Subgroups — C₃×#3173 #85037

Order 384.

362 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#542	C₂	1	2	yes	maximal, complemented
2	C₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#489	C₂	1	2	yes	maximal, complemented
5	C₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#469	C₂	1	1	yes	maximal, characteristic
7	C₃×#465	C₂	1	2	yes	maximal, complemented
8	C₃×#442	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
11	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	#3173	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₃×#126	—	2	2	no	—
20	C₃×#124	—	2	2	no	—
21	C₃×#124	C₂²	1	2	yes	—
22	C₃×#123	—	2	2	no	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×#121	C₂²	1	2	yes	—
26	C₃×C₄×Q₈	C₂²	1	2	yes	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×#117	C₂²	1	1	yes	characteristic
29	C₃×#106	C₂²	1	2	yes	—
30	C₃×#106	C₂²	1	1	yes	characteristic
31	C₃×#106	C₂²	1	1	yes	characteristic
32	C₃×#103	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#97	C₂²	1	1	yes	characteristic
37	C₃×#97	C₂²	1	1	yes	characteristic
38	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
40	#542	C₂×C₃	1	2	yes	complemented
41	#498	C₂×C₃	1	1	yes	characteristic, complemented
42	#495	C₂×C₃	1	1	yes	characteristic, complemented
43	#489	C₂×C₃	1	2	yes	complemented
44	#470	C₂×C₃	1	1	yes	characteristic, complemented
45	#469	C₂×C₃	1	1	yes	characteristic
46	#465	C₂×C₃	1	2	yes	complemented
47	#442	C₂×C₃	1	1	yes	characteristic, complemented
48	#441	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
50	#405	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×C₃×Q₈	—	2	2	no	—
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	—	2	2	no	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×Q₁₆	—	2	2	no	—
58	C₃×SD₁₆	—	2	4	no	—
59	C₃×D₈	—	2	2	no	—
60	C₃×M₁₆	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	2	no	—
67	C₃×C₄⋊C₄	—	2	2	no	—
68	C₃×C₄⋊C₄	—	2	2	no	—
69	C₃×C₄⋊C₄	D₄	1	2	yes	—
70	C₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₂²⋊C₄	—	2	2	no	—
73	C₃×C₂²⋊C₄	—	2	2	no	—
74	C₃×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×C₂²⋊C₄	D₄	1	2	yes	—
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	C₂³	1	1	yes	characteristic
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
83	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
84	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
86	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
87	#126	—	2	2	no	—
88	#124	—	2	2	no	—
89	#124	C₂²×C₃	1	2	yes	—
90	#123	—	2	2	no	—
91	C₂²⋊Q₈	—	2	2	no	—
92	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
93	#121	C₂²×C₃	1	2	yes	—
94	C₄×Q₈	C₂²×C₃	1	2	yes	—
95	C₄×D₄	C₂²×C₃	1	2	yes	—
96	#117	C₂²×C₃	1	1	yes	characteristic
97	#106	C₂²×C₃	1	2	yes	—
98	#106	C₂²×C₃	1	1	yes	characteristic
99	#106	C₂²×C₃	1	1	yes	characteristic
100	#103	C₂²×C₃	1	2	yes	—
101	#103	C₂²×C₃	1	2	yes	—
102	#102	C₂²×C₃	1	2	yes	—
103	#102	C₂²×C₃	1	2	yes	—
104	#97	C₂²×C₃	1	1	yes	characteristic
105	#97	C₂²×C₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₂³×C₃	—	2	2	no	—
109	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
110	C₃×Q₈	—	2	2	no	—
111	C₃×Q₈	—	2	2	no	—
112	C₃×Q₈	—	8	2	no	—
113	C₃×D₄	—	2	2	no	—
114	C₃×D₄	—	2	2	no	—
115	C₃×D₄	—	8	2	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	8	2	no	—
122	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
123	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
126	C₃×C₈	—	2	1	no	—
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	#41	1	2	yes	—
129	C₃×C₈	#41	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	C₂³×C₃	1	2	yes	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	C₂³×C₃	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
136	Q₁₆	—	2	2	no	—
137	SD₁₆	—	2	4	no	—
138	D₈	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	C₃×D₄	1	2	yes	—
149	C₄⋊C₄	C₂³×C₃	1	2	yes	—
150	C₄⋊C₄	C₂³×C₃	1	2	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	C₃×D₄	1	2	yes	—
157	C₄²	—	2	2	no	—
158	C₄²	C₂³×C₃	1	1	yes	characteristic
159	C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	4	2	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	4	2	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	—	4	2	no	—
172	C₃×C₄	C₂×#41	1	1	yes	characteristic
173	C₃×C₄	C₂×#41	1	1	yes	characteristic
174	C₂³	—	2	2	no	—
175	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	8	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	8	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
190	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	C₃×#41	1	2	yes	—
195	C₈	C₃×#41	1	2	yes	—
196	C₂×C₃	—	4	1	no	—
197	C₂×C₃	—	8	2	no	—
198	C₂×C₃	#532	1	1	yes	characteristic
199	C₂×C₃	#577	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	C₂×C₃×#41	1	1	yes	characteristic
213	C₄	C₂×C₃×#41	1	1	yes	characteristic
214	C₃	#3173	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	4	1	no	—
216	C₂	—	8	2	no	—
217	C₂	C₃×#532	1	1	yes	characteristic
218	C₂	C₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#469	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×#542	C₂	semidirect
2. C₃×#498	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#470	C₂	semidirect
7. C₃×#465	C₂	semidirect
8. C₃×#442	C₂	semidirect
9. C₃×#441	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×#405	C₂	semidirect
12. #3173	C₃	direct (N × H)
40. #542	C₂×C₃	semidirect
41. #498	C₂×C₃	semidirect
42. #495	C₂×C₃	semidirect
43. #489	C₂×C₃	semidirect
44. #470	C₂×C₃	semidirect
46. #465	C₂×C₃	semidirect
47. #442	C₂×C₃	semidirect
48. #441	C₂×C₃	semidirect
49. C₄×SD₁₆	C₂×C₃	semidirect
50. #405	C₂×C₃	semidirect
214. C₃	#3173	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×#542	2
C₃×#498	2
C₃×#495	2
C₃×#489	2
C₃×#470	2
C₃×#469	2
C₃×#465	2
C₃×#442	2
C₃×#441	2
C₃×C₄×SD₁₆	2
C₃×#405	2
#3173	3

Subgroups — C3×#3173 #85037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3173 #85037