Subgroups — C₃×#2969 #84833

Order 384.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#436	C₂	1	2	yes	maximal, complemented
6	C₃×#435	C₂	1	2	yes	maximal, complemented
7	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#433	C₂	1	2	yes	maximal, complemented
9	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#405	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#2969	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₁₆	C₂²	1	2	yes	—
16	C₂²×C₃×C₈	C₂²	1	2	yes	—
17	C₃×#126	C₄	1	2	yes	—
18	C₃×#125	C₄	1	1	yes	characteristic
19	C₃×#124	C₄	1	1	yes	characteristic
20	C₃×#123	C₄	1	2	yes	—
21	C₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
22	C₃×#121	C₄	1	1	yes	characteristic
23	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	C₂²	1	2	yes	—
26	C₃×#117	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×#97	C₂²	1	1	yes	characteristic
36	C₃×#97	C₂²	1	1	yes	characteristic
37	C₃×#97	C₂²	1	2	yes	—
38	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
40	C₃×C₄×C₈	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	2	yes	—
42	#529	C₂×C₃	1	1	yes	characteristic
43	#446	C₂×C₃	1	1	yes	characteristic, complemented
44	C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
45	#436	C₂×C₃	1	1	yes	characteristic, complemented
46	#436	C₂×C₃	1	2	yes	complemented
47	#435	C₂×C₃	1	2	yes	complemented
48	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
49	#433	C₂×C₃	1	2	yes	complemented
50	#431	C₂×C₃	1	1	yes	characteristic, complemented
51	#405	C₂×C₃	1	2	yes	complemented
52	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₃×M₁₆	—	2	2	no	—
61	C₃×M₁₆	—	2	2	no	—
62	C₃×M₁₆	—	2	4	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂³	1	2	yes	—
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
71	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
72	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
74	C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
76	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
77	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
78	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
79	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
80	C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
81	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
82	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
83	C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₃×C₄²	C₂³	1	1	yes	characteristic
85	C₃×C₄²	C₂³	1	2	yes	—
86	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₂×M₁₆	C₂²×C₃	1	2	yes	—
89	C₂²×C₈	C₂²×C₃	1	2	yes	—
90	#126	C₃×C₄	1	2	yes	—
91	#125	C₃×C₄	1	1	yes	characteristic
92	#124	C₃×C₄	1	1	yes	characteristic
93	#123	C₃×C₄	1	2	yes	—
94	C₂²⋊Q₈	C₃×C₄	1	1	yes	characteristic
95	#121	C₃×C₄	1	1	yes	characteristic
96	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₄×D₄	C₂²×C₃	1	2	yes	—
99	#117	C₂²×C₃	1	2	yes	—
100	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
101	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
102	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
103	C₄⋊C₈	C₂²×C₃	1	2	yes	—
104	C₄⋊C₈	C₂²×C₃	1	2	yes	—
105	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
106	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
107	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
108	#97	C₂²×C₃	1	1	yes	characteristic
109	#97	C₂²×C₃	1	1	yes	characteristic
110	#97	C₂²×C₃	1	2	yes	—
111	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×C₃	1	2	yes	—
114	C₄×C₈	C₂²×C₃	1	2	yes	—
115	C₂³×C₃	C₂²×C₄	1	2	yes	—
116	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
117	C₃×Q₈	—	2	2	no	—
118	C₃×D₄	—	2	2	no	—
119	C₃×D₄	—	4	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	4	1	no	—
126	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
132	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
133	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
135	C₃×C₈	—	2	2	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	—	2	1	no	—
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	#41	1	2	yes	—
140	C₃×C₈	#41	1	2	yes	—
141	C₂×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×D₄	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
144	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	C₂³×C₃	1	2	yes	—
153	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
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	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
159	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
160	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
162	C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
163	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
164	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
165	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
166	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
167	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
168	C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
169	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
170	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
171	C₄²	C₂³×C₃	1	1	yes	characteristic
172	C₄²	C₂³×C₃	1	1	yes	characteristic
173	C₄²	C₂³×C₃	1	2	yes	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	#131	1	2	yes	—
189	C₃×C₄	C₂×#41	1	1	yes	characteristic
190	C₃×C₄	C₂×#41	1	1	yes	characteristic
191	C₂³	C₂²×C₃×C₄	1	2	yes	—
192	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
193	Q₈	—	2	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
207	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
208	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
209	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	2	no	—
215	C₈	C₃×#41	1	2	yes	—
216	C₈	C₃×#41	1	2	yes	—
217	C₂×C₃	—	4	1	no	—
218	C₂×C₃	—	4	2	no	—
219	C₂×C₃	C₄×#41	1	1	yes	characteristic
220	C₂×C₃	C₂×#131	1	1	yes	characteristic
221	C₂×C₃	#568	1	1	yes	characteristic
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	C₃×#131	1	2	yes	—
237	C₄	C₂×C₃×#41	1	1	yes	characteristic
238	C₄	C₂×C₃×#41	1	1	yes	characteristic
239	C₃	#2969	1	1	yes	characteristic, direct product factor, Sylow
240	C₂	—	4	1	no	—
241	C₂	—	4	2	no	—
242	C₂	C₃×C₄×#41	1	1	yes	characteristic
243	C₂	C₂×C₃×#131	1	1	yes	characteristic
244	C₂	C₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₄
prime radical	C₃×#121	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#446	C₂	semidirect
3. C₃×C₈×Q₈	C₂	semidirect
4. C₃×#436	C₂	semidirect
5. C₃×#436	C₂	semidirect
6. C₃×#435	C₂	semidirect
7. C₃×C₈×D₄	C₂	semidirect
8. C₃×#433	C₂	semidirect
9. C₃×#431	C₂	semidirect
10. C₃×#405	C₂	semidirect
11. C₃×C₄×M₁₆	C₂	semidirect
12. #2969	C₃	direct (N × H)
43. #446	C₂×C₃	semidirect
44. C₈×Q₈	C₂×C₃	semidirect
45. #436	C₂×C₃	semidirect
46. #436	C₂×C₃	semidirect
47. #435	C₂×C₃	semidirect
48. C₈×D₄	C₂×C₃	semidirect
49. #433	C₂×C₃	semidirect
50. #431	C₂×C₃	semidirect
51. #405	C₂×C₃	semidirect
52. C₄×M₁₆	C₂×C₃	semidirect
239. C₃	#2969	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×#529	2
C₃×#446	2
C₃×C₈×Q₈	2
C₃×#436	2
C₃×#435	2
C₃×C₈×D₄	2
C₃×#433	2
C₃×#431	2
C₃×#405	2
C₃×C₄×M₁₆	2
#2969	3

Subgroups — C3×#2969 #84833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2969 #84833