Subgroups — C₃×#3154 #85018

Order 384.

564 conjugacy classes of subgroups.

The subgroup lattice has 298 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×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#497	C₂	1	2	yes	maximal, complemented
8	C₃×#496	C₂	1	2	yes	maximal, complemented
9	C₃×#495	C₂	1	2	yes	maximal, complemented
10	C₃×#492	C₂	1	2	yes	maximal, complemented
11	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	#3154	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	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₃×#137	—	2	4	no	—
18	C₃×AGL(1,ℤ/8)	—	2	4	no	—
19	C₃×#135	—	2	4	no	—
20	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
22	C₂×C₃×SD₁₆	—	2	4	no	—
23	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
24	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
25	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
26	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
29	C₂×C₃×D₈	C₂²	1	2	yes	—
30	C₂×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
33	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
34	C₃×#127	C₂²	1	2	yes	—
35	C₃×#124	C₂²	1	2	yes	—
36	C₃×#124	C₂²	1	2	yes	complemented
37	C₃×#123	—	2	2	no	—
38	C₃×C₂²⋊Q₈	—	2	2	no	—
39	C₃×#121	—	2	2	no	—
40	C₃×#120	—	2	2	no	—
41	C₃×#117	C₂²	1	1	yes	characteristic
42	C₃×#97	C₂²	1	2	yes	complemented
43	C₃×C₄×C₈	C₂²	1	2	yes	complemented
44	C₂×#137	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#135	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂²×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
48	#536	C₂×C₃	1	1	yes	characteristic, complemented
49	#535	C₂×C₃	1	1	yes	characteristic, complemented
50	#497	C₂×C₃	1	2	yes	complemented
51	#496	C₂×C₃	1	2	yes	complemented
52	#495	C₂×C₃	1	2	yes	complemented
53	#492	C₂×C₃	1	2	yes	complemented
54	#405	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	—	2	1	no	—
56	C₃×#41	—	2	2	no	—
57	C₃×#41	—	2	2	no	—
58	C₃×#41	—	2	2	no	—
59	C₃×#41	—	2	2	no	—
60	C₂×C₃×Q₈	—	2	1	no	—
61	C₂×C₃×Q₈	—	2	2	no	—
62	C₂×C₃×Q₈	—	2	2	no	—
63	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×Q₈	C₂³	1	2	yes	—
65	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×D₄	—	2	1	no	—
67	C₂×C₃×D₄	—	2	1	no	—
68	C₂×C₃×D₄	—	2	2	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	2	1	no	—
71	C₂×C₃×D₄	—	2	4	no	—
72	C₂×C₃×D₄	C₂³	1	2	yes	—
73	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
74	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
75	C₂²×C₃×C₄	—	2	1	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
79	C₃×Q₁₆	—	2	2	no	—
80	C₃×Q₁₆	—	2	4	no	—
81	C₃×SD₁₆	—	2	4	no	—
82	C₃×SD₁₆	—	2	4	no	—
83	C₃×SD₁₆	—	2	4	no	—
84	C₃×SD₁₆	—	2	2	no	—
85	C₃×SD₁₆	—	2	2	no	—
86	C₃×SD₁₆	—	2	4	no	—
87	C₃×D₈	—	2	4	no	—
88	C₃×D₈	—	2	2	no	—
89	C₃×M₁₆	D₄	1	4	yes	complemented
90	C₂×C₃×C₈	D₄	1	4	yes	complemented
91	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂³	1	2	yes	—
93	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	D₄	1	2	yes	—
96	C₃×C₂²⋊C₄	—	2	4	no	—
97	C₃×C₂²⋊C₄	—	2	4	no	—
98	C₃×C₂²⋊C₄	D₄	1	2	yes	—
99	C₃×C₄²	C₂³	1	2	yes	—
100	C₂×#41	C₂²×C₃	1	1	yes	characteristic
101	C₂×#41	C₂²×C₃	1	1	yes	characteristic
102	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
103	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
104	#137	—	2	4	no	—
105	AGL(1,ℤ/8)	—	2	4	no	—
106	#135	—	2	4	no	—
107	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
108	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
109	C₂×SD₁₆	—	2	4	no	—
110	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
111	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
112	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
113	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
114	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
115	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
116	C₂×D₈	C₂²×C₃	1	2	yes	—
117	C₂×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
120	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
121	#127	C₂²×C₃	1	2	yes	—
122	#124	C₂²×C₃	1	2	yes	—
123	#124	C₂²×C₃	1	2	yes	complemented
124	#123	—	2	2	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	#121	—	2	2	no	—
127	#120	—	2	2	no	—
128	#117	C₂²×C₃	1	1	yes	characteristic
129	#97	C₂²×C₃	1	2	yes	complemented
130	C₄×C₈	C₂²×C₃	1	2	yes	complemented
131	C₂³×C₃	—	2	1	no	—
132	C₂³×C₃	—	2	2	no	—
133	C₂³×C₃	—	2	1	no	—
134	C₂³×C₃	—	4	2	no	—
135	C₂³×C₃	—	8	2	no	—
136	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	1	no	—
139	C₃×Q₈	—	2	2	no	—
140	C₃×Q₈	—	2	1	no	—
141	C₃×Q₈	—	2	1	no	—
142	C₃×Q₈	—	2	1	no	—
143	C₃×Q₈	—	4	2	no	—
144	C₃×D₄	—	2	1	no	—
145	C₃×D₄	—	2	2	no	—
146	C₃×D₄	—	2	1	no	—
147	C₃×D₄	—	2	1	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	2	1	no	—
150	C₃×D₄	—	4	2	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	4	2	no	—
153	C₃×D₄	—	8	4	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	4	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
161	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
163	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
164	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
165	C₃×C₈	C₂×D₄	1	2	yes	—
166	C₃×C₈	C₂×D₄	1	2	yes	—
167	C₃×C₈	C₂×D₄	1	4	yes	—
168	C₂⁴	—	2	1	no	—
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
177	C₂×Q₈	C₂³×C₃	1	2	yes	—
178	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	C₂³×C₃	1	2	yes	—
186	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
192	Q₁₆	—	2	2	no	—
193	Q₁₆	—	2	4	no	—
194	SD₁₆	—	2	4	no	—
195	SD₁₆	—	2	4	no	—
196	SD₁₆	—	2	4	no	—
197	SD₁₆	—	2	2	no	—
198	SD₁₆	—	2	2	no	—
199	SD₁₆	—	2	4	no	—
200	D₈	—	2	4	no	—
201	D₈	—	2	2	no	—
202	M₁₆	C₃×D₄	1	4	yes	complemented
203	C₂×C₈	C₃×D₄	1	4	yes	complemented
204	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
205	C₂×C₈	C₂³×C₃	1	2	yes	—
206	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	C₃×D₄	1	2	yes	—
209	C₂²⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	C₃×D₄	1	2	yes	—
212	C₄²	C₂³×C₃	1	2	yes	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	2	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	4	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	8	4	no	—
221	C₂²×C₃	#127	1	2	yes	—
222	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	#127	1	2	yes	—
228	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
229	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	2	2	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	1	no	—
241	Q₈	—	2	1	no	—
242	Q₈	—	4	2	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	1	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
260	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
262	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
263	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
264	C₈	C₂×C₃×D₄	1	2	yes	—
265	C₈	C₂×C₃×D₄	1	2	yes	—
266	C₈	C₂×C₃×D₄	1	4	yes	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	8	1	no	—
269	C₂×C₃	—	8	1	no	—
270	C₂×C₃	—	8	2	no	—
271	C₂×C₃	C₂×#127	1	1	yes	characteristic
272	C₂×C₃	#577	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	C₃×#127	1	2	yes	—
282	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	2	no	—
287	C₄	C₃×#127	1	2	yes	—
288	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
289	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
290	C₃	#3154	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	—	8	1	no	—
294	C₂	—	8	1	no	—
295	C₂	C₂×C₃×#127	1	1	yes	characteristic
296	C₂	C₃×#577	1	2	yes	—

	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 ◁ 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
2. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₃×#135	C₂	semidirect
4. C₂²×C₃×SD₁₆	C₂	semidirect
5. C₃×#536	C₂	semidirect
6. C₃×#535	C₂	semidirect
7. C₃×#497	C₂	semidirect
8. C₃×#496	C₂	semidirect
9. C₃×#495	C₂	semidirect
10. C₃×#492	C₂	semidirect
11. C₃×#405	C₂	semidirect
12. #3154	C₃	direct (N × H)
20. C₂×C₃×Q₁₆	C₂²	semidirect
21. C₂×C₃×Q₁₆	C₂²	semidirect
26. C₂×C₃×SD₁₆	C₂²	semidirect
27. C₂×C₃×SD₁₆	C₂²	semidirect
28. C₂×C₃×SD₁₆	C₂²	semidirect
30. C₂×C₃×D₈	C₂²	semidirect
33. C₃×C₄⋊Q₈	C₂²	semidirect
36. C₃×#124	C₂²	semidirect
42. C₃×#97	C₂²	semidirect
43. C₃×C₄×C₈	C₂²	semidirect
44. C₂×#137	C₂×C₃	semidirect
45. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
46. C₂×#135	C₂×C₃	semidirect
47. C₂²×SD₁₆	C₂×C₃	semidirect
48. #536	C₂×C₃	semidirect
49. #535	C₂×C₃	semidirect
50. #497	C₂×C₃	semidirect
51. #496	C₂×C₃	semidirect
52. #495	C₂×C₃	semidirect
53. #492	C₂×C₃	semidirect
54. #405	C₂×C₃	semidirect
89. C₃×M₁₆	D₄	semidirect
90. C₂×C₃×C₈	D₄	semidirect
107. C₂×Q₁₆	C₂²×C₃	semidirect
108. C₂×Q₁₆	C₂²×C₃	semidirect
113. C₂×SD₁₆	C₂²×C₃	semidirect
114. C₂×SD₁₆	C₂²×C₃	semidirect
115. C₂×SD₁₆	C₂²×C₃	semidirect
117. C₂×D₈	C₂²×C₃	semidirect
120. C₄⋊Q₈	C₂²×C₃	semidirect
123. #124	C₂²×C₃	semidirect
129. #97	C₂²×C₃	semidirect
130. C₄×C₈	C₂²×C₃	semidirect
202. M₁₆	C₃×D₄	semidirect
203. C₂×C₈	C₃×D₄	semidirect
290. C₃	#3154	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×AGL(1,ℤ/8)	2
C₂×C₃×#135	2
C₂²×C₃×SD₁₆	2
C₃×#536	2
C₃×#535	2
C₃×#497	2
C₃×#496	2
C₃×#495	2
C₃×#492	2
C₃×#405	2
#3154	3

Subgroups — C3×#3154 #85018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3154 #85018