Subgroups — C₃×#3009 #84873

Order 384.

752 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×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#452	C₂	1	4	yes	maximal, complemented
6	C₃×#450	C₂	1	4	yes	maximal, complemented
7	C₃×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	#3009	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×H(5,𝔽₂)	—	2	4	no	—
12	C₃×H(5,𝔽₂)	—	2	4	no	—
13	C₂×C₃×#41	—	2	1	no	—
14	C₂×C₃×#41	C₂²	1	2	yes	—
15	C₂×C₃×#41	C₂²	1	1	yes	characteristic
16	C₂×C₃×#41	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₃×D₄	—	2	2	no	—
19	C₂²×C₃×D₄	—	2	1	no	—
20	C₂²×C₃×D₄	C₂²	1	2	yes	—
21	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
22	C₃×#137	—	2	4	no	—
23	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
24	C₂×C₃×SD₁₆	—	2	4	no	—
25	C₂×C₃×SD₁₆	C₂²	1	4	yes	complemented
26	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
29	C₃×C₄⋊Q₈	—	2	1	no	—
30	C₃×#124	—	2	1	no	—
31	C₃×#123	—	2	2	no	—
32	C₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
33	C₃×#117	C₂²	1	1	yes	characteristic, complemented
34	C₃×#103	C₂²	1	2	yes	complemented
35	C₃×#102	C₂²	1	2	yes	complemented
36	C₃×#102	C₂²	1	4	yes	complemented
37	C₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
38	C₂×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic
39	C₂×#137	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂²×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
41	#536	C₂×C₃	1	1	yes	characteristic, complemented
42	#452	C₂×C₃	1	4	yes	complemented
43	#450	C₂×C₃	1	4	yes	complemented
44	#418	C₂×C₃	1	1	yes	characteristic, complemented
45	#416	C₂×C₃	1	1	yes	characteristic, complemented
46	#408	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂⁴×C₃	—	2	2	no	—
48	C₂⁴×C₃	—	2	1	no	—
49	C₃×#41	—	2	8	no	—
50	C₃×#41	—	2	4	no	—
51	C₃×#41	—	2	4	no	—
52	C₃×#41	—	2	2	no	—
53	C₃×#41	—	2	2	no	—
54	C₃×#41	—	2	2	no	—
55	C₃×#41	D₄	1	4	yes	—
56	C₂×C₃×Q₈	—	2	1	no	—
57	C₂×C₃×Q₈	—	2	2	no	—
58	C₂×C₃×Q₈	—	2	2	no	—
59	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
60	C₂×C₃×Q₈	C₂³	1	2	yes	—
61	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×D₄	—	2	16	no	—
63	C₂×C₃×D₄	—	2	8	no	—
64	C₂×C₃×D₄	—	2	2	no	—
65	C₂×C₃×D₄	—	2	1	no	—
66	C₂×C₃×D₄	—	2	4	no	—
67	C₂×C₃×D₄	—	2	4	no	—
68	C₂×C₃×D₄	—	2	2	no	—
69	C₂×C₃×D₄	—	2	1	no	—
70	C₂×C₃×D₄	—	2	2	no	—
71	C₂×C₃×D₄	D₄	1	2	yes	—
72	C₂×C₃×D₄	D₄	1	1	yes	characteristic
73	C₂×C₃×D₄	D₄	1	4	yes	—
74	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
75	C₂×C₃×D₄	C₂³	1	2	yes	—
76	C₂²×C₃×C₄	—	2	2	no	—
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	C₂³	1	2	yes	—
80	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
81	C₃×Q₁₆	—	2	4	no	—
82	C₃×SD₁₆	—	2	8	no	—
83	C₃×SD₁₆	—	2	4	no	—
84	C₃×M₁₆	—	2	2	no	—
85	C₂×C₃×C₈	—	2	2	no	—
86	C₂×C₃×C₈	C₂³	1	2	yes	—
87	C₂×C₃×C₈	C₂³	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	C₂³	1	2	yes	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	—	2	4	no	—
92	C₃×C₄²	—	2	1	no	—
93	H(5,𝔽₂)	—	2	4	no	—
94	H(5,𝔽₂)	—	2	4	no	—
95	C₂×#41	—	2	1	no	—
96	C₂×#41	C₂²×C₃	1	2	yes	—
97	C₂×#41	C₂²×C₃	1	1	yes	characteristic
98	C₂×#41	C₂²×C₃	1	1	yes	characteristic
99	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
100	C₂²×D₄	—	2	2	no	—
101	C₂²×D₄	—	2	1	no	—
102	C₂²×D₄	C₂²×C₃	1	2	yes	—
103	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
104	#137	—	2	4	no	—
105	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
106	C₂×SD₁₆	—	2	4	no	—
107	C₂×SD₁₆	C₂²×C₃	1	4	yes	complemented
108	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
109	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
111	C₄⋊Q₈	—	2	1	no	—
112	#124	—	2	1	no	—
113	#123	—	2	2	no	—
114	C₂²⋊Q₈	C₂²×C₃	1	4	yes	complemented
115	#117	C₂²×C₃	1	1	yes	characteristic, complemented
116	#103	C₂²×C₃	1	2	yes	complemented
117	#102	C₂²×C₃	1	2	yes	complemented
118	#102	C₂²×C₃	1	4	yes	complemented
119	C₂²⋊C₈	C₂²×C₃	1	4	yes	complemented
120	C₂³×C₃	—	2	8	no	—
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	—	2	1	no	—
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	8	2	no	—
125	C₂³×C₃	—	8	4	no	—
126	C₂³×C₃	C₂×D₄	1	2	yes	—
127	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	2	1	no	—
130	C₃×Q₈	—	2	2	no	—
131	C₃×Q₈	—	2	2	no	—
132	C₃×Q₈	—	4	2	no	—
133	C₃×Q₈	C₂×D₄	1	2	yes	—
134	C₃×D₄	—	2	8	no	—
135	C₃×D₄	—	2	8	no	—
136	C₃×D₄	—	2	4	no	—
137	C₃×D₄	—	2	4	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	2	4	no	—
140	C₃×D₄	—	2	2	no	—
141	C₃×D₄	—	2	1	no	—
142	C₃×D₄	C₂×D₄	1	2	yes	—
143	C₃×D₄	C₂×D₄	1	4	yes	—
144	C₂×C₃×C₄	—	2	8	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
153	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₂⁴	—	2	2	no	—
160	C₂⁴	—	2	1	no	—
161	#41	—	2	8	no	—
162	#41	—	2	4	no	—
163	#41	—	2	4	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	C₃×D₄	1	1	yes	characteristic
172	C₂×Q₈	C₂³×C₃	1	2	yes	—
173	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
174	C₂×D₄	—	2	16	no	—
175	C₂×D₄	—	2	8	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	C₃×D₄	1	2	yes	—
184	C₂×D₄	C₃×D₄	1	1	yes	characteristic
185	C₂×D₄	C₃×D₄	1	4	yes	—
186	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₃	1	2	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	C₂³×C₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	SD₁₆	—	2	8	no	—
195	SD₁₆	—	2	4	no	—
196	M₁₆	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂³×C₃	1	2	yes	—
199	C₂×C₈	C₂³×C₃	1	2	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	C₂³×C₃	1	2	yes	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	1	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	2	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	4	4	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	8	4	no	—
213	C₂²×C₃	—	8	8	no	—
214	C₂²×C₃	#120	1	2	yes	—
215	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	#120	1	2	yes	—
221	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
222	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
223	C₂³	—	2	8	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	4	no	—
229	C₂³	C₂×C₃×D₄	1	2	yes	—
230	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	—	2	1	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	4	2	no	—
236	Q₈	C₂×C₃×D₄	1	2	yes	—
237	D₄	—	2	8	no	—
238	D₄	—	2	8	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	1	no	—
245	D₄	C₂×C₃×D₄	1	2	yes	—
246	D₄	C₂×C₃×D₄	1	4	yes	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
256	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
258	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
259	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	4	2	no	—
264	C₂×C₃	—	4	4	no	—
265	C₂×C₃	—	4	2	no	—
266	C₂×C₃	C₂×#120	1	1	yes	characteristic
267	C₂×C₃	#577	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	8	no	—
277	C₂²	C₃×#120	1	2	yes	—
278	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	2	no	—
283	C₄	C₃×#120	1	2	yes	—
284	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
285	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
286	C₃	#3009	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	4	2	no	—
289	C₂	—	4	4	no	—
290	C₂	—	4	2	no	—
291	C₂	C₂×C₃×#120	1	1	yes	characteristic
292	C₂	C₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#137	C₂	semidirect
3. C₂²×C₃×SD₁₆	C₂	semidirect
4. C₃×#536	C₂	semidirect
5. C₃×#452	C₂	semidirect
6. C₃×#450	C₂	semidirect
7. C₃×#418	C₂	semidirect
8. C₃×#416	C₂	semidirect
9. C₃×#408	C₂	semidirect
10. #3009	C₃	direct (N × H)
17. C₂²×C₃×Q₈	C₂²	semidirect
23. C₂×C₃×Q₁₆	C₂²	semidirect
25. C₂×C₃×SD₁₆	C₂²	semidirect
26. C₂×C₃×SD₁₆	C₂²	semidirect
27. C₂×C₃×M₁₆	C₂²	semidirect
28. C₂²×C₃×C₈	C₂²	semidirect
32. C₃×C₂²⋊Q₈	C₂²	semidirect
33. C₃×#117	C₂²	semidirect
34. C₃×#103	C₂²	semidirect
35. C₃×#102	C₂²	semidirect
36. C₃×#102	C₂²	semidirect
37. C₃×C₂²⋊C₈	C₂²	semidirect
39. C₂×#137	C₂×C₃	semidirect
40. C₂²×SD₁₆	C₂×C₃	semidirect
41. #536	C₂×C₃	semidirect
42. #452	C₂×C₃	semidirect
43. #450	C₂×C₃	semidirect
44. #418	C₂×C₃	semidirect
45. #416	C₂×C₃	semidirect
46. #408	C₂×C₃	semidirect
99. C₂²×Q₈	C₂²×C₃	semidirect
105. C₂×Q₁₆	C₂²×C₃	semidirect
107. C₂×SD₁₆	C₂²×C₃	semidirect
108. C₂×SD₁₆	C₂²×C₃	semidirect
109. C₂×M₁₆	C₂²×C₃	semidirect
110. C₂²×C₈	C₂²×C₃	semidirect
114. C₂²⋊Q₈	C₂²×C₃	semidirect
115. #117	C₂²×C₃	semidirect
116. #103	C₂²×C₃	semidirect
117. #102	C₂²×C₃	semidirect
118. #102	C₂²×C₃	semidirect
119. C₂²⋊C₈	C₂²×C₃	semidirect
286. C₃	#3009	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×H(5,𝔽₂)	2
C₂×C₃×#137	2
C₂²×C₃×SD₁₆	2
C₃×#536	2
C₃×#452	2
C₃×#450	2
C₃×#418	2
C₃×#416	2
C₃×#408	2
#3009	3

Subgroups — C3×#3009 #84873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3009 #84873