Subgroups — C₂×C₃×#64364 #13181739

Order 1728.

5388 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64364	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×#3832	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×Q₈×C₃⋊D₃	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#3828	C₂	1	6	yes	maximal, complemented
6	C₂×#64364	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#1007	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	4	yes	—
9	C₂×C₃³×Q₈	C₂²	1	8	yes	complemented
10	C₂²×C₃³×C₄	C₂²	1	3	yes	—
11	C₂×C₃×#638	C₂²	1	4	yes	—
12	C₃×#3832	C₂²	1	32	yes	complemented
13	C₃×Q₈×C₃⋊D₃	C₂²	1	32	yes	complemented
14	C₃×#3828	C₂²	1	48	yes	complemented
15	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	6	yes	—
16	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	12	yes	—
17	C₂×C₃×#634	C₂²	1	6	yes	—
18	C₂²×C₃²×Q₈	—	2	4	no	—
19	C₂²×C₃²×Q₈	D₃	1	4	yes	complemented
20	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	#64364	C₂×C₃	1	16	yes	direct product factor
22	C₂×#3832	C₂×C₃	1	4	yes	complemented
23	C₂×Q₈×C₃⋊D₃	C₂×C₃	1	4	yes	complemented
24	C₂×#3828	C₂×C₃	1	6	yes	complemented
25	C₃×#1007	—	3	64	no	—
26	C₂×C₃×#239	—	3	16	no	—
27	C₂×C₃×D₃×Q₈	—	3	16	no	—
28	C₂×C₃×#235	—	3	24	no	—
29	C₂³×C₃³	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
31	C₃³×Q₈	C₂³	1	16	yes	—
32	C₂×C₃³×C₄	C₂³	1	6	yes	—
33	C₂×C₃³×C₄	C₂³	1	12	yes	—
34	C₃×#638	C₂³	1	16	yes	—
35	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
36	C₃×C₄⋊(C₃⋊D₃)	C₂³	1	24	yes	—
37	C₃×C₄×C₃⋊D₃	C₂³	1	48	yes	—
38	C₃×#634	C₂³	1	24	yes	—
39	C₂×C₃×2₋¹⁺⁴	—	9	1	no	—
40	C₂×#1007	—	3	4	no	—
41	C₂×C₃²×Q₈	—	2	16	no	—
42	C₂×C₃²×Q₈	—	2	32	no	—
43	C₂×C₃²×Q₈	C₂×D₃	1	16	yes	—
44	C₂×C₃²×Q₈	C₂×D₃	1	32	yes	complemented
45	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
46	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	complemented
47	C₂²×C₃²×C₄	—	2	12	no	—
48	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
49	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
50	C₂×#638	C₂²×C₃	1	4	yes	—
51	#3832	C₂²×C₃	1	32	yes	complemented
52	Q₈×C₃⋊D₃	C₂²×C₃	1	32	yes	complemented
53	#3828	C₂²×C₃	1	48	yes	complemented
54	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	6	yes	—
55	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	12	yes	—
56	C₂×#634	C₂²×C₃	1	6	yes	—
57	C₂×C₃×#67	—	3	16	no	—
58	C₃×#239	—	3	128	no	—
59	C₃×D₃×Q₈	—	3	128	no	—
60	C₃×#235	—	3	192	no	—
61	C₂×C₃×D₁₂	—	3	24	no	—
62	C₂×C₃×C₄×D₃	—	3	48	no	—
63	C₂×C₃×Dic₆	—	3	24	no	—
64	C₂²×C₃³	—	2	2	no	—
65	C₂²×C₃³	C₂⁴	1	2	yes	—
66	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
67	C₂×C₃×C₃⋊D₃	—	2	8	no	—
68	C₂×C₃×C₃⋊D₃	C₂⁴	1	8	yes	—
69	C₃³×C₄	C₂⁴	1	12	yes	—
70	C₃×C₃⋊Dic₃	C₂⁴	1	8	yes	—
71	C₃×2₋¹⁺⁴	—	9	16	no	—
72	C₂×C₃×#41	—	9	4	no	—
73	C₂×C₃×#41	—	9	6	no	—
74	C₂²×C₃×Q₈	—	2	4	no	—
75	C₂²×C₃×Q₈	—	9	4	no	—
76	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
77	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
78	#1007	—	3	64	no	—
79	C₂×#239	—	3	16	no	—
80	C₂×D₃×Q₈	—	3	16	no	—
81	C₂×#235	—	3	24	no	—
82	C₂³×C₃²	—	2	4	no	—
83	C₂³×C₃²	C₂²×D₃	1	4	yes	—
84	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₃⋊D₃	C₂³×C₃	1	4	yes	—
86	C₂²×C₃×D₃	—	3	16	no	—
87	C₃²×Q₈	—	2	64	no	—
88	C₃²×Q₈	C₂²×D₃	1	64	yes	—
89	C₃²×Q₈	C₂³×C₃	1	16	yes	—
90	C₂×C₃²×C₄	—	2	24	no	—
91	C₂×C₃²×C₄	—	2	48	no	—
92	C₂×C₃²×C₄	C₂²×D₃	1	24	yes	—
93	C₂×C₃²×C₄	C₂²×D₃	1	48	yes	—
94	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
95	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
96	#638	C₂³×C₃	1	16	yes	—
97	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
98	C₄⋊(C₃⋊D₃)	C₂³×C₃	1	24	yes	—
99	C₄×C₃⋊D₃	C₂³×C₃	1	48	yes	—
100	#634	C₂³×C₃	1	24	yes	—
101	C₃×#67	—	3	64	no	—
102	C₂×C₃×Dic₃	—	3	16	no	—
103	C₃×D₁₂	—	3	96	no	—
104	C₃×C₄×D₃	—	3	192	no	—
105	C₃×Dic₆	—	3	96	no	—
106	C₂×2₋¹⁺⁴	—	9	1	no	Sylow
107	C₂×C₃³	—	2	2	no	—
108	C₂×C₃³	2₋¹⁺⁴	1	2	yes	complemented
109	C₂×C₃³	C₂⁵	1	1	yes	characteristic
110	C₃×C₃⋊D₃	—	2	8	no	—
111	C₃×#41	—	9	32	no	—
112	C₃×#41	—	9	48	no	—
113	C₂×C₃×Q₈	—	2	16	no	—
114	C₂×C₃×Q₈	—	2	32	no	—
115	C₂×C₃×Q₈	—	9	32	no	—
116	C₂×C₃×Q₈	—	9	6	no	—
117	C₂×C₃×Q₈	C₂×C₃×D₃	1	16	yes	—
118	C₂×C₃×Q₈	C₂×C₃×D₃	1	32	yes	complemented
119	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	4	yes	—
120	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	8	yes	complemented
121	C₂×C₃×D₄	—	9	4	no	—
122	C₂×C₃×D₄	—	9	6	no	—
123	C₂²×C₃×C₄	—	2	12	no	—
124	C₂²×C₃×C₄	—	9	12	no	—
125	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
126	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
127	C₂×#67	—	3	16	no	—
128	#239	—	3	128	no	—
129	D₃×Q₈	—	3	128	no	—
130	#235	—	3	192	no	—
131	C₂×D₁₂	—	3	24	no	—
132	C₂×C₄×D₃	—	3	48	no	—
133	C₂×Dic₆	—	3	24	no	—
134	C₂²×C₃²	—	2	16	no	—
135	C₂²×C₃²	—	2	8	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	8	no	—
138	C₂²×C₃²	—	2	4	no	—
139	C₂²×C₃²	C₂³×D₃	1	8	yes	—
140	C₂²×C₃²	C₂³×D₃	1	4	yes	—
141	C₂²×C₃²	C₂⁴×C₃	1	2	yes	—
142	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
143	C₂×C₃⋊D₃	—	2	8	no	—
144	C₂×C₃⋊D₃	C₂⁴×C₃	1	8	yes	—
145	C₂×C₃×D₃	—	3	32	no	—
146	C₂×C₃×D₃	—	6	32	no	—
147	C₃²×C₄	—	2	48	no	—
148	C₃²×C₄	C₂³×D₃	1	48	yes	—
149	C₃²×C₄	C₂⁴×C₃	1	12	yes	—
150	C₃⋊Dic₃	C₂⁴×C₃	1	8	yes	—
151	C₃×Dic₃	—	3	32	no	—
152	2₋¹⁺⁴	—	9	16	no	—
153	C₂×#41	—	9	4	no	—
154	C₂×#41	—	9	6	no	—
155	C₂²×Q₈	—	9	4	no	—
156	C₂²×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
157	C₃³	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₃	—	2	4	no	—
159	C₂³×C₃	—	9	4	no	—
160	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
161	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₂²×D₃	—	3	16	no	—
163	C₃×Q₈	—	2	64	no	—
164	C₃×Q₈	—	9	24	no	—
165	C₃×Q₈	C₂²×C₃×D₃	1	64	yes	—
166	C₃×Q₈	C₂²×C₃⋊D₃	1	16	yes	—
167	C₃×D₄	—	9	16	no	—
168	C₃×D₄	—	9	24	no	—
169	C₂×C₃×C₄	—	2	24	no	—
170	C₂×C₃×C₄	—	2	48	no	—
171	C₂×C₃×C₄	—	9	4	no	—
172	C₂×C₃×C₄	—	9	48	no	—
173	C₂×C₃×C₄	C₂²×C₃×D₃	1	24	yes	—
174	C₂×C₃×C₄	C₂²×C₃×D₃	1	48	yes	—
175	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	12	yes	—
176	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	6	yes	—
177	#67	—	3	64	no	—
178	C₂×Dic₃	—	3	16	no	—
179	D₁₂	—	3	96	no	—
180	C₄×D₃	—	3	192	no	—
181	Dic₆	—	3	96	no	—
182	C₂×C₃²	—	2	16	no	—
183	C₂×C₃²	—	2	8	no	—
184	C₂×C₃²	—	2	2	no	—
185	C₂×C₃²	—	2	8	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	#1007	1	8	yes	complemented
188	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	complemented
189	C₂×C₃²	C₂⁴×D₃	1	4	yes	—
190	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
191	C₃⋊D₃	—	2	8	no	—
192	C₃×D₃	—	6	32	no	—
193	#41	—	9	32	no	—
194	#41	—	9	48	no	—
195	C₂×Q₈	—	9	32	no	—
196	C₂×Q₈	—	9	6	no	—
197	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	4	yes	—
198	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	8	yes	complemented
199	C₂×D₄	—	9	4	no	—
200	C₂×D₄	—	9	6	no	—
201	C₂²×C₄	—	9	12	no	—
202	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
203	C₂²×C₃	—	2	16	no	—
204	C₂²×C₃	—	2	8	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	8	no	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	9	8	no	—
209	C₂²×C₃	—	18	8	no	—
210	C₂²×C₃	C₂³×C₃×D₃	1	8	yes	—
211	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
212	C₂²×C₃	C₂³×C₃⋊D₃	1	2	yes	—
213	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
214	C₂×D₃	—	3	32	no	—
215	C₂×D₃	—	6	32	no	—
216	C₃×C₄	—	2	48	no	—
217	C₃×C₄	—	9	8	no	—
218	C₃×C₄	C₂³×C₃×D₃	1	48	yes	—
219	C₃×C₄	C₂³×C₃⋊D₃	1	12	yes	—
220	Dic₃	—	3	32	no	—
221	C₃²	—	2	4	no	—
222	C₃²	C₂×#1007	1	4	yes	complemented
223	C₃²	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented
224	C₂³	—	9	4	no	—
225	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
226	Q₈	—	9	24	no	—
227	Q₈	C₂²×C₃×C₃⋊D₃	1	16	yes	—
228	D₄	—	9	16	no	—
229	D₄	—	9	24	no	—
230	C₂×C₄	—	9	4	no	—
231	C₂×C₄	—	9	48	no	—
232	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	6	yes	—
233	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	12	yes	—
234	C₂×C₃	—	2	16	no	—
235	C₂×C₃	—	2	8	no	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	2	8	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	18	8	no	—
240	C₂×C₃	C₃×#1007	1	8	yes	complemented
241	C₂×C₃	#64364	1	2	yes	direct product factor
242	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
243	C₂×C₃	C₂⁴×C₃⋊D₃	1	1	yes	characteristic
244	D₃	—	6	32	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	9	8	no	—
247	C₂²	—	18	8	no	—
248	C₂²	C₂³×C₃×C₃⋊D₃	1	2	yes	—
249	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
250	C₄	—	9	8	no	—
251	C₄	C₂³×C₃×C₃⋊D₃	1	12	yes	—
252	C₃	—	2	4	no	—
253	C₃	C₂×C₃×#1007	1	4	yes	complemented
254	C₃	C₂×#64364	1	1	yes	characteristic, direct product factor
255	C₂	—	2	2	no	—
256	C₂	—	18	8	no	—
257	C₂	C₃×#64364	1	2	yes	direct product factor
258	C₂	C₂⁴×C₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁵×C₃
Frattini	C₂	C₂⁴×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₂²×C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
2. C₃×#64364	C₂	direct (N × H)
3. C₂×C₃×#3832	C₂	semidirect
4. C₂×C₃×Q₈×C₃⋊D₃	C₂	semidirect
5. C₂×C₃×#3828	C₂	semidirect
6. C₂×#64364	C₃	direct (N × H)
9. C₂×C₃³×Q₈	C₂²	semidirect
12. C₃×#3832	C₂²	semidirect
13. C₃×Q₈×C₃⋊D₃	C₂²	semidirect
14. C₃×#3828	C₂²	semidirect
19. C₂²×C₃²×Q₈	D₃	semidirect
20. C₂²×C₃²×Q₈	C₂×C₃	semidirect
21. #64364	C₂×C₃	direct (N × H)
22. C₂×#3832	C₂×C₃	semidirect
23. C₂×Q₈×C₃⋊D₃	C₂×C₃	semidirect
24. C₂×#3828	C₂×C₃	semidirect
44. C₂×C₃²×Q₈	C₂×D₃	semidirect
46. C₂×C₃²×Q₈	C₂²×C₃	semidirect
51. #3832	C₂²×C₃	semidirect
52. Q₈×C₃⋊D₃	C₂²×C₃	semidirect
53. #3828	C₂²×C₃	semidirect
76. C₂²×C₃×Q₈	C₃×D₃	semidirect
77. C₂²×C₃×Q₈	C₃⋊D₃	semidirect
108. C₂×C₃³	2₋¹⁺⁴	semidirect
118. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
120. C₂×C₃×Q₈	C₂×C₃⋊D₃	semidirect
156. C₂²×Q₈	C₃×C₃⋊D₃	semidirect
157. C₃³	C₂×2₋¹⁺⁴	semidirect
187. C₂×C₃²	#1007	semidirect
188. C₂×C₃²	C₃×2₋¹⁺⁴	semidirect
198. C₂×Q₈	C₂×C₃×C₃⋊D₃	semidirect
222. C₃²	C₂×#1007	semidirect
223. C₃²	C₂×C₃×2₋¹⁺⁴	semidirect
240. C₂×C₃	C₃×#1007	semidirect
241. C₂×C₃	#64364	direct (N × H)
253. C₃	C₂×C₃×#1007	semidirect
254. C₃	C₂×#64364	direct (N × H)
257. C₂	C₃×#64364	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×Q₈	2
C₃×#64364	2
C₂×C₃×#3832	2
C₂×C₃×Q₈×C₃⋊D₃	2
C₂×C₃×#3828	2
C₂×#64364	3
C₂×C₃×#1007	3

Subgroups — C2×C3×#64364 #13181739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64364 #13181739