Subgroups — C₃×#3212 #85076

Order 384.

400 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#495	C₂	1	2	yes	maximal, complemented
4	C₃×#488	C₂	1	2	yes	maximal, complemented
5	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#452	C₂	1	2	yes	maximal, complemented
10	C₃×#449	C₂	1	2	yes	maximal, complemented
11	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#3212	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₃×D₈	C₂²	1	2	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₃×#127	C₂²	1	1	yes	characteristic
20	C₃×#126	—	2	1	no	—
21	C₃×#125	C₂²	1	1	yes	characteristic
22	C₃×#124	C₂²	1	1	yes	characteristic
23	C₃×#124	C₂²	1	2	yes	—
24	C₃×#123	—	2	1	no	—
25	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₃×#121	—	2	1	no	—
27	C₃×#121	—	2	1	no	—
28	C₃×#121	C₂²	1	1	yes	characteristic
29	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	—	2	1	no	—
32	C₃×C₄×D₄	—	2	1	no	—
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₃×#117	—	2	1	no	—
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#103	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×#102	C₂²	1	2	yes	—
42	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₃×C₄×C₈	C₂²	1	2	yes	—
44	#532	C₂×C₃	1	1	yes	characteristic, complemented
45	#529	C₂×C₃	1	1	yes	characteristic, complemented
46	#495	C₂×C₃	1	2	yes	complemented
47	#488	C₂×C₃	1	2	yes	complemented
48	#462	C₂×C₃	1	1	yes	characteristic, complemented
49	#461	C₂×C₃	1	1	yes	characteristic, complemented
50	#460	C₂×C₃	1	1	yes	characteristic, complemented
51	#459	C₂×C₃	1	1	yes	characteristic, complemented
52	#452	C₂×C₃	1	2	yes	complemented
53	#449	C₂×C₃	1	2	yes	complemented
54	#431	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×#41	—	4	2	no	—
56	C₃×#41	—	4	2	no	—
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	2	1	no	—
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	—	2	2	no	—
67	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₃×Q₁₆	—	4	2	no	—
71	C₃×SD₁₆	—	4	2	no	—
72	C₃×SD₁₆	—	4	2	no	—
73	C₃×D₈	—	4	2	no	—
74	C₂×C₃×C₈	C₂³	1	2	yes	—
75	C₂×C₃×C₈	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	—	2	1	no	—
81	C₃×C₄⋊C₄	—	2	1	no	—
82	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	D₄	1	1	yes	characteristic
93	C₃×C₄²	D₄	1	1	yes	characteristic
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₃×C₄²	C₂³	1	1	yes	characteristic
96	C₂×#41	C₂²×C₃	1	2	yes	—
97	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
98	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×D₈	C₂²×C₃	1	2	yes	—
101	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
102	#127	C₂²×C₃	1	1	yes	characteristic
103	#126	—	2	1	no	—
104	#125	C₂²×C₃	1	1	yes	characteristic
105	#124	C₂²×C₃	1	1	yes	characteristic
106	#124	C₂²×C₃	1	2	yes	—
107	#123	—	2	1	no	—
108	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
109	#121	—	2	1	no	—
110	#121	—	2	1	no	—
111	#121	C₂²×C₃	1	1	yes	characteristic
112	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
113	C₄×D₄	—	2	1	no	—
114	C₄×D₄	—	2	1	no	—
115	C₄×D₄	—	2	1	no	—
116	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
117	#117	—	2	1	no	—
118	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
119	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
120	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
121	#103	C₂²×C₃	1	2	yes	—
122	#103	C₂²×C₃	1	2	yes	—
123	#102	C₂²×C₃	1	2	yes	—
124	#102	C₂²×C₃	1	2	yes	—
125	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
126	C₄×C₈	C₂²×C₃	1	2	yes	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
130	C₃×Q₈	—	2	1	no	—
131	C₃×Q₈	—	2	2	no	—
132	C₃×Q₈	—	2	1	no	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×D₄	—	2	1	no	—
135	C₃×D₄	—	2	2	no	—
136	C₃×D₄	—	2	1	no	—
137	C₃×D₄	—	2	2	no	—
138	C₃×D₄	—	4	2	no	—
139	C₃×D₄	—	4	2	no	—
140	C₃×D₄	—	4	4	no	—
141	C₃×D₄	—	4	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	4	1	no	—
150	C₂×C₃×C₄	—	4	4	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	2	2	no	—
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	C₂×Q₈	C₂³×C₃	1	2	yes	—
163	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₂³×C₃	1	2	yes	—
169	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
175	Q₁₆	—	4	2	no	—
176	SD₁₆	—	4	2	no	—
177	SD₁₆	—	4	2	no	—
178	D₈	—	4	2	no	—
179	C₂×C₈	C₂³×C₃	1	2	yes	—
180	C₂×C₈	C₂³×C₃	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₃×D₄	1	1	yes	characteristic
198	C₄²	C₃×D₄	1	1	yes	characteristic
199	C₄²	C₂³×C₃	1	1	yes	characteristic
200	C₄²	C₂³×C₃	1	1	yes	characteristic
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	4	2	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	4	1	no	—
210	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	4	2	no	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	—	4	1	no	—
219	C₃×C₄	#135	1	2	yes	—
220	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
221	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	2	2	no	—
227	Q₈	—	2	1	no	—
228	Q₈	—	2	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	1	no	—
232	D₄	—	2	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₂×C₃	—	4	1	no	—
256	C₂×C₃	—	8	2	no	—
257	C₂×C₃	—	8	1	no	—
258	C₂×C₃	#535	1	1	yes	characteristic
259	C₂×C₃	C₂×#135	1	1	yes	characteristic
260	C₂×C₃	#575	1	1	yes	characteristic
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	C₃×#135	1	2	yes	—
280	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
281	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₃	#3212	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	4	1	no	—
284	C₂	—	8	1	no	—
285	C₂	—	8	2	no	—
286	C₂	C₃×#535	1	1	yes	characteristic
287	C₂	C₂×C₃×#135	1	1	yes	characteristic
288	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#532	C₂	semidirect
2. C₃×#529	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#488	C₂	semidirect
5. C₃×#462	C₂	semidirect
6. C₃×#461	C₂	semidirect
7. C₃×#460	C₂	semidirect
8. C₃×#459	C₂	semidirect
9. C₃×#452	C₂	semidirect
10. C₃×#449	C₂	semidirect
11. C₃×#431	C₂	semidirect
12. #3212	C₃	direct (N × H)
44. #532	C₂×C₃	semidirect
45. #529	C₂×C₃	semidirect
46. #495	C₂×C₃	semidirect
47. #488	C₂×C₃	semidirect
48. #462	C₂×C₃	semidirect
49. #461	C₂×C₃	semidirect
50. #460	C₂×C₃	semidirect
51. #459	C₂×C₃	semidirect
52. #452	C₂×C₃	semidirect
53. #449	C₂×C₃	semidirect
54. #431	C₂×C₃	semidirect
282. C₃	#3212	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#532	2
C₃×#529	2
C₃×#495	2
C₃×#488	2
C₃×#462	2
C₃×#461	2
C₃×#460	2
C₃×#459	2
C₃×#452	2
C₃×#449	2
C₃×#431	2
#3212	3

Subgroups — C3×#3212 #85076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3212 #85076