Subgroups — C₃×#3046 #84910

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#464	C₂	1	2	yes	maximal, complemented
6	C₃×#463	C₂	1	2	yes	maximal, complemented
7	C₃×#462	C₂	1	2	yes	maximal, complemented
8	C₃×#461	C₂	1	2	yes	maximal, complemented
9	C₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	#3046	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×#41	C₂²	1	1	yes	characteristic
15	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
16	C₃×#137	—	2	4	no	—
17	C₃×#135	—	2	4	no	—
18	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
20	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
23	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
24	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
25	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
27	C₃×#124	C₂²	1	2	yes	—
28	C₃×#123	—	2	2	no	—
29	C₃×C₂²⋊Q₈	—	2	1	no	—
30	C₃×C₂²⋊Q₈	—	2	1	no	—
31	C₃×C₄×Q₈	C₂²	1	2	yes	—
32	C₃×C₄×D₄	—	2	2	no	—
33	C₃×C₄×D₄	C₂²	1	2	yes	—
34	C₃×#117	—	2	1	no	—
35	C₃×#117	C₂²	1	1	yes	characteristic
36	C₂×C₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂²	1	2	yes	—
39	C₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃×#103	C₂²	1	2	yes	—
41	C₃×#103	C₂²	1	2	yes	—
42	C₃×#103	C₂²	1	2	yes	—
43	C₃×#102	C₂²	1	2	yes	—
44	C₂×#137	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#135	C₂×C₃	1	1	yes	characteristic
46	#536	C₂×C₃	1	1	yes	characteristic, complemented
47	#520	C₂×C₃	1	1	yes	characteristic, complemented
48	#464	C₂×C₃	1	2	yes	complemented
49	#463	C₂×C₃	1	2	yes	complemented
50	#462	C₂×C₃	1	2	yes	complemented
51	#461	C₂×C₃	1	2	yes	complemented
52	#424	C₂×C₃	1	1	yes	characteristic, complemented
53	#417	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×#103	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×#41	—	2	2	no	—
56	C₃×#41	—	2	2	no	—
57	C₃×#41	—	2	2	no	—
58	C₃×#41	D₄	1	4	yes	—
59	C₂×C₃×Q₈	—	2	1	no	—
60	C₂×C₃×Q₈	—	2	2	no	—
61	C₂×C₃×Q₈	—	2	2	no	—
62	C₂×C₃×Q₈	C₂³	1	2	yes	—
63	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×D₄	—	2	1	no	—
66	C₂×C₃×D₄	—	2	1	no	—
67	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
68	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	1	no	—
70	C₂²×C₃×C₄	—	2	1	no	—
71	C₂²×C₃×C₄	—	2	1	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	—	2	2	no	—
74	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
75	C₃×Q₁₆	—	2	2	no	—
76	C₃×Q₁₆	—	2	4	no	—
77	C₃×SD₁₆	—	2	2	no	—
78	C₃×SD₁₆	—	2	2	no	—
79	C₃×SD₁₆	—	2	4	no	—
80	C₃×D₈	—	2	2	no	—
81	C₃×M₁₆	—	2	2	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	C₂³	1	2	yes	—
84	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
91	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
92	C₃×C₄⋊C₄	C₂³	1	2	yes	—
93	C₃×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₂²⋊C₄	—	2	2	no	—
95	C₃×C₂²⋊C₄	—	2	2	no	—
96	C₃×C₂²⋊C₄	D₄	1	2	yes	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	C₂³	1	2	yes	—
99	C₂×#41	C₂²×C₃	1	1	yes	characteristic
100	C₂×#41	C₂²×C₃	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
102	#137	—	2	4	no	—
103	#135	—	2	4	no	—
104	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
105	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
106	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
107	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
108	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
109	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
110	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
112	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
113	#124	C₂²×C₃	1	2	yes	—
114	#123	—	2	2	no	—
115	C₂²⋊Q₈	—	2	1	no	—
116	C₂²⋊Q₈	—	2	1	no	—
117	C₄×Q₈	C₂²×C₃	1	2	yes	—
118	C₄×D₄	—	2	2	no	—
119	C₄×D₄	C₂²×C₃	1	2	yes	—
120	#117	—	2	1	no	—
121	#117	C₂²×C₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
124	C₄⋊C₈	C₂²×C₃	1	2	yes	—
125	C₄⋊C₈	C₂²×C₃	1	2	yes	—
126	#103	C₂²×C₃	1	2	yes	—
127	#103	C₂²×C₃	1	2	yes	—
128	#103	C₂²×C₃	1	2	yes	—
129	#102	C₂²×C₃	1	2	yes	—
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	—	2	1	no	—
132	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	2	1	no	—
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	2	1	no	—
136	C₃×Q₈	—	2	2	no	—
137	C₃×Q₈	—	2	1	no	—
138	C₃×Q₈	—	4	2	no	—
139	C₃×Q₈	C₂×D₄	1	2	yes	—
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	—	2	1	no	—
142	C₃×D₄	—	2	1	no	—
143	C₃×D₄	—	2	1	no	—
144	C₃×D₄	—	4	2	no	—
145	C₃×D₄	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	—	8	2	no	—
154	C₂×C₃×C₄	—	8	1	no	—
155	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
159	C₂×C₃×C₄	#41	1	2	yes	—
160	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	2	2	no	—
164	#41	—	2	2	no	—
165	#41	—	2	2	no	—
166	#41	—	2	2	no	—
167	#41	C₃×D₄	1	4	yes	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	C₂³×C₃	1	2	yes	—
172	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
173	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
177	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
184	Q₁₆	—	2	2	no	—
185	Q₁₆	—	2	4	no	—
186	SD₁₆	—	2	2	no	—
187	SD₁₆	—	2	2	no	—
188	SD₁₆	—	2	4	no	—
189	D₈	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	C₂³×C₃	1	2	yes	—
193	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
200	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₃	1	2	yes	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	C₃×D₄	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂³×C₃	1	2	yes	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	4	1	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	#121	1	2	yes	—
217	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	4	2	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	#121	1	2	yes	—
225	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
226	C₃×C₄	C₂×#41	1	1	yes	characteristic
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
230	Q₈	—	2	1	no	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	1	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	1	no	—
235	Q₈	—	4	2	no	—
236	Q₈	C₂×C₃×D₄	1	2	yes	—
237	D₄	—	2	4	no	—
238	D₄	—	2	1	no	—
239	D₄	—	2	1	no	—
240	D₄	—	2	1	no	—
241	D₄	—	4	2	no	—
242	D₄	C₂×C₃×D₄	1	2	yes	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	1	no	—
252	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
253	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
254	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
256	C₂×C₄	C₃×#41	1	2	yes	—
257	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	4	2	no	—
263	C₂×C₃	—	8	1	no	—
264	C₂×C₃	C₂×#121	1	1	yes	characteristic
265	C₂×C₃	#577	1	1	yes	characteristic
266	C₂×C₃	#578	1	1	yes	characteristic
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	1	no	—
275	C₂²	C₃×#121	1	2	yes	—
276	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	2	no	—
283	C₄	C₃×#121	1	2	yes	—
284	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
285	C₄	C₂×C₃×#41	1	1	yes	characteristic
286	C₃	#3046	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×C₃×#121	1	1	yes	characteristic
291	C₂	C₃×#577	1	1	yes	characteristic
292	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#137	C₂	semidirect
3. C₃×#536	C₂	semidirect
4. C₃×#520	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#463	C₂	semidirect
7. C₃×#462	C₂	semidirect
8. C₃×#461	C₂	semidirect
9. C₃×#424	C₂	semidirect
10. C₃×#417	C₂	semidirect
11. C₂×C₃×#103	C₂	semidirect
12. #3046	C₃	direct (N × H)
44. C₂×#137	C₂×C₃	semidirect
46. #536	C₂×C₃	semidirect
47. #520	C₂×C₃	semidirect
48. #464	C₂×C₃	semidirect
49. #463	C₂×C₃	semidirect
50. #462	C₂×C₃	semidirect
51. #461	C₂×C₃	semidirect
52. #424	C₂×C₃	semidirect
53. #417	C₂×C₃	semidirect
54. C₂×#103	C₂×C₃	semidirect
286. C₃	#3046	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#137	2
C₂×C₃×#135	2
C₃×#536	2
C₃×#520	2
C₃×#464	2
C₃×#463	2
C₃×#462	2
C₃×#461	2
C₃×#424	2
C₃×#417	2
C₂×C₃×#103	2
#3046	3

Subgroups — C3×#3046 #84910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3046 #84910