Subgroups — C₃×#3188 #85052

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#482	C₂	1	2	yes	maximal, complemented
5	C₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#466	C₂	1	2	yes	maximal, complemented
7	C₃×#465	C₂	1	2	yes	maximal, complemented
8	C₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#449	C₂	1	2	yes	maximal, complemented
10	C₃×#447	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#3188	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×D₄	—	2	1	no	—
15	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×D₈	C₂²	1	2	yes	complemented
19	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
20	C₃×#123	—	2	1	no	—
21	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₃×#121	—	2	1	no	—
24	C₃×#121	—	2	1	no	—
25	C₃×#121	C₂²	1	1	yes	characteristic
26	C₃×#121	C₂²	1	2	yes	—
27	C₃×#121	C₂²	1	2	yes	—
28	C₃×#121	C₂²	1	1	yes	characteristic, complemented
29	C₃×#120	—	2	1	no	—
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	—	2	1	no	—
32	C₃×#117	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
35	C₃×#107	C₂²	1	2	yes	complemented
36	C₃×#106	C₂²	1	2	yes	complemented
37	C₃×#103	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	2	yes	complemented
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×#102	C₂²	1	2	yes	complemented
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
44	#525	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#121	C₂×C₃	1	1	yes	characteristic, complemented
46	#484	C₂×C₃	1	1	yes	characteristic, complemented
47	#482	C₂×C₃	1	2	yes	complemented
48	#481	C₂×C₃	1	1	yes	characteristic, complemented
49	#466	C₂×C₃	1	2	yes	complemented
50	#465	C₂×C₃	1	2	yes	complemented
51	#450	C₂×C₃	1	1	yes	characteristic, complemented
52	#449	C₂×C₃	1	2	yes	complemented
53	#447	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	—	2	1	no	—
56	C₂⁴×C₃	D₄	1	1	yes	characteristic
57	C₃×#41	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	—	2	2	no	—
64	C₂×C₃×D₄	—	2	1	no	—
65	C₂×C₃×D₄	—	4	2	no	—
66	C₂×C₃×D₄	—	4	4	no	—
67	C₂×C₃×D₄	C₂³	1	2	yes	—
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	—	4	2	no	—
72	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
73	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
78	C₃×SD₁₆	—	4	2	no	—
79	C₃×D₈	—	4	2	no	—
80	C₂×C₃×C₈	—	4	2	no	—
81	C₂×C₃×C₈	C₂³	1	2	yes	—
82	C₂×C₃×C₈	C₂³	1	2	yes	—
83	C₃×C₄⋊C₄	—	2	1	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₄⋊C₄	C₂³	1	2	yes	—
89	C₃×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₂²⋊C₄	—	2	2	no	—
93	C₃×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₂²⋊C₄	—	2	1	no	—
95	C₃×C₂²⋊C₄	—	2	1	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₂×#41	C₂²×C₃	1	1	yes	characteristic, complemented
98	C₂²×D₄	—	2	1	no	—
99	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
100	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
101	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
102	C₂×D₈	C₂²×C₃	1	2	yes	complemented
103	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
104	#123	—	2	1	no	—
105	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
106	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
107	#121	—	2	1	no	—
108	#121	—	2	1	no	—
109	#121	C₂²×C₃	1	2	yes	—
110	#121	C₂²×C₃	1	2	yes	—
111	#121	C₂²×C₃	1	1	yes	characteristic
112	#121	C₂²×C₃	1	1	yes	characteristic, complemented
113	#120	—	2	1	no	—
114	C₄×D₄	—	2	1	no	—
115	C₄×D₄	—	2	1	no	—
116	#117	C₂²×C₃	1	1	yes	characteristic, complemented
117	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	—	2	1	no	—
119	#107	C₂²×C₃	1	2	yes	complemented
120	#106	C₂²×C₃	1	2	yes	complemented
121	#103	C₂²×C₃	1	2	yes	complemented
122	#102	C₂²×C₃	1	2	yes	complemented
123	#102	C₂²×C₃	1	2	yes	complemented
124	#102	C₂²×C₃	1	2	yes	complemented
125	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
126	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
127	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	—	2	2	no	—
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	—	2	2	no	—
132	C₂³×C₃	—	4	1	no	—
133	C₂³×C₃	—	4	2	no	—
134	C₂³×C₃	—	4	2	no	—
135	C₂³×C₃	—	4	2	no	—
136	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
137	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
139	C₃×Q₈	—	2	2	no	—
140	C₃×D₄	—	2	2	no	—
141	C₃×D₄	—	2	2	no	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	—	4	2	no	—
144	C₃×D₄	—	4	4	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	—	4	2	no	—
147	C₃×D₄	—	4	2	no	—
148	C₃×D₄	—	4	4	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	1	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
165	C₃×C₈	—	2	2	no	—
166	C₃×C₈	—	2	2	no	—
167	C₂⁴	—	2	1	no	—
168	C₂⁴	C₃×D₄	1	1	yes	characteristic
169	#41	—	4	2	no	—
170	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	2	1	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	C₂³×C₃	1	2	yes	—
180	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
186	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
190	SD₁₆	—	4	2	no	—
191	D₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂³×C₃	1	2	yes	—
194	C₂×C₈	C₂³×C₃	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₃	1	2	yes	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	4	4	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	4	2	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	8	4	no	—
224	C₂²×C₃	#135	1	2	yes	—
225	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
226	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	4	1	no	—
231	C₃×C₄	—	4	1	no	—
232	C₃×C₄	—	4	2	no	—
233	C₃×C₄	—	4	1	no	—
234	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	1	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
244	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
245	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
271	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	4	1	no	—
277	C₂×C₃	—	8	1	no	—
278	C₂×C₃	—	8	2	no	—
279	C₂×C₃	#534	1	1	yes	characteristic
280	C₂×C₃	C₂×#135	1	1	yes	characteristic
281	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	8	4	no	—
297	C₂²	C₃×#135	1	2	yes	—
298	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
299	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
308	C₃	#3188	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	2	2	no	—
310	C₂	—	2	2	no	—
311	C₂	—	4	1	no	—
312	C₂	—	8	2	no	—
313	C₂	—	8	1	no	—
314	C₂	C₃×#534	1	1	yes	characteristic
315	C₂	C₂×C₃×#135	1	1	yes	characteristic
316	C₂	C₂×C₃×AGL(1,ℤ/8)	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 ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#525	C₂	semidirect
2. C₂×C₃×#121	C₂	semidirect
3. C₃×#484	C₂	semidirect
4. C₃×#482	C₂	semidirect
5. C₃×#481	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#465	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×#449	C₂	semidirect
10. C₃×#447	C₂	semidirect
11. C₂×C₃×C₂²⋊C₈	C₂	semidirect
12. #3188	C₃	direct (N × H)
13. C₂×C₃×#41	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
18. C₂×C₃×D₈	C₂²	semidirect
19. C₂²×C₃×C₈	C₂²	semidirect
22. C₃×C₂²⋊Q₈	C₂²	semidirect
28. C₃×#121	C₂²	semidirect
32. C₃×#117	C₂²	semidirect
35. C₃×#107	C₂²	semidirect
36. C₃×#106	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×#102	C₂²	semidirect
43. C₃×C₂²⋊C₈	C₂²	semidirect
44. #525	C₂×C₃	semidirect
45. C₂×#121	C₂×C₃	semidirect
46. #484	C₂×C₃	semidirect
47. #482	C₂×C₃	semidirect
48. #481	C₂×C₃	semidirect
49. #466	C₂×C₃	semidirect
50. #465	C₂×C₃	semidirect
51. #450	C₂×C₃	semidirect
52. #449	C₂×C₃	semidirect
53. #447	C₂×C₃	semidirect
54. C₂×C₂²⋊C₈	C₂×C₃	semidirect
97. C₂×#41	C₂²×C₃	semidirect
101. C₂×SD₁₆	C₂²×C₃	semidirect
102. C₂×D₈	C₂²×C₃	semidirect
103. C₂²×C₈	C₂²×C₃	semidirect
106. C₂²⋊Q₈	C₂²×C₃	semidirect
112. #121	C₂²×C₃	semidirect
116. #117	C₂²×C₃	semidirect
119. #107	C₂²×C₃	semidirect
120. #106	C₂²×C₃	semidirect
121. #103	C₂²×C₃	semidirect
122. #102	C₂²×C₃	semidirect
123. #102	C₂²×C₃	semidirect
124. #102	C₂²×C₃	semidirect
127. C₂²⋊C₈	C₂²×C₃	semidirect
308. C₃	#3188	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×#525	2
C₂×C₃×#121	2
C₃×#484	2
C₃×#482	2
C₃×#481	2
C₃×#466	2
C₃×#465	2
C₃×#450	2
C₃×#449	2
C₃×#447	2
C₂×C₃×C₂²⋊C₈	2
#3188	3

Subgroups — C3×#3188 #85052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3188 #85052