Subgroups — C₃×#3290 #85154

Order 384.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#577	C₂	1	2	yes	maximal, complemented
3	C₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#473	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#471	C₂	1	2	yes	maximal, complemented
8	C₃×#455	C₂	1	2	yes	maximal, complemented
9	C₃×#454	C₂	1	2	yes	maximal, complemented
10	C₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#3290	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×2₋¹⁺⁴	C₂²	1	2	yes	complemented
14	C₃×H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₃×#137	—	2	2	no	—
16	C₃×#137	—	2	2	no	—
17	C₃×#137	C₂²	1	2	yes	—
18	C₃×#137	C₂²	1	2	yes	complemented
19	C₃×AGL(1,ℤ/8)	—	2	2	no	—
20	C₃×AGL(1,ℤ/8)	—	2	2	no	—
21	C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
22	C₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
23	C₃×#135	—	2	1	no	—
24	C₃×#135	—	2	2	no	—
25	C₃×#135	—	2	1	no	—
26	C₃×#135	C₂²	1	1	yes	characteristic
27	C₃×#135	C₂²	1	1	yes	characteristic
28	C₂×C₃×Q₁₆	—	2	1	no	—
29	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×SD₁₆	—	2	2	no	—
31	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
32	C₂×C₃×D₈	—	2	1	no	—
33	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
34	C₃×#131	C₂²	1	1	yes	characteristic
35	C₃×#131	C₂²	1	2	yes	complemented
36	C₃×#131	C₂²	1	1	yes	characteristic, complemented
37	C₃×#124	C₂²	1	2	yes	complemented
38	C₃×#108	C₂²	1	1	yes	characteristic, complemented
39	C₃×#108	C₂²	1	1	yes	characteristic, complemented
40	C₃×#104	C₂²	1	1	yes	characteristic
41	C₃×#104	C₂²	1	1	yes	characteristic, complemented
42	C₃×#104	C₂²	1	2	yes	complemented
43	C₃×#101	C₂²	1	2	yes	complemented
44	C₃×#100	C₂²	1	2	yes	complemented
45	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
46	#578	C₂×C₃	1	1	yes	characteristic, complemented
47	#577	C₂×C₃	1	2	yes	complemented
48	#576	C₂×C₃	1	1	yes	characteristic, complemented
49	#495	C₂×C₃	1	1	yes	characteristic, complemented
50	#473	C₂×C₃	1	1	yes	characteristic, complemented
51	#472	C₂×C₃	1	1	yes	characteristic, complemented
52	#471	C₂×C₃	1	2	yes	complemented
53	#455	C₂×C₃	1	2	yes	complemented
54	#454	C₂×C₃	1	2	yes	complemented
55	#443	C₂×C₃	1	1	yes	characteristic, complemented
56	#443	C₂×C₃	1	1	yes	characteristic, complemented
57	C₃×#41	—	2	2	no	—
58	C₃×#41	—	2	2	no	—
59	C₃×#41	—	2	2	no	—
60	C₃×#41	—	2	2	no	—
61	C₃×#41	—	4	2	no	—
62	C₃×#41	C₂³	1	1	yes	characteristic
63	C₃×#41	C₂³	1	2	yes	—
64	C₃×#41	C₂³	1	1	yes	characteristic
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂×C₃×Q₈	C₂³	1	2	yes	—
68	C₂×C₃×D₄	—	2	2	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	4	2	no	—
71	C₂×C₃×D₄	C₂³	1	2	yes	—
72	C₃×Q₁₆	—	2	2	no	—
73	C₃×Q₁₆	—	2	2	no	—
74	C₃×Q₁₆	—	2	1	no	—
75	C₃×Q₁₆	—	2	1	no	—
76	C₃×Q₁₆	D₄	1	1	yes	characteristic
77	C₃×Q₁₆	D₄	1	1	yes	characteristic
78	C₃×SD₁₆	—	2	2	no	—
79	C₃×SD₁₆	—	2	2	no	—
80	C₃×SD₁₆	—	2	2	no	—
81	C₃×SD₁₆	—	2	2	no	—
82	C₃×SD₁₆	—	2	2	no	—
83	C₃×SD₁₆	—	2	2	no	—
84	C₃×SD₁₆	D₄	1	1	yes	characteristic
85	C₃×SD₁₆	D₄	1	1	yes	characteristic
86	C₃×SD₁₆	D₄	1	2	yes	—
87	C₃×D₈	—	2	2	no	—
88	C₃×D₈	—	2	2	no	—
89	C₃×D₈	—	2	1	no	—
90	C₃×D₈	—	2	1	no	—
91	C₃×D₈	D₄	1	1	yes	characteristic
92	C₃×D₈	D₄	1	1	yes	characteristic
93	C₃×M₁₆	—	2	1	no	—
94	C₃×M₁₆	—	2	2	no	—
95	C₃×M₁₆	—	2	1	no	—
96	C₃×M₁₆	C₂³	1	1	yes	characteristic
97	C₃×M₁₆	C₂³	1	2	yes	—
98	C₃×M₁₆	C₂³	1	1	yes	characteristic
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
104	C₃×C₂²⋊C₄	—	4	2	no	—
105	C₃×C₄²	C₂³	1	1	yes	characteristic
106	2₋¹⁺⁴	C₂²×C₃	1	2	yes	complemented
107	H(5,𝔽₂)	C₂²×C₃	1	2	yes	—
108	#137	—	2	2	no	—
109	#137	—	2	2	no	—
110	#137	C₂²×C₃	1	2	yes	—
111	#137	C₂²×C₃	1	2	yes	complemented
112	AGL(1,ℤ/8)	—	2	2	no	—
113	AGL(1,ℤ/8)	—	2	2	no	—
114	AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	—
115	AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
116	#135	—	2	1	no	—
117	#135	—	2	2	no	—
118	#135	—	2	1	no	—
119	#135	C₂²×C₃	1	1	yes	characteristic
120	#135	C₂²×C₃	1	1	yes	characteristic
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
123	C₂×SD₁₆	—	2	2	no	—
124	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
125	C₂×D₈	—	2	1	no	—
126	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
127	#131	C₂²×C₃	1	1	yes	characteristic
128	#131	C₂²×C₃	1	2	yes	complemented
129	#131	C₂²×C₃	1	1	yes	characteristic, complemented
130	#124	C₂²×C₃	1	2	yes	complemented
131	#108	C₂²×C₃	1	1	yes	characteristic, complemented
132	#108	C₂²×C₃	1	1	yes	characteristic, complemented
133	#104	C₂²×C₃	1	1	yes	characteristic
134	#104	C₂²×C₃	1	1	yes	characteristic, complemented
135	#104	C₂²×C₃	1	2	yes	complemented
136	#101	C₂²×C₃	1	2	yes	complemented
137	#100	C₂²×C₃	1	2	yes	complemented
138	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
139	C₂³×C₃	—	2	2	no	—
140	C₂³×C₃	—	4	2	no	—
141	C₃×Q₈	—	2	2	no	—
142	C₃×Q₈	—	2	2	no	—
143	C₃×Q₈	—	4	2	no	—
144	C₃×Q₈	C₂×D₄	1	1	yes	characteristic
145	C₃×Q₈	C₂×D₄	1	2	yes	—
146	C₃×Q₈	C₂×D₄	1	1	yes	characteristic
147	C₃×D₄	—	2	1	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	2	1	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	2	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₃×D₄	—	4	2	no	—
154	C₃×D₄	—	4	2	no	—
155	C₃×D₄	C₂×D₄	1	1	yes	characteristic
156	C₃×D₄	C₂×D₄	1	2	yes	—
157	C₃×D₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	C₂×D₄	1	1	yes	characteristic
171	C₃×C₈	C₂×D₄	1	1	yes	characteristic
172	C₃×C₈	C₂×D₄	1	1	yes	characteristic
173	C₃×C₈	C₂×D₄	1	1	yes	characteristic
174	#41	—	2	2	no	—
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	—	4	2	no	—
179	#41	C₂³×C₃	1	1	yes	characteristic
180	#41	C₂³×C₃	1	2	yes	—
181	#41	C₂³×C₃	1	1	yes	characteristic
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂³×C₃	1	2	yes	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	C₂³×C₃	1	2	yes	—
189	Q₁₆	—	2	2	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	1	no	—
192	Q₁₆	—	2	1	no	—
193	Q₁₆	C₃×D₄	1	1	yes	characteristic
194	Q₁₆	C₃×D₄	1	1	yes	characteristic
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	—	2	2	no	—
199	SD₁₆	—	2	2	no	—
200	SD₁₆	—	2	2	no	—
201	SD₁₆	C₃×D₄	1	1	yes	characteristic
202	SD₁₆	C₃×D₄	1	1	yes	characteristic
203	SD₁₆	C₃×D₄	1	2	yes	—
204	D₈	—	2	2	no	—
205	D₈	—	2	2	no	—
206	D₈	—	2	1	no	—
207	D₈	—	2	1	no	—
208	D₈	C₃×D₄	1	1	yes	characteristic
209	D₈	C₃×D₄	1	1	yes	characteristic
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	C₂³×C₃	1	1	yes	characteristic
214	M₁₆	C₂³×C₃	1	2	yes	—
215	M₁₆	C₂³×C₃	1	1	yes	characteristic
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	C₂³×C₃	1	1	yes	characteristic
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	—	8	2	no	—
228	C₂²×C₃	—	16	2	no	—
229	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	4	1	no	—
234	C₃×C₄	—	4	2	no	—
235	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
236	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
237	C₂³	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	Q₈	—	2	2	no	—
240	Q₈	—	2	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	C₂×C₃×D₄	1	1	yes	characteristic
243	Q₈	C₂×C₃×D₄	1	2	yes	—
244	Q₈	C₂×C₃×D₄	1	1	yes	characteristic
245	D₄	—	2	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	C₂×C₃×D₄	1	1	yes	characteristic
254	D₄	C₂×C₃×D₄	1	2	yes	—
255	D₄	C₂×C₃×D₄	1	1	yes	characteristic
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	1	no	—
268	C₈	C₂×C₃×D₄	1	1	yes	characteristic
269	C₈	C₂×C₃×D₄	1	1	yes	characteristic
270	C₈	C₂×C₃×D₄	1	1	yes	characteristic
271	C₈	C₂×C₃×D₄	1	1	yes	characteristic
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	—	4	2	no	—
275	C₂×C₃	—	4	1	no	—
276	C₂×C₃	—	8	2	no	—
277	C₂×C₃	D₄²	1	1	yes	characteristic
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	16	2	no	—
284	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	2	no	—
290	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
291	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₃	#3290	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	8	2	no	—
298	C₂	C₃×D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#578	C₂	semidirect
2. C₃×#577	C₂	semidirect
3. C₃×#576	C₂	semidirect
4. C₃×#495	C₂	semidirect
5. C₃×#473	C₂	semidirect
6. C₃×#472	C₂	semidirect
7. C₃×#471	C₂	semidirect
8. C₃×#455	C₂	semidirect
9. C₃×#454	C₂	semidirect
10. C₃×#443	C₂	semidirect
11. C₃×#443	C₂	semidirect
12. #3290	C₃	direct (N × H)
13. C₃×2₋¹⁺⁴	C₂²	semidirect
18. C₃×#137	C₂²	semidirect
22. C₃×AGL(1,ℤ/8)	C₂²	semidirect
29. C₂×C₃×Q₁₆	C₂²	semidirect
31. C₂×C₃×SD₁₆	C₂²	semidirect
35. C₃×#131	C₂²	semidirect
36. C₃×#131	C₂²	semidirect
37. C₃×#124	C₂²	semidirect
38. C₃×#108	C₂²	semidirect
39. C₃×#108	C₂²	semidirect
41. C₃×#104	C₂²	semidirect
42. C₃×#104	C₂²	semidirect
43. C₃×#101	C₂²	semidirect
44. C₃×#100	C₂²	semidirect
45. C₃×C₄×C₈	C₂²	semidirect
46. #578	C₂×C₃	semidirect
47. #577	C₂×C₃	semidirect
48. #576	C₂×C₃	semidirect
49. #495	C₂×C₃	semidirect
50. #473	C₂×C₃	semidirect
51. #472	C₂×C₃	semidirect
52. #471	C₂×C₃	semidirect
53. #455	C₂×C₃	semidirect
54. #454	C₂×C₃	semidirect
55. #443	C₂×C₃	semidirect
56. #443	C₂×C₃	semidirect
106. 2₋¹⁺⁴	C₂²×C₃	semidirect
111. #137	C₂²×C₃	semidirect
115. AGL(1,ℤ/8)	C₂²×C₃	semidirect
122. C₂×Q₁₆	C₂²×C₃	semidirect
124. C₂×SD₁₆	C₂²×C₃	semidirect
128. #131	C₂²×C₃	semidirect
129. #131	C₂²×C₃	semidirect
130. #124	C₂²×C₃	semidirect
131. #108	C₂²×C₃	semidirect
132. #108	C₂²×C₃	semidirect
134. #104	C₂²×C₃	semidirect
135. #104	C₂²×C₃	semidirect
136. #101	C₂²×C₃	semidirect
137. #100	C₂²×C₃	semidirect
138. C₄×C₈	C₂²×C₃	semidirect
292. C₃	#3290	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×#578	2
C₃×#577	2
C₃×#576	2
C₃×#495	2
C₃×#473	2
C₃×#472	2
C₃×#471	2
C₃×#455	2
C₃×#454	2
C₃×#443	2
#3290	3

Subgroups — C3×#3290 #85154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3290 #85154