Subgroups — C₃²×C₄×C₄⋊Dic₃ #13162368

Order 1728.

1596 conjugacy classes of subgroups.

The subgroup lattice has 282 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	C₂×C₃²×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂×C₃²×C₄×Dic₃	C₂	1	2	yes	maximal
4	C₃²×C₃⋊#95	C₂	1	2	yes	maximal
5	C₃²×C₄×C₄⋊C₄	—	3	1	no	maximal
6	C₃×C₄×C₄⋊Dic₃	C₃	1	4	yes	maximal, direct product factor
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×Dic₃	C₂²	1	4	yes	—
11	C₃³×C₄²	C₄	1	4	yes	complemented
12	C₃²×C₄⋊Dic₃	C₄	1	8	yes	direct product factor
13	C₃²×C₄×Dic₃	—	2	4	no	—
14	C₂×C₃²×C₄⋊C₄	—	3	2	no	—
15	C₂×C₃²×C₄²	—	2	4	no	—
16	C₂×C₃²×C₄²	—	3	2	no	—
17	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
18	C₂×C₃²×C₄²	C₂×C₃	1	4	yes	—
19	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	8	yes	—
20	C₂×C₃×C₄×Dic₃	C₂×C₃	1	8	yes	—
21	C₃²×#95	—	3	2	no	—
22	C₃×C₃⋊#95	C₂×C₃	1	8	yes	—
23	C₂³×C₃³	C₂³	1	1	yes	characteristic
24	C₂×C₃³×C₄	—	2	2	no	—
25	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
29	C₂×C₃³×C₄	D₄	1	2	yes	—
30	C₂×C₃³×C₄	Q₈	1	2	yes	—
31	C₂×C₃²×Dic₃	—	2	4	no	—
32	C₂×C₃²×Dic₃	—	2	4	no	—
33	C₂×C₃²×Dic₃	C₂×C₄	1	8	yes	—
34	C₃×C₄×C₄⋊C₄	—	3	4	no	—
35	C₄×C₄⋊Dic₃	C₃²	1	1	yes	characteristic, direct product factor
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	—	2	4	no	—
39	C₂²×C₃²×C₄	—	3	4	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
45	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
46	C₂²×C₃×Dic₃	C₂²×C₃	1	16	yes	—
47	C₃²×C₄⋊C₄	—	3	8	no	—
48	C₃²×C₄²	—	2	16	no	—
49	C₃²×C₄²	—	6	4	no	—
50	C₃²×C₄²	Dic₃	1	4	yes	complemented
51	C₃²×C₄²	C₃×C₄	1	16	yes	complemented
52	C₃×C₄⋊Dic₃	C₃×C₄	1	32	yes	direct product factor
53	C₃×C₄×Dic₃	—	2	16	no	—
54	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₃³	C₂²×C₄	1	2	yes	—
56	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₃³	#41	1	2	yes	—
59	C₃³×C₄	—	2	2	no	—
60	C₃³×C₄	C₄²	1	4	yes	complemented
61	C₃³×C₄	C₄⋊C₄	1	4	yes	complemented
62	C₃²×Dic₃	—	2	8	no	—
63	C₂×C₃×C₄⋊C₄	—	3	8	no	—
64	C₂×C₃×C₄²	—	2	4	no	—
65	C₂×C₃×C₄²	—	3	8	no	—
66	C₂×C₃×C₄²	C₃×D₃	1	4	yes	—
67	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	C₂×C₃²	1	2	yes	—
69	C₂×C₄×Dic₃	C₂×C₃²	1	2	yes	—
70	C₃×#95	—	3	8	no	—
71	C₃⋊#95	C₂×C₃²	1	2	yes	—
72	C₂³×C₃²	—	2	4	no	—
73	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
74	C₂³×C₃²	C₂³×C₃	1	4	yes	—
75	C₂×C₃²×C₄	—	2	16	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	8	no	—
78	C₂×C₃²×C₄	—	2	8	no	—
79	C₂×C₃²×C₄	—	2	16	no	—
80	C₂×C₃²×C₄	—	2	8	no	—
81	C₂×C₃²×C₄	—	2	8	no	—
82	C₂×C₃²×C₄	—	2	8	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	3	8	no	—
85	C₂×C₃²×C₄	—	6	4	no	—
86	C₂×C₃²×C₄	—	6	4	no	—
87	C₂×C₃²×C₄	Dic₆	1	2	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
89	C₂×C₃²×C₄	D₁₂	1	2	yes	—
90	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
91	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
92	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	16	yes	—
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
95	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
96	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
98	C₂×C₃²×C₄	C₃×Q₈	1	8	yes	—
99	C₂×C₃×Dic₃	—	2	16	no	—
100	C₂×C₃×Dic₃	—	2	16	no	—
101	C₂×C₃×Dic₃	C₂×C₃×C₄	1	32	yes	—
102	C₄×C₄⋊C₄	—	3	1	no	Sylow
103	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₃³	#117	1	1	yes	characteristic
106	C₂×C₃³	C₄×D₄	1	2	yes	—
107	C₂×C₃³	C₄×Q₈	1	2	yes	—
108	C₂²×C₃×C₄	—	2	4	no	—
109	C₂²×C₃×C₄	—	2	4	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	3	16	no	—
112	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
113	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
114	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
115	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
118	C₂²×Dic₃	C₂²×C₃²	1	4	yes	—
119	C₃×C₄⋊C₄	—	3	32	no	—
120	C₃×C₄²	—	2	16	no	—
121	C₃×C₄²	—	6	16	no	—
122	C₃×C₄²	C₃×Dic₃	1	16	yes	complemented
123	C₃×C₄²	C₃²×C₄	1	4	yes	complemented
124	C₄⋊Dic₃	C₃²×C₄	1	8	yes	direct product factor
125	C₄×Dic₃	—	2	4	no	—
126	C₂²×C₃²	—	2	4	no	—
127	C₂²×C₃²	—	2	8	no	—
128	C₂²×C₃²	—	2	4	no	—
129	C₂²×C₃²	—	2	8	no	—
130	C₂²×C₃²	—	2	4	no	—
131	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
132	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
133	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
134	C₂²×C₃²	#235	1	2	yes	—
135	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
136	C₂²×C₃²	C₂²×C₃×C₄	1	8	yes	—
137	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
138	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
139	C₂²×C₃²	C₂×C₃×Q₈	1	4	yes	—
140	C₂²×C₃²	C₃×#41	1	8	yes	—
141	C₃²×C₄	—	2	16	no	—
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	—	2	8	no	—
144	C₃²×C₄	—	2	16	no	—
145	C₃²×C₄	—	2	16	no	—
146	C₃²×C₄	—	6	8	no	—
147	C₃²×C₄	C₄×Dic₃	1	4	yes	complemented
148	C₃²×C₄	C₄⋊Dic₃	1	4	yes	direct product factor
149	C₃²×C₄	C₃×C₄²	1	16	yes	complemented
150	C₃²×C₄	C₃×C₄⋊C₄	1	16	yes	complemented
151	C₃×Dic₃	—	2	32	no	—
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄²	—	3	2	no	—
154	C₂×C₄²	C₃²×D₃	1	1	yes	characteristic
155	#95	—	3	2	no	—
156	C₃³	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₃	—	2	4	no	—
158	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
159	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	16	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	16	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	—	3	32	no	—
170	C₂×C₃×C₄	—	6	16	no	—
171	C₂×C₃×C₄	—	6	16	no	—
172	C₂×C₃×C₄	C₃×Dic₆	1	8	yes	—
173	C₂×C₃×C₄	C₃×C₄×D₃	1	16	yes	—
174	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
175	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
176	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
177	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
178	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
179	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
182	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
183	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
184	C₂×Dic₃	—	2	4	no	—
185	C₂×Dic₃	—	2	4	no	—
186	C₂×Dic₃	C₂×C₃²×C₄	1	8	yes	—
187	C₂×C₃²	—	2	4	no	—
188	C₂×C₃²	—	2	4	no	—
189	C₂×C₃²	—	2	4	no	—
190	C₂×C₃²	—	2	8	no	—
191	C₂×C₃²	—	2	8	no	—
192	C₂×C₃²	C₄×Dic₆	1	2	yes	—
193	C₂×C₃²	C₄×D₁₂	1	2	yes	—
194	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
195	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
196	C₂×C₃²	#926	1	1	yes	characteristic
197	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
198	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	4	yes	—
199	C₂×C₃²	C₃×#117	1	4	yes	—
200	C₂×C₃²	C₃×C₄×D₄	1	8	yes	—
201	C₂×C₃²	C₃×C₄×Q₈	1	8	yes	—
202	C₂²×C₄	—	3	4	no	—
203	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
206	C₄⋊C₄	—	3	8	no	—
207	C₄²	—	6	4	no	—
208	C₄²	C₃²×Dic₃	1	4	yes	complemented
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	8	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	C₂×C₃×Dic₆	1	4	yes	—
215	C₂²×C₃	C₂×C₃×C₄×D₃	1	8	yes	—
216	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
217	C₂²×C₃	C₃×#235	1	8	yes	—
218	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
219	C₂²×C₃	C₂²×C₃²×C₄	1	2	yes	—
220	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
222	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
223	C₂²×C₃	C₃²×#41	1	2	yes	—
224	C₃×C₄	—	2	16	no	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	16	no	—
228	C₃×C₄	—	2	16	no	—
229	C₃×C₄	—	6	32	no	—
230	C₃×C₄	C₃×C₄×Dic₃	1	16	yes	complemented
231	C₃×C₄	C₃×C₄⋊Dic₃	1	16	yes	direct product factor
232	C₃×C₄	C₃²×C₄²	1	4	yes	complemented
233	C₃×C₄	C₃²×C₄⋊C₄	1	4	yes	complemented
234	Dic₃	—	2	8	no	—
235	C₃²	—	2	4	no	—
236	C₃²	C₄×C₄⋊Dic₃	1	1	yes	characteristic, direct product factor
237	C₃²	C₃×C₄×C₄⋊C₄	1	4	yes	complemented
238	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	3	8	no	—
241	C₂×C₄	—	6	4	no	—
242	C₂×C₄	—	6	4	no	—
243	C₂×C₄	C₃²×Dic₆	1	2	yes	—
244	C₂×C₄	C₃²×C₄×D₃	1	4	yes	—
245	C₂×C₄	C₃²×D₁₂	1	2	yes	—
246	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
247	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
248	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	2	4	no	—
252	C₂×C₃	—	2	8	no	—
253	C₂×C₃	—	2	8	no	—
254	C₂×C₃	C₃×C₄×Dic₆	1	8	yes	—
255	C₂×C₃	C₃×C₄×D₁₂	1	8	yes	—
256	C₂×C₃	C₂×C₃×C₄×Dic₃	1	4	yes	—
257	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
258	C₂×C₃	C₃×#926	1	4	yes	—
259	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
260	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₃	C₃²×#117	1	1	yes	characteristic
262	C₂×C₃	C₃²×C₄×D₄	1	2	yes	—
263	C₂×C₃	C₃²×C₄×Q₈	1	2	yes	—
264	C₂²	C₂×C₃²×Dic₆	1	1	yes	characteristic
265	C₂²	C₂×C₃²×C₄×D₃	1	2	yes	—
266	C₂²	C₂×C₃²×D₁₂	1	1	yes	characteristic
267	C₂²	C₃²×#235	1	2	yes	—
268	C₂²	C₂²×C₃²×Dic₃	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	6	8	no	—
271	C₄	C₃²×C₄×Dic₃	1	4	yes	complemented
272	C₄	C₃²×C₄⋊Dic₃	1	4	yes	direct product factor
273	C₃	—	2	4	no	—
274	C₃	C₃×C₄×C₄⋊Dic₃	1	4	yes	direct product factor
275	C₃	C₃²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented
276	C₂	C₃²×C₄×Dic₆	1	2	yes	—
277	C₂	C₃²×C₄×D₁₂	1	2	yes	—
278	C₂	C₂×C₃²×C₄×Dic₃	1	1	yes	characteristic
279	C₂	C₂×C₃²×C₄⋊Dic₃	1	1	yes	characteristic
280	C₂	C₃²×#926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	C₂×D₃
commutator	C₂×C₃	C₂×C₃²×C₄²
Frattini	C₂³	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄²	C₂
socle, prime radical	C₂³×C₃³	C₂³
cosocle	C₂³×C₃	C₂³×C₃²
hypercenter	C₂×C₃²×C₄²	D₃
hypocenter	C₃	C₃²×C₄×C₄⋊C₄
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
6. C₃×C₄×C₄⋊Dic₃	C₃	direct (N × H)
11. C₃³×C₄²	C₄	semidirect
12. C₃²×C₄⋊Dic₃	C₄	direct (N × H)
35. C₄×C₄⋊Dic₃	C₃²	direct (N × H)
50. C₃²×C₄²	Dic₃	semidirect
51. C₃²×C₄²	C₃×C₄	semidirect
52. C₃×C₄⋊Dic₃	C₃×C₄	direct (N × H)
60. C₃³×C₄	C₄²	semidirect
61. C₃³×C₄	C₄⋊C₄	semidirect
122. C₃×C₄²	C₃×Dic₃	semidirect
123. C₃×C₄²	C₃²×C₄	semidirect
124. C₄⋊Dic₃	C₃²×C₄	direct (N × H)
147. C₃²×C₄	C₄×Dic₃	semidirect
148. C₃²×C₄	C₄⋊Dic₃	direct (N × H)
149. C₃²×C₄	C₃×C₄²	semidirect
150. C₃²×C₄	C₃×C₄⋊C₄	semidirect
156. C₃³	C₄×C₄⋊C₄	semidirect
208. C₄²	C₃²×Dic₃	semidirect
230. C₃×C₄	C₃×C₄×Dic₃	semidirect
231. C₃×C₄	C₃×C₄⋊Dic₃	direct (N × H)
232. C₃×C₄	C₃²×C₄²	semidirect
233. C₃×C₄	C₃²×C₄⋊C₄	semidirect
236. C₃²	C₄×C₄⋊Dic₃	direct (N × H)
237. C₃²	C₃×C₄×C₄⋊C₄	semidirect
271. C₄	C₃²×C₄×Dic₃	semidirect
272. C₄	C₃²×C₄⋊Dic₃	direct (N × H)
274. C₃	C₃×C₄×C₄⋊Dic₃	direct (N × H)
275. C₃	C₃²×C₄×C₄⋊C₄	semidirect

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
C₂×C₃²×C₄⋊Dic₃	2
C₂×C₃²×C₄×Dic₃	2
C₃²×C₃⋊#95	2
C₃²×C₄×C₄⋊C₄	3
C₃×C₄×C₄⋊Dic₃	3

Subgroups — C32×C4×C4⋊Dic3 #13162368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₄⋊Dic₃ #13162368