Subgroups — C₄×#3672⋊C₃ #13142757

Order 1728.

413 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×#3672⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₄×3₋¹⁺²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#63392⋊C₃	C₂	1	1	yes	maximal, characteristic
5	#63368⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#63367⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×#214	—	3	1	no	maximal
9	C₄×#3672	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×3₋¹⁺²	—	2	1	no	—
12	C₂×3₋¹⁺²⋊Q₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
15	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	#3672⋊C₃	C₄	1	4	yes	direct product factor
17	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	1	yes	characteristic
18	(3₋¹⁺²⋊C₄)⋊C₄	—	2	1	no	—
19	C₄×3₋¹⁺²⋊C₄	—	2	1	no	—
20	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₈	C₂²	1	1	yes	characteristic
22	C₃²×C₄×D₄	D₃	1	1	yes	characteristic
23	C₂×C₃×#214	—	3	1	no	—
24	C₃×C₄×Dic₆	—	3	1	no	—
25	C₃×#832	—	3	1	no	—
26	C₃×#810	—	3	1	no	—
27	C₃×#808	—	3	1	no	—
28	C₃×C₄×C₃⋊C₈	—	3	1	no	—
29	C₄×D₄×C₉	—	2	1	no	—
30	C₄×D₄×C₉	C₂×C₃	1	1	yes	characteristic
31	C₂×#3672	C₂×C₃	1	1	yes	characteristic
32	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#63392	C₂×C₃	1	1	yes	characteristic
34	#63368	C₂×C₃	1	1	yes	characteristic, complemented
35	#63367	C₂×C₃	1	1	yes	characteristic, complemented
36	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂³×3₋¹⁺²	—	2	1	no	—
38	D₄×3₋¹⁺²	—	2	1	no	—
39	D₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
40	C₂×C₄×3₋¹⁺²	—	2	1	no	—
41	C₂×C₄×3₋¹⁺²	—	4	2	no	—
42	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
43	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
44	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
45	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
46	C₂×3₋¹⁺²⋊C₄	—	2	1	no	—
47	3₋¹⁺²⋊Q₈	—	2	1	no	—
48	3₋¹⁺²⋊Q₈	C₂×C₄	1	2	yes	complemented
49	3₋¹⁺²⋊C₈	—	2	1	no	—
50	3₋¹⁺²⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₃×C₄×SD₁₆	—	9	1	no	—
52	C₄×#214	—	3	1	no	—
53	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃²×C₄	—	2	1	no	—
55	C₂×C₃×Dic₆	—	3	1	no	—
56	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₃²×C₂²⋊C₄	—	2	1	no	—
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₃×#214	—	3	4	no	—
60	C₃×C₄⋊Dic₃	—	3	1	no	—
61	C₃×Dic₃⋊C₄	—	6	1	no	—
62	C₃×C₄×Dic₃	—	6	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	—	3	1	no	—
65	C₂×D₄×C₉	—	2	1	no	—
66	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₄×C₉	—	2	2	no	—
68	C₂²×C₄×C₉	—	2	1	no	—
69	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₉×C₂²⋊C₄	—	2	2	no	—
73	C₉×C₂²⋊C₄	—	2	1	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
76	#3672	C₃×C₄	1	4	yes	complemented
77	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
78	Dic₉⋊C₄	—	2	1	no	—
79	C₄×Dic₉	—	2	1	no	—
80	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×3₋¹⁺²	—	2	2	no	—
83	C₂²×3₋¹⁺²	—	4	1	no	—
84	C₂²×3₋¹⁺²	—	4	1	no	—
85	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
86	C₄×3₋¹⁺²	—	2	1	no	—
87	C₄×3₋¹⁺²	—	4	1	no	—
88	C₄×3₋¹⁺²	SD₁₆	1	2	yes	complemented
89	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
90	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
91	3₋¹⁺²⋊C₄	—	2	2	no	—
92	3₋¹⁺²⋊C₄	—	4	1	no	—
93	C₂×C₃×SD₁₆	—	9	1	no	—
94	C₃×C₄×Q₈	—	9	1	no	—
95	C₃×C₄×D₄	—	3	1	no	—
96	C₃×C₄×D₄	C₃×D₃	1	1	yes	characteristic
97	C₂×#214	—	3	1	no	—
98	C₄×Dic₆	—	3	1	no	—
99	C₃×#106	—	9	1	no	—
100	C₃×#103	—	9	1	no	—
101	C₃×#102	—	9	1	no	—
102	C₃×C₄×C₈	—	9	1	no	—
103	#832	—	3	1	no	—
104	#810	—	3	1	no	—
105	#808	—	3	1	no	—
106	C₄×C₃⋊C₈	—	3	1	no	—
107	C₂³×C₃²	—	2	1	no	—
108	C₃²×D₄	—	2	1	no	—
109	C₃²×D₄	C₄×D₃	1	2	yes	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	4	2	no	—
112	C₂×C₃²×C₄	#67	1	1	yes	characteristic
113	C₂×C₃²×C₄	#67	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
115	C₂×C₃×Dic₃	—	6	1	no	—
116	C₂×C₃×Dic₃	—	6	1	no	—
117	C₃×Dic₆	—	3	2	no	—
118	C₃×Dic₆	—	6	1	no	—
119	C₂³×C₉	—	2	2	no	—
120	C₂³×C₉	—	2	1	no	—
121	C₃×C₃⋊C₈	—	3	2	no	—
122	C₃×C₃⋊C₈	—	6	1	no	—
123	D₄×C₉	—	2	2	no	—
124	D₄×C₉	—	2	1	no	—
125	D₄×C₉	—	2	2	no	—
126	D₄×C₉	C₂×C₃×C₄	1	2	yes	—
127	C₂×C₄×C₉	—	2	2	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	4	4	no	—
133	C₂×C₄×C₉	—	4	2	no	—
134	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
137	C₂×Dic₉	—	2	1	no	—
138	C₂×Dic₉	—	2	1	no	—
139	Dic₁₈	—	2	1	no	—
140	Dic₁₈	C₂×C₃×C₄	1	2	yes	complemented
141	C₉⋊C₈	—	2	1	no	—
142	C₉⋊C₈	C₂×C₃×C₄	1	2	yes	complemented
143	C₄×SD₁₆	—	9	1	no	Sylow
144	C₂×3₋¹⁺²	—	4	2	no	—
145	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
146	C₂×3₋¹⁺²	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
148	C₂×C₃×Q₈	—	9	1	no	—
149	C₂×C₃×D₄	—	3	1	no	—
150	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂²×C₃×C₄	—	2	1	no	—
152	C₂²×C₃×C₄	—	6	1	no	—
153	C₂×Dic₆	—	3	1	no	—
154	C₃×SD₁₆	—	9	4	no	—
155	C₂×C₃×C₈	—	9	1	no	—
156	C₂×C₃×C₈	—	9	1	no	—
157	C₃×C₄⋊C₄	—	3	1	no	—
158	C₃×C₄⋊C₄	—	9	1	no	—
159	C₃×C₄⋊C₄	—	18	1	no	—
160	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
161	C₃×C₂²⋊C₄	—	2	1	no	—
162	C₃×C₂²⋊C₄	—	6	1	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	18	1	no	—
165	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
166	#214	—	3	4	no	—
167	C₄⋊Dic₃	—	3	1	no	—
168	Dic₃⋊C₄	—	6	1	no	—
169	C₄×Dic₃	—	6	1	no	—
170	C₂×C₃⋊C₈	—	3	1	no	—
171	C₂×C₃⋊C₈	—	3	1	no	—
172	C₂²×C₃²	—	2	2	no	—
173	C₂²×C₃²	—	4	1	no	—
174	C₂²×C₃²	—	4	1	no	—
175	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
176	C₃²×C₄	—	2	1	no	—
177	C₃²×C₄	—	4	1	no	—
178	C₃²×C₄	#214	1	2	yes	—
179	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₃²×C₄	#235	1	1	yes	characteristic
181	C₃×Dic₃	—	6	2	no	—
182	C₃×Dic₃	—	12	1	no	—
183	C₂²×C₉	—	2	4	no	—
184	C₂²×C₉	—	2	2	no	—
185	C₂²×C₉	—	2	1	no	—
186	C₂²×C₉	—	4	2	no	—
187	C₂²×C₉	—	4	2	no	—
188	C₂²×C₉	—	4	1	no	—
189	C₂²×C₉	—	4	1	no	—
190	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	2	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	2	1	no	—
195	C₄×C₉	—	4	1	no	—
196	C₄×C₉	—	4	2	no	—
197	C₄×C₉	—	4	1	no	—
198	C₄×C₉	C₃×SD₁₆	1	2	yes	complemented
199	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₄×C₉	C₃×#41	1	1	yes	characteristic
201	Dic₉	—	2	2	no	—
202	Dic₉	—	4	1	no	—
203	C₂×SD₁₆	—	9	1	no	—
204	C₄×Q₈	—	9	1	no	—
205	C₄×D₄	AGL(1,ℤ/9)	1	1	yes	characteristic
206	#106	—	9	1	no	—
207	#103	—	9	1	no	—
208	#102	—	9	1	no	—
209	C₄×C₈	—	9	1	no	—
210	3₋¹⁺²	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₃	—	2	1	no	—
212	C₂³×C₃	—	6	1	no	—
213	C₃×Q₈	—	9	2	no	—
214	C₃×Q₈	—	18	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₃×D₄	—	3	2	no	—
217	C₃×D₄	—	6	1	no	—
218	C₃×D₄	C₃×C₄×D₃	1	2	yes	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	3	1	no	—
221	C₂×C₃×C₄	—	3	1	no	—
222	C₂×C₃×C₄	—	3	1	no	—
223	C₂×C₃×C₄	—	4	2	no	—
224	C₂×C₃×C₄	—	6	1	no	—
225	C₂×C₃×C₄	—	12	2	no	—
226	C₂×C₃×C₄	—	18	1	no	—
227	C₂×C₃×C₄	—	18	1	no	—
228	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
229	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
231	C₂×Dic₃	—	6	1	no	—
232	C₂×Dic₃	—	6	1	no	—
233	Dic₆	—	3	2	no	—
234	Dic₆	—	6	1	no	—
235	C₃×C₈	—	9	2	no	—
236	C₃×C₈	—	18	1	no	—
237	C₃⋊C₈	—	3	2	no	—
238	C₃⋊C₈	—	6	1	no	—
239	C₂×C₃²	—	4	2	no	—
240	C₂×C₃²	C₄×#67	1	1	yes	characteristic
241	C₂×C₃²	C₂×#214	1	1	yes	characteristic
242	C₂×C₃²	#950	1	1	yes	characteristic
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	—	4	4	no	—
247	C₂×C₉	—	4	2	no	—
248	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
249	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
250	C₂×C₉	C₃×#135	1	1	yes	characteristic
251	C₂×Q₈	—	9	1	no	—
252	C₂×D₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	SD₁₆	—	9	4	no	—
255	C₂×C₈	—	9	1	no	—
256	C₂×C₈	—	9	1	no	—
257	C₄⋊C₄	—	9	1	no	—
258	C₄⋊C₄	—	18	1	no	—
259	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	18	1	no	—
262	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	3	1	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	6	2	no	—
268	C₂²×C₃	—	12	1	no	—
269	C₂²×C₃	—	12	1	no	—
270	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	3	1	no	—
273	C₃×C₄	—	3	1	no	—
274	C₃×C₄	—	3	2	no	—
275	C₃×C₄	—	4	1	no	—
276	C₃×C₄	—	6	1	no	—
277	C₃×C₄	—	12	1	no	—
278	C₃×C₄	—	18	2	no	—
279	C₃×C₄	—	36	1	no	—
280	C₃×C₄	C₃×#214	1	2	yes	—
281	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
282	C₃×C₄	C₃×#235	1	1	yes	characteristic
283	Dic₃	—	6	2	no	—
284	Dic₃	—	12	1	no	—
285	C₃²	C₄×#214	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	C₃×C₄×SD₁₆	1	1	yes	characteristic, complemented
288	C₂³	—	2	1	no	—
289	Q₈	—	9	2	no	—
290	Q₈	—	18	1	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄×AGL(1,ℤ/9)	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	18	1	no	—
296	C₂×C₄	—	18	1	no	—
297	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
298	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
300	C₈	—	9	2	no	—
301	C₈	—	18	1	no	—
302	C₂×C₃	—	3	1	no	—
303	C₂×C₃	—	3	1	no	—
304	C₂×C₃	—	3	1	no	—
305	C₂×C₃	—	4	2	no	—
306	C₂×C₃	—	12	2	no	—
307	C₂×C₃	C₃×C₄×#67	1	1	yes	characteristic
308	C₂×C₃	C₂×C₃×#214	1	1	yes	characteristic
309	C₂×C₃	C₃×#950	1	1	yes	characteristic
310	C₂²	—	2	2	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	C₂×#611⋊C₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	18	2	no	—
317	C₄	—	36	1	no	—
318	C₄	#3672⋊C₃	1	2	yes	direct product factor
319	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
320	C₄	#3697⋊C₃	1	1	yes	characteristic
321	C₃	—	3	1	no	—
322	C₃	C₃×C₄×#214	1	1	yes	characteristic
323	C₂	—	4	2	no	—
324	C₂	C₄×#611⋊C₃	1	1	yes	characteristic
325	C₂	C₂×#3672⋊C₃	1	1	yes	characteristic
326	C₂	#63513⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#611⋊C₃
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×D₄×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₄×D₄	AGL(1,ℤ/9)
hypocenter	C₉	C₃×C₄×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₄×3₋¹⁺²⋊Q₈	C₂	semidirect
5. #63368⋊C₃	C₂	semidirect
6. #63367⋊C₃	C₂	semidirect
7. C₄×3₋¹⁺²⋊C₈	C₂	semidirect
9. C₄×#3672	C₃	semidirect
16. #3672⋊C₃	C₄	direct (N × H)
32. C₄×Dic₁₈	C₂×C₃	semidirect
34. #63368	C₂×C₃	semidirect
35. #63367	C₂×C₃	semidirect
36. C₄×C₉⋊C₈	C₂×C₃	semidirect
48. 3₋¹⁺²⋊Q₈	C₂×C₄	semidirect
50. 3₋¹⁺²⋊C₈	C₂×C₄	semidirect
76. #3672	C₃×C₄	semidirect
88. C₄×3₋¹⁺²	SD₁₆	semidirect
140. Dic₁₈	C₂×C₃×C₄	semidirect
142. C₉⋊C₈	C₂×C₃×C₄	semidirect
198. C₄×C₉	C₃×SD₁₆	semidirect
210. 3₋¹⁺²	C₄×SD₁₆	semidirect
287. C₉	C₃×C₄×SD₁₆	semidirect
318. C₄	#3672⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×D₄×3₋¹⁺²	2
C₂×#3672⋊C₃	2
C₄×3₋¹⁺²⋊Q₈	2
#63392⋊C₃	2
#63368⋊C₃	2
#63367⋊C₃	2
C₄×3₋¹⁺²⋊C₈	2
C₃×C₄×#214	3
C₄×#3672	3

Subgroups — C4×#3672⋊C3 #13142757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#3672⋊C₃ #13142757