Subgroups — C₄×#88795 #13158989

Order 1728.

1108 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⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88795	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11722054	C₂	1	1	yes	maximal, characteristic, complemented
6	#11722038	C₂	1	1	yes	maximal, characteristic, complemented
7	#11722035	C₂	1	1	yes	maximal, characteristic
8	C₄×#3755	—	3	1	no	maximal
9	C₄×#3717	—	3	4	no	maximal
10	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₃⋊(C₃⋊D₃)	—	2	1	no	—
12	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	#88859	—	2	1	no	—
15	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₃²×Dic₃⋊C₄	—	2	1	no	—
20	C₃²×C₄×Dic₃	—	2	1	no	—
21	#88795	C₄	1	4	yes	direct product factor
22	C₃²×C₄×Q₈	—	3	1	no	—
23	C₂×#3755	—	3	1	no	—
24	C₄×C₄⋊(C₃⋊D₃)	—	3	1	no	—
25	C₄×C₄⋊(C₃⋊D₃)	—	3	4	no	—
26	C₄×C₄⋊(C₃⋊D₃)	—	6	4	no	—
27	C₃×C₄×Dic₆	D₃	1	4	yes	complemented
28	C₂×#3717	—	3	4	no	—
29	#63662	—	3	1	no	—
30	#63636	—	3	1	no	—
31	#63635	—	3	1	no	—
32	C₄×C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
33	C₃×C₄×C₃⋊C₈	—	3	4	no	—
34	#63573	—	3	4	no	—
35	#63566	—	3	4	no	—
36	#63564	—	3	4	no	—
37	C₂²×C₃⋊(C₃⋊D₃)	—	2	1	no	—
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
42	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
43	C₄⋊(C₃⋊(C₃⋊D₃))	C₂×C₄	1	2	yes	—
44	C₄×C₃⋊(C₃⋊D₃)	—	4	2	no	—
45	C₂×C₃²×Dic₃	—	2	1	no	—
46	C₂×C₃²×Dic₃	—	2	1	no	—
47	C₃²×Dic₆	—	2	1	no	—
48	C₃²×Dic₆	C₂×C₄	1	2	yes	complemented
49	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
50	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
51	C₄×#215	—	9	4	no	—
52	C₄×#204	—	9	1	no	—
53	C₂×C₃²×Q₈	—	3	1	no	—
54	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
55	C₂×C₄⋊(C₃⋊D₃)	—	3	4	no	—
56	C₂×C₄⋊(C₃⋊D₃)	—	6	4	no	—
57	C₂×C₄×C₃⋊D₃	—	6	8	no	—
58	C₂×C₄×C₃⋊D₃	—	6	1	no	—
59	C₂×C₄×C₃⋊D₃	—	6	4	no	—
60	C₂×C₃×Dic₆	C₂×D₃	1	4	yes	—
61	C₃²×C₄⋊C₄	—	3	1	no	—
62	C₃²×C₄⋊C₄	—	6	1	no	—
63	C₃²×C₄²	—	2	4	no	—
64	C₃²×C₄²	—	6	1	no	—
65	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
66	C₃²×C₄²	C₂×D₃	1	4	yes	—
67	#3755	—	3	4	no	—
68	#3752	—	6	8	no	—
69	#3752	—	6	1	no	—
70	#3752	—	6	4	no	—
71	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
72	C₄⋊(C₃⋊Dic₃)	—	3	4	no	—
73	C₄⋊(C₃⋊Dic₃)	—	6	4	no	—
74	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
76	C₃×C₄⋊Dic₃	C₂×D₃	1	4	yes	—
77	C₃×Dic₃⋊C₄	—	2	4	no	—
78	C₃×C₄×Dic₃	—	2	4	no	—
79	C₂×C₃×C₃⋊C₈	—	3	4	no	—
80	C₂×C₃×C₃⋊C₈	—	3	4	no	—
81	#3717	—	3	16	no	—
82	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
84	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
85	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
86	C₃³×C₄	—	2	1	no	—
87	C₃³×C₄	SD₁₆	1	2	yes	complemented
88	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₃³×C₄	#41	1	1	yes	characteristic
90	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
91	C₃²×Dic₃	—	2	2	no	—
92	C₃²×Dic₃	—	4	1	no	—
93	C₃×C₄×Q₈	—	3	4	no	—
94	C₂×#215	—	9	4	no	—
95	C₂×#204	—	9	1	no	—
96	C₄×D₁₂	—	9	4	no	—
97	C₄×D₁₂	—	9	1	no	—
98	C₄×D₁₂	—	18	4	no	—
99	C₄×Dic₆	C₃⋊D₃	1	1	yes	characteristic, complemented
100	C₃×C₄×C₈	—	9	1	no	—
101	#835	—	9	4	no	—
102	#821	—	9	1	no	—
103	#817	—	9	1	no	—
104	#816	—	9	1	no	—
105	#809	—	9	4	no	—
106	#808	—	9	4	no	—
107	C₄×C₃⋊C₈	—	3	4	no	—
108	C₂²×C₃⋊D₃	—	6	8	no	—
109	C₂²×C₃⋊D₃	—	6	1	no	—
110	C₂²×C₃⋊D₃	—	6	4	no	—
111	C₃²×Q₈	—	3	2	no	—
112	C₃²×Q₈	—	6	1	no	—
113	C₂×C₃²×C₄	—	2	4	no	—
114	C₂×C₃²×C₄	—	2	4	no	—
115	C₂×C₃²×C₄	—	2	4	no	—
116	C₂×C₃²×C₄	—	6	1	no	—
117	C₂×C₃²×C₄	—	6	1	no	—
118	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
119	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
120	C₂×C₃²×C₄	#67	1	4	yes	—
121	C₂×C₃²×C₄	#67	1	4	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₃⋊Dic₃	—	6	8	no	—
125	C₂×C₃⋊Dic₃	—	6	1	no	—
126	C₂×C₃⋊Dic₃	—	6	4	no	—
127	C₄⋊(C₃⋊D₃)	—	3	2	no	—
128	C₄⋊(C₃⋊D₃)	—	3	8	no	—
129	C₄⋊(C₃⋊D₃)	—	6	8	no	—
130	C₄⋊(C₃⋊D₃)	—	6	8	no	—
131	C₄⋊(C₃⋊D₃)	—	6	4	no	—
132	C₄⋊(C₃⋊D₃)	—	6	1	no	—
133	C₄×C₃⋊D₃	—	12	16	no	—
134	C₄×C₃⋊D₃	—	12	2	no	—
135	C₄×C₃⋊D₃	—	12	8	no	—
136	C₂×C₃×Dic₃	—	2	4	no	—
137	C₂×C₃×Dic₃	—	2	4	no	—
138	C₃×Dic₆	—	2	4	no	—
139	C₃×Dic₆	C₄×D₃	1	8	yes	complemented
140	C₃⋊(C₃⋊C₈)	—	2	1	no	—
141	C₃⋊(C₃⋊C₈)	C₄×D₃	1	2	yes	complemented
142	C₃×C₃⋊C₈	—	3	8	no	—
143	C₃×C₃⋊C₈	—	6	4	no	—
144	C₄×SD₁₆	—	27	1	no	Sylow
145	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
146	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₃³	#135	1	1	yes	characteristic
148	C₃⋊(C₃⋊D₃)	—	4	2	no	—
149	C₂×C₃×Q₈	—	3	4	no	—
150	C₂×D₁₂	—	9	4	no	—
151	C₂×D₁₂	—	9	1	no	—
152	C₂×D₁₂	—	18	4	no	—
153	C₂×C₄×D₃	—	18	8	no	—
154	C₂×C₄×D₃	—	18	4	no	—
155	C₂×C₄×D₃	—	18	1	no	—
156	C₂×Dic₆	C₂×C₃⋊D₃	1	1	yes	characteristic
157	C₂×C₃×C₈	—	9	1	no	—
158	C₂×C₃×C₈	—	9	1	no	—
159	C₃×C₄⋊C₄	—	3	4	no	—
160	C₃×C₄⋊C₄	—	6	4	no	—
161	C₃×C₄²	—	2	4	no	—
162	C₃×C₄²	—	6	4	no	—
163	C₃×C₄²	D₃²	1	4	yes	—
164	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
165	#215	—	9	16	no	—
166	(C₂×D₃)⋊C₄	—	18	8	no	—
167	(C₂×D₃)⋊C₄	—	18	4	no	—
168	(C₂×D₃)⋊C₄	—	18	1	no	—
169	C₄⋊Dic₃	—	9	4	no	—
170	C₄⋊Dic₃	—	9	1	no	—
171	C₄⋊Dic₃	—	18	4	no	—
172	C₄⋊Dic₃	C₂×C₃⋊D₃	1	1	yes	characteristic
173	Dic₃⋊C₄	—	2	1	no	—
174	C₄×Dic₃	—	2	1	no	—
175	C₂×C₃⋊C₈	—	3	4	no	—
176	C₂×C₃⋊C₈	—	3	4	no	—
177	#204	—	9	4	no	—
178	C₂²×C₃²	—	2	4	no	—
179	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
180	C₂²×C₃²	C₂×#67	1	4	yes	—
181	C₂×C₃⋊D₃	—	6	16	no	—
182	C₂×C₃⋊D₃	—	6	2	no	—
183	C₂×C₃⋊D₃	—	6	8	no	—
184	C₂×C₃⋊D₃	—	12	8	no	—
185	C₂×C₃⋊D₃	—	12	8	no	—
186	C₂×C₃⋊D₃	—	12	1	no	—
187	C₂×C₃⋊D₃	—	12	1	no	—
188	C₂×C₃⋊D₃	—	12	4	no	—
189	C₂×C₃⋊D₃	—	12	4	no	—
190	C₃²×C₄	—	2	1	no	—
191	C₃²×C₄	—	2	8	no	—
192	C₃²×C₄	—	2	4	no	—
193	C₃²×C₄	—	2	4	no	—
194	C₃²×C₄	—	2	4	no	—
195	C₃²×C₄	—	4	4	no	—
196	C₃²×C₄	—	6	2	no	—
197	C₃²×C₄	—	12	1	no	—
198	C₃²×C₄	#204	1	2	yes	complemented
199	C₃²×C₄	#215	1	8	yes	complemented
200	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
201	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₃²×C₄	#235	1	4	yes	—
203	C₃²×C₄	#235	1	1	yes	characteristic
204	C₃⋊Dic₃	—	12	8	no	—
205	C₃⋊Dic₃	—	12	1	no	—
206	C₃⋊Dic₃	—	12	4	no	—
207	C₃×Dic₃	—	2	8	no	—
208	C₃×Dic₃	—	4	4	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	C₃³	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
217	C₂²×D₃	—	18	8	no	—
218	C₂²×D₃	—	18	4	no	—
219	C₂²×D₃	—	18	1	no	—
220	C₃×Q₈	—	3	8	no	—
221	C₃×Q₈	—	6	4	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	6	4	no	—
226	C₂×C₃×C₄	—	6	4	no	—
227	C₂×C₃×C₄	#626	1	4	yes	—
228	C₂×C₃×C₄	#626	1	4	yes	—
229	C₂×C₃×C₄	#638	1	1	yes	characteristic
230	C₂×C₃×C₄	#638	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
232	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
233	C₂×Dic₃	—	2	1	no	—
234	C₂×Dic₃	—	2	1	no	—
235	C₂×Dic₃	—	18	8	no	—
236	C₂×Dic₃	—	18	4	no	—
237	C₂×Dic₃	—	18	1	no	—
238	D₁₂	—	9	8	no	—
239	D₁₂	—	9	2	no	—
240	D₁₂	—	18	8	no	—
241	D₁₂	—	18	8	no	—
242	D₁₂	—	18	4	no	—
243	D₁₂	—	18	1	no	—
244	C₄×D₃	—	36	16	no	—
245	C₄×D₃	—	36	8	no	—
246	C₄×D₃	—	36	2	no	—
247	Dic₆	—	2	1	no	—
248	Dic₆	C₄×C₃⋊D₃	1	2	yes	complemented
249	C₃×C₈	—	9	2	no	—
250	C₃×C₈	—	18	1	no	—
251	C₃⋊C₈	—	3	8	no	—
252	C₃⋊C₈	—	6	4	no	—
253	C₂×C₃²	—	2	4	no	—
254	C₂×C₃²	—	2	4	no	—
255	C₂×C₃²	—	2	4	no	—
256	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
257	C₂×C₃²	C₂×#204	1	1	yes	characteristic
258	C₂×C₃²	#904	1	1	yes	characteristic
259	C₂×C₃²	C₄×#67	1	4	yes	—
260	C₂×C₃²	C₂×#215	1	4	yes	—
261	C₂×C₃²	#950	1	4	yes	—
262	C₃⋊D₃	—	12	16	no	—
263	C₃⋊D₃	—	12	2	no	—
264	C₃⋊D₃	—	12	8	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×C₃⋊D₃	1	1	yes	characteristic
277	C₂²×C₃	—	2	4	no	—
278	C₂²×C₃	C₂×#626	1	4	yes	—
279	C₂²×C₃	C₂×#638	1	1	yes	characteristic
280	C₂×D₃	—	18	16	no	—
281	C₂×D₃	—	18	2	no	—
282	C₂×D₃	—	18	8	no	—
283	C₂×D₃	—	36	8	no	—
284	C₂×D₃	—	36	8	no	—
285	C₂×D₃	—	36	4	no	—
286	C₂×D₃	—	36	4	no	—
287	C₂×D₃	—	36	1	no	—
288	C₂×D₃	—	36	1	no	—
289	C₃×C₄	—	2	4	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	4	no	—
292	C₃×C₄	—	2	4	no	—
293	C₃×C₄	—	2	8	no	—
294	C₃×C₄	—	4	4	no	—
295	C₃×C₄	—	6	8	no	—
296	C₃×C₄	—	12	4	no	—
297	C₃×C₄	#3717	1	8	yes	complemented
298	C₃×C₄	#3755	1	2	yes	complemented
299	C₃×C₄	#3798	1	4	yes	—
300	C₃×C₄	C₄×D₃²	1	4	yes	—
301	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
302	C₃×C₄	#3828	1	1	yes	characteristic
303	Dic₃	—	2	2	no	—
304	Dic₃	—	4	1	no	—
305	Dic₃	—	36	8	no	—
306	Dic₃	—	36	4	no	—
307	Dic₃	—	36	1	no	—
308	C₃²	—	2	4	no	—
309	C₃²	C₄×#204	1	1	yes	characteristic, complemented
310	C₃²	C₄×#215	1	4	yes	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	#6291	1	1	yes	characteristic
321	C₂×C₄	#6291	1	1	yes	characteristic
322	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
323	C₈	—	9	2	no	—
324	C₈	—	18	1	no	—
325	C₂×C₃	—	2	4	no	—
326	C₂×C₃	—	2	4	no	—
327	C₂×C₃	—	2	4	no	—
328	C₂×C₃	#63829	1	4	yes	—
329	C₂×C₃	C₂×#3717	1	4	yes	—
330	C₂×C₃	C₄×#626	1	4	yes	—
331	C₂×C₃	C₄×#638	1	1	yes	characteristic
332	C₂×C₃	C₂×#3755	1	1	yes	characteristic
333	C₂×C₃	#64159	1	1	yes	characteristic
334	D₃	—	36	16	no	—
335	D₃	—	36	8	no	—
336	D₃	—	36	2	no	—
337	C₂²	—	54	2	no	—
338	C₂²	—	108	1	no	—
339	C₂²	—	108	1	no	—
340	C₂²	C₂×#6291	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	#88795	1	2	yes	direct product factor
346	C₄	#89020	1	1	yes	characteristic
347	C₄	C₄×D₃×C₃⋊D₃	1	1	yes	characteristic
348	C₃	—	2	4	no	—
349	C₃	C₄×#3717	1	4	yes	complemented
350	C₃	C₄×#3755	1	1	yes	characteristic, complemented
351	C₂	—	108	2	no	—
352	C₂	#11723377	1	1	yes	characteristic
353	C₂	C₂×#88795	1	1	yes	characteristic
354	C₂	C₄×#6291	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6291
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₃³×C₄²	C₂²
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄²	D₃×C₃⋊D₃
hypocenter	C₃³	C₄×SD₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×Dic₆	C₂	semidirect
4. C₃×C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
5. #11722054	C₂	semidirect
6. #11722038	C₂	semidirect
21. #88795	C₄	direct (N × H)
27. C₃×C₄×Dic₆	D₃	semidirect
32. C₄×C₃⋊(C₃⋊C₈)	D₃	semidirect
48. C₃²×Dic₆	C₂×C₄	semidirect
50. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
87. C₃³×C₄	SD₁₆	semidirect
99. C₄×Dic₆	C₃⋊D₃	semidirect
139. C₃×Dic₆	C₄×D₃	semidirect
141. C₃⋊(C₃⋊C₈)	C₄×D₃	semidirect
198. C₃²×C₄	#204	semidirect
199. C₃²×C₄	#215	semidirect
216. C₃³	C₄×SD₁₆	semidirect
248. Dic₆	C₄×C₃⋊D₃	semidirect
297. C₃×C₄	#3717	semidirect
298. C₃×C₄	#3755	semidirect
309. C₃²	C₄×#204	semidirect
310. C₃²	C₄×#215	semidirect
345. C₄	#88795	direct (N × H)
349. C₃	C₄×#3717	semidirect
350. C₃	C₄×#3755	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⋊(C₃⋊D₃))	2
C₃²×C₄×Dic₆	2
C₂×#88795	2
C₃×C₄×C₃⋊(C₃⋊C₈)	2
#11722054	2
#11722038	2
#11722035	2
C₄×#3755	3
C₄×#3717	3

Subgroups — C4×#88795 #13158989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88795 #13158989