Subgroups — C₃×#3175 #85039

Order 384.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#500	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×#468	C₂	1	2	yes	maximal, complemented
6	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#466	C₂	1	1	yes	maximal, characteristic
8	C₃×#440	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	#3175	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×#107	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×#107	C₂²	1	2	yes	—
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	#500	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	#468	C₂×C₃	1	2	yes	complemented
45	#467	C₂×C₃	1	1	yes	characteristic, complemented
46	#466	C₂×C₃	1	1	yes	characteristic
47	#440	C₂×C₃	1	2	yes	complemented
48	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄×D₈	C₂×C₃	1	1	yes	characteristic, 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	1	yes	characteristic
70	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	C₂³	1	2	yes	—
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	—	2	2	no	—
78	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
79	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
80	C₃×C₄²	—	2	2	no	—
81	C₃×C₄²	C₂³	1	1	yes	characteristic
82	C₃×C₄²	C₂³	1	1	yes	characteristic
83	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
85	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
86	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
88	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
89	#126	—	2	2	no	—
90	#124	—	2	2	no	—
91	#124	C₂²×C₃	1	2	yes	—
92	#123	—	2	2	no	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
95	#121	C₂²×C₃	1	2	yes	—
96	C₄×Q₈	C₂²×C₃	1	2	yes	—
97	C₄×D₄	C₂²×C₃	1	2	yes	—
98	#117	C₂²×C₃	1	1	yes	characteristic
99	#107	C₂²×C₃	1	1	yes	characteristic
100	#107	C₂²×C₃	1	1	yes	characteristic
101	#107	C₂²×C₃	1	2	yes	—
102	#103	C₂²×C₃	1	2	yes	—
103	#103	C₂²×C₃	1	2	yes	—
104	#102	C₂²×C₃	1	2	yes	—
105	#102	C₂²×C₃	1	2	yes	—
106	#97	C₂²×C₃	1	1	yes	characteristic
107	#97	C₂²×C₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
110	C₂³×C₃	—	2	2	no	—
111	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
112	C₃×Q₈	—	2	2	no	—
113	C₃×Q₈	—	2	2	no	—
114	C₃×Q₈	—	8	2	no	—
115	C₃×D₄	—	2	2	no	—
116	C₃×D₄	—	2	2	no	—
117	C₃×D₄	—	8	2	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	8	2	no	—
124	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	#41	1	2	yes	—
133	C₃×C₈	#41	1	2	yes	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	C₂³×C₃	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
140	Q₁₆	—	2	2	no	—
141	SD₁₆	—	2	4	no	—
142	D₈	—	2	2	no	—
143	M₁₆	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
146	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
153	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
154	C₄⋊C₄	C₂³×C₃	1	2	yes	—
155	C₄⋊C₄	C₂³×C₃	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
162	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
163	C₄²	—	2	2	no	—
164	C₄²	C₂³×C₃	1	1	yes	characteristic
165	C₄²	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	2	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	C₂×#41	1	1	yes	characteristic
182	C₃×C₄	C₂×#41	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	8	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	2	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	C₃×#41	1	2	yes	—
206	C₈	C₃×#41	1	2	yes	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	—	8	2	no	—
209	C₂×C₃	#532	1	1	yes	characteristic
210	C₂×C₃	#576	1	1	yes	characteristic
211	C₂×C₃	#578	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₃×#41	1	1	yes	characteristic
228	C₄	C₂×C₃×#41	1	1	yes	characteristic
229	C₃	#3175	1	1	yes	characteristic, direct product factor, Sylow
230	C₂	—	4	1	no	—
231	C₂	—	8	2	no	—
232	C₂	C₃×#532	1	1	yes	characteristic
233	C₂	C₃×#576	1	1	yes	characteristic
234	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#466	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×#500	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#489	C₂	semidirect
5. C₃×#468	C₂	semidirect
6. C₃×#467	C₂	semidirect
8. C₃×#440	C₂	semidirect
9. C₃×C₄×Q₁₆	C₂	semidirect
10. C₃×C₄×D₈	C₂	semidirect
11. C₃×#405	C₂	semidirect
12. #3175	C₃	direct (N × H)
40. #542	C₂×C₃	semidirect
41. #500	C₂×C₃	semidirect
42. #495	C₂×C₃	semidirect
43. #489	C₂×C₃	semidirect
44. #468	C₂×C₃	semidirect
45. #467	C₂×C₃	semidirect
47. #440	C₂×C₃	semidirect
48. C₄×Q₁₆	C₂×C₃	semidirect
49. C₄×D₈	C₂×C₃	semidirect
50. #405	C₂×C₃	semidirect
229. C₃	#3175	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×#500	2
C₃×#495	2
C₃×#489	2
C₃×#468	2
C₃×#467	2
C₃×#466	2
C₃×#440	2
C₃×C₄×Q₁₆	2
C₃×C₄×D₈	2
C₃×#405	2
#3175	3

Subgroups — C3×#3175 #85039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3175 #85039