Subgroups — C₃×#3172 #85036

Order 384.

364 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#470	C₂	1	2	yes	maximal, complemented
8	C₃×#468	C₂	1	2	yes	maximal
9	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#3172	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×M₁₆	C₂²	1	2	yes	—
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#126	—	2	1	no	—
19	C₃×#125	—	2	1	no	—
20	C₃×#125	C₂²	1	1	yes	characteristic
21	C₃×#124	C₂²	1	1	yes	characteristic
22	C₃×#123	—	2	1	no	—
23	C₃×C₂²⋊Q₈	—	2	1	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₃×#121	C₂²	1	1	yes	characteristic
27	C₃×C₄×Q₈	—	2	1	no	—
28	C₃×C₄×Q₈	C₂²	1	2	yes	—
29	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
32	C₃×#117	—	2	1	no	—
33	C₃×#117	—	2	1	no	—
34	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
35	C₃×#107	C₂²	1	2	yes	—
36	C₃×#106	C₂²	1	2	yes	—
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#103	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×#97	C₂²	1	1	yes	characteristic
42	C₃×#97	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₈	C₂²	1	2	yes	—
44	#533	C₂×C₃	1	1	yes	characteristic, complemented
45	#529	C₂×C₃	1	1	yes	characteristic
46	#501	C₂×C₃	1	1	yes	characteristic, complemented
47	#497	C₂×C₃	1	1	yes	characteristic, complemented
48	#491	C₂×C₃	1	1	yes	characteristic, complemented
49	#489	C₂×C₃	1	1	yes	characteristic, complemented
50	#470	C₂×C₃	1	2	yes	complemented
51	#468	C₂×C₃	1	2	yes	—
52	C₄×Q₁₆	C₂×C₃	1	2	yes	complemented
53	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
54	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	2	yes	—
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₃×Q₁₆	—	2	4	no	—
65	C₃×SD₁₆	—	2	4	no	—
66	C₃×M₁₆	—	2	4	no	—
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	—	2	1	no	—
72	C₃×C₄⋊C₄	—	2	1	no	—
73	C₃×C₄⋊C₄	—	2	1	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	2	no	—
76	C₃×C₄⋊C₄	—	2	2	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	C₂³	1	2	yes	—
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	1	no	—
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	1	no	—
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₄²	—	2	1	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	D₄	1	1	yes	characteristic
90	C₃×C₄²	D₄	1	1	yes	characteristic
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
94	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
95	C₂×M₁₆	C₂²×C₃	1	2	yes	—
96	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
98	#126	—	2	1	no	—
99	#125	—	2	1	no	—
100	#125	C₂²×C₃	1	1	yes	characteristic
101	#124	C₂²×C₃	1	1	yes	characteristic
102	#123	—	2	1	no	—
103	C₂²⋊Q₈	—	2	1	no	—
104	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
105	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
106	#121	C₂²×C₃	1	1	yes	characteristic
107	C₄×Q₈	—	2	1	no	—
108	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
109	C₄×Q₈	C₂²×C₃	1	2	yes	—
110	C₄×D₄	—	2	1	no	—
111	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
112	#117	—	2	1	no	—
113	#117	—	2	1	no	—
114	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
115	#107	C₂²×C₃	1	2	yes	—
116	#106	C₂²×C₃	1	2	yes	—
117	#103	C₂²×C₃	1	2	yes	—
118	#103	C₂²×C₃	1	2	yes	—
119	#103	C₂²×C₃	1	2	yes	—
120	#102	C₂²×C₃	1	2	yes	—
121	#97	C₂²×C₃	1	1	yes	characteristic
122	#97	C₂²×C₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×C₃	1	2	yes	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
126	C₃×Q₈	—	2	2	no	—
127	C₃×Q₈	—	2	2	no	—
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	4	2	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	4	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	4	1	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
147	C₃×C₈	—	2	2	no	—
148	C₃×C₈	#41	1	4	yes	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×Q₈	C₂³×C₃	1	2	yes	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
156	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
158	Q₁₆	—	2	4	no	—
159	SD₁₆	—	2	4	no	—
160	M₁₆	—	2	4	no	—
161	C₂×C₈	C₂³×C₃	1	2	yes	—
162	C₂×C₈	C₂³×C₃	1	2	yes	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	C₂³×C₃	1	2	yes	—
173	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
174	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	1	no	—
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	1	no	—
180	C₂²⋊C₄	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₃×D₄	1	1	yes	characteristic
184	C₄²	C₃×D₄	1	1	yes	characteristic
185	C₄²	C₂³×C₃	1	1	yes	characteristic
186	C₄²	C₂³×C₃	1	1	yes	characteristic
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	4	1	no	—
191	C₂²×C₃	—	4	1	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	1	no	—
199	C₃×C₄	—	4	2	no	—
200	C₃×C₄	—	4	1	no	—
201	C₃×C₄	—	4	1	no	—
202	C₃×C₄	—	4	2	no	—
203	C₃×C₄	#137	1	2	yes	—
204	C₃×C₄	C₂×#41	1	2	yes	—
205	C₂³	—	2	1	no	—
206	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
207	Q₈	—	2	2	no	—
208	Q₈	—	2	2	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	4	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
226	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	C₃×#41	1	4	yes	—
230	C₂×C₃	—	4	1	no	—
231	C₂×C₃	—	8	1	no	—
232	C₂×C₃	#532	1	1	yes	characteristic
233	C₂×C₃	C₂×#137	1	1	yes	characteristic
234	C₂×C₃	#575	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	C₃×#137	1	2	yes	—
252	C₄	C₂×C₃×#41	1	2	yes	—
253	C₃	#3172	1	1	yes	characteristic, direct product factor, Sylow
254	C₂	—	4	1	no	—
255	C₂	—	8	1	no	—
256	C₂	C₃×#532	1	1	yes	characteristic
257	C₂	C₂×C₃×#137	1	1	yes	characteristic
258	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₄
prime radical	C₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#533	C₂	semidirect
3. C₃×#501	C₂	semidirect
4. C₃×#497	C₂	semidirect
5. C₃×#491	C₂	semidirect
6. C₃×#489	C₂	semidirect
7. C₃×#470	C₂	semidirect
9. C₃×C₄×Q₁₆	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×C₄×M₁₆	C₂	semidirect
12. #3172	C₃	direct (N × H)
44. #533	C₂×C₃	semidirect
46. #501	C₂×C₃	semidirect
47. #497	C₂×C₃	semidirect
48. #491	C₂×C₃	semidirect
49. #489	C₂×C₃	semidirect
50. #470	C₂×C₃	semidirect
52. C₄×Q₁₆	C₂×C₃	semidirect
53. C₄×SD₁₆	C₂×C₃	semidirect
54. C₄×M₁₆	C₂×C₃	semidirect
253. C₃	#3172	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×#533	2
C₃×#529	2
C₃×#501	2
C₃×#497	2
C₃×#491	2
C₃×#489	2
C₃×#470	2
C₃×#468	2
C₃×C₄×Q₁₆	2
C₃×C₄×SD₁₆	2
C₃×C₄×M₁₆	2
#3172	3

Subgroups — C3×#3172 #85036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3172 #85036