Subgroups — C₄×#88426 #13137256

Order 1728.

460 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88426	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720298	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720267	C₂	1	1	yes	maximal, characteristic
7	#11720264	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#3717	—	3	1	no	maximal
9	C₄×#3675	—	3	1	no	maximal
10	C₂×C₄⋊(C₃⋊D₉)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃⋊D₉	—	2	1	no	—
12	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	#88536	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×Dic₃⋊C₄	—	2	1	no	—
18	C₄×C₉×Dic₃	—	2	1	no	—
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	#88426	C₄	1	4	yes	direct product factor
22	C₄×C₄⋊(C₃⋊D₃)	—	3	1	no	—
23	C₃×C₄×Dic₆	D₃	1	1	yes	characteristic
24	C₂×#3717	—	3	1	no	—
25	C₃×C₄×C₃⋊C₈	—	3	1	no	—
26	#63573	—	3	1	no	—
27	#63566	—	3	1	no	—
28	#63564	—	3	1	no	—
29	C₄×Q₈×C₉	—	3	1	no	—
30	C₂×#3675	—	3	1	no	—
31	C₄×D₃₆	—	3	1	no	—
32	C₄×D₃₆	—	6	1	no	—
33	#63395	—	3	1	no	—
34	#63369	—	3	1	no	—
35	#63367	—	3	1	no	—
36	C₄×C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
37	C₂²×C₃⋊D₉	—	2	1	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	—	2	1	no	—
42	C₄⋊(C₃⋊D₉)	—	2	1	no	—
43	C₄⋊(C₃⋊D₉)	C₂×C₄	1	2	yes	—
44	C₄×C₃⋊D₉	—	4	2	no	—
45	C₂×C₉×Dic₃	—	2	1	no	—
46	C₂×C₉×Dic₃	—	2	1	no	—
47	C₉×Dic₆	—	2	1	no	—
48	C₉×Dic₆	C₂×C₄	1	2	yes	complemented
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
51	C₄×#215	—	9	1	no	—
52	C₄×#204	—	9	1	no	—
53	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
54	C₂×C₄×C₃⋊D₃	—	6	1	no	—
55	C₂×C₃×Dic₆	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	#3752	—	6	1	no	—
58	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
59	C₃×C₄⋊Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₃×Dic₃⋊C₄	—	2	1	no	—
61	C₃×C₄×Dic₃	—	2	1	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	#3717	—	3	4	no	—
65	C₂×Q₈×C₉	—	3	1	no	—
66	C₂×D₃₆	—	3	1	no	—
67	C₂×D₃₆	—	6	1	no	—
68	C₂×C₄×D₉	—	6	2	no	—
69	C₂×C₄×D₉	—	6	1	no	—
70	C₉×C₄⋊C₄	—	3	1	no	—
71	C₉×C₄⋊C₄	—	6	1	no	—
72	C₄²×C₉	—	2	1	no	—
73	C₄²×C₉	—	6	1	no	—
74	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
75	#3675	—	3	4	no	—
76	(C₂×D₉)⋊C₄	—	6	2	no	—
77	(C₂×D₉)⋊C₄	—	6	1	no	—
78	C₄⋊Dic₉	—	3	1	no	—
79	C₄⋊Dic₉	—	6	1	no	—
80	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃⋊D₉	—	2	2	no	—
84	C₂×C₃⋊D₉	—	4	1	no	—
85	C₂×C₃⋊D₉	—	4	1	no	—
86	C₃×C₄×C₉	—	2	1	no	—
87	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
88	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
89	C₃×C₄×C₉	#41	1	1	yes	characteristic
90	C₃⋊Dic₉	—	4	1	no	—
91	C₉×Dic₃	—	2	2	no	—
92	C₉×Dic₃	—	4	1	no	—
93	C₃×C₄×Q₈	—	3	1	no	—
94	C₂×#215	—	9	1	no	—
95	C₂×#204	—	9	1	no	—
96	C₄×D₁₂	—	9	1	no	—
97	C₄×D₁₂	—	9	1	no	—
98	C₄×D₁₂	—	18	1	no	—
99	C₄×Dic₆	D₉	1	1	yes	characteristic, complemented
100	C₃×C₄×C₈	—	9	1	no	—
101	#835	—	9	1	no	—
102	#821	—	9	1	no	—
103	#817	—	9	1	no	—
104	#816	—	9	1	no	—
105	#809	—	9	1	no	—
106	#808	—	9	1	no	—
107	C₄×C₃⋊C₈	—	3	1	no	—
108	C₂²×C₃⋊D₃	—	6	1	no	—
109	C₂×C₃²×C₄	#67	1	1	yes	characteristic
110	C₂×C₃²×C₄	#67	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃⋊Dic₃	—	6	1	no	—
113	C₄⋊(C₃⋊D₃)	—	3	2	no	—
114	C₄⋊(C₃⋊D₃)	—	6	1	no	—
115	C₄×C₃⋊D₃	—	12	2	no	—
116	C₂×C₃×Dic₃	—	2	1	no	—
117	C₂×C₃×Dic₃	—	2	1	no	—
118	C₃×Dic₆	—	2	1	no	—
119	C₃×Dic₆	C₄×D₃	1	2	yes	—
120	C₂²×D₉	—	6	2	no	—
121	C₂²×D₉	—	6	1	no	—
122	C₃×C₃⋊C₈	—	3	2	no	—
123	C₃×C₃⋊C₈	—	6	1	no	—
124	Q₈×C₉	—	3	2	no	—
125	Q₈×C₉	—	6	1	no	—
126	C₂×C₄×C₉	—	2	1	no	—
127	C₂×C₄×C₉	—	2	1	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	6	1	no	—
130	C₂×C₄×C₉	—	6	1	no	—
131	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
132	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
133	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
134	C₂×Dic₉	—	6	2	no	—
135	C₂×Dic₉	—	6	1	no	—
136	D₃₆	—	3	2	no	—
137	D₃₆	—	6	2	no	—
138	D₃₆	—	6	2	no	—
139	D₃₆	—	6	1	no	—
140	C₄×D₉	—	12	4	no	—
141	C₄×D₉	—	12	2	no	—
142	C₉⋊C₈	—	2	1	no	—
143	C₉⋊C₈	C₄×D₃	1	2	yes	complemented
144	C₄×SD₁₆	—	27	1	no	Sylow
145	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₃×C₉	#135	1	1	yes	characteristic
148	C₃⋊D₉	—	4	2	no	—
149	C₂×C₃×Q₈	—	3	1	no	—
150	C₂×D₁₂	—	9	1	no	—
151	C₂×D₁₂	—	9	1	no	—
152	C₂×D₁₂	—	18	1	no	—
153	C₂×C₄×D₃	—	18	2	no	—
154	C₂×C₄×D₃	—	18	1	no	—
155	C₂×C₄×D₃	—	18	1	no	—
156	C₂×Dic₆	C₂×D₉	1	1	yes	characteristic
157	C₂×C₃×C₈	—	9	1	no	—
158	C₂×C₃×C₈	—	9	1	no	—
159	C₃×C₄⋊C₄	—	3	1	no	—
160	C₃×C₄⋊C₄	—	6	1	no	—
161	C₃×C₄²	—	2	1	no	—
162	C₃×C₄²	—	6	1	no	—
163	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
164	C₃×C₄²	D₃²	1	1	yes	characteristic
165	#215	—	9	4	no	—
166	(C₂×D₃)⋊C₄	—	18	2	no	—
167	(C₂×D₃)⋊C₄	—	18	1	no	—
168	(C₂×D₃)⋊C₄	—	18	1	no	—
169	C₄⋊Dic₃	—	9	1	no	—
170	C₄⋊Dic₃	—	9	1	no	—
171	C₄⋊Dic₃	—	18	1	no	—
172	C₄⋊Dic₃	C₂×D₉	1	1	yes	characteristic
173	Dic₃⋊C₄	—	2	1	no	—
174	C₄×Dic₃	—	2	1	no	—
175	C₂×C₃⋊C₈	—	3	1	no	—
176	C₂×C₃⋊C₈	—	3	1	no	—
177	#204	—	9	4	no	—
178	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
179	C₂×C₃⋊D₃	—	6	2	no	—
180	C₂×C₃⋊D₃	—	12	1	no	—
181	C₂×C₃⋊D₃	—	12	1	no	—
182	C₃²×C₄	—	2	1	no	—
183	C₃²×C₄	#215	1	2	yes	—
184	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₃²×C₄	#235	1	1	yes	characteristic
186	C₃⋊Dic₃	—	12	1	no	—
187	C₃×Dic₃	—	2	2	no	—
188	C₃×Dic₃	—	4	1	no	—
189	C₂²×C₉	—	2	1	no	—
190	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
191	C₂×D₉	—	6	4	no	—
192	C₂×D₉	—	6	2	no	—
193	C₂×D₉	—	12	2	no	—
194	C₂×D₉	—	12	2	no	—
195	C₂×D₉	—	12	1	no	—
196	C₂×D₉	—	12	1	no	—
197	C₄×C₉	—	2	1	no	—
198	C₄×C₉	—	2	2	no	—
199	C₄×C₉	—	2	1	no	—
200	C₄×C₉	—	2	1	no	—
201	C₄×C₉	—	4	1	no	—
202	C₄×C₉	—	6	2	no	—
203	C₄×C₉	—	12	1	no	—
204	C₄×C₉	#204	1	2	yes	complemented
205	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
206	C₄×C₉	#235	1	1	yes	characteristic
207	Dic₉	—	12	2	no	—
208	Dic₉	—	12	1	no	—
209	C₂×SD₁₆	—	27	1	no	—
210	C₄×Q₈	—	3	1	no	—
211	C₄×D₄	—	27	1	no	—
212	#106	—	27	1	no	—
213	#103	—	27	1	no	—
214	#102	—	27	1	no	—
215	C₄×C₈	—	9	1	no	—
216	C₃×C₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
217	C₂²×D₃	—	18	2	no	—
218	C₂²×D₃	—	18	1	no	—
219	C₂²×D₃	—	18	1	no	—
220	C₃×Q₈	—	3	2	no	—
221	C₃×Q₈	—	6	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	6	1	no	—
226	C₂×C₃×C₄	—	6	1	no	—
227	C₂×C₃×C₄	#611	1	1	yes	characteristic
228	C₂×C₃×C₄	#611	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
230	C₂×C₃×C₄	#626	1	1	yes	characteristic
231	C₂×C₃×C₄	#626	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
233	C₂×Dic₃	—	2	1	no	—
234	C₂×Dic₃	—	2	1	no	—
235	C₂×Dic₃	—	18	2	no	—
236	C₂×Dic₃	—	18	1	no	—
237	C₂×Dic₃	—	18	1	no	—
238	D₁₂	—	9	2	no	—
239	D₁₂	—	9	2	no	—
240	D₁₂	—	18	2	no	—
241	D₁₂	—	18	2	no	—
242	D₁₂	—	18	1	no	—
243	D₁₂	—	18	1	no	—
244	C₄×D₃	—	36	4	no	—
245	C₄×D₃	—	36	2	no	—
246	C₄×D₃	—	36	2	no	—
247	Dic₆	—	2	1	no	—
248	Dic₆	C₄×D₉	1	2	yes	complemented
249	C₃×C₈	—	9	2	no	—
250	C₃×C₈	—	18	1	no	—
251	C₃⋊C₈	—	3	2	no	—
252	C₃⋊C₈	—	6	1	no	—
253	C₂×C₃²	C₄×#67	1	1	yes	characteristic
254	C₂×C₃²	C₂×#215	1	1	yes	characteristic
255	C₂×C₃²	#950	1	1	yes	characteristic
256	C₃⋊D₃	—	12	2	no	—
257	C₂×C₉	—	2	1	no	—
258	C₂×C₉	—	2	1	no	—
259	C₂×C₉	—	2	1	no	—
260	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
261	C₂×C₉	C₂×#204	1	1	yes	characteristic
262	C₂×C₉	#904	1	1	yes	characteristic
263	D₉	—	12	4	no	—
264	D₉	—	12	2	no	—
265	C₂×Q₈	—	3	1	no	—
266	C₂×D₄	—	27	1	no	—
267	C₂²×C₄	—	54	1	no	—
268	SD₁₆	—	27	4	no	—
269	C₂×C₈	—	9	1	no	—
270	C₂×C₈	—	9	1	no	—
271	C₄⋊C₄	—	3	1	no	—
272	C₄⋊C₄	—	6	1	no	—
273	C₄⋊C₄	—	27	1	no	—
274	C₂²⋊C₄	—	54	1	no	—
275	C₄²	—	6	1	no	—
276	C₄²	D₃×D₉	1	1	yes	characteristic
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	C₂×#611	1	1	yes	characteristic
279	C₂²×C₃	C₂×#626	1	1	yes	characteristic
280	C₂×D₃	—	18	4	no	—
281	C₂×D₃	—	18	2	no	—
282	C₂×D₃	—	18	2	no	—
283	C₂×D₃	—	36	2	no	—
284	C₂×D₃	—	36	2	no	—
285	C₂×D₃	—	36	1	no	—
286	C₂×D₃	—	36	1	no	—
287	C₂×D₃	—	36	1	no	—
288	C₂×D₃	—	36	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	4	1	no	—
295	C₃×C₄	—	6	2	no	—
296	C₃×C₄	—	12	1	no	—
297	C₃×C₄	#3675	1	2	yes	complemented
298	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
299	C₃×C₄	#3697	1	1	yes	characteristic
300	C₃×C₄	#3717	1	2	yes	—
301	C₃×C₄	#3798	1	1	yes	characteristic
302	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
303	Dic₃	—	2	2	no	—
304	Dic₃	—	4	1	no	—
305	Dic₃	—	36	2	no	—
306	Dic₃	—	36	1	no	—
307	Dic₃	—	36	1	no	—
308	C₃²	C₄×#215	1	1	yes	characteristic
309	C₉	—	2	1	no	—
310	C₉	C₄×#204	1	1	yes	characteristic, complemented
311	C₂³	—	54	1	no	—
312	Q₈	—	3	2	no	—
313	Q₈	—	6	1	no	—
314	D₄	—	27	2	no	—
315	D₄	—	54	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	6	1	no	—
318	C₂×C₄	—	54	1	no	—
319	C₂×C₄	—	108	2	no	—
320	C₂×C₄	#6195	1	1	yes	characteristic
321	C₂×C₄	#6195	1	1	yes	characteristic
322	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
323	C₈	—	9	2	no	—
324	C₈	—	18	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	C₄×#611	1	1	yes	characteristic
329	C₂×C₃	C₂×#3675	1	1	yes	characteristic
330	C₂×C₃	#63513	1	1	yes	characteristic
331	C₂×C₃	#63829	1	1	yes	characteristic
332	C₂×C₃	C₂×#3717	1	1	yes	characteristic
333	C₂×C₃	C₄×#626	1	1	yes	characteristic
334	D₃	—	36	4	no	—
335	D₃	—	36	2	no	—
336	D₃	—	36	2	no	—
337	C₂²	—	54	2	no	—
338	C₂²	—	108	1	no	—
339	C₂²	—	108	1	no	—
340	C₂²	C₂×#6195	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	6	2	no	—
343	C₄	—	12	1	no	—
344	C₄	—	108	1	no	—
345	C₄	#88426	1	2	yes	direct product factor
346	C₄	#88640	1	1	yes	characteristic
347	C₄	C₄×D₃×D₉	1	1	yes	characteristic
348	C₃	—	2	1	no	—
349	C₃	C₄×#3675	1	1	yes	characteristic, complemented
350	C₃	C₄×#3717	1	1	yes	characteristic
351	C₂	—	108	2	no	—
352	C₂	#11720889	1	1	yes	characteristic
353	C₂	C₂×#88426	1	1	yes	characteristic
354	C₂	C₄×#6195	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6195
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₄⋊(C₃⋊D₉)	C₂²
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	C₄×SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₉×Dic₆	C₂	semidirect
4. C₃×C₄×C₉⋊C₈	C₂	semidirect
5. #11720298	C₂	semidirect
7. #11720264	C₂	semidirect
21. #88426	C₄	direct (N × H)
36. C₄×C₉⋊C₈	D₃	semidirect
48. C₉×Dic₆	C₂×C₄	semidirect
50. C₃×C₉⋊C₈	C₂×C₄	semidirect
87. C₃×C₄×C₉	SD₁₆	semidirect
99. C₄×Dic₆	D₉	semidirect
143. C₉⋊C₈	C₄×D₃	semidirect
204. C₄×C₉	#204	semidirect
216. C₃×C₉	C₄×SD₁₆	semidirect
248. Dic₆	C₄×D₉	semidirect
297. C₃×C₄	#3675	semidirect
310. C₉	C₄×#204	semidirect
345. C₄	#88426	direct (N × H)
349. C₃	C₄×#3675	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₃⋊D₉)	2
C₄×C₉×Dic₆	2
C₂×#88426	2
C₃×C₄×C₉⋊C₈	2
#11720298	2
#11720267	2
#11720264	2
C₄×#3717	3
C₄×#3675	3

Subgroups — C4×#88426 #13137256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88426 #13137256