Subgroups — C₃×#3180 #85044

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#466	C₂	1	2	yes	maximal, complemented
8	C₃×#465	C₂	1	2	yes	maximal, complemented
9	C₃×#442	C₂	1	2	yes	maximal, complemented
10	C₃×#440	C₂	1	2	yes	maximal, complemented
11	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	#3180	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	—	2	1	no	—
14	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	2	yes	—
16	C₂²×C₃×C₈	C₂²	1	2	yes	—
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#127	C₂²	1	1	yes	characteristic
19	C₃×#126	—	2	1	no	—
20	C₃×#125	C₂²	1	1	yes	characteristic
21	C₃×#124	—	2	1	no	—
22	C₃×#124	C₂²	1	1	yes	characteristic
23	C₃×#123	—	2	1	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃×#121	—	2	1	no	—
26	C₃×#121	C₂²	1	2	yes	—
27	C₃×#121	C₂²	1	1	yes	characteristic
28	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
29	C₃×C₄×D₄	—	2	1	no	—
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	C₂²	1	2	yes	—
32	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
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×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×#97	C₂²	1	2	yes	—
42	C₃×#97	C₂²	1	1	yes	characteristic
43	C₃×#97	C₂²	1	1	yes	characteristic
44	#532	C₂×C₃	1	1	yes	characteristic, complemented
45	#529	C₂×C₃	1	1	yes	characteristic, complemented
46	#501	C₂×C₃	1	1	yes	characteristic, complemented
47	#496	C₂×C₃	1	1	yes	characteristic, complemented
48	#490	C₂×C₃	1	1	yes	characteristic, complemented
49	#489	C₂×C₃	1	1	yes	characteristic, complemented
50	#466	C₂×C₃	1	2	yes	complemented
51	#465	C₂×C₃	1	2	yes	complemented
52	#442	C₂×C₃	1	2	yes	complemented
53	#440	C₂×C₃	1	2	yes	complemented
54	C₂×#97	C₂×C₃	1	1	yes	characteristic, complemented
55	C₃×#41	—	4	4	no	—
56	C₂×C₃×Q₈	—	2	1	no	—
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	1	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	—	2	1	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
69	C₃×SD₁₆	—	2	4	no	—
70	C₃×D₈	—	2	4	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	C₂³	1	2	yes	—
73	C₂×C₃×C₈	C₂³	1	2	yes	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	2	yes	—
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	2	no	—
86	C₃×C₂²⋊C₄	—	2	1	no	—
87	C₃×C₂²⋊C₄	—	2	2	no	—
88	C₃×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₄²	—	2	1	no	—
90	C₃×C₄²	D₄	1	1	yes	characteristic
91	C₃×C₄²	D₄	1	1	yes	characteristic
92	C₃×C₄²	C₂³	1	1	yes	characteristic
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	C₂×#41	—	2	1	no	—
95	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
96	C₂×D₈	C₂²×C₃	1	2	yes	—
97	C₂²×C₈	C₂²×C₃	1	2	yes	—
98	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
99	#127	C₂²×C₃	1	1	yes	characteristic
100	#126	—	2	1	no	—
101	#125	C₂²×C₃	1	1	yes	characteristic
102	#124	—	2	1	no	—
103	#124	C₂²×C₃	1	1	yes	characteristic
104	#123	—	2	1	no	—
105	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
106	#121	—	2	1	no	—
107	#121	C₂²×C₃	1	1	yes	characteristic
108	#121	C₂²×C₃	1	2	yes	—
109	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
110	C₄×D₄	—	2	1	no	—
111	C₄×D₄	—	2	1	no	—
112	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
113	C₄×D₄	C₂²×C₃	1	2	yes	—
114	#117	—	2	1	no	—
115	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
116	#107	C₂²×C₃	1	2	yes	—
117	#106	C₂²×C₃	1	2	yes	—
118	#103	C₂²×C₃	1	2	yes	—
119	#102	C₂²×C₃	1	2	yes	—
120	#102	C₂²×C₃	1	2	yes	—
121	#102	C₂²×C₃	1	2	yes	—
122	#97	C₂²×C₃	1	2	yes	—
123	#97	C₂²×C₃	1	1	yes	characteristic
124	#97	C₂²×C₃	1	1	yes	characteristic
125	C₂³×C₃	—	2	1	no	—
126	C₂³×C₃	—	2	2	no	—
127	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	4	2	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	2	2	no	—
133	C₃×D₄	—	4	2	no	—
134	C₃×D₄	—	4	2	no	—
135	C₃×D₄	—	4	2	no	—
136	C₃×D₄	—	4	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	#41	1	4	yes	complemented
154	#41	—	4	4	no	—
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₃	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
168	SD₁₆	—	2	4	no	—
169	D₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	C₂³×C₃	1	2	yes	—
172	C₂×C₈	C₂³×C₃	1	2	yes	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
180	C₄⋊C₄	C₂³×C₃	1	2	yes	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₃×D₄	1	1	yes	characteristic
190	C₄²	C₃×D₄	1	1	yes	characteristic
191	C₄²	C₂³×C₃	1	1	yes	characteristic
192	C₄²	C₂³×C₃	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
202	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	4	2	no	—
210	C₃×C₄	—	4	1	no	—
211	C₃×C₄	—	4	1	no	—
212	C₃×C₄	C₂×#41	1	2	yes	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	4	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	C₃×#41	1	4	yes	complemented
242	C₂×C₃	—	2	2	no	—
243	C₂×C₃	—	8	2	no	—
244	C₂×C₃	—	8	1	no	—
245	C₂×C₃	#532	1	1	yes	characteristic
246	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₃	#575	1	1	yes	characteristic
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	1	no	—
267	C₄	C₂×C₃×#41	1	2	yes	—
268	C₃	#3180	1	1	yes	characteristic, direct product factor, Sylow
269	C₂	—	2	2	no	—
270	C₂	—	8	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₃×#532	1	1	yes	characteristic
273	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂	C₃×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃×#532	C₂	semidirect
2. C₃×#529	C₂	semidirect
3. C₃×#501	C₂	semidirect
4. C₃×#496	C₂	semidirect
5. C₃×#490	C₂	semidirect
6. C₃×#489	C₂	semidirect
7. C₃×#466	C₂	semidirect
8. C₃×#465	C₂	semidirect
9. C₃×#442	C₂	semidirect
10. C₃×#440	C₂	semidirect
11. C₂×C₃×#97	C₂	semidirect
12. #3180	C₃	direct (N × H)
44. #532	C₂×C₃	semidirect
45. #529	C₂×C₃	semidirect
46. #501	C₂×C₃	semidirect
47. #496	C₂×C₃	semidirect
48. #490	C₂×C₃	semidirect
49. #489	C₂×C₃	semidirect
50. #466	C₂×C₃	semidirect
51. #465	C₂×C₃	semidirect
52. #442	C₂×C₃	semidirect
53. #440	C₂×C₃	semidirect
54. C₂×#97	C₂×C₃	semidirect
153. C₃×C₈	#41	semidirect
241. C₈	C₃×#41	semidirect
268. C₃	#3180	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×#532	2
C₃×#529	2
C₃×#501	2
C₃×#496	2
C₃×#490	2
C₃×#489	2
C₃×#466	2
C₃×#465	2
C₃×#442	2
C₃×#440	2
C₂×C₃×#97	2
#3180	3

Subgroups — C3×#3180 #85044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3180 #85044