Subgroups — C₃×#3537 #87421

Order 384.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#560	C₂	1	4	yes	maximal, complemented
2	C₃×#553	C₂	1	4	yes	maximal, complemented
3	C₃×D₄²	C₂	1	4	yes	maximal, complemented
4	C₃×D₄²	C₂	1	4	yes	maximal, complemented
5	C₃×#540	C₂	1	2	yes	maximal, complemented
6	C₃×#538	C₂	1	4	yes	maximal, complemented
7	C₃×#535	C₂	1	2	yes	maximal, complemented
8	C₃×#534	C₂	1	4	yes	maximal, complemented
9	C₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×#121	C₂	1	2	yes	maximal, complemented
11	#3537	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂×C₃×#41	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
14	C₂²×C₃×D₄	C₂²	1	4	yes	complemented
15	C₂²×C₃×D₄	C₂²	1	8	yes	complemented
16	C₂²×C₃×D₄	C₂²	1	2	yes	complemented
17	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₃×#127	C₂²	1	4	yes	complemented
19	C₃×#127	C₂²	1	2	yes	complemented
20	C₃×#126	C₂²	1	8	yes	complemented
21	C₃×#125	C₂²	1	2	yes	complemented
22	C₃×#124	C₂²	1	4	yes	complemented
23	C₃×#123	C₂²	1	2	yes	complemented
24	C₃×#123	C₂²	1	8	yes	complemented
25	C₃×#123	C₂²	1	8	yes	complemented
26	C₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₃×#121	C₂²	1	4	yes	complemented
28	C₃×#121	C₂²	1	8	yes	complemented
29	C₃×#121	C₂²	1	8	yes	complemented
30	C₃×#121	C₂²	1	4	yes	complemented
31	C₃×#121	C₂²	1	4	yes	complemented
32	C₃×#121	C₂²	1	4	yes	complemented
33	C₃×#121	C₂²	1	2	yes	complemented
34	C₃×#121	C₂²	1	8	yes	complemented
35	C₃×#121	C₂²	1	4	yes	complemented
36	C₃×#120	C₂²	1	8	yes	complemented
37	C₃×#120	C₂²	1	4	yes	complemented
38	C₃×#120	C₂²	1	2	yes	complemented
39	C₃×#120	C₂²	1	4	yes	complemented
40	C₃×#120	C₂²	1	8	yes	complemented
41	C₃×C₄×D₄	C₂²	1	4	yes	complemented
42	C₃×C₄×D₄	C₂²	1	8	yes	complemented
43	C₃×C₄×D₄	C₂²	1	4	yes	complemented
44	C₃×#117	C₂²	1	1	yes	characteristic, complemented
45	C₃×#117	C₂²	1	2	yes	complemented
46	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
47	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
48	#560	C₂×C₃	1	4	yes	complemented
49	#553	C₂×C₃	1	4	yes	complemented
50	D₄²	C₂×C₃	1	4	yes	complemented
51	D₄²	C₂×C₃	1	4	yes	complemented
52	#540	C₂×C₃	1	2	yes	complemented
53	#538	C₂×C₃	1	4	yes	complemented
54	#535	C₂×C₃	1	2	yes	complemented
55	#534	C₂×C₃	1	4	yes	complemented
56	#525	C₂×C₃	1	1	yes	characteristic, complemented
57	C₂×#121	C₂×C₃	1	2	yes	complemented
58	C₂⁴×C₃	C₂³	1	1	yes	characteristic
59	C₂⁴×C₃	C₂³	1	4	yes	—
60	C₂⁴×C₃	C₂³	1	4	yes	—
61	C₃×#41	—	4	2	no	—
62	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×D₄	—	4	4	no	—
64	C₂×C₃×D₄	—	4	4	no	—
65	C₂×C₃×D₄	—	4	8	no	—
66	C₂×C₃×D₄	—	4	16	no	—
67	C₂×C₃×D₄	C₂³	1	4	yes	—
68	C₂×C₃×D₄	C₂³	1	8	yes	—
69	C₂×C₃×D₄	C₂³	1	4	yes	—
70	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
71	C₂×C₃×D₄	C₂³	1	2	yes	—
72	C₂×C₃×D₄	C₂³	1	4	yes	—
73	C₂×C₃×D₄	C₂³	1	8	yes	—
74	C₂×C₃×D₄	C₂³	1	8	yes	—
75	C₂×C₃×D₄	C₂³	1	4	yes	—
76	C₂×C₃×D₄	C₂³	1	8	yes	—
77	C₂²×C₃×C₄	—	4	2	no	—
78	C₂²×C₃×C₄	C₂³	1	2	yes	—
79	C₂²×C₃×C₄	C₂³	1	2	yes	—
80	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³	1	2	yes	—
82	C₂²×C₃×C₄	C₂³	1	4	yes	—
83	C₂²×C₃×C₄	C₂³	1	8	yes	—
84	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
85	C₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₃×C₄⋊C₄	C₂³	1	4	yes	—
87	C₃×C₄⋊C₄	C₂³	1	2	yes	—
88	C₃×C₄⋊C₄	C₂³	1	4	yes	—
89	C₃×C₄⋊C₄	C₂³	1	8	yes	—
90	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
91	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
93	C₃×C₂²⋊C₄	C₂³	1	2	yes	—
94	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
96	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
97	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
98	C₃×C₄²	C₂³	1	2	yes	—
99	C₃×C₄²	C₂³	1	4	yes	—
100	C₂×#41	C₂²×C₃	1	1	yes	characteristic, complemented
101	C₂²×D₄	C₂²×C₃	1	2	yes	complemented
102	C₂²×D₄	C₂²×C₃	1	4	yes	complemented
103	C₂²×D₄	C₂²×C₃	1	8	yes	complemented
104	C₂²×D₄	C₂²×C₃	1	2	yes	complemented
105	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
106	#127	C₂²×C₃	1	4	yes	complemented
107	#127	C₂²×C₃	1	2	yes	complemented
108	#126	C₂²×C₃	1	8	yes	complemented
109	#125	C₂²×C₃	1	2	yes	complemented
110	#124	C₂²×C₃	1	4	yes	complemented
111	#123	C₂²×C₃	1	2	yes	complemented
112	#123	C₂²×C₃	1	8	yes	complemented
113	#123	C₂²×C₃	1	8	yes	complemented
114	C₂²⋊Q₈	C₂²×C₃	1	2	yes	complemented
115	#121	C₂²×C₃	1	4	yes	complemented
116	#121	C₂²×C₃	1	4	yes	complemented
117	#121	C₂²×C₃	1	4	yes	complemented
118	#121	C₂²×C₃	1	4	yes	complemented
119	#121	C₂²×C₃	1	2	yes	complemented
120	#121	C₂²×C₃	1	8	yes	complemented
121	#121	C₂²×C₃	1	4	yes	complemented
122	#121	C₂²×C₃	1	8	yes	complemented
123	#121	C₂²×C₃	1	8	yes	complemented
124	#120	C₂²×C₃	1	2	yes	complemented
125	#120	C₂²×C₃	1	4	yes	complemented
126	#120	C₂²×C₃	1	8	yes	complemented
127	#120	C₂²×C₃	1	4	yes	complemented
128	#120	C₂²×C₃	1	8	yes	complemented
129	C₄×D₄	C₂²×C₃	1	4	yes	complemented
130	C₄×D₄	C₂²×C₃	1	8	yes	complemented
131	C₄×D₄	C₂²×C₃	1	4	yes	complemented
132	#117	C₂²×C₃	1	1	yes	characteristic, complemented
133	#117	C₂²×C₃	1	2	yes	complemented
134	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
135	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	complemented
136	C₂³×C₃	—	2	4	no	—
137	C₂³×C₃	—	2	8	no	—
138	C₂³×C₃	—	4	1	no	—
139	C₂³×C₃	—	4	4	no	—
140	C₂³×C₃	—	4	4	no	—
141	C₂³×C₃	—	4	4	no	—
142	C₂³×C₃	—	4	8	no	—
143	C₂³×C₃	C₂⁴	1	2	yes	—
144	C₂³×C₃	C₂⁴	1	8	yes	—
145	C₂³×C₃	C₂⁴	1	1	yes	characteristic
146	C₂³×C₃	C₂⁴	1	2	yes	—
147	C₃×Q₈	—	4	1	no	—
148	C₃×D₄	—	2	16	no	—
149	C₃×D₄	—	2	8	no	—
150	C₃×D₄	—	2	8	no	—
151	C₃×D₄	—	4	4	no	—
152	C₃×D₄	—	4	4	no	—
153	C₃×D₄	—	4	8	no	—
154	C₃×D₄	—	4	8	no	—
155	C₃×D₄	—	4	8	no	—
156	C₃×D₄	—	4	1	no	—
157	C₃×D₄	—	4	2	no	—
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	4	no	—
164	C₂×C₃×C₄	—	4	8	no	—
165	C₂×C₃×C₄	C₂⁴	1	2	yes	—
166	C₂×C₃×C₄	C₂⁴	1	4	yes	—
167	C₂×C₃×C₄	C₂⁴	1	8	yes	—
168	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂⁴	1	2	yes	—
170	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
171	C₂⁴	C₂³×C₃	1	1	yes	characteristic
172	C₂⁴	C₂³×C₃	1	4	yes	—
173	C₂⁴	C₂³×C₃	1	4	yes	—
174	#41	—	4	2	no	—
175	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	16	no	—
180	C₂×D₄	C₂³×C₃	1	4	yes	—
181	C₂×D₄	C₂³×C₃	1	8	yes	—
182	C₂×D₄	C₂³×C₃	1	4	yes	—
183	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
184	C₂×D₄	C₂³×C₃	1	2	yes	—
185	C₂×D₄	C₂³×C₃	1	4	yes	—
186	C₂×D₄	C₂³×C₃	1	8	yes	—
187	C₂×D₄	C₂³×C₃	1	8	yes	—
188	C₂×D₄	C₂³×C₃	1	4	yes	—
189	C₂×D₄	C₂³×C₃	1	8	yes	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂³×C₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₃	1	2	yes	—
193	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₃	1	4	yes	—
196	C₂²×C₄	C₂³×C₃	1	8	yes	—
197	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₃	1	4	yes	—
199	C₄⋊C₄	C₂³×C₃	1	4	yes	—
200	C₄⋊C₄	C₂³×C₃	1	2	yes	—
201	C₄⋊C₄	C₂³×C₃	1	4	yes	—
202	C₄⋊C₄	C₂³×C₃	1	8	yes	—
203	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
204	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
205	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
206	C₂²⋊C₄	C₂³×C₃	1	2	yes	—
207	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
208	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
209	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
210	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
211	C₄²	C₂³×C₃	1	2	yes	—
212	C₄²	C₂³×C₃	1	4	yes	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	8	no	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	8	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	—	8	8	no	—
224	C₂²×C₃	—	8	8	no	—
225	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
226	C₂²×C₃	C₂⁵	1	1	yes	characteristic
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	8	no	—
231	C₃×C₄	—	2	4	no	—
232	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	4	1	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	8	no	—
240	C₂³	C₂⁴×C₃	1	2	yes	—
241	C₂³	C₂⁴×C₃	1	8	yes	—
242	C₂³	C₂⁴×C₃	1	1	yes	characteristic
243	C₂³	C₂⁴×C₃	1	2	yes	—
244	Q₈	—	4	1	no	—
245	D₄	—	2	16	no	—
246	D₄	—	2	8	no	—
247	D₄	—	2	8	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	8	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₂⁴×C₃	1	2	yes	—
263	C₂×C₄	C₂⁴×C₃	1	4	yes	—
264	C₂×C₄	C₂⁴×C₃	1	8	yes	—
265	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
266	C₂×C₄	C₂⁴×C₃	1	2	yes	—
267	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	—	4	2	no	—
271	C₂×C₃	—	4	8	no	—
272	C₂×C₃	C₂×H(5,𝔽₂)	1	2	yes	—
273	C₂×C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	8	8	no	—
285	C₂²	—	8	8	no	—
286	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
287	C₂²	C₂⁵×C₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	8	no	—
292	C₄	—	2	4	no	—
293	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
294	C₃	#3537	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	8	no	—
299	C₂	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
300	C₂	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#560	C₂	semidirect
2. C₃×#553	C₂	semidirect
3. C₃×D₄²	C₂	semidirect
4. C₃×D₄²	C₂	semidirect
5. C₃×#540	C₂	semidirect
6. C₃×#538	C₂	semidirect
7. C₃×#535	C₂	semidirect
8. C₃×#534	C₂	semidirect
9. C₃×#525	C₂	semidirect
10. C₂×C₃×#121	C₂	semidirect
11. #3537	C₃	direct (N × H)
12. C₂×C₃×#41	C₂²	semidirect
13. C₂²×C₃×D₄	C₂²	semidirect
14. C₂²×C₃×D₄	C₂²	semidirect
15. C₂²×C₃×D₄	C₂²	semidirect
16. C₂²×C₃×D₄	C₂²	semidirect
17. C₂³×C₃×C₄	C₂²	semidirect
18. C₃×#127	C₂²	semidirect
19. C₃×#127	C₂²	semidirect
20. C₃×#126	C₂²	semidirect
21. C₃×#125	C₂²	semidirect
22. C₃×#124	C₂²	semidirect
23. C₃×#123	C₂²	semidirect
24. C₃×#123	C₂²	semidirect
25. C₃×#123	C₂²	semidirect
26. C₃×C₂²⋊Q₈	C₂²	semidirect
27. C₃×#121	C₂²	semidirect
28. C₃×#121	C₂²	semidirect
29. C₃×#121	C₂²	semidirect
30. C₃×#121	C₂²	semidirect
31. C₃×#121	C₂²	semidirect
32. C₃×#121	C₂²	semidirect
33. C₃×#121	C₂²	semidirect
34. C₃×#121	C₂²	semidirect
35. C₃×#121	C₂²	semidirect
36. C₃×#120	C₂²	semidirect
37. C₃×#120	C₂²	semidirect
38. C₃×#120	C₂²	semidirect
39. C₃×#120	C₂²	semidirect
40. C₃×#120	C₂²	semidirect
41. C₃×C₄×D₄	C₂²	semidirect
42. C₃×C₄×D₄	C₂²	semidirect
43. C₃×C₄×D₄	C₂²	semidirect
44. C₃×#117	C₂²	semidirect
45. C₃×#117	C₂²	semidirect
46. C₂×C₃×C₄⋊C₄	C₂²	semidirect
47. C₂×C₃×C₂²⋊C₄	C₂²	semidirect
48. #560	C₂×C₃	semidirect
49. #553	C₂×C₃	semidirect
50. D₄²	C₂×C₃	semidirect
51. D₄²	C₂×C₃	semidirect
52. #540	C₂×C₃	semidirect
53. #538	C₂×C₃	semidirect
54. #535	C₂×C₃	semidirect
55. #534	C₂×C₃	semidirect
56. #525	C₂×C₃	semidirect
57. C₂×#121	C₂×C₃	semidirect
100. C₂×#41	C₂²×C₃	semidirect
101. C₂²×D₄	C₂²×C₃	semidirect
102. C₂²×D₄	C₂²×C₃	semidirect
103. C₂²×D₄	C₂²×C₃	semidirect
104. C₂²×D₄	C₂²×C₃	semidirect
105. C₂³×C₄	C₂²×C₃	semidirect
106. #127	C₂²×C₃	semidirect
107. #127	C₂²×C₃	semidirect
108. #126	C₂²×C₃	semidirect
109. #125	C₂²×C₃	semidirect
110. #124	C₂²×C₃	semidirect
111. #123	C₂²×C₃	semidirect
112. #123	C₂²×C₃	semidirect
113. #123	C₂²×C₃	semidirect
114. C₂²⋊Q₈	C₂²×C₃	semidirect
115. #121	C₂²×C₃	semidirect
116. #121	C₂²×C₃	semidirect
117. #121	C₂²×C₃	semidirect
118. #121	C₂²×C₃	semidirect
119. #121	C₂²×C₃	semidirect
120. #121	C₂²×C₃	semidirect
121. #121	C₂²×C₃	semidirect
122. #121	C₂²×C₃	semidirect
123. #121	C₂²×C₃	semidirect
124. #120	C₂²×C₃	semidirect
125. #120	C₂²×C₃	semidirect
126. #120	C₂²×C₃	semidirect
127. #120	C₂²×C₃	semidirect
128. #120	C₂²×C₃	semidirect
129. C₄×D₄	C₂²×C₃	semidirect
130. C₄×D₄	C₂²×C₃	semidirect
131. C₄×D₄	C₂²×C₃	semidirect
132. #117	C₂²×C₃	semidirect
133. #117	C₂²×C₃	semidirect
134. C₂×C₄⋊C₄	C₂²×C₃	semidirect
135. C₂×C₂²⋊C₄	C₂²×C₃	semidirect
294. C₃	#3537	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×#560	2
C₃×#553	2
C₃×D₄²	2
C₃×#540	2
C₃×#538	2
C₃×#535	2
C₃×#534	2
C₃×#525	2
C₂×C₃×#121	2
#3537	3

Subgroups — C3×#3537 #87421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3537 #87421