Subgroups — C₃×#3277 #85141

Order 384.

476 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	C₂×C₃×#137	C₂	1	2	yes	maximal, complemented
2	C₃×#548	C₂	1	2	yes	maximal, complemented
3	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#468	C₂	1	2	yes	maximal, complemented
5	C₃×#464	C₂	1	2	yes	maximal, complemented
6	C₃×#451	C₂	1	2	yes	maximal, complemented
7	C₃×#449	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	#3277	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#41	C₂²	1	2	yes	—
12	C₂²×C₃×Q₈	C₂²	1	2	yes	—
13	C₃×#137	—	2	8	no	—
14	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×M₁₆	C₂²	1	2	yes	—
21	C₃×#124	—	2	2	no	—
22	C₃×#124	C₂²	1	2	yes	—
23	C₃×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₃×#121	—	2	2	no	—
26	C₃×#121	C₂²	1	2	yes	—
27	C₃×C₄×Q₈	C₂²	1	2	yes	—
28	C₃×C₄×D₄	—	2	2	no	—
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₃×#103	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
37	C₂×#137	C₂×C₃	1	2	yes	complemented
38	#548	C₂×C₃	1	2	yes	complemented
39	#495	C₂×C₃	1	1	yes	characteristic, complemented
40	#468	C₂×C₃	1	2	yes	complemented
41	#464	C₂×C₃	1	2	yes	complemented
42	#451	C₂×C₃	1	2	yes	complemented
43	#449	C₂×C₃	1	2	yes	complemented
44	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
45	#436	C₂×C₃	1	1	yes	characteristic, complemented
46	C₃×#41	—	2	4	no	—
47	C₃×#41	—	4	2	no	—
48	C₂×C₃×Q₈	—	2	2	no	—
49	C₂×C₃×Q₈	—	2	4	no	—
50	C₂×C₃×Q₈	—	4	2	no	—
51	C₂×C₃×Q₈	C₂³	1	2	yes	—
52	C₂×C₃×Q₈	C₂³	1	2	yes	—
53	C₂×C₃×D₄	—	2	2	no	—
54	C₂×C₃×D₄	—	2	2	no	—
55	C₂×C₃×D₄	D₄	1	1	yes	characteristic
56	C₂×C₃×D₄	C₂³	1	2	yes	—
57	C₂²×C₃×C₄	—	2	2	no	—
58	C₂²×C₃×C₄	—	2	2	no	—
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₃×Q₁₆	—	2	4	no	—
61	C₃×Q₁₆	—	4	1	no	—
62	C₃×Q₁₆	D₄	1	4	yes	complemented
63	C₃×SD₁₆	—	2	4	no	—
64	C₃×SD₁₆	—	2	4	no	—
65	C₃×D₈	—	4	1	no	—
66	C₃×M₁₆	—	2	4	no	—
67	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂³	1	2	yes	—
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
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	C₂³	1	2	yes	—
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	2	yes	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	C₂³	1	1	yes	characteristic
81	C₂×#41	C₂²×C₃	1	2	yes	—
82	C₂²×Q₈	C₂²×C₃	1	2	yes	—
83	#137	—	2	8	no	—
84	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
85	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
86	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
88	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
89	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
90	C₂×M₁₆	C₂²×C₃	1	2	yes	—
91	#124	—	2	2	no	—
92	#124	C₂²×C₃	1	2	yes	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
95	#121	—	2	2	no	—
96	#121	C₂²×C₃	1	2	yes	—
97	C₄×Q₈	C₂²×C₃	1	2	yes	—
98	C₄×D₄	—	2	2	no	—
99	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
100	#107	C₂²×C₃	1	1	yes	characteristic
101	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
102	#103	C₂²×C₃	1	2	yes	—
103	#103	C₂²×C₃	1	2	yes	—
104	#102	C₂²×C₃	1	2	yes	—
105	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
106	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
107	C₂³×C₃	—	2	2	no	—
108	C₂³×C₃	C₂×D₄	1	2	yes	—
109	C₃×Q₈	—	2	2	no	—
110	C₃×Q₈	—	2	2	no	—
111	C₃×Q₈	—	2	2	no	—
112	C₃×Q₈	—	4	4	no	—
113	C₃×Q₈	C₂×D₄	1	4	yes	—
114	C₃×D₄	—	2	2	no	—
115	C₃×D₄	—	2	2	no	—
116	C₃×D₄	—	4	1	no	—
117	C₃×D₄	—	4	4	no	—
118	C₃×D₄	—	4	4	no	—
119	C₂×C₃×C₄	—	2	2	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	—	2	2	no	—
124	C₂×C₃×C₄	—	4	4	no	—
125	C₂×C₃×C₄	—	4	4	no	—
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	2	yes	—
129	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	C₂×D₄	1	2	yes	—
133	#41	—	2	4	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	C₂³×C₃	1	2	yes	—
139	C₂×Q₈	C₂³×C₃	1	2	yes	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₃×D₄	1	1	yes	characteristic
143	C₂×D₄	C₂³×C₃	1	2	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	Q₁₆	—	2	4	no	—
148	Q₁₆	—	4	1	no	—
149	Q₁₆	C₃×D₄	1	4	yes	complemented
150	SD₁₆	—	2	4	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	4	1	no	—
153	M₁₆	—	2	4	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	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
160	C₄⋊C₄	C₂³×C₃	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	C₃×D₄	1	2	yes	—
166	C₄²	—	2	2	no	—
167	C₄²	C₂³×C₃	1	1	yes	characteristic
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	4	2	no	—
172	C₂²×C₃	—	4	2	no	—
173	C₂²×C₃	—	4	2	no	—
174	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
182	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	C₂×C₃×D₄	1	2	yes	—
185	Q₈	—	2	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	4	4	no	—
189	Q₈	C₂×C₃×D₄	1	4	yes	—
190	D₄	—	2	2	no	—
191	D₄	—	2	2	no	—
192	D₄	—	4	1	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
205	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	1	no	—
208	C₈	C₂×C₃×D₄	1	2	yes	—
209	C₂×C₃	—	4	2	no	—
210	C₂×C₃	—	8	2	no	—
211	C₂×C₃	D₄²	1	1	yes	characteristic
212	C₂×C₃	#575	1	1	yes	characteristic
213	C₂×C₃	#578	1	1	yes	characteristic
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
229	C₃	#3277	1	1	yes	characteristic, direct product factor, Sylow
230	C₂	—	4	2	no	—
231	C₂	—	8	2	no	—
232	C₂	C₃×D₄²	1	1	yes	characteristic
233	C₂	C₃×#575	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
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#137	C₂	semidirect
2. C₃×#548	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#468	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#451	C₂	semidirect
7. C₃×#449	C₂	semidirect
8. C₃×C₄×Q₁₆	C₂	semidirect
9. C₃×#436	C₂	semidirect
10. #3277	C₃	direct (N × H)
37. C₂×#137	C₂×C₃	semidirect
38. #548	C₂×C₃	semidirect
39. #495	C₂×C₃	semidirect
40. #468	C₂×C₃	semidirect
41. #464	C₂×C₃	semidirect
42. #451	C₂×C₃	semidirect
43. #449	C₂×C₃	semidirect
44. C₄×Q₁₆	C₂×C₃	semidirect
45. #436	C₂×C₃	semidirect
62. C₃×Q₁₆	D₄	semidirect
149. Q₁₆	C₃×D₄	semidirect
229. C₃	#3277	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#137	2
C₃×#548	2
C₃×#495	2
C₃×#468	2
C₃×#464	2
C₃×#451	2
C₃×#449	2
C₃×C₄×Q₁₆	2
C₃×#436	2
#3277	3

Subgroups — C3×#3277 #85141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3277 #85141