Subgroups — C₂²×C₃²×#213 #13179175

Order 1728.

2972 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#213	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃²×D₈	—	3	1	no	maximal
6	C₂²×C₃×#213	C₃	1	4	yes	maximal, direct product factor
7	C₂⁴×C₃³	—	2	1	no	—
8	C₂³×C₃²×D₃	—	2	1	no	—
9	C₂×C₃³×D₄	—	2	3	no	—
10	C₂×C₃³×D₄	C₂²	1	6	yes	complemented
11	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃²×D₁₂	—	2	3	no	—
13	C₂×C₃²×D₁₂	C₂²	1	6	yes	complemented
14	C₃²×#213	C₂²	1	16	yes	direct product factor
15	C₂×C₃²×C₃⋊C₈	C₂²	1	6	yes	complemented
16	C₂²×C₃²×D₄	—	2	4	no	—
17	C₂²×C₃²×D₄	—	3	1	no	—
18	C₂²×C₃²×D₄	D₃	1	1	yes	characteristic, complemented
19	C₂²×C₃²×D₄	C₂×C₃	1	4	yes	complemented
20	C₂²×C₃×D₁₂	C₂×C₃	1	4	yes	complemented
21	C₂×C₃²×D₈	—	3	12	no	—
22	C₂²×C₃²×C₈	—	3	1	no	—
23	C₂×C₃×#213	C₂×C₃	1	48	yes	direct product factor
24	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	4	yes	complemented
25	C₂³×C₃³	—	2	6	no	—
26	C₂³×C₃³	—	4	4	no	—
27	C₂³×C₃³	D₄	1	1	yes	characteristic
28	C₂²×C₃²×D₃	—	2	6	no	—
29	C₂²×C₃²×D₃	—	4	4	no	—
30	C₃³×D₄	—	2	6	no	—
31	C₃³×D₄	C₂³	1	4	yes	complemented
32	C₂×C₃³×C₄	D₄	1	3	yes	—
33	C₂×C₃³×C₄	C₂³	1	3	yes	—
34	C₃²×D₁₂	—	2	6	no	—
35	C₃²×D₁₂	C₂³	1	4	yes	complemented
36	C₃²×C₃⋊C₈	C₂³	1	4	yes	complemented
37	C₂²×C₃×D₈	—	3	4	no	—
38	C₂²×#213	C₃²	1	1	yes	characteristic, direct product factor
39	C₂⁴×C₃²	—	2	8	no	—
40	C₂⁴×C₃²	—	2	1	no	—
41	C₂⁴×C₃²	—	2	4	no	—
42	C₂⁴×C₃²	—	6	1	no	—
43	C₂³×C₃×D₃	—	2	4	no	—
44	C₂×C₃²×D₄	—	2	24	no	—
45	C₂×C₃²×D₄	—	2	24	no	—
46	C₂×C₃²×D₄	—	2	3	no	—
47	C₂×C₃²×D₄	—	2	12	no	—
48	C₂×C₃²×D₄	—	3	6	no	—
49	C₂×C₃²×D₄	—	6	3	no	—
50	C₂×C₃²×D₄	C₂×D₃	1	6	yes	complemented
51	C₂×C₃²×D₄	C₂²×C₃	1	24	yes	complemented
52	C₂²×C₃²×C₄	—	2	4	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
55	C₂×C₃×D₁₂	—	2	12	no	—
56	C₂×C₃×D₁₂	C₂²×C₃	1	24	yes	complemented
57	C₃²×D₈	—	3	16	no	—
58	C₂×C₃²×C₈	—	3	6	no	—
59	C₃×#213	C₂²×C₃	1	64	yes	direct product factor
60	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	24	yes	complemented
61	C₂²×C₃³	—	2	4	no	—
62	C₂²×C₃³	—	4	12	no	—
63	C₂²×C₃³	D₈	1	4	yes	complemented
64	C₂²×C₃³	C₂×D₄	1	3	yes	—
65	C₂×C₃²×D₃	—	2	4	no	—
66	C₂×C₃²×D₃	—	4	12	no	—
67	C₃³×C₄	C₂×D₄	1	3	yes	—
68	C₃³×C₄	C₂⁴	1	1	yes	characteristic
69	C₂²×C₃×D₄	—	2	4	no	—
70	C₂²×C₃×D₄	—	3	4	no	—
71	C₂²×C₃×D₄	C₃×D₃	1	4	yes	complemented
72	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
73	C₂²×D₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₂×C₃×D₈	—	3	48	no	—
75	C₂²×C₃×C₈	—	3	4	no	—
76	C₂×#213	C₂×C₃²	1	12	yes	direct product factor
77	C₂²×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
78	C₂³×C₃²	—	2	48	no	—
79	C₂³×C₃²	—	2	6	no	—
80	C₂³×C₃²	—	2	24	no	—
81	C₂³×C₃²	—	2	4	no	—
82	C₂³×C₃²	—	4	32	no	—
83	C₂³×C₃²	—	4	4	no	—
84	C₂³×C₃²	—	4	16	no	—
85	C₂³×C₃²	—	6	6	no	—
86	C₂³×C₃²	—	12	4	no	—
87	C₂³×C₃²	#67	1	1	yes	characteristic
88	C₂³×C₃²	C₃×D₄	1	4	yes	—
89	C₂²×C₃×D₃	—	2	24	no	—
90	C₂²×C₃×D₃	—	4	16	no	—
91	C₃²×D₄	—	2	48	no	—
92	C₃²×D₄	—	2	6	no	—
93	C₃²×D₄	—	2	24	no	—
94	C₃²×D₄	—	2	16	no	—
95	C₃²×D₄	—	3	4	no	—
96	C₃²×D₄	—	6	6	no	—
97	C₃²×D₄	C₂²×D₃	1	4	yes	complemented
98	C₃²×D₄	C₂³×C₃	1	16	yes	complemented
99	C₂×C₃²×C₄	—	2	12	no	—
100	C₂×C₃²×C₄	—	2	12	no	—
101	C₂×C₃²×C₄	#67	1	3	yes	—
102	C₂×C₃²×C₄	C₃×D₄	1	12	yes	—
103	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
104	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
105	C₃×D₁₂	—	2	24	no	—
106	C₃×D₁₂	C₂³×C₃	1	16	yes	complemented
107	C₃²×C₈	—	3	4	no	—
108	C₃×C₃⋊C₈	C₂³×C₃	1	16	yes	complemented
109	C₂²×D₈	—	3	1	no	Sylow
110	C₂×C₃³	—	4	4	no	—
111	C₂×C₃³	C₂×D₈	1	6	yes	complemented
112	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
113	C₃²×D₃	—	4	4	no	—
114	C₂⁴×C₃	—	2	8	no	—
115	C₂⁴×C₃	—	2	4	no	—
116	C₂⁴×C₃	—	2	1	no	—
117	C₂⁴×C₃	—	6	4	no	—
118	C₂³×D₃	—	2	1	no	—
119	C₂×C₃×D₄	—	2	24	no	—
120	C₂×C₃×D₄	—	2	12	no	—
121	C₂×C₃×D₄	—	2	3	no	—
122	C₂×C₃×D₄	—	2	24	no	—
123	C₂×C₃×D₄	—	3	24	no	—
124	C₂×C₃×D₄	—	6	12	no	—
125	C₂×C₃×D₄	C₂×C₃×D₃	1	24	yes	complemented
126	C₂×C₃×D₄	C₂²×C₃²	1	6	yes	complemented
127	C₂²×C₃×C₄	—	2	4	no	—
128	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
129	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
130	C₂×D₁₂	—	2	3	no	—
131	C₂×D₁₂	C₂²×C₃²	1	6	yes	complemented
132	C₃×D₈	—	3	64	no	—
133	C₂×C₃×C₈	—	3	24	no	—
134	#213	C₂²×C₃²	1	16	yes	direct product factor
135	C₂×C₃⋊C₈	C₂²×C₃²	1	6	yes	complemented
136	C₂²×C₃²	—	2	32	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	2	16	no	—
139	C₂²×C₃²	—	2	16	no	—
140	C₂²×C₃²	—	2	12	no	—
141	C₂²×C₃²	—	4	96	no	—
142	C₂²×C₃²	—	4	12	no	—
143	C₂²×C₃²	—	4	48	no	—
144	C₂²×C₃²	—	6	4	no	—
145	C₂²×C₃²	—	12	12	no	—
146	C₂²×C₃²	#213	1	4	yes	direct product factor
147	C₂²×C₃²	C₃×D₈	1	16	yes	complemented
148	C₂²×C₃²	C₂×#67	1	3	yes	—
149	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
150	C₂×C₃×D₃	—	2	16	no	—
151	C₂×C₃×D₃	—	4	48	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	—	2	12	no	—
154	C₃²×C₄	C₂×#67	1	3	yes	—
155	C₃²×C₄	C₂×C₃×D₄	1	12	yes	—
156	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
157	C₃²×C₄	C₂⁴×C₃	1	4	yes	—
158	C₂²×D₄	—	3	1	no	—
159	C₂²×D₄	C₃²×D₃	1	1	yes	characteristic, complemented
160	C₂×D₈	—	3	12	no	—
161	C₂²×C₈	—	3	1	no	—
162	C₃³	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₃	—	2	48	no	—
164	C₂³×C₃	—	2	24	no	—
165	C₂³×C₃	—	2	6	no	—
166	C₂³×C₃	—	2	4	no	—
167	C₂³×C₃	—	4	32	no	—
168	C₂³×C₃	—	4	16	no	—
169	C₂³×C₃	—	4	4	no	—
170	C₂³×C₃	—	6	24	no	—
171	C₂³×C₃	—	12	16	no	—
172	C₂³×C₃	C₃×#67	1	4	yes	—
173	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
174	C₂²×D₃	—	2	6	no	—
175	C₂²×D₃	—	4	4	no	—
176	C₃×D₄	—	2	48	no	—
177	C₃×D₄	—	2	24	no	—
178	C₃×D₄	—	2	6	no	—
179	C₃×D₄	—	2	16	no	—
180	C₃×D₄	—	3	16	no	—
181	C₃×D₄	—	6	24	no	—
182	C₃×D₄	C₂²×C₃×D₃	1	16	yes	complemented
183	C₃×D₄	C₂³×C₃²	1	4	yes	complemented
184	C₂×C₃×C₄	—	2	12	no	—
185	C₂×C₃×C₄	—	2	12	no	—
186	C₂×C₃×C₄	C₃×#67	1	12	yes	—
187	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
188	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
189	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
190	D₁₂	—	2	6	no	—
191	D₁₂	C₂³×C₃²	1	4	yes	complemented
192	C₃×C₈	—	3	16	no	—
193	C₃⋊C₈	C₂³×C₃²	1	4	yes	complemented
194	C₂×C₃²	—	2	24	no	—
195	C₂×C₃²	—	2	4	no	—
196	C₂×C₃²	—	4	32	no	—
197	C₂×C₃²	—	4	16	no	—
198	C₂×C₃²	—	4	4	no	—
199	C₂×C₃²	—	12	4	no	—
200	C₂×C₃²	C₂×#213	1	6	yes	direct product factor
201	C₂×C₃²	C₂×C₃×D₈	1	24	yes	complemented
202	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
203	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
204	C₃×D₃	—	4	16	no	—
205	C₂⁴	—	2	1	no	—
206	C₂⁴	—	6	1	no	—
207	C₂×D₄	—	2	3	no	—
208	C₂×D₄	—	3	6	no	—
209	C₂×D₄	—	6	3	no	—
210	C₂×D₄	C₂×C₃²×D₃	1	6	yes	complemented
211	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
212	D₈	—	3	16	no	—
213	C₂×C₈	—	3	6	no	—
214	C₂²×C₃	—	2	32	no	—
215	C₂²×C₃	—	2	16	no	—
216	C₂²×C₃	—	2	4	no	—
217	C₂²×C₃	—	2	12	no	—
218	C₂²×C₃	—	2	16	no	—
219	C₂²×C₃	—	4	96	no	—
220	C₂²×C₃	—	4	48	no	—
221	C₂²×C₃	—	4	12	no	—
222	C₂²×C₃	—	6	16	no	—
223	C₂²×C₃	—	12	48	no	—
224	C₂²×C₃	C₃×#213	1	16	yes	direct product factor
225	C₂²×C₃	C₃²×D₈	1	4	yes	complemented
226	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
227	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
228	C₂×D₃	—	2	4	no	—
229	C₂×D₃	—	4	12	no	—
230	C₃×C₄	—	2	4	no	—
231	C₃×C₄	—	2	12	no	—
232	C₃×C₄	C₂×C₃×#67	1	12	yes	—
233	C₃×C₄	C₂×C₃²×D₄	1	3	yes	—
234	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
235	C₃×C₄	C₂⁴×C₃²	1	1	yes	characteristic
236	C₃²	—	2	4	no	—
237	C₃²	C₂²×#213	1	1	yes	characteristic, direct product factor
238	C₃²	C₂²×C₃×D₈	1	4	yes	complemented
239	C₂³	—	2	6	no	—
240	C₂³	—	4	4	no	—
241	C₂³	—	6	6	no	—
242	C₂³	—	12	4	no	—
243	C₂³	C₃²×#67	1	1	yes	characteristic
244	D₄	—	2	6	no	—
245	D₄	—	3	4	no	—
246	D₄	—	6	6	no	—
247	D₄	C₂²×C₃²×D₃	1	4	yes	complemented
248	C₂×C₄	C₃²×#67	1	3	yes	—
249	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
250	C₈	—	3	4	no	—
251	C₂×C₃	—	2	24	no	—
252	C₂×C₃	—	2	4	no	—
253	C₂×C₃	—	4	32	no	—
254	C₂×C₃	—	4	4	no	—
255	C₂×C₃	—	4	16	no	—
256	C₂×C₃	—	12	16	no	—
257	C₂×C₃	C₂×C₃×#213	1	24	yes	direct product factor
258	C₂×C₃	C₂×C₃²×D₈	1	6	yes	complemented
259	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
260	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
261	D₃	—	4	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	4	12	no	—
264	C₂²	—	6	4	no	—
265	C₂²	—	12	12	no	—
266	C₂²	C₃²×#213	1	4	yes	direct product factor
267	C₂²	C₂×C₃²×#67	1	3	yes	—
268	C₄	C₂×C₃²×#67	1	3	yes	—
269	C₄	C₂³×C₃²×D₃	1	1	yes	characteristic
270	C₃	—	2	4	no	—
271	C₃	C₂²×C₃×#213	1	4	yes	direct product factor
272	C₃	C₂²×C₃²×D₈	1	1	yes	characteristic, complemented
273	C₂	—	4	4	no	—
274	C₂	—	12	4	no	—
275	C₂	C₂×C₃²×#213	1	6	yes	direct product factor
276	C₂	C₂²×C₃²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃²	#67
commutator, cosocle	C₃×C₄	C₂⁴×C₃²
Frattini	C₄	C₂³×C₃²×D₃
Fitting	C₂²×C₃³×D₄	C₂
socle	C₂³×C₃³	D₄
hypercenter	C₂²×C₃²×D₄	D₃
hypocenter	C₃	C₂²×C₃²×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₂²×C₃²×D₁₂	C₂	semidirect
3. C₂×C₃²×#213	C₂	direct (N × H)
4. C₂²×C₃²×C₃⋊C₈	C₂	semidirect
6. C₂²×C₃×#213	C₃	direct (N × H)
10. C₂×C₃³×D₄	C₂²	semidirect
13. C₂×C₃²×D₁₂	C₂²	semidirect
14. C₃²×#213	C₂²	direct (N × H)
15. C₂×C₃²×C₃⋊C₈	C₂²	semidirect
18. C₂²×C₃²×D₄	D₃	semidirect
19. C₂²×C₃²×D₄	C₂×C₃	semidirect
20. C₂²×C₃×D₁₂	C₂×C₃	semidirect
23. C₂×C₃×#213	C₂×C₃	direct (N × H)
24. C₂²×C₃×C₃⋊C₈	C₂×C₃	semidirect
31. C₃³×D₄	C₂³	semidirect
35. C₃²×D₁₂	C₂³	semidirect
36. C₃²×C₃⋊C₈	C₂³	semidirect
38. C₂²×#213	C₃²	direct (N × H)
50. C₂×C₃²×D₄	C₂×D₃	semidirect
51. C₂×C₃²×D₄	C₂²×C₃	semidirect
56. C₂×C₃×D₁₂	C₂²×C₃	semidirect
59. C₃×#213	C₂²×C₃	direct (N × H)
60. C₂×C₃×C₃⋊C₈	C₂²×C₃	semidirect
63. C₂²×C₃³	D₈	semidirect
71. C₂²×C₃×D₄	C₃×D₃	semidirect
72. C₂²×C₃×D₄	C₂×C₃²	semidirect
73. C₂²×D₁₂	C₂×C₃²	semidirect
76. C₂×#213	C₂×C₃²	direct (N × H)
77. C₂²×C₃⋊C₈	C₂×C₃²	semidirect
97. C₃²×D₄	C₂²×D₃	semidirect
98. C₃²×D₄	C₂³×C₃	semidirect
106. C₃×D₁₂	C₂³×C₃	semidirect
108. C₃×C₃⋊C₈	C₂³×C₃	semidirect
111. C₂×C₃³	C₂×D₈	semidirect
125. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
126. C₂×C₃×D₄	C₂²×C₃²	semidirect
131. C₂×D₁₂	C₂²×C₃²	semidirect
134. #213	C₂²×C₃²	direct (N × H)
135. C₂×C₃⋊C₈	C₂²×C₃²	semidirect
146. C₂²×C₃²	#213	direct (N × H)
147. C₂²×C₃²	C₃×D₈	semidirect
159. C₂²×D₄	C₃²×D₃	semidirect
162. C₃³	C₂²×D₈	semidirect
182. C₃×D₄	C₂²×C₃×D₃	semidirect
183. C₃×D₄	C₂³×C₃²	semidirect
191. D₁₂	C₂³×C₃²	semidirect
193. C₃⋊C₈	C₂³×C₃²	semidirect
200. C₂×C₃²	C₂×#213	direct (N × H)
201. C₂×C₃²	C₂×C₃×D₈	semidirect
210. C₂×D₄	C₂×C₃²×D₃	semidirect
224. C₂²×C₃	C₃×#213	direct (N × H)
225. C₂²×C₃	C₃²×D₈	semidirect
237. C₃²	C₂²×#213	direct (N × H)
238. C₃²	C₂²×C₃×D₈	semidirect
247. D₄	C₂²×C₃²×D₃	semidirect
257. C₂×C₃	C₂×C₃×#213	direct (N × H)
258. C₂×C₃	C₂×C₃²×D₈	semidirect
266. C₂²	C₃²×#213	direct (N × H)
271. C₃	C₂²×C₃×#213	direct (N × H)
272. C₃	C₂²×C₃²×D₈	semidirect
275. C₂	C₂×C₃²×#213	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×D₄	2
C₂²×C₃²×D₁₂	2
C₂×C₃²×#213	2
C₂²×C₃²×C₃⋊C₈	2
C₂²×C₃²×D₈	3
C₂²×C₃×#213	3

Subgroups — C22×C32×#213 #13179175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×#213 #13179175