Subgroups — C₂×C₃²×C₄⋊(C₃⋊C₈) #13162359

Order 1728.

1260 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₃²×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄⋊C₈	—	3	1	no	maximal
5	C₂×C₃×C₄⋊(C₃⋊C₈)	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₃³×C₄²	C₄	1	2	yes	—
10	C₃³×C₄²	C₂²	1	2	yes	—
11	C₂×C₃²×C₃⋊C₈	—	2	4	no	—
12	C₂×C₃²×C₃⋊C₈	C₂²	1	4	yes	—
13	C₂²×C₃²×C₈	—	3	2	no	—
14	C₂×C₃²×C₄²	—	2	4	no	—
15	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
16	C₂×C₃²×C₄²	C₂×C₃	1	4	yes	—
17	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	8	yes	—
18	C₃²×C₄⋊C₈	—	3	4	no	—
19	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	16	yes	direct product factor
20	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃³×C₄	—	2	2	no	—
22	C₂×C₃³×C₄	C₈	1	4	yes	complemented
23	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
24	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃³×C₄	D₄	1	2	yes	—
27	C₂×C₃³×C₄	Q₈	1	2	yes	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₃²×C₃⋊C₈	—	2	4	no	—
30	C₂×C₃×C₄⋊C₈	—	3	4	no	—
31	C₂×C₄⋊(C₃⋊C₈)	C₃²	1	1	yes	characteristic, direct product factor
32	C₂²×C₃²×C₄	—	2	4	no	—
33	C₂²×C₃²×C₄	—	2	4	no	—
34	C₂²×C₃²×C₄	—	2	4	no	—
35	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
36	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
38	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
41	C₂×C₃²×C₈	—	3	4	no	—
42	C₂×C₃²×C₈	—	6	4	no	—
43	C₃²×C₄²	—	2	8	no	—
44	C₃²×C₄²	—	2	8	no	—
45	C₃²×C₄²	Dic₃	1	2	yes	—
46	C₃²×C₄²	C₃×C₄	1	8	yes	—
47	C₃²×C₄²	C₂×D₃	1	2	yes	—
48	C₃²×C₄²	C₂²×C₃	1	8	yes	—
49	C₂×C₃×C₃⋊C₈	—	2	16	no	—
50	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	16	yes	—
51	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
52	C₂²×C₃³	C₂×C₈	1	2	yes	—
53	C₂²×C₃³	M₁₆	1	2	yes	—
54	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
55	C₃³×C₄	—	2	2	no	—
56	C₃³×C₄	C₄⋊C₄	1	2	yes	—
57	C₃³×C₄	C₂×C₈	1	4	yes	complemented
58	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
59	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₃×C₈	—	3	8	no	—
61	C₂×C₃×C₄²	—	2	4	no	—
62	C₂×C₃×C₄²	C₃×D₃	1	4	yes	—
63	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	C₂×C₃²	1	2	yes	—
65	C₃×C₄⋊C₈	—	3	16	no	—
66	C₄⋊(C₃⋊C₈)	C₂×C₃²	1	4	yes	direct product factor
67	C₂³×C₃²	—	2	4	no	—
68	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
69	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
70	C₂×C₃²×C₄	—	2	16	no	—
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	8	no	—
73	C₂×C₃²×C₄	—	2	16	no	—
74	C₂×C₃²×C₄	—	2	8	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	8	no	—
77	C₂×C₃²×C₄	—	2	8	no	—
78	C₂×C₃²×C₄	—	2	4	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	complemented
81	C₂×C₃²×C₄	C₃×C₈	1	16	yes	complemented
82	C₂×C₃²×C₄	Dic₆	1	2	yes	—
83	C₂×C₃²×C₄	D₁₂	1	2	yes	—
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
86	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
88	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
90	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
91	C₂×C₃²×C₄	C₃×Q₈	1	8	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
94	C₃²×C₈	—	6	4	no	—
95	C₃×C₃⋊C₈	—	2	16	no	—
96	C₂×C₄⋊C₈	—	3	1	no	Sylow
97	C₂×C₃³	C₄⋊C₈	1	4	yes	complemented
98	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	2	4	no	—
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	—	2	4	no	—
104	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
105	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
106	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
109	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
110	C₂×C₃×C₈	—	3	16	no	—
111	C₂×C₃×C₈	—	6	16	no	—
112	C₃×C₄²	—	2	8	no	—
113	C₃×C₄²	—	2	8	no	—
114	C₃×C₄²	C₃×Dic₃	1	8	yes	—
115	C₃×C₄²	C₃²×C₄	1	2	yes	—
116	C₃×C₄²	C₂×C₃×D₃	1	8	yes	—
117	C₃×C₄²	C₂²×C₃²	1	2	yes	—
118	C₂×C₃⋊C₈	—	2	4	no	—
119	C₂×C₃⋊C₈	C₂²×C₃²	1	4	yes	—
120	C₂²×C₃²	—	2	8	no	—
121	C₂²×C₃²	—	2	8	no	—
122	C₂²×C₃²	—	2	8	no	—
123	C₂²×C₃²	—	2	4	no	—
124	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
125	C₂²×C₃²	#208	1	2	yes	—
126	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
127	C₂²×C₃²	C₃×C₄⋊C₄	1	8	yes	—
128	C₂²×C₃²	C₂×C₃×C₈	1	8	yes	—
129	C₂²×C₃²	C₃×M₁₆	1	8	yes	—
130	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
131	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
132	C₃²×C₄	—	2	16	no	—
133	C₃²×C₄	—	2	2	no	—
134	C₃²×C₄	—	2	8	no	—
135	C₃²×C₄	—	2	16	no	—
136	C₃²×C₄	—	2	8	no	—
137	C₃²×C₄	—	2	4	no	—
138	C₃²×C₄	—	2	4	no	—
139	C₃²×C₄	C₂×C₃⋊C₈	1	4	yes	complemented
140	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
141	C₃²×C₄	C₃×C₄⋊C₄	1	8	yes	—
142	C₃²×C₄	C₂×C₃×C₈	1	16	yes	complemented
143	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
144	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
145	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
146	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
147	C₂²×C₈	—	3	2	no	—
148	C₂×C₄²	C₃²×D₃	1	1	yes	characteristic
149	C₄⋊C₈	—	3	4	no	—
150	C₃³	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	—	2	4	no	—
152	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
153	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
154	C₂×C₃×C₄	—	2	16	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	16	no	—
158	C₂×C₃×C₄	—	2	8	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	C₃×C₃⋊C₈	1	16	yes	complemented
165	C₂×C₃×C₄	C₃²×C₈	1	4	yes	complemented
166	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
167	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
168	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
169	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
170	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
171	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
174	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
175	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
177	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
178	C₃×C₈	—	6	16	no	—
179	C₃⋊C₈	—	2	4	no	—
180	C₂×C₃²	—	2	4	no	—
181	C₂×C₃²	—	2	4	no	—
182	C₂×C₃²	—	2	4	no	—
183	C₂×C₃²	—	2	16	no	—
184	C₂×C₃²	C₄⋊(C₃⋊C₈)	1	4	yes	direct product factor
185	C₂×C₃²	C₃×C₄⋊C₈	1	16	yes	complemented
186	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
187	C₂×C₃²	C₂×#208	1	1	yes	characteristic
188	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
189	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
190	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
191	C₂×C₃²	C₂×C₃×M₁₆	1	4	yes	—
192	C₂²×C₄	C₃²×Dic₃	1	1	yes	characteristic
193	C₂²×C₄	C₃²×Dic₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
195	C₂×C₈	—	3	4	no	—
196	C₂×C₈	—	6	4	no	—
197	C₄²	C₃²×Dic₃	1	2	yes	—
198	C₄²	C₂×C₃²×D₃	1	2	yes	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	—	2	8	no	—
203	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	8	yes	—
204	C₂²×C₃	C₃×#208	1	8	yes	—
205	C₂²×C₃	C₃×C₄⋊Dic₃	1	8	yes	—
206	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
207	C₂²×C₃	C₂×C₃²×C₈	1	2	yes	—
208	C₂²×C₃	C₃²×M₁₆	1	2	yes	—
209	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
210	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
211	C₃×C₄	—	2	16	no	—
212	C₃×C₄	—	2	8	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	16	no	—
217	C₃×C₄	—	2	8	no	—
218	C₃×C₄	C₂×C₃×C₃⋊C₈	1	16	yes	complemented
219	C₃×C₄	C₃×C₄⋊Dic₃	1	8	yes	—
220	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
221	C₃×C₄	C₂×C₃²×C₈	1	4	yes	complemented
222	C₃×C₄	C₂×C₃×Dic₆	1	4	yes	—
223	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
224	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
225	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
226	C₃²	—	2	4	no	—
227	C₃²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor
228	C₃²	C₂×C₃×C₄⋊C₈	1	4	yes	complemented
229	C₂³	C₂×C₃²×Dic₃	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	C₃²×C₃⋊C₈	1	4	yes	complemented
232	C₂×C₄	C₃²×Dic₆	1	2	yes	—
233	C₂×C₄	C₃²×D₁₂	1	2	yes	—
234	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
235	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
236	C₂×C₄	C₂×C₃²×Dic₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
238	C₈	—	6	4	no	—
239	C₂×C₃	—	2	4	no	—
240	C₂×C₃	—	2	4	no	—
241	C₂×C₃	—	2	4	no	—
242	C₂×C₃	—	2	16	no	—
243	C₂×C₃	C₃×C₄⋊(C₃⋊C₈)	1	16	yes	direct product factor
244	C₂×C₃	C₃²×C₄⋊C₈	1	4	yes	complemented
245	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	4	yes	—
246	C₂×C₃	C₂×C₃×#208	1	4	yes	—
247	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
248	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
250	C₂×C₃	C₂×C₃²×M₁₆	1	1	yes	characteristic
251	C₂²	C₂×C₃²×C₃⋊C₈	1	2	yes	—
252	C₂²	C₃²×#208	1	2	yes	—
253	C₂²	C₃²×C₄⋊Dic₃	1	2	yes	—
254	C₂²	C₂²×C₃²×Dic₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	C₂×C₃²×C₃⋊C₈	1	4	yes	complemented
257	C₄	C₃²×C₄⋊Dic₃	1	2	yes	—
258	C₄	C₂×C₃²×Dic₆	1	1	yes	characteristic
259	C₄	C₂×C₃²×D₁₂	1	1	yes	characteristic
260	C₃	—	2	4	no	—
261	C₃	C₂×C₃×C₄⋊(C₃⋊C₈)	1	4	yes	direct product factor
262	C₃	C₂×C₃²×C₄⋊C₈	1	1	yes	characteristic, complemented
263	C₂	C₃²×C₄⋊(C₃⋊C₈)	1	4	yes	direct product factor
264	C₂	C₂²×C₃²×C₃⋊C₈	1	1	yes	characteristic
265	C₂	C₂×C₃²×#208	1	1	yes	characteristic
266	C₂	C₂×C₃²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₃²×C₈
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄²	C₂
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₂×C₃²×C₄²	D₃
hypocenter	C₃	C₂×C₃²×C₄⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄⋊(C₃⋊C₈)	C₂	direct (N × H)
5. C₂×C₃×C₄⋊(C₃⋊C₈)	C₃	direct (N × H)
19. C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	direct (N × H)
22. C₂×C₃³×C₄	C₈	semidirect
31. C₂×C₄⋊(C₃⋊C₈)	C₃²	direct (N × H)
57. C₃³×C₄	C₂×C₈	semidirect
66. C₄⋊(C₃⋊C₈)	C₂×C₃²	direct (N × H)
80. C₂×C₃²×C₄	C₃⋊C₈	semidirect
81. C₂×C₃²×C₄	C₃×C₈	semidirect
97. C₂×C₃³	C₄⋊C₈	semidirect
139. C₃²×C₄	C₂×C₃⋊C₈	semidirect
142. C₃²×C₄	C₂×C₃×C₈	semidirect
150. C₃³	C₂×C₄⋊C₈	semidirect
164. C₂×C₃×C₄	C₃×C₃⋊C₈	semidirect
165. C₂×C₃×C₄	C₃²×C₈	semidirect
184. C₂×C₃²	C₄⋊(C₃⋊C₈)	direct (N × H)
185. C₂×C₃²	C₃×C₄⋊C₈	semidirect
218. C₃×C₄	C₂×C₃×C₃⋊C₈	semidirect
221. C₃×C₄	C₂×C₃²×C₈	semidirect
227. C₃²	C₂×C₄⋊(C₃⋊C₈)	direct (N × H)
228. C₃²	C₂×C₃×C₄⋊C₈	semidirect
231. C₂×C₄	C₃²×C₃⋊C₈	semidirect
243. C₂×C₃	C₃×C₄⋊(C₃⋊C₈)	direct (N × H)
244. C₂×C₃	C₃²×C₄⋊C₈	semidirect
256. C₄	C₂×C₃²×C₃⋊C₈	semidirect
261. C₃	C₂×C₃×C₄⋊(C₃⋊C₈)	direct (N × H)
262. C₃	C₂×C₃²×C₄⋊C₈	semidirect
263. C₂	C₃²×C₄⋊(C₃⋊C₈)	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×C₃³×C₄²	2
C₂²×C₃²×C₃⋊C₈	2
C₃²×C₄⋊(C₃⋊C₈)	2
C₂×C₃²×C₄⋊C₈	3
C₂×C₃×C₄⋊(C₃⋊C₈)	3

Subgroups — C2×C32×C4⋊(C3⋊C8) #13162359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄⋊(C₃⋊C₈) #13162359