₃²×Q₈	—	3	2	no	—
19	C₂×C₃×D₃×Q₈	C₂×C₃	1	8	yes	complemented
20	C₃²×C₄⋊Q₈	—	3	3	no	—
21	C₃²×C₂²⋊Q₈	—	3	6	no	—
22	C₃²×C₄×Q₈	—	2	4	no	—
23	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic, complemented
24	C₃²×C₄×Q₈	C₂×C₃	1	4	yes	complemented
25	C₃²×C₄×D₄	—	3	3	no	—
26	C₃×#897	C₂×C₃	1	24	yes	complemented
27	C₃×C₄×D₁₂	C₂×C₃	1	12	yes	—
28	C₃×#869	C₂×C₃	1	12	yes	complemented
29	C₂²×C₃²×D₃	C₂³	1	2	yes	—
30	C₃³×Q₈	D₄	1	4	yes	complemented
31	C₂×C₃³×C₄	C₂³	1	3	yes	—
32	C₂×C₃³×C₄	C₂³	1	3	yes	—
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃²×Dic₃	C₂³	1	6	yes	—
35	C₃²×D₁₂	Q₈	1	4	yes	direct product factor
36	C₃²×C₄×D₃	—	2	12	no	—
37	C₃²×Dic₆	—	2	12	no	—
38	C₃×D₄×Q₈	—	3	4	no	—
39	Q₈×D₁₂	C₃²	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	—	2	4	no	—
41	C₂×C₃²×Q₈	—	3	6	no	—
42	C₂×C₃²×Q₈	—	6	8	no	—
43	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
45	C₂×C₃²×D₄	—	3	1	no	—
46	C₂²×C₃²×C₄	—	3	6	no	—
47	C₃×D₃×Q₈	—	2	32	no	—
48	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
49	C₂×C₃×C₄×D₃	C₂²×C₃	1	24	yes	—
50	C₂×C₃×Dic₆	C₂²×C₃	1	24	yes	—
51	C₃²×C₄⋊C₄	—	2	12	no	—
52	C₃²×C₄⋊C₄	—	3	3	no	—
53	C₃²×C₄⋊C₄	—	3	6	no	—
54	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
55	C₃²×C₄⋊C₄	C₂²×C₃	1	12	yes	—
56	C₃²×C₂²⋊C₄	—	3	6	no	—
57	C₃²×C₄²	—	2	12	no	—
58	C₃²×C₄²	C₂×D₃	1	3	yes	—
59	C₃²×C₄²	C₂²×C₃	1	12	yes	—
60	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	24	yes	—
61	C₃×C₄⋊Dic₃	C₂²×C₃	1	24	yes	—
62	C₃×C₄⋊Dic₃	C₂²×C₃	1	12	yes	—
63	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
64	C₂×C₃²×D₃	—	2	2	no	—
65	C₂×C₃²×D₃	—	2	2	no	—
66	C₂×C₃²×D₃	C₂×Q₈	1	4	yes	complemented
67	C₃³×C₄	—	2	3	no	—
68	C₃³×C₄	C₂×D₄	1	6	yes	—
69	C₃³×C₄	C₂×Q₈	1	2	yes	complemented
70	C₃²×Dic₃	—	2	6	no	—
71	C₂²×C₃×Q₈	—	3	8	no	—
72	C₂×D₃×Q₈	C₂×C₃²	1	2	yes	complemented
73	C₃×C₄⋊Q₈	—	3	12	no	—
74	C₃×C₂²⋊Q₈	—	3	24	no	—
75	C₃×C₄×Q₈	—	2	4	no	—
76	C₃×C₄×Q₈	C₃×D₃	1	4	yes	complemented
77	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
78	C₃×C₄×D₄	—	3	12	no	—
79	#897	C₂×C₃²	1	6	yes	complemented
80	C₄×D₁₂	C₂×C₃²	1	3	yes	—
81	#869	C₂×C₃²	1	3	yes	complemented
82	C₂³×C₃²	—	3	2	no	—
83	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
84	C₃²×Q₈	—	2	16	no	—
85	C₃²×Q₈	—	6	12	no	—
86	C₃²×Q₈	D₁₂	1	4	yes	direct product factor
87	C₃²×Q₈	C₃×D₄	1	16	yes	complemented
88	C₃²×D₄	—	3	4	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	12	no	—
91	C₂×C₃²×C₄	—	2	12	no	—
92	C₂×C₃²×C₄	—	3	6	no	—
93	C₂×C₃²×C₄	—	6	12	no	—
94	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
96	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
98	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
100	C₂×C₃×Dic₃	C₂³×C₃	1	24	yes	—
101	C₃×D₁₂	C₃×Q₈	1	16	yes	direct product factor
102	C₃×C₄×D₃	—	2	48	no	—
103	C₃×Dic₆	—	2	48	no	—
104	D₄×Q₈	—	3	1	no	Sylow
105	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
108	C₃²×D₃	—	2	4	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	—	3	24	no	—
111	C₂×C₃×Q₈	—	6	32	no	—
112	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
113	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
114	C₂×C₃×D₄	—	3	4	no	—
115	C₂²×C₃×C₄	—	3	24	no	—
116	D₃×Q₈	—	2	8	no	—
117	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
118	C₂×C₄×D₃	C₂²×C₃²	1	6	yes	—
119	C₂×Dic₆	C₂²×C₃²	1	6	yes	—
120	C₃×C₄⋊C₄	—	2	12	no	—
121	C₃×C₄⋊C₄	—	3	12	no	—
122	C₃×C₄⋊C₄	—	3	24	no	—
123	C₃×C₄⋊C₄	C₂×C₃×D₃	1	12	yes	—
124	C₃×C₄⋊C₄	C₂²×C₃²	1	3	yes	—
125	C₃×C₂²⋊C₄	—	3	24	no	—
126	C₃×C₄²	—	2	12	no	—
127	C₃×C₄²	C₂×C₃×D₃	1	12	yes	—
128	C₃×C₄²	C₂²×C₃²	1	3	yes	—
129	(C₂×D₃)⋊C₄	C₂²×C₃²	1	6	yes	—
130	C₄⋊Dic₃	C₂²×C₃²	1	3	yes	—
131	C₄⋊Dic₃	C₂²×C₃²	1	6	yes	—
132	C₂²×C₃²	—	2	4	no	—
133	C₂²×C₃²	—	3	4	no	—
134	C₂²×C₃²	—	6	2	no	—
135	C₂²×C₃²	—	6	2	no	—
136	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
137	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
138	C₂×C₃×D₃	—	2	8	no	—
139	C₂×C₃×D₃	—	2	8	no	—
140	C₂×C₃×D₃	C₂×C₃×Q₈	1	16	yes	complemented
141	C₃²×C₄	—	2	3	no	—
142	C₃²×C₄	—	2	24	no	—
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	12	no	—
145	C₃²×C₄	—	4	12	no	—
146	C₃²×C₄	—	6	6	no	—
147	C₃²×C₄	C₂×D₁₂	1	6	yes	—
148	C₃²×C₄	D₃×Q₈	1	2	yes	complemented
149	C₃²×C₄	C₂×C₃×D₄	1	24	yes	—
150	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	complemented
151	C₃×Dic₃	—	2	24	no	—
152	C₂²×Q₈	—	3	2	no	—
153	C₄⋊Q₈	—	3	3	no	—
154	C₂²⋊Q₈	—	3	6	no	—
155	C₄×Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
156	C₄×D₄	—	3	3	no	—
157	C₃³	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	3	8	no	—
159	C₂²×D₃	C₂³×C₃²	1	2	yes	—
160	C₃×Q₈	—	2	16	no	—
161	C₃×Q₈	—	6	48	no	—
162	C₃×Q₈	C₃×D₁₂	1	16	yes	direct product factor
163	C₃×Q₈	C₃²×D₄	1	4	yes	complemented
164	C₃×D₄	—	3	16	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	12	no	—
167	C₂×C₃×C₄	—	2	12	no	—
168	C₂×C₃×C₄	—	3	24	no	—
169	C₂×C₃×C₄	—	6	48	no	—
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
171	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
172	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
173	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
174	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
175	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
176	C₂×Dic₃	C₂³×C₃²	1	6	yes	—
177	D₁₂	C₃²×Q₈	1	4	yes	direct product factor
178	C₄×D₃	—	2	12	no	—
179	Dic₆	—	2	12	no	—
180	C₂×C₃²	—	2	4	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	6	4	no	—
184	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
185	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
186	C₂×C₃²	#1010	1	1	yes	characteristic
187	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
188	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
189	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
190	C₃×D₃	—	2	16	no	—
191	C₂×Q₈	—	3	6	no	—
192	C₂×Q₈	—	6	8	no	—
193	C₂×Q₈	C₂×C₃²×D₃	1	1	yes	characteristic
194	C₂×D₄	—	3	1	no	—
195	C₂²×C₄	—	3	6	no	—
196	C₄⋊C₄	—	3	3	no	—
197	C₄⋊C₄	—	3	6	no	—
198	C₄⋊C₄	C₂×C₃²×D₃	1	3	yes	—
199	C₂²⋊C₄	—	3	6	no	—
200	C₄²	C₂×C₃²×D₃	1	3	yes	—
201	C₂²×C₃	—	2	4	no	—
202	C₂²×C₃	—	3	16	no	—
203	C₂²×C₃	—	6	8	no	—
204	C₂²×C₃	—	6	8	no	—
205	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
206	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
207	C₂×D₃	—	2	2	no	—
208	C₂×D₃	—	2	2	no	—
209	C₂×D₃	C₂×C₃²×Q₈	1	4	yes	complemented
210	C₃×C₄	—	2	12	no	—
211	C₃×C₄	—	2	3	no	—
212	C₃×C₄	—	2	24	no	—
213	C₃×C₄	—	2	8	no	—
214	C₃×C₄	—	4	12	no	—
215	C₃×C₄	—	6	24	no	—
216	C₃×C₄	C₂×C₃×D₁₂	1	24	yes	—
217	C₃×C₄	C₃×D₃×Q₈	1	8	yes	complemented
218	C₃×C₄	C₂×C₃²×D₄	1	6	yes	—
219	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	complemented
220	Dic₃	—	2	6	no	—
221	C₃²	—	2	4	no	—
222	C₃²	Q₈×D₁₂	1	1	yes	characteristic, direct product factor
223	C₃²	C₃×D₄×Q₈	1	4	yes	complemented
224	C₂³	—	3	2	no	—
225	Q₈	—	6	12	no	—
226	Q₈	C₃²×D₁₂	1	4	yes	direct product factor
227	D₄	—	3	4	no	—
228	C₂×C₄	—	3	6	no	—
229	C₂×C₄	—	6	12	no	—
230	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
231	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
232	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	2	4	no	—
236	C₂×C₃	—	6	16	no	—
237	C₂×C₃	C₂²×C₃×D₁₂	1	4	yes	—
238	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
239	C₂×C₃	C₃×#1010	1	4	yes	—
240	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
241	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
242	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
243	D₃	—	2	4	no	—
244	C₂²	—	3	4	no	—
245	C₂²	—	6	2	no	—
246	C₂²	—	6	2	no	—
247	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
248	C₄	—	2	3	no	—
249	C₄	—	6	6	no	—
250	C₄	C₂×C₃²×D₁₂	1	6	yes	—
251	C₄	C₃²×D₃×Q₈	1	2	yes	complemented
252	C₃	—	2	4	no	—
253	C₃	C₃×Q₈×D₁₂	1	4	yes	direct product factor
254	C₃	C₃²×D₄×Q₈	1	1	yes	characteristic, complemented
255	C₂	—	6	4	no	—
256	C₂	C₂²×C₃²×D₁₂	1	1	yes	characteristic
257	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
258	C₂	C₃²×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄×Q₈	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂×C₃²×D₁₂	C₂²
hypercenter	C₃²×C₄×Q₈	D₃
hypocenter	C₃	C₃²×D₄×Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₃×Q₈	C₂	semidirect
2. C₃³×C₄×Q₈	C₂	semidirect
3. C₃²×#897	C₂	semidirect
5. C₃²×#869	C₂	semidirect
7. C₃×Q₈×D₁₂	C₃	direct (N × H)
19. C₂×C₃×D₃×Q₈	C₂×C₃	semidirect
23. C₃²×C₄×Q₈	D₃	semidirect
24. C₃²×C₄×Q₈	C₂×C₃	semidirect
26. C₃×#897	C₂×C₃	semidirect
28. C₃×#869	C₂×C₃	semidirect
30. C₃³×Q₈	D₄	semidirect
35. C₃²×D₁₂	Q₈	direct (N × H)
39. Q₈×D₁₂	C₃²	direct (N × H)
66. C₂×C₃²×D₃	C₂×Q₈	semidirect
69. C₃³×C₄	C₂×Q₈	semidirect
72. C₂×D₃×Q₈	C₂×C₃²	semidirect
76. C₃×C₄×Q₈	C₃×D₃	semidirect
77. C₃×C₄×Q₈	C₂×C₃²	semidirect
79. #897	C₂×C₃²	semidirect
81. #869	C₂×C₃²	semidirect
86. C₃²×Q₈	D₁₂	direct (N × H)
87. C₃²×Q₈	C₃×D₄	semidirect
101. C₃×D₁₂	C₃×Q₈	direct (N × H)
140. C₂×C₃×D₃	C₂×C₃×Q₈	semidirect
148. C₃²×C₄	D₃×Q₈	semidirect
150. C₃²×C₄	C₂×C₃×Q₈	semidirect
155. C₄×Q₈	C₃²×D₃	semidirect
157. C₃³	D₄×Q₈	semidirect
162. C₃×Q₈	C₃×D₁₂	direct (N × H)
163. C₃×Q₈	C₃²×D₄	semidirect
177. D₁₂	C₃²×Q₈	direct (N × H)
209. C₂×D₃	C₂×C₃²×Q₈	semidirect
217. C₃×C₄	C₃×D₃×Q₈	semidirect
219. C₃×C₄	C₂×C₃²×Q₈	semidirect
222. C₃²	Q₈×D₁₂	direct (N × H)
223. C₃²	C₃×D₄×Q₈	semidirect
226. Q₈	C₃²×D₁₂	direct (N × H)
251. C₄	C₃²×D₃×Q₈	semidirect
253. C₃	C₃×Q₈×D₁₂	direct (N × H)
254. C₃	C₃²×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×C₃²×D₃×Q₈	2
C₃³×C₄×Q₈	2
C₃²×#897	2
C₃²×C₄×D₁₂	2
C₃²×#869	2
C₃²×D₄×Q₈	3
C₃×Q₈×D₁₂	3

Subgroups — C32×Q8×D12 #13178952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Q₈×D₁₂ #13178952