Subgroups — C₃×#82015 #11899766

Order 1152.

742 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5753	C₂	1	2	yes	maximal, complemented
2	C₃²×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4909	C₂	1	6	yes	maximal, complemented
4	C₂×C₃×Dic₂₄	C₂	1	3	yes	maximal, complemented
5	C₃×#4904	C₂	1	3	yes	maximal, complemented
6	C₃×#3417	—	3	1	no	maximal
7	#82015	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×#1010	—	2	2	no	—
9	C₃²×#131	C₂²	1	1	yes	characteristic
10	C₃×#909	—	2	6	no	—
11	C₂×C₃×Dic₁₂	C₂²	1	6	yes	—
12	C₃×#904	C₂²	1	6	yes	—
13	C₃²×M₃₂	C₂²	1	3	yes	—
14	C₂×C₃²×C₁₆	C₂²	1	3	yes	—
15	C₃×Dic₂₄	C₂²	1	3	yes	—
16	C₃×Dic₂₄	C₂²	1	6	yes	—
17	C₃×#800	C₂²	1	6	yes	—
18	C₃×D₄₈	C₂²	1	1	yes	characteristic
19	C₃×#578	—	3	2	no	—
20	#5753	C₂×C₃	1	2	yes	complemented
21	C₃×#510	—	3	6	no	—
22	C₃×#508	—	3	3	no	—
23	C₂×C₃×Q₃₂	—	3	3	no	—
24	C₃×#504	—	2	1	no	—
25	C₃×#504	D₃	1	1	yes	characteristic, complemented
26	C₃×#504	C₂×C₃	1	1	yes	characteristic, complemented
27	#4909	C₂×C₃	1	6	yes	complemented
28	C₂×Dic₂₄	C₂×C₃	1	3	yes	complemented
29	#4904	C₂×C₃	1	3	yes	complemented
30	C₃²×#41	D₄	1	1	yes	characteristic
31	C₃×D₃×Q₈	—	4	2	no	—
32	C₃×#237	—	4	6	no	—
33	C₃×#235	—	2	6	no	—
34	C₂×C₃×Dic₆	—	2	6	no	—
35	C₃²×M₁₆	D₄	1	3	yes	—
36	C₂×C₃²×C₈	C₂³	1	3	yes	—
37	C₃×Dic₁₂	—	2	6	no	—
38	C₃×Dic₁₂	C₂³	1	6	yes	—
39	C₃×D₂₄	C₂³	1	2	yes	—
40	C₃×#204	—	2	6	no	—
41	C₃²×C₁₆	C₂³	1	1	yes	characteristic
42	C₃²×C₁₆	C₂³	1	3	yes	—
43	#3417	—	3	1	no	Sylow
44	C₃×2₋¹⁺⁴	—	6	2	no	—
45	#1010	—	2	2	no	—
46	C₃×#137	—	6	6	no	—
47	C₃×#135	—	3	6	no	—
48	C₂×C₃×Q₁₆	—	3	6	no	—
49	C₃×#131	—	2	1	no	—
50	C₃×#131	C₂×D₃	1	1	yes	characteristic
51	C₃×#131	C₂²×C₃	1	1	yes	characteristic
52	#909	—	2	6	no	—
53	C₂×Dic₁₂	C₂²×C₃	1	6	yes	—
54	#904	C₂²×C₃	1	6	yes	—
55	C₃×Q₃₂	—	3	3	no	—
56	C₃×Q₃₂	—	3	6	no	—
57	C₃×SD₃₂	—	3	6	no	—
58	C₃×D₁₆	—	3	1	no	—
59	C₃×M₃₂	—	2	3	no	—
60	C₃×M₃₂	C₂×D₃	1	3	yes	—
61	C₃×M₃₂	C₂²×C₃	1	3	yes	—
62	C₂×C₃×C₁₆	—	2	3	no	—
63	C₂×C₃×C₁₆	C₂×D₃	1	3	yes	—
64	C₂×C₃×C₁₆	C₂²×C₃	1	3	yes	—
65	Dic₂₄	C₂²×C₃	1	3	yes	—
66	Dic₂₄	C₂²×C₃	1	6	yes	—
67	#800	C₂²×C₃	1	6	yes	—
68	D₄₈	C₂²×C₃	1	1	yes	characteristic
69	C₃²×Q₈	D₈	1	1	yes	characteristic
70	C₃²×D₄	D₈	1	3	yes	—
71	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
72	C₃×#67	—	4	6	no	—
73	C₂×C₃×Dic₃	—	4	6	no	—
74	C₃×D₁₂	—	2	2	no	—
75	C₃×C₄×D₃	—	4	6	no	—
76	C₃×Dic₆	—	2	6	no	—
77	C₃×Dic₆	—	4	6	no	—
78	C₃²×C₈	C₂×D₄	1	3	yes	—
79	C₃²×C₈	C₂⁴	1	1	yes	characteristic
80	#578	—	3	2	no	—
81	#510	—	3	6	no	—
82	#508	—	3	3	no	—
83	C₂×Q₃₂	—	3	3	no	—
84	#504	C₃×D₃	1	1	yes	characteristic, complemented
85	C₃×#41	—	2	1	no	—
86	C₃×#41	—	6	6	no	—
87	C₃×#41	—	12	6	no	—
88	C₃×#41	D₁₂	1	1	yes	characteristic
89	C₃×#41	C₃×D₄	1	1	yes	characteristic
90	C₂×C₃×Q₈	—	6	6	no	—
91	C₂×C₃×Q₈	—	12	2	no	—
92	D₃×Q₈	—	4	2	no	—
93	#237	—	4	6	no	—
94	#235	—	2	6	no	—
95	C₂×Dic₆	—	2	6	no	—
96	C₃×Q₁₆	—	3	6	no	—
97	C₃×Q₁₆	—	6	6	no	—
98	C₃×SD₁₆	—	6	6	no	—
99	C₃×D₈	—	3	2	no	—
100	C₃×M₁₆	—	2	3	no	—
101	C₃×M₁₆	D₁₂	1	3	yes	—
102	C₃×M₁₆	C₃×D₄	1	3	yes	—
103	C₂×C₃×C₈	—	2	3	no	—
104	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
105	C₂×C₃×C₈	C₂³×C₃	1	3	yes	—
106	Dic₁₂	—	2	6	no	—
107	Dic₁₂	C₂³×C₃	1	6	yes	—
108	D₂₄	C₂³×C₃	1	2	yes	—
109	#204	—	2	6	no	—
110	C₃×C₁₆	—	2	1	no	—
111	C₃×C₁₆	—	2	3	no	—
112	C₃×C₁₆	C₂²×D₃	1	1	yes	characteristic
113	C₃×C₁₆	C₂²×D₃	1	3	yes	—
114	C₃×C₁₆	C₂³×C₃	1	1	yes	characteristic
115	C₃×C₁₆	C₂³×C₃	1	3	yes	—
116	C₂²×C₃²	C₂×D₈	1	3	yes	—
117	C₂×C₃×D₃	—	4	2	no	—
118	C₃²×C₄	C₂×D₈	1	3	yes	—
119	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₃×Dic₃	—	4	6	no	—
121	2₋¹⁺⁴	—	6	2	no	—
122	#137	—	6	6	no	—
123	#135	—	3	6	no	—
124	C₂×Q₁₆	—	3	6	no	—
125	#131	C₂×C₃×D₃	1	1	yes	characteristic
126	Q₃₂	—	3	3	no	—
127	Q₃₂	—	3	6	no	—
128	SD₃₂	—	3	6	no	—
129	D₁₆	—	3	1	no	—
130	M₃₂	C₂×C₃×D₃	1	3	yes	—
131	C₂×C₁₆	C₂×C₃×D₃	1	3	yes	—
132	C₃×Q₈	—	2	1	no	—
133	C₃×Q₈	—	6	6	no	—
134	C₃×Q₈	—	12	6	no	—
135	C₃×Q₈	D₂₄	1	1	yes	characteristic
136	C₃×Q₈	C₃×D₈	1	1	yes	characteristic
137	C₃×D₄	—	2	3	no	—
138	C₃×D₄	—	6	2	no	—
139	C₃×D₄	—	12	6	no	—
140	C₃×D₄	D₂₄	1	3	yes	—
141	C₃×D₄	C₃×D₈	1	3	yes	—
142	C₂×C₃×C₄	—	2	3	no	—
143	C₂×C₃×C₄	—	12	6	no	—
144	C₂×C₃×C₄	—	12	6	no	—
145	C₂×C₃×C₄	C₂×D₁₂	1	3	yes	—
146	C₂×C₃×C₄	C₂×C₃×D₄	1	3	yes	—
147	#67	—	4	6	no	—
148	C₂×Dic₃	—	4	6	no	—
149	D₁₂	—	2	2	no	—
150	C₄×D₃	—	4	6	no	—
151	Dic₆	—	2	6	no	—
152	Dic₆	—	4	6	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	2	3	no	—
155	C₃×C₈	C₂×D₁₂	1	3	yes	—
156	C₃×C₈	C₂×C₃×D₄	1	3	yes	—
157	C₃×C₈	C₂³×D₃	1	1	yes	characteristic
158	C₃×C₈	C₂⁴×C₃	1	1	yes	characteristic
159	C₂×C₃²	—	2	3	no	—
160	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
161	C₃×D₃	—	8	2	no	—
162	#41	—	6	6	no	—
163	#41	—	12	6	no	—
164	#41	C₃×D₁₂	1	1	yes	characteristic
165	C₂×Q₈	—	6	6	no	—
166	C₂×Q₈	—	12	2	no	—
167	Q₁₆	—	3	6	no	—
168	Q₁₆	—	6	6	no	—
169	SD₁₆	—	6	6	no	—
170	D₈	—	3	2	no	—
171	M₁₆	C₃×D₁₂	1	3	yes	—
172	C₂×C₈	C₂²×C₃×D₃	1	3	yes	—
173	C₁₆	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₁₆	C₂²×C₃×D₃	1	3	yes	—
175	C₂²×C₃	—	2	3	no	—
176	C₂²×C₃	—	12	2	no	—
177	C₂²×C₃	C₂×D₂₄	1	3	yes	—
178	C₂²×C₃	C₂×C₃×D₈	1	3	yes	—
179	C₂×D₃	—	4	2	no	—
180	C₃×C₄	—	2	3	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	12	6	no	—
183	C₃×C₄	C₂×D₂₄	1	3	yes	—
184	C₃×C₄	C₂×C₃×D₈	1	3	yes	—
185	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
186	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
187	Dic₃	—	4	6	no	—
188	C₃²	#3417	1	1	yes	characteristic, complemented, Sylow
189	Q₈	—	6	6	no	—
190	Q₈	—	12	6	no	—
191	Q₈	C₃×D₂₄	1	1	yes	characteristic
192	D₄	—	6	2	no	—
193	D₄	—	12	6	no	—
194	D₄	C₃×D₂₄	1	3	yes	—
195	C₂×C₄	—	12	6	no	—
196	C₂×C₄	—	12	6	no	—
197	C₂×C₄	C₂×C₃×D₁₂	1	3	yes	—
198	C₈	C₂×C₃×D₁₂	1	3	yes	—
199	C₈	C₂³×C₃×D₃	1	1	yes	characteristic
200	C₂×C₃	—	2	3	no	—
201	C₂×C₃	—	2	3	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	4	3	no	—
204	C₂×C₃	—	24	2	no	—
205	C₂×C₃	C₂²×D₂₄	1	1	yes	characteristic
206	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
207	D₃	—	8	2	no	—
208	C₂²	—	12	2	no	—
209	C₂²	C₂×C₃×D₂₄	1	3	yes	—
210	C₄	—	12	6	no	—
211	C₄	C₂×C₃×D₂₄	1	3	yes	—
212	C₄	C₂²×C₃×D₁₂	1	1	yes	characteristic
213	C₃	—	2	1	no	—
214	C₃	#82015	1	1	yes	characteristic, direct product factor
215	C₃	C₃×#3417	1	1	yes	characteristic, complemented
216	C₂	—	2	3	no	—
217	C₂	—	24	2	no	—
218	C₂	C₂²×C₃×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₂₄
commutator, cosocle	C₃×C₈	C₂⁴×C₃
Frattini	C₈	C₂³×C₃×D₃
Fitting	C₃²×#504	C₂
socle	C₂×C₃²	C₂²×D₈
hypercenter	C₃×#504	D₃
hypocenter	C₃	C₃×#3417
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×#131 ◁ C₃×#504

Complements

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

N	H ≅ G/N	type
1. C₃×#5753	C₂	semidirect
2. C₃²×#504	C₂	semidirect
3. C₃×#4909	C₂	semidirect
4. C₂×C₃×Dic₂₄	C₂	semidirect
5. C₃×#4904	C₂	semidirect
7. #82015	C₃	direct (N × H)
20. #5753	C₂×C₃	semidirect
25. C₃×#504	D₃	semidirect
26. C₃×#504	C₂×C₃	semidirect
27. #4909	C₂×C₃	semidirect
28. C₂×Dic₂₄	C₂×C₃	semidirect
29. #4904	C₂×C₃	semidirect
84. #504	C₃×D₃	semidirect
188. C₃²	#3417	semidirect
214. C₃	#82015	direct (N × H)
215. C₃	C₃×#3417	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5753	2
C₃²×#504	2
C₃×#4909	2
C₂×C₃×Dic₂₄	2
C₃×#4904	2
C₃×#3417	3
#82015	3

Subgroups — C3×#82015 #11899766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#82015 #11899766