Subgroups — C₃×#3153 #85017

Order 384.

572 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×C₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#497	C₂	1	2	yes	maximal, complemented
6	C₃×#496	C₂	1	2	yes	maximal, complemented
7	C₃×#495	C₂	1	4	yes	maximal, complemented
8	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#3153	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	C₂²	1	2	yes	—
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₃×#137	—	2	8	no	—
14	C₃×AGL(1,ℤ/8)	—	2	8	no	—
15	C₂×C₃×Q₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	4	yes	complemented
18	C₂×C₃×D₈	C₂²	1	4	yes	—
19	C₂×C₃×M₁₆	C₂²	1	2	yes	—
20	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₃×#127	C₂²	1	1	yes	characteristic, complemented
22	C₃×#124	—	2	2	no	—
23	C₃×#124	C₂²	1	2	yes	—
24	C₃×#124	C₂²	1	2	yes	—
25	C₃×#124	C₂²	1	2	yes	complemented
26	C₃×#121	—	2	2	no	—
27	C₃×C₄×D₄	—	2	2	no	—
28	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₃×#97	C₂²	1	2	yes	complemented
31	C₃×C₄×C₈	C₂²	1	2	yes	complemented
32	C₂×#137	C₂×C₃	1	2	yes	complemented
33	C₂×AGL(1,ℤ/8)	C₂×C₃	1	2	yes	complemented
34	#532	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
36	#497	C₂×C₃	1	2	yes	complemented
37	#496	C₂×C₃	1	2	yes	complemented
38	#495	C₂×C₃	1	4	yes	complemented
39	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂⁴×C₃	—	2	1	no	—
41	C₃×#41	—	2	4	no	—
42	C₃×#41	—	2	4	no	—
43	C₂×C₃×Q₈	—	2	1	no	—
44	C₂×C₃×Q₈	—	2	4	no	—
45	C₂×C₃×Q₈	C₂³	1	2	yes	—
46	C₂×C₃×Q₈	C₂³	1	2	yes	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	2	4	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	1	no	—
51	C₂×C₃×D₄	C₂³	1	2	yes	—
52	C₂×C₃×D₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	—	2	1	no	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂²×C₃×C₄	D₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×Q₁₆	—	2	8	no	—
58	C₃×SD₁₆	—	2	8	no	—
59	C₃×SD₁₆	—	2	8	no	—
60	C₃×D₈	—	2	8	no	—
61	C₃×M₁₆	D₄	1	8	yes	complemented
62	C₂×C₃×C₈	C₂³	1	4	yes	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₂²⋊C₄	—	2	4	no	—
65	C₃×C₂²⋊C₄	—	2	4	no	—
66	C₃×C₂²⋊C₄	—	2	4	no	—
67	C₃×C₄²	D₄	1	2	yes	—
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	C₃×C₄²	C₂³	1	1	yes	characteristic
70	C₂×#41	C₂²×C₃	1	2	yes	—
71	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
72	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
73	#137	—	2	8	no	—
74	AGL(1,ℤ/8)	—	2	8	no	—
75	C₂×Q₁₆	C₂²×C₃	1	4	yes	complemented
76	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
77	C₂×SD₁₆	C₂²×C₃	1	4	yes	complemented
78	C₂×D₈	C₂²×C₃	1	4	yes	—
79	C₂×M₁₆	C₂²×C₃	1	2	yes	—
80	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
81	#127	C₂²×C₃	1	1	yes	characteristic, complemented
82	#124	—	2	2	no	—
83	#124	C₂²×C₃	1	2	yes	—
84	#124	C₂²×C₃	1	2	yes	—
85	#124	C₂²×C₃	1	2	yes	complemented
86	#121	—	2	2	no	—
87	C₄×D₄	—	2	2	no	—
88	C₂×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
90	#97	C₂²×C₃	1	2	yes	complemented
91	C₄×C₈	C₂²×C₃	1	2	yes	complemented
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	4	4	no	—
95	C₂³×C₃	—	4	2	no	—
96	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
97	C₃×Q₈	—	2	2	no	—
98	C₃×Q₈	—	2	4	no	—
99	C₃×Q₈	—	2	2	no	—
100	C₃×Q₈	—	4	2	no	—
101	C₃×D₄	—	2	2	no	—
102	C₃×D₄	—	2	4	no	—
103	C₃×D₄	—	2	2	no	—
104	C₃×D₄	—	4	4	no	—
105	C₃×D₄	—	4	4	no	—
106	C₃×D₄	—	4	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	4	2	no	—
110	C₂×C₃×C₄	—	4	2	no	—
111	C₂×C₃×C₄	—	4	4	no	—
112	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
113	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
116	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
117	C₃×C₈	C₂×D₄	1	8	yes	—
118	C₂⁴	—	2	1	no	—
119	#41	—	2	4	no	—
120	#41	—	2	4	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	C₂³×C₃	1	2	yes	—
124	C₂×Q₈	C₂³×C₃	1	2	yes	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	C₂³×C₃	1	2	yes	—
130	C₂×D₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₃×D₄	1	2	yes	—
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	Q₁₆	—	2	8	no	—
136	SD₁₆	—	2	8	no	—
137	SD₁₆	—	2	8	no	—
138	D₈	—	2	8	no	—
139	M₁₆	C₃×D₄	1	8	yes	complemented
140	C₂×C₈	C₂³×C₃	1	4	yes	—
141	C₄⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₄²	C₃×D₄	1	2	yes	—
146	C₄²	C₂³×C₃	1	1	yes	characteristic
147	C₄²	C₂³×C₃	1	1	yes	characteristic
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	—	4	4	no	—
152	C₂²×C₃	—	4	4	no	—
153	C₂²×C₃	—	8	4	no	—
154	C₂²×C₃	#127	1	2	yes	—
155	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	—	4	2	no	—
160	C₃×C₄	#127	1	2	yes	—
161	C₃×C₄	C₂²×D₄	1	2	yes	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	4	4	no	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	Q₈	—	2	4	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	4	2	no	—
171	D₄	—	2	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
186	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
187	C₈	C₂×C₃×D₄	1	8	yes	—
188	C₂×C₃	—	2	2	no	—
189	C₂×C₃	—	8	2	no	—
190	C₂×C₃	—	8	2	no	—
191	C₂×C₃	C₂×#127	1	1	yes	characteristic
192	C₂×C₃	#575	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	8	4	no	—
199	C₂²	C₃×#127	1	2	yes	—
200	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	C₃×#127	1	2	yes	—
206	C₄	C₂²×C₃×D₄	1	2	yes	—
207	C₃	#3153	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	—	8	2	no	—
211	C₂	C₂×C₃×#127	1	1	yes	characteristic
212	C₂	C₃×#575	1	2	yes	—

	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. C₂×C₃×#137	C₂	semidirect
2. C₂×C₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₃×#532	C₂	semidirect
4. C₂×C₃×#124	C₂	semidirect
5. C₃×#497	C₂	semidirect
6. C₃×#496	C₂	semidirect
7. C₃×#495	C₂	semidirect
8. C₃×C₄×M₁₆	C₂	semidirect
9. #3153	C₃	direct (N × H)
15. C₂×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#127	C₂²	semidirect
25. C₃×#124	C₂²	semidirect
30. C₃×#97	C₂²	semidirect
31. C₃×C₄×C₈	C₂²	semidirect
32. C₂×#137	C₂×C₃	semidirect
33. C₂×AGL(1,ℤ/8)	C₂×C₃	semidirect
34. #532	C₂×C₃	semidirect
35. C₂×#124	C₂×C₃	semidirect
36. #497	C₂×C₃	semidirect
37. #496	C₂×C₃	semidirect
38. #495	C₂×C₃	semidirect
39. C₄×M₁₆	C₂×C₃	semidirect
61. C₃×M₁₆	D₄	semidirect
75. C₂×Q₁₆	C₂²×C₃	semidirect
77. C₂×SD₁₆	C₂²×C₃	semidirect
80. C₄⋊Q₈	C₂²×C₃	semidirect
81. #127	C₂²×C₃	semidirect
85. #124	C₂²×C₃	semidirect
90. #97	C₂²×C₃	semidirect
91. C₄×C₈	C₂²×C₃	semidirect
139. M₁₆	C₃×D₄	semidirect
207. C₃	#3153	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
C₂×C₃×AGL(1,ℤ/8)	2
C₃×#532	2
C₂×C₃×#124	2
C₃×#497	2
C₃×#496	2
C₃×#495	2
C₃×C₄×M₁₆	2
#3153	3

Subgroups — C3×#3153 #85017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3153 #85017