Subgroups — C₃×#3136 #85000

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#535	C₂	1	1	yes	maximal, characteristic
3	C₃×#490	C₂	1	2	yes	maximal, complemented
4	C₃×#489	C₂	1	2	yes	maximal, complemented
5	C₃×#488	C₂	1	2	yes	maximal, complemented
6	C₃×#486	C₂	1	2	yes	maximal, complemented
7	C₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	#3136	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	—	2	1	no	—
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₃×#127	C₂²	1	2	yes	—
20	C₃×#125	—	2	1	no	—
21	C₃×#125	C₂²	1	2	yes	complemented
22	C₃×#124	C₂²	1	2	yes	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×#121	—	2	1	no	—
25	C₃×#121	—	2	1	no	—
26	C₃×#120	—	2	2	no	—
27	C₃×#117	C₂²	1	1	yes	characteristic
28	C₃×#117	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×#103	C₂²	1	1	yes	characteristic, complemented
31	C₃×#103	C₂²	1	1	yes	characteristic, complemented
32	C₃×#103	C₂²	1	2	yes	complemented
33	C₃×#102	C₂²	1	2	yes	—
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	—
37	C₃×#102	C₂²	1	1	yes	characteristic, complemented
38	C₃×#102	C₂²	1	1	yes	characteristic, complemented
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×#97	C₂²	1	2	yes	complemented
41	C₃×C₄×C₈	C₂²	1	2	yes	complemented
42	#544	C₂×C₃	1	1	yes	characteristic, complemented
43	#535	C₂×C₃	1	1	yes	characteristic
44	#490	C₂×C₃	1	2	yes	complemented
45	#489	C₂×C₃	1	2	yes	complemented
46	#488	C₂×C₃	1	2	yes	complemented
47	#486	C₂×C₃	1	2	yes	complemented
48	#418	C₂×C₃	1	1	yes	characteristic, complemented
49	#417	C₂×C₃	1	1	yes	characteristic, complemented
50	#416	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×#102	C₂×C₃	1	1	yes	characteristic, complemented
52	#405	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂⁴×C₃	—	2	1	no	—
54	C₃×#41	—	2	2	no	—
55	C₃×#41	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	2	no	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	—	2	2	no	—
64	C₂×C₃×D₄	C₂³	1	2	yes	—
65	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₃×M₁₆	—	2	2	no	—
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂³	1	2	yes	—
73	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	2	2	no	—
78	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
79	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	2	yes	—
81	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	D₄	1	2	yes	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	C₂³	1	2	yes	—
89	C₂×#41	C₂²×C₃	1	1	yes	characteristic
90	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
91	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	2	1	no	—
94	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
95	#127	C₂²×C₃	1	2	yes	—
96	#125	—	2	1	no	—
97	#125	C₂²×C₃	1	2	yes	complemented
98	#124	C₂²×C₃	1	2	yes	—
99	C₂²⋊Q₈	—	2	2	no	—
100	#121	—	2	1	no	—
101	#121	—	2	1	no	—
102	#120	—	2	2	no	—
103	#117	C₂²×C₃	1	1	yes	characteristic
104	#117	C₂²×C₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
106	#103	C₂²×C₃	1	1	yes	characteristic, complemented
107	#103	C₂²×C₃	1	1	yes	characteristic, complemented
108	#103	C₂²×C₃	1	2	yes	complemented
109	#102	C₂²×C₃	1	2	yes	—
110	#102	C₂²×C₃	1	2	yes	—
111	#102	C₂²×C₃	1	2	yes	—
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	1	yes	characteristic, complemented
114	#102	C₂²×C₃	1	1	yes	characteristic, complemented
115	#102	C₂²×C₃	1	2	yes	complemented
116	#97	C₂²×C₃	1	2	yes	complemented
117	C₄×C₈	C₂²×C₃	1	2	yes	complemented
118	C₂³×C₃	—	2	2	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	4	2	no	—
121	C₂³×C₃	—	8	1	no	—
122	C₂³×C₃	—	8	1	no	—
123	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
124	C₃×Q₈	—	2	1	no	—
125	C₃×Q₈	—	2	1	no	—
126	C₃×Q₈	—	8	2	no	—
127	C₃×D₄	—	2	2	no	—
128	C₃×D₄	—	2	2	no	—
129	C₃×D₄	—	2	1	no	—
130	C₃×D₄	—	2	1	no	—
131	C₃×D₄	—	4	2	no	—
132	C₃×D₄	—	4	2	no	—
133	C₃×D₄	—	8	2	no	—
134	C₃×D₄	—	8	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	8	1	no	—
141	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
142	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
143	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	#41	1	2	yes	—
145	C₂×C₃×C₄	#41	1	2	yes	—
146	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
147	C₃×C₈	—	2	1	no	—
148	C₃×C₈	—	2	1	no	—
149	C₃×C₈	—	2	2	no	—
150	C₂⁴	—	2	1	no	—
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₂³×C₃	1	2	yes	—
162	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₃	1	2	yes	—
170	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
176	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	C₃×D₄	1	2	yes	—
184	C₄²	—	2	1	no	—
185	C₄²	C₂³×C₃	1	2	yes	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	4	1	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	—	4	1	no	—
191	C₂²×C₃	—	4	2	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	2	no	—
194	C₂²×C₃	—	8	4	no	—
195	C₂²×C₃	#124	1	2	yes	—
196	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	—	4	1	no	—
202	C₃×C₄	—	4	1	no	—
203	C₃×C₄	#124	1	2	yes	—
204	C₃×C₄	C₂×#41	1	1	yes	characteristic
205	C₃×C₄	C₂×#41	1	1	yes	characteristic
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	8	1	no	—
210	C₂³	—	8	1	no	—
211	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	2	1	no	—
214	Q₈	—	8	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	1	no	—
229	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
230	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
231	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₄	C₃×#41	1	2	yes	—
233	C₂×C₄	C₃×#41	1	2	yes	—
234	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	—	8	1	no	—
240	C₂×C₃	—	8	2	no	—
241	C₂×C₃	C₂×#124	1	1	yes	characteristic
242	C₂×C₃	#576	1	1	yes	characteristic
243	C₂×C₃	#577	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	C₃×#124	1	2	yes	—
254	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	C₃×#124	1	2	yes	—
262	C₄	C₂×C₃×#41	1	1	yes	characteristic
263	C₄	C₂×C₃×#41	1	1	yes	characteristic
264	C₃	#3136	1	1	yes	characteristic, direct product factor, Sylow
265	C₂	—	2	2	no	—
266	C₂	—	8	2	no	—
267	C₂	—	8	1	no	—
268	C₂	C₂×C₃×#124	1	1	yes	characteristic
269	C₂	C₃×#576	1	1	yes	characteristic
270	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#535	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. C₃×#544	C₂	semidirect
3. C₃×#490	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#488	C₂	semidirect
6. C₃×#486	C₂	semidirect
7. C₃×#418	C₂	semidirect
8. C₃×#417	C₂	semidirect
9. C₃×#416	C₂	semidirect
10. C₂×C₃×#102	C₂	semidirect
11. C₃×#405	C₂	semidirect
12. #3136	C₃	direct (N × H)
18. C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
30. C₃×#103	C₂²	semidirect
31. C₃×#103	C₂²	semidirect
32. C₃×#103	C₂²	semidirect
37. C₃×#102	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×#97	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
42. #544	C₂×C₃	semidirect
44. #490	C₂×C₃	semidirect
45. #489	C₂×C₃	semidirect
46. #488	C₂×C₃	semidirect
47. #486	C₂×C₃	semidirect
48. #418	C₂×C₃	semidirect
49. #417	C₂×C₃	semidirect
50. #416	C₂×C₃	semidirect
51. C₂×#102	C₂×C₃	semidirect
52. #405	C₂×C₃	semidirect
94. C₄⋊Q₈	C₂²×C₃	semidirect
97. #125	C₂²×C₃	semidirect
106. #103	C₂²×C₃	semidirect
107. #103	C₂²×C₃	semidirect
108. #103	C₂²×C₃	semidirect
113. #102	C₂²×C₃	semidirect
114. #102	C₂²×C₃	semidirect
115. #102	C₂²×C₃	semidirect
116. #97	C₂²×C₃	semidirect
117. C₄×C₈	C₂²×C₃	semidirect
264. C₃	#3136	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×#544	2
C₃×#535	2
C₃×#490	2
C₃×#489	2
C₃×#488	2
C₃×#486	2
C₃×#418	2
C₃×#417	2
C₃×#416	2
C₂×C₃×#102	2
C₃×#405	2
#3136	3

Subgroups — C3×#3136 #85000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3136 #85000