Subgroups — C₃×#87040 #11904240

Order 1152.

2510 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#5734	C₂	1	9	yes	maximal, complemented
3	C₃×#5725	C₂	1	9	yes	maximal, complemented
4	C₃×#5577	C₂	1	6	yes	maximal, complemented
5	C₃×C₄×#239	C₂	1	6	yes	maximal, complemented
6	C₃×#3483	—	3	1	no	maximal
7	#87040	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#239	C₂²	1	6	yes	—
9	C₂×C₃×#239	C₂²	1	6	yes	—
10	C₃²×C₄⋊Q₈	C₂²	1	9	yes	—
11	C₃²×C₄×Q₈	C₂²	1	6	yes	—
12	C₃×#947	C₂²	1	18	yes	—
13	C₃×Q₈×Dic₃	C₂²	1	2	yes	—
14	C₃×#895	C₂²	1	36	yes	—
15	C₃×#893	C₂²	1	18	yes	—
16	C₃×#892	C₂²	1	18	yes	—
17	C₃×#874	C₂²	1	9	yes	—
18	C₃×C₄×D₁₂	C₂²	1	18	yes	—
19	C₃×C₄²×D₃	C₂²	1	9	yes	—
20	C₃×Q₈²	—	2	1	no	—
21	C₃×Q₈²	D₃	1	1	yes	characteristic, complemented
22	C₃×Q₈²	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×#555	—	3	9	no	—
24	C₃×#550	—	3	6	no	—
25	C₃×#532	—	3	9	no	—
26	C₃×C₄×#41	—	3	6	no	—
27	#5734	C₂×C₃	1	9	yes	complemented
28	#5725	C₂×C₃	1	9	yes	complemented
29	#5577	C₂×C₃	1	6	yes	complemented
30	C₄×#239	C₂×C₃	1	6	yes	complemented
31	C₂×C₃²×Q₈	C₂³	1	6	yes	—
32	C₂×C₃²×Q₈	C₂³	1	2	yes	—
33	C₃×#239	—	2	24	no	—
34	C₃×#239	—	4	12	no	—
35	C₂×C₃×D₁₂	C₂³	1	18	yes	—
36	C₂×C₃×D₁₂	C₂³	1	18	yes	—
37	C₂×C₃×C₄×D₃	C₂³	1	18	yes	—
38	C₂×C₃×C₄×D₃	C₂³	1	9	yes	—
39	C₃²×C₄⋊C₄	C₂³	1	18	yes	—
40	C₃²×C₄²	C₂³	1	9	yes	—
41	C₃×(C₂×D₃)⋊C₄	C₂³	1	36	yes	—
42	C₃×C₄⋊Dic₃	C₂³	1	6	yes	—
43	C₃×C₄×Dic₃	C₂³	1	9	yes	—
44	C₃×C₄×Dic₃	C₂³	1	6	yes	—
45	#3483	—	3	1	no	Sylow
46	C₂×C₃×#41	—	3	6	no	—
47	C₂×C₃×#41	—	3	6	no	—
48	C₂×#239	C₂²×C₃	1	6	yes	—
49	C₂×#239	C₂²×C₃	1	6	yes	—
50	C₃×C₄⋊Q₈	—	2	9	no	—
51	C₃×C₄⋊Q₈	C₂×D₃	1	9	yes	—
52	C₃×C₄⋊Q₈	C₂²×C₃	1	9	yes	—
53	C₃×#127	—	3	9	no	—
54	C₃×#124	—	3	18	no	—
55	C₃×#121	—	3	36	no	—
56	C₃×C₄×Q₈	—	2	6	no	—
57	C₃×C₄×Q₈	—	3	2	no	—
58	C₃×C₄×Q₈	C₂×D₃	1	6	yes	—
59	C₃×C₄×Q₈	C₂²×C₃	1	6	yes	—
60	C₃×C₄×D₄	—	3	18	no	—
61	C₃×C₄×D₄	—	3	18	no	—
62	C₃×#117	—	3	18	no	—
63	C₂×C₃×C₄²	—	3	9	no	—
64	#947	C₂²×C₃	1	18	yes	—
65	Q₈×Dic₃	C₂²×C₃	1	2	yes	—
66	#895	C₂²×C₃	1	36	yes	—
67	#893	C₂²×C₃	1	18	yes	—
68	#892	C₂²×C₃	1	18	yes	—
69	#874	C₂²×C₃	1	9	yes	—
70	C₄×D₁₂	C₂²×C₃	1	18	yes	—
71	C₄²×D₃	C₂²×C₃	1	9	yes	—
72	C₂²×C₃×D₃	C₂⁴	1	9	yes	—
73	C₃²×Q₈	—	4	6	no	—
74	C₃²×Q₈	#41	1	8	yes	complemented
75	C₂×C₃²×C₄	C₂⁴	1	9	yes	—
76	C₂×C₃²×C₄	C₂⁴	1	6	yes	—
77	C₂×C₃×Dic₃	C₂⁴	1	6	yes	—
78	C₂×C₃×Dic₃	C₂⁴	1	1	yes	characteristic
79	C₃×D₁₂	—	2	36	no	—
80	C₃×D₁₂	—	4	18	no	—
81	C₃×C₄×D₃	—	2	36	no	—
82	C₃×C₄×D₃	—	2	18	no	—
83	Q₈²	C₃×D₃	1	1	yes	characteristic, complemented
84	#555	—	3	9	no	—
85	#550	—	3	6	no	—
86	#532	—	3	9	no	—
87	C₄×#41	—	3	6	no	—
88	C₃×#41	—	6	24	no	—
89	C₃×#41	—	12	12	no	—
90	C₂×C₃×Q₈	—	2	2	no	—
91	C₂×C₃×Q₈	—	2	6	no	—
92	C₂×C₃×Q₈	C₂²×D₃	1	6	yes	—
93	C₂×C₃×Q₈	C₂²×D₃	1	2	yes	—
94	C₂×C₃×Q₈	C₂³×C₃	1	6	yes	—
95	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
96	C₂×C₃×D₄	—	3	18	no	—
97	C₂×C₃×D₄	—	3	18	no	—
98	C₂²×C₃×C₄	—	3	18	no	—
99	C₂²×C₃×C₄	—	3	9	no	—
100	#239	—	2	24	no	—
101	#239	—	4	12	no	—
102	C₂×D₁₂	C₂³×C₃	1	18	yes	—
103	C₂×D₁₂	C₂³×C₃	1	18	yes	—
104	C₂×C₄×D₃	C₂³×C₃	1	18	yes	—
105	C₂×C₄×D₃	C₂³×C₃	1	9	yes	—
106	C₃×C₄⋊C₄	—	2	18	no	—
107	C₃×C₄⋊C₄	—	3	6	no	—
108	C₃×C₄⋊C₄	C₂²×D₃	1	18	yes	—
109	C₃×C₄⋊C₄	C₂³×C₃	1	18	yes	—
110	C₃×C₂²⋊C₄	—	3	36	no	—
111	C₃×C₄²	—	2	9	no	—
112	C₃×C₄²	—	3	6	no	—
113	C₃×C₄²	—	3	9	no	—
114	C₃×C₄²	C₂²×D₃	1	9	yes	—
115	C₃×C₄²	C₂³×C₃	1	9	yes	—
116	(C₂×D₃)⋊C₄	C₂³×C₃	1	36	yes	—
117	C₄⋊Dic₃	C₂³×C₃	1	6	yes	—
118	C₄×Dic₃	C₂³×C₃	1	6	yes	—
119	C₄×Dic₃	C₂³×C₃	1	9	yes	—
120	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
121	C₂×C₃×D₃	—	2	18	no	—
122	C₂×C₃×D₃	—	2	9	no	—
123	C₃²×C₄	—	2	9	no	—
124	C₃²×C₄	C₂×#41	1	12	yes	—
125	C₃×Dic₃	—	2	6	no	—
126	C₃×Dic₃	H(5,𝔽₂)	1	2	yes	—
127	C₂×#41	—	3	6	no	—
128	C₂×#41	—	3	6	no	—
129	C₄⋊Q₈	C₂×C₃×D₃	1	9	yes	—
130	#127	—	3	9	no	—
131	#124	—	3	18	no	—
132	#121	—	3	36	no	—
133	C₄×Q₈	—	3	2	no	—
134	C₄×Q₈	C₂×C₃×D₃	1	6	yes	—
135	C₄×D₄	—	3	18	no	—
136	C₄×D₄	—	3	18	no	—
137	#117	—	3	18	no	—
138	C₂×C₄²	—	3	9	no	—
139	C₂³×C₃	—	3	9	no	—
140	C₂²×D₃	C₂⁴×C₃	1	9	yes	—
141	C₃×Q₈	—	2	8	no	—
142	C₃×Q₈	—	4	6	no	—
143	C₃×Q₈	—	4	6	no	—
144	C₃×Q₈	—	8	6	no	—
145	C₃×Q₈	#239	1	8	yes	complemented
146	C₃×Q₈	C₃×#41	1	8	yes	complemented
147	C₃×D₄	—	6	36	no	—
148	C₃×D₄	—	12	18	no	—
149	C₂×C₃×C₄	—	2	6	no	—
150	C₂×C₃×C₄	—	2	9	no	—
151	C₂×C₃×C₄	—	3	1	no	—
152	C₂×C₃×C₄	—	3	6	no	—
153	C₂×C₃×C₄	—	6	18	no	—
154	C₂×C₃×C₄	—	6	36	no	—
155	C₂×C₃×C₄	C₂³×D₃	1	9	yes	—
156	C₂×C₃×C₄	C₂³×D₃	1	6	yes	—
157	C₂×C₃×C₄	C₂⁴×C₃	1	9	yes	—
158	C₂×C₃×C₄	C₂⁴×C₃	1	6	yes	—
159	C₂×Dic₃	C₂⁴×C₃	1	6	yes	—
160	C₂×Dic₃	C₂⁴×C₃	1	1	yes	characteristic
161	D₁₂	—	2	36	no	—
162	D₁₂	—	4	18	no	—
163	C₄×D₃	—	2	36	no	—
164	C₄×D₃	—	2	18	no	—
165	C₂×C₃²	C₂²×#41	1	2	yes	—
166	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
167	C₃×D₃	—	4	9	no	—
168	#41	—	6	24	no	—
169	#41	—	12	12	no	—
170	C₂×Q₈	C₂²×C₃×D₃	1	6	yes	—
171	C₂×Q₈	C₂²×C₃×D₃	1	2	yes	—
172	C₂×D₄	—	3	18	no	—
173	C₂×D₄	—	3	18	no	—
174	C₂²×C₄	—	3	18	no	—
175	C₂²×C₄	—	3	9	no	—
176	C₄⋊C₄	—	3	6	no	—
177	C₄⋊C₄	C₂²×C₃×D₃	1	18	yes	—
178	C₂²⋊C₄	—	3	36	no	—
179	C₄²	—	3	6	no	—
180	C₄²	—	3	9	no	—
181	C₄²	C₂²×C₃×D₃	1	9	yes	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	6	18	no	—
184	C₂²×C₃	—	6	9	no	—
185	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
186	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
187	C₂×D₃	—	2	18	no	—
188	C₂×D₃	—	2	9	no	—
189	C₃×C₄	—	2	9	no	—
190	C₃×C₄	—	2	9	no	—
191	C₃×C₄	—	2	12	no	—
192	C₃×C₄	—	3	2	no	—
193	C₃×C₄	—	4	9	no	—
194	C₃×C₄	—	6	6	no	—
195	C₃×C₄	C₂×#239	1	12	yes	—
196	C₃×C₄	C₂×C₃×#41	1	12	yes	—
197	Dic₃	—	2	6	no	—
198	Dic₃	C₃×H(5,𝔽₂)	1	2	yes	—
199	C₃²	#3483	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	3	9	no	—
201	Q₈	—	4	6	no	—
202	Q₈	C₃×#239	1	8	yes	complemented
203	D₄	—	6	36	no	—
204	D₄	—	12	18	no	—
205	C₂×C₄	—	3	1	no	—
206	C₂×C₄	—	3	6	no	—
207	C₂×C₄	—	6	18	no	—
208	C₂×C₄	—	6	36	no	—
209	C₂×C₄	C₂³×C₃×D₃	1	9	yes	—
210	C₂×C₄	C₂³×C₃×D₃	1	6	yes	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	12	9	no	—
214	C₂×C₃	C₂²×#239	1	2	yes	—
215	C₂×C₃	D₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×C₃	C₂²×C₃×#41	1	2	yes	—
217	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
218	D₃	—	4	9	no	—
219	C₂²	—	6	18	no	—
220	C₂²	—	6	9	no	—
221	C₂²	C₂⁴×C₃×D₃	1	1	yes	characteristic
222	C₄	—	2	9	no	—
223	C₄	—	3	2	no	—
224	C₄	—	6	6	no	—
225	C₄	C₂×C₃×#239	1	12	yes	—
226	C₃	—	2	1	no	—
227	C₃	#87040	1	1	yes	characteristic, direct product factor
228	C₃	C₃×#3483	1	1	yes	characteristic, complemented
229	C₂	—	12	9	no	—
230	C₂	C₂²×C₃×#239	1	2	yes	—
231	C₂	C₃×D₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂⁴×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₃
Fitting	C₃²×Q₈²	C₂
socle	C₂²×C₃²	C₂⁵
hypercenter	C₃×Q₈²	D₃
hypocenter	C₃	C₃×#3483
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×Q₈²

Complements

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

N	H ≅ G/N	type
1. C₃²×Q₈²	C₂	semidirect
2. C₃×#5734	C₂	semidirect
3. C₃×#5725	C₂	semidirect
4. C₃×#5577	C₂	semidirect
5. C₃×C₄×#239	C₂	semidirect
7. #87040	C₃	direct (N × H)
21. C₃×Q₈²	D₃	semidirect
22. C₃×Q₈²	C₂×C₃	semidirect
27. #5734	C₂×C₃	semidirect
28. #5725	C₂×C₃	semidirect
29. #5577	C₂×C₃	semidirect
30. C₄×#239	C₂×C₃	semidirect
74. C₃²×Q₈	#41	semidirect
83. Q₈²	C₃×D₃	semidirect
145. C₃×Q₈	#239	semidirect
146. C₃×Q₈	C₃×#41	semidirect
199. C₃²	#3483	semidirect
202. Q₈	C₃×#239	semidirect
227. C₃	#87040	direct (N × H)
228. C₃	C₃×#3483	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈²	2
C₃×#5734	2
C₃×#5725	2
C₃×#5577	2
C₃×C₄×#239	2
C₃×#3483	3
#87040	3

Subgroups — C3×#87040 #11904240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#87040 #11904240