Subgroups — C₃×#141754 #72225866

Order 1536.

1016 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	C₂×C₃×#1712	C₂	1	1	yes	maximal, characteristic
2	C₃×#13213	C₂	1	4	yes	maximal, complemented
3	C₃×#11402	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#8822	C₂	1	1	yes	maximal, characteristic, complemented
5	#141754	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#493	—	2	2	no	—
7	C₂×C₃×#493	C₂²	1	2	yes	—
8	C₂×C₃×#361	—	2	8	no	—
9	C₃×C₄×M₃₂	C₂²	1	2	yes	—
10	C₃×#1848	C₂²	1	2	yes	—
11	C₃×#1712	—	2	2	no	—
12	C₃×#1712	C₄	1	2	yes	—
13	C₃×#1712	C₄	1	2	yes	—
14	C₃×#1566	—	2	2	no	—
15	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
16	C₃×#1312	—	2	2	no	—
17	C₂×#1712	C₂×C₃	1	1	yes	characteristic
18	#13213	C₂×C₃	1	4	yes	complemented
19	#11402	C₂×C₃	1	1	yes	characteristic, complemented
20	#8822	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₃×D₈	—	2	4	no	—
22	C₂²×C₃×D₈	—	4	4	no	—
23	C₂×C₃×#127	—	2	2	no	—
24	C₂×C₃×M₃₂	D₄	1	4	yes	—
25	C₃×#493	—	2	4	no	—
26	C₃×#493	—	2	4	no	—
27	C₃×#493	—	2	2	no	—
28	C₃×#493	—	2	4	no	—
29	C₃×#493	—	4	4	no	—
30	C₃×#493	C₂×C₄	1	4	yes	—
31	C₂×C₃×#108	—	2	4	no	—
32	C₃×#423	—	2	2	no	—
33	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₃×#361	—	4	16	no	—
36	C₃×#346	—	2	2	no	—
37	C₃×C₄×C₁₆	—	2	2	no	—
38	C₃×#334	—	4	4	no	—
39	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈²	D₄	1	2	yes	—
42	C₂×#493	—	2	2	no	—
43	C₂×#493	C₂²×C₃	1	2	yes	—
44	C₂×#361	—	2	8	no	—
45	C₄×M₃₂	C₂²×C₃	1	2	yes	—
46	#1848	C₂²×C₃	1	2	yes	—
47	#1712	—	2	2	no	—
48	#1712	C₃×C₄	1	2	yes	—
49	#1712	C₃×C₄	1	2	yes	—
50	#1566	—	2	2	no	—
51	C₂×C₈²	C₂²×C₃	1	1	yes	characteristic
52	#1312	—	2	2	no	—
53	C₂²×C₃×D₄	—	4	2	no	—
54	C₂²×C₃×D₄	—	4	4	no	—
55	C₂×C₃×D₈	—	2	8	no	—
56	C₂×C₃×D₈	—	4	8	no	—
57	C₂×C₃×D₈	—	4	8	no	—
58	C₂×C₃×D₈	—	4	4	no	—
59	C₂×C₃×D₈	—	4	8	no	—
60	C₂×C₃×D₈	—	4	8	no	—
61	C₂×C₃×D₈	—	8	16	no	—
62	C₂×C₃×M₁₆	—	4	2	no	—
63	C₂²×C₃×C₈	SD₁₆	1	4	yes	—
64	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
65	C₃×#127	—	2	4	no	—
66	C₃×#127	—	4	2	no	—
67	C₃×#127	—	4	4	no	—
68	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₃×M₃₂	—	2	8	no	—
70	C₂×C₃×C₁₆	—	2	4	no	—
71	C₃×#108	—	4	8	no	—
72	C₃×C₄⋊C₈	—	4	4	no	—
73	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄×C₈	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄×C₈	D₈	1	4	yes	—
76	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×C₈	#41	1	2	yes	—
79	C₂²×D₈	—	2	4	no	—
80	C₂²×D₈	—	4	4	no	—
81	C₂×#127	—	2	2	no	—
82	C₂×M₃₂	C₃×D₄	1	4	yes	—
83	#493	—	2	4	no	—
84	#493	—	2	4	no	—
85	#493	—	2	2	no	—
86	#493	—	2	4	no	—
87	#493	—	4	4	no	—
88	#493	C₂×C₃×C₄	1	4	yes	—
89	C₂×#108	—	2	4	no	—
90	#423	—	2	2	no	—
91	C₂×C₄×C₈	C₃×D₄	1	2	yes	—
92	C₂×C₄×C₈	C₂³×C₃	1	1	yes	characteristic
93	#361	—	4	16	no	—
94	#346	—	2	2	no	—
95	C₄×C₁₆	—	2	2	no	—
96	#334	—	4	4	no	—
97	C₈²	C₂×C₃×C₄	1	1	yes	characteristic
98	C₈²	C₂×C₃×C₄	1	1	yes	characteristic
99	C₈²	C₃×D₄	1	2	yes	—
100	C₂⁴×C₃	—	8	2	no	—
101	C₂×C₃×D₄	—	4	4	no	—
102	C₂×C₃×D₄	—	4	8	no	—
103	C₂×C₃×D₄	—	8	16	no	—
104	C₂×C₃×D₄	—	8	4	no	—
105	C₂×C₃×D₄	—	8	2	no	—
106	C₂×C₃×D₄	—	8	4	no	—
107	C₂×C₃×D₄	—	8	4	no	—
108	C₂²×C₃×C₄	#127	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×SD₁₆	1	2	yes	—
110	C₃×D₈	—	4	16	no	—
111	C₃×D₈	—	8	16	no	—
112	C₃×D₈	—	8	16	no	—
113	C₃×D₈	—	8	8	no	—
114	C₃×M₁₆	—	8	4	no	—
115	C₂×C₃×C₈	—	2	8	no	—
116	C₂×C₃×C₈	—	8	2	no	—
117	C₂×C₃×C₈	#102	1	4	yes	—
118	C₂×C₃×C₈	#102	1	4	yes	—
119	C₂×C₃×C₈	C₄×D₄	1	2	yes	—
120	C₂×C₃×C₈	#121	1	2	yes	—
121	C₂×C₃×C₈	AGL(1,ℤ/8)	1	8	yes	—
122	C₃×C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄²	#124	1	1	yes	characteristic
124	C₃×C₄²	C₂×D₈	1	2	yes	—
125	C₃×C₁₆	—	4	4	no	—
126	C₂²×D₄	—	4	2	no	—
127	C₂²×D₄	—	4	4	no	—
128	C₂×D₈	—	2	8	no	—
129	C₂×D₈	—	4	8	no	—
130	C₂×D₈	—	4	8	no	—
131	C₂×D₈	—	4	4	no	—
132	C₂×D₈	—	4	8	no	—
133	C₂×D₈	—	4	8	no	—
134	C₂×D₈	—	8	16	no	—
135	C₂×M₁₆	—	4	2	no	—
136	C₂²×C₈	C₃×SD₁₆	1	4	yes	—
137	C₂²×C₈	C₂×C₃×D₄	1	2	yes	—
138	#127	—	2	4	no	—
139	#127	—	4	2	no	—
140	#127	—	4	4	no	—
141	C₂×C₄²	C₂×C₃×D₄	1	1	yes	characteristic
142	M₃₂	—	2	8	no	—
143	C₂×C₁₆	—	2	4	no	—
144	#108	—	4	8	no	—
145	C₄⋊C₈	—	4	4	no	—
146	C₄×C₈	C₃×C₂²⋊C₄	1	2	yes	—
147	C₄×C₈	C₃×C₂²⋊C₄	1	2	yes	—
148	C₄×C₈	C₃×D₈	1	4	yes	—
149	C₄×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₄×C₈	C₂×C₃×D₄	1	1	yes	characteristic
151	C₄×C₈	C₃×#41	1	2	yes	—
152	C₂³×C₃	—	8	4	no	—
153	C₂³×C₃	—	16	8	no	—
154	C₂³×C₃	—	16	2	no	—
155	C₂³×C₃	—	16	2	no	—
156	C₂³×C₃	#492	1	1	yes	characteristic
157	C₃×D₄	—	8	16	no	—
158	C₃×D₄	—	8	8	no	—
159	C₃×D₄	—	16	8	no	—
160	C₃×D₄	—	16	4	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	#390	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×#102	1	2	yes	—
164	C₂×C₃×C₄	#486	1	2	yes	—
165	C₂×C₃×C₄	#493	1	1	yes	characteristic
166	C₂×C₃×C₄	#496	1	4	yes	—
167	C₃×C₈	—	2	4	no	—
168	C₃×C₈	—	16	2	no	—
169	C₃×C₈	#361	1	8	yes	complemented
170	C₃×C₈	#442	1	4	yes	—
171	C₃×C₈	#469	1	4	yes	—
172	C₂⁴	—	8	2	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	8	no	—
175	C₂×D₄	—	8	16	no	—
176	C₂×D₄	—	8	4	no	—
177	C₂×D₄	—	8	2	no	—
178	C₂×D₄	—	8	4	no	—
179	C₂×D₄	—	8	4	no	—
180	C₂²×C₄	C₃×#127	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₃×SD₁₆	1	2	yes	—
182	D₈	—	4	16	no	—
183	D₈	—	8	16	no	—
184	D₈	—	8	16	no	—
185	D₈	—	8	8	no	—
186	M₁₆	—	8	4	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	8	2	no	—
189	C₂×C₈	C₃×#102	1	4	yes	—
190	C₂×C₈	C₃×#102	1	4	yes	—
191	C₂×C₈	C₃×C₄×D₄	1	2	yes	—
192	C₂×C₈	C₃×#121	1	2	yes	—
193	C₂×C₈	C₃×AGL(1,ℤ/8)	1	8	yes	—
194	C₄²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₄²	C₃×#124	1	1	yes	characteristic
196	C₄²	C₂×C₃×D₈	1	2	yes	—
197	C₁₆	—	4	4	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	16	4	no	—
200	C₂²×C₃	—	16	8	no	—
201	C₂²×C₃	—	32	4	no	—
202	C₂²×C₃	—	32	4	no	—
203	C₂²×C₃	#1715	1	2	yes	—
204	C₂²×C₃	#1957	1	1	yes	characteristic
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	#1721	1	2	yes	—
207	C₃×C₄	#1972	1	2	yes	—
208	C₃×C₄	C₂×#361	1	4	yes	—
209	C₂³	—	8	4	no	—
210	C₂³	—	16	8	no	—
211	C₂³	—	16	2	no	—
212	C₂³	—	16	2	no	—
213	C₂³	C₃×#492	1	1	yes	characteristic
214	D₄	—	8	16	no	—
215	D₄	—	8	8	no	—
216	D₄	—	16	8	no	—
217	D₄	—	16	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	C₃×#390	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×#102	1	2	yes	—
221	C₂×C₄	C₃×#486	1	2	yes	—
222	C₂×C₄	C₃×#493	1	1	yes	characteristic
223	C₂×C₄	C₃×#496	1	4	yes	—
224	C₈	—	2	4	no	—
225	C₈	—	16	2	no	—
226	C₈	C₃×#361	1	8	yes	complemented
227	C₈	C₃×#442	1	4	yes	—
228	C₈	C₃×#469	1	4	yes	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	32	4	no	—
231	C₂×C₃	#9837	1	1	yes	characteristic
232	C₂×C₃	#13213	1	2	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	16	4	no	—
235	C₂²	—	16	8	no	—
236	C₂²	—	32	4	no	—
237	C₂²	—	32	4	no	—
238	C₂²	C₃×#1715	1	2	yes	—
239	C₂²	C₃×#1957	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	C₃×#1721	1	2	yes	—
242	C₄	C₃×#1972	1	2	yes	—
243	C₄	C₂×C₃×#361	1	4	yes	—
244	C₃	#141754	1	1	yes	characteristic, direct product factor, Sylow
245	C₂	—	2	2	no	—
246	C₂	—	32	4	no	—
247	C₂	C₃×#9837	1	1	yes	characteristic
248	C₂	C₃×#13213	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#1957
commutator	C₄×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³×C₃
prime radical	C₂×C₃×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#13213	C₂	semidirect
3. C₃×#11402	C₂	semidirect
4. C₃×#8822	C₂	semidirect
5. #141754	C₃	direct (N × H)
18. #13213	C₂×C₃	semidirect
19. #11402	C₂×C₃	semidirect
20. #8822	C₂×C₃	semidirect
169. C₃×C₈	#361	semidirect
226. C₈	C₃×#361	semidirect
244. C₃	#141754	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×#1712	2
C₃×#13213	2
C₃×#11402	2
C₃×#8822	2
#141754	3

Subgroups — C3×#141754 #72225866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#141754 #72225866