Subgroups — C₃×#3081 #84945

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#518	C₂	1	1	yes	maximal, characteristic
3	C₃×#478	C₂	1	2	yes	maximal, complemented
4	C₃×#478	C₂	1	2	yes	maximal, complemented
5	C₃×#476	C₂	1	2	yes	maximal, complemented
6	C₃×#474	C₂	1	2	yes	maximal, complemented
7	C₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#418	C₂	1	1	yes	maximal, characteristic
11	C₂×C₃×#102	C₂	1	1	yes	maximal, characteristic
12	#3081	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	—	2	1	no	—
17	C₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₃×#125	—	2	1	no	—
19	C₃×#125	C₂²	1	2	yes	complemented
20	C₃×C₂²⋊Q₈	—	2	2	no	—
21	C₃×C₄×D₄	—	2	1	no	—
22	C₃×C₄×D₄	—	2	1	no	—
23	C₃×C₄×D₄	C₂²	1	2	yes	—
24	C₃×C₄×D₄	C₂²	1	2	yes	—
25	C₃×#117	C₂²	1	1	yes	characteristic
26	C₃×#117	C₂²	1	1	yes	characteristic
27	C₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
30	C₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₃×#107	C₂²	1	1	yes	characteristic, complemented
32	C₃×#107	C₂²	1	2	yes	complemented
33	C₃×#106	C₂²	1	1	yes	characteristic, complemented
34	C₃×#106	C₂²	1	1	yes	characteristic, complemented
35	C₃×#106	C₂²	1	2	yes	complemented
36	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×#102	C₂²	1	2	yes	—
42	#544	C₂×C₃	1	1	yes	characteristic, complemented
43	#518	C₂×C₃	1	1	yes	characteristic
44	#478	C₂×C₃	1	2	yes	complemented
45	#478	C₂×C₃	1	2	yes	complemented
46	#476	C₂×C₃	1	2	yes	complemented
47	#474	C₂×C₃	1	2	yes	complemented
48	#428	C₂×C₃	1	1	yes	characteristic, complemented
49	#427	C₂×C₃	1	1	yes	characteristic, complemented
50	#424	C₂×C₃	1	1	yes	characteristic, complemented
51	#418	C₂×C₃	1	1	yes	characteristic
52	C₂×#102	C₂×C₃	1	1	yes	characteristic
53	C₂⁴×C₃	—	2	1	no	—
54	C₂×C₃×Q₈	—	2	2	no	—
55	C₂×C₃×D₄	—	2	1	no	—
56	C₂×C₃×D₄	—	2	2	no	—
57	C₂×C₃×D₄	Q₈	1	4	yes	complemented
58	C₂×C₃×D₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	—	2	2	no	—
62	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
63	C₃×M₁₆	—	2	2	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂³	1	2	yes	—
67	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	2	1	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	2	2	no	—
72	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₃×C₄⋊C₄	C₂³	1	2	yes	—
75	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₂²⋊C₄	—	2	1	no	—
80	C₃×C₂²⋊C₄	—	2	1	no	—
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	D₄	1	2	yes	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	C₂³	1	2	yes	—
87	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
88	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
92	#125	—	2	1	no	—
93	#125	C₂²×C₃	1	2	yes	complemented
94	C₂²⋊Q₈	—	2	2	no	—
95	C₄×D₄	—	2	1	no	—
96	C₄×D₄	—	2	1	no	—
97	C₄×D₄	C₂²×C₃	1	2	yes	—
98	C₄×D₄	C₂²×C₃	1	2	yes	—
99	#117	C₂²×C₃	1	1	yes	characteristic
100	#117	C₂²×C₃	1	1	yes	characteristic
101	#117	C₂²×C₃	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	#107	C₂²×C₃	1	1	yes	characteristic, complemented
105	#107	C₂²×C₃	1	1	yes	characteristic, complemented
106	#107	C₂²×C₃	1	2	yes	complemented
107	#106	C₂²×C₃	1	1	yes	characteristic, complemented
108	#106	C₂²×C₃	1	1	yes	characteristic, complemented
109	#106	C₂²×C₃	1	2	yes	complemented
110	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
111	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	2	yes	—
114	#102	C₂²×C₃	1	2	yes	—
115	#102	C₂²×C₃	1	2	yes	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	2	4	no	—
118	C₂³×C₃	—	8	1	no	—
119	C₂³×C₃	—	8	1	no	—
120	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
121	C₃×Q₈	—	8	2	no	—
122	C₃×D₄	—	2	4	no	—
123	C₃×D₄	—	2	2	no	—
124	C₃×D₄	C₂×Q₈	1	4	yes	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	8	2	no	—
131	C₂×C₃×C₄	—	8	2	no	—
132	C₂×C₃×C₄	—	8	1	no	—
133	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
137	C₂×C₃×C₄	#41	1	2	yes	—
138	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	2	no	—
142	C₂⁴	—	2	1	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	C₃×Q₈	1	4	yes	complemented
147	C₂×D₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₈	C₂³×C₃	1	2	yes	—
156	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
162	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
163	C₄⋊C₄	C₂³×C₃	1	2	yes	—
164	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
165	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
166	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	C₃×D₄	1	2	yes	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	C₂³×C₃	1	2	yes	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	8	4	no	—
182	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
183	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	2	no	—
186	C₃×C₄	—	4	1	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	4	1	no	—
189	C₃×C₄	—	4	1	no	—
190	C₃×C₄	—	4	1	no	—
191	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
192	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
193	C₃×C₄	C₂×#41	1	1	yes	characteristic
194	C₂³	—	2	2	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	8	1	no	—
197	C₂³	—	8	1	no	—
198	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
199	Q₈	—	8	2	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	2	no	—
202	D₄	C₂×C₃×Q₈	1	4	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	1	no	—
211	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
215	C₂×C₄	C₃×#41	1	2	yes	—
216	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	4	4	no	—
222	C₂×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
223	C₂×C₃	#576	1	1	yes	characteristic
224	C₂×C₃	#577	1	1	yes	characteristic
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	C₃×C₂²⋊Q₈	1	2	yes	—
232	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	C₃×C₂²⋊Q₈	1	2	yes	—
241	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
242	C₄	C₂×C₃×#41	1	1	yes	characteristic
243	C₃	#3081	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	—	2	2	no	—
245	C₂	—	4	4	no	—
246	C₂	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂	C₃×#576	1	1	yes	characteristic
248	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂²
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×#544	C₂	semidirect
3. C₃×#478	C₂	semidirect
4. C₃×#478	C₂	semidirect
5. C₃×#476	C₂	semidirect
6. C₃×#474	C₂	semidirect
7. C₃×#428	C₂	semidirect
8. C₃×#427	C₂	semidirect
9. C₃×#424	C₂	semidirect
12. #3081	C₃	direct (N × H)
17. C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#125	C₂²	semidirect
30. C₃×#107	C₂²	semidirect
31. C₃×#107	C₂²	semidirect
32. C₃×#107	C₂²	semidirect
33. C₃×#106	C₂²	semidirect
34. C₃×#106	C₂²	semidirect
35. C₃×#106	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×C₄⋊C₈	C₂²	semidirect
42. #544	C₂×C₃	semidirect
44. #478	C₂×C₃	semidirect
45. #478	C₂×C₃	semidirect
46. #476	C₂×C₃	semidirect
47. #474	C₂×C₃	semidirect
48. #428	C₂×C₃	semidirect
49. #427	C₂×C₃	semidirect
50. #424	C₂×C₃	semidirect
57. C₂×C₃×D₄	Q₈	semidirect
91. C₄⋊Q₈	C₂²×C₃	semidirect
93. #125	C₂²×C₃	semidirect
104. #107	C₂²×C₃	semidirect
105. #107	C₂²×C₃	semidirect
106. #107	C₂²×C₃	semidirect
107. #106	C₂²×C₃	semidirect
108. #106	C₂²×C₃	semidirect
109. #106	C₂²×C₃	semidirect
110. C₄⋊C₈	C₂²×C₃	semidirect
111. C₄⋊C₈	C₂²×C₃	semidirect
146. C₂×D₄	C₃×Q₈	semidirect
243. C₃	#3081	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×#544	2
C₃×#518	2
C₃×#478	2
C₃×#476	2
C₃×#474	2
C₃×#428	2
C₃×#427	2
C₃×#424	2
C₃×#418	2
C₂×C₃×#102	2
#3081	3

Subgroups — C3×#3081 #84945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3081 #84945