Subgroups — C₄×#63575 #11784474

Order 1152.

657 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#63575	C₂	1	2	yes	maximal
5	#10589164	C₂	1	2	yes	maximal
6	C₄×#818	—	3	1	no	maximal
7	C₄×#807	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₄×C₃⋊Dic₃	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	—	2	1	no	—
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	#63658	—	2	1	no	—
16	C₃×C₃⋊#95	—	2	1	no	—
17	C₃×C₄×C₃⋊C₈	C₄	1	4	yes	complemented
18	#63575	C₄	1	8	yes	direct product factor
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	C₃×C₄×C₄⋊C₄	—	3	1	no	—
21	C₂×#818	—	3	2	no	—
22	C₂×#807	—	3	2	no	—
23	C₄×C₄⋊Dic₃	—	3	1	no	—
24	C₄×C₄⋊Dic₃	—	3	1	no	—
25	C₄×C₄⋊Dic₃	—	6	1	no	—
26	C₄×C₄⋊Dic₃	D₃	1	1	yes	characteristic
27	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
28	#4551	—	3	2	no	—
29	#4529	—	3	2	no	—
30	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃⋊Dic₃	—	2	2	no	—
34	C₂²×C₃×Dic₃	—	2	2	no	—
35	C₃²×C₄²	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	Q₈	1	2	yes	—
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	C₂×C₄	1	4	yes	—
39	C₄×C₃⋊Dic₃	—	4	2	no	—
40	C₃×C₄⋊Dic₃	—	2	2	no	—
41	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
42	C₃×C₄×Dic₃	—	4	2	no	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
47	C₄×#107	—	9	1	no	Sylow
48	C₂²×C₃×C₈	—	3	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	—	6	1	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×C₄⋊Dic₃	—	3	2	no	—
57	C₂×C₄⋊Dic₃	—	6	2	no	—
58	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
59	C₂×C₄×Dic₃	—	2	1	no	—
60	C₂×C₄×Dic₃	—	6	2	no	—
61	C₂×C₄×Dic₃	—	6	1	no	—
62	C₂×C₄×Dic₃	—	6	1	no	—
63	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄×C₈	—	3	4	no	—
66	C₃×#95	—	6	1	no	—
67	C₃⋊#95	—	2	1	no	—
68	C₃⋊#95	—	6	2	no	—
69	C₃⋊#95	—	6	1	no	—
70	C₃⋊#95	—	6	1	no	—
71	#818	—	3	8	no	—
72	#807	—	3	8	no	—
73	C₄×C₃⋊C₈	Dic₃	1	4	yes	complemented
74	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
77	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
78	C₂×C₃²×C₄	D₈	1	2	yes	—
79	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
80	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	2	yes	—
84	C₂×C₃⋊Dic₃	—	2	4	no	—
85	C₂×C₃⋊Dic₃	—	4	2	no	—
86	C₂×C₃⋊Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	2	4	no	—
88	C₂×C₃×Dic₃	—	4	2	no	—
89	C₂×C₃×Dic₃	—	4	2	no	—
90	C₃×C₃⋊C₈	—	2	2	no	—
91	C₃×C₃⋊C₈	C₄²	1	4	yes	complemented
92	C₂×#107	—	9	2	no	—
93	C₂×C₄×C₈	—	3	1	no	—
94	C₄×C₄⋊C₄	—	3	1	no	—
95	C₄×C₄⋊C₄	—	9	1	no	—
96	#340	—	9	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	—	6	2	no	—
101	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
102	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
103	C₂²×C₃×C₄	#67	1	1	yes	characteristic
104	C₂²×C₃×C₄	#67	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	2	2	no	—
108	C₂²×Dic₃	—	6	4	no	—
109	C₂²×Dic₃	—	6	2	no	—
110	C₂²×Dic₃	—	6	2	no	—
111	C₂×C₃×C₈	—	3	4	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	3	2	no	—
114	C₂×C₃×C₈	—	6	2	no	—
115	C₃×C₄⋊C₄	—	3	4	no	—
116	C₃×C₄⋊C₄	—	6	2	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	—	12	2	no	—
120	C₃×C₄²	Dic₆	1	2	yes	—
121	C₃×C₄²	Dic₆	1	2	yes	—
122	C₃×C₄²	C₄×D₃	1	2	yes	—
123	C₃×C₄²	C₂×Dic₃	1	2	yes	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₄⋊Dic₃	—	3	4	no	—
126	C₄⋊Dic₃	—	3	4	no	—
127	C₄⋊Dic₃	—	6	4	no	—
128	C₄⋊Dic₃	—	6	4	no	—
129	C₄⋊Dic₃	—	6	2	no	—
130	C₄⋊Dic₃	—	6	2	no	—
131	C₄⋊Dic₃	C₄×D₃	1	4	yes	—
132	C₄×Dic₃	—	4	2	no	—
133	C₄×Dic₃	—	12	4	no	—
134	C₄×Dic₃	—	12	2	no	—
135	C₄×Dic₃	—	12	2	no	—
136	C₂×C₃⋊C₈	—	2	2	no	—
137	C₂×C₃⋊C₈	C₄×D₃	1	4	yes	—
138	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
139	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
140	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₃²	C₄×D₄	1	2	yes	—
142	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
143	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
144	C₂²×C₃²	#135	1	2	yes	—
145	C₃²×C₄	—	2	2	no	—
146	C₃²×C₄	#107	1	4	yes	complemented
147	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
148	C₃²×C₄	#117	1	1	yes	characteristic
149	C₃²×C₄	C₄×Q₈	1	2	yes	—
150	C₃⋊Dic₃	—	4	4	no	—
151	C₃×Dic₃	—	4	4	no	—
152	C₂²×C₈	—	3	1	no	—
153	C₂²×C₈	—	3	1	no	—
154	C₂×C₄⋊C₄	—	3	2	no	—
155	C₂×C₄⋊C₄	—	9	2	no	—
156	C₂×C₄²	—	6	1	no	—
157	C₂×C₄²	—	18	1	no	—
158	C₂×C₄²	D₃²	1	1	yes	characteristic
159	#107	—	9	8	no	—
160	C₄×C₈	—	3	4	no	—
161	#95	—	6	1	no	—
162	#95	—	18	1	no	—
163	C₂³×C₃	—	2	1	no	—
164	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
165	C₂³×C₃	C₂×#67	1	1	yes	characteristic
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	6	4	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	D₂₄	1	2	yes	—
181	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
182	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
183	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
184	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
185	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
186	C₂×C₃×C₄	#213	1	2	yes	—
187	C₂×C₃×C₄	#216	1	2	yes	—
188	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
191	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
192	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	#235	1	2	yes	—
194	C₂×C₃×C₄	#235	1	2	yes	—
195	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	6	8	no	—
200	C₂×Dic₃	—	6	4	no	—
201	C₂×Dic₃	—	6	4	no	—
202	C₂×Dic₃	—	12	4	no	—
203	C₂×Dic₃	—	12	4	no	—
204	C₂×Dic₃	—	12	2	no	—
205	C₂×Dic₃	—	12	2	no	—
206	C₂×Dic₃	—	12	2	no	—
207	C₂×Dic₃	—	12	2	no	—
208	C₃×C₈	—	3	4	no	—
209	C₃×C₈	—	6	2	no	—
210	C₃⋊C₈	—	2	2	no	—
211	C₃⋊C₈	C₄×Dic₃	1	4	yes	complemented
212	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₃²	C₂×#107	1	1	yes	characteristic
214	C₂×C₃²	#427	1	1	yes	characteristic
215	C₂×C₃²	C₄×D₈	1	2	yes	—
216	C₂×C₃²	C₄×Q₁₆	1	2	yes	—
217	C₂²×C₄	—	6	2	no	—
218	C₂²×C₄	—	18	2	no	—
219	C₂²×C₄	#626	1	1	yes	characteristic
220	C₂²×C₄	#626	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
222	C₂×C₈	—	3	4	no	—
223	C₂×C₈	—	3	2	no	—
224	C₂×C₈	—	3	2	no	—
225	C₂×C₈	—	6	2	no	—
226	C₄⋊C₄	—	3	4	no	—
227	C₄⋊C₄	—	6	2	no	—
228	C₄⋊C₄	—	9	4	no	—
229	C₄⋊C₄	—	18	2	no	—
230	C₄²	—	12	2	no	—
231	C₄²	—	36	2	no	—
232	C₄²	D₃×Dic₃	1	2	yes	—
233	C₄²	#627	1	2	yes	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	C₄×D₁₂	1	2	yes	—
240	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
241	C₂²×C₃	#904	1	2	yes	—
242	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
243	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
244	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
245	C₂²×C₃	C₄×#67	1	2	yes	—
246	C₂²×C₃	C₂×#213	1	1	yes	characteristic
247	C₂²×C₃	C₂×#216	1	1	yes	characteristic
248	C₂²×C₃	#950	1	2	yes	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	12	4	no	—
257	C₃×C₄	#807	1	4	yes	complemented
258	C₃×C₄	#818	1	4	yes	complemented
259	C₃×C₄	C₄×Dic₆	1	2	yes	—
260	C₃×C₄	C₄×Dic₆	1	2	yes	—
261	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
262	C₃×C₄	#872	1	1	yes	characteristic
263	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
264	C₃×C₄	#926	1	1	yes	characteristic
265	Dic₃	—	4	4	no	—
266	Dic₃	—	12	8	no	—
267	Dic₃	—	12	4	no	—
268	Dic₃	—	12	4	no	—
269	C₃²	C₄×#107	1	1	yes	characteristic, complemented, Sylow
270	C₂³	C₂×#626	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	6	4	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	—	12	2	no	—
275	C₂×C₄	—	18	4	no	—
276	C₂×C₄	—	36	2	no	—
277	C₂×C₄	—	36	2	no	—
278	C₂×C₄	#3714	1	2	yes	—
279	C₂×C₄	#3719	1	2	yes	—
280	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
281	C₂×C₄	Dic₃⋊Dic₃	1	2	yes	—
282	C₂×C₄	#3798	1	2	yes	—
283	C₂×C₄	C₄×D₃²	1	2	yes	—
284	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
285	C₂×C₄	C₂×#627	1	1	yes	characteristic
286	C₈	—	3	4	no	—
287	C₈	—	6	2	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	C₄×D₂₄	1	2	yes	—
294	C₂×C₃	C₄×Dic₁₂	1	2	yes	—
295	C₂×C₃	C₄×Dic₃⋊C₄	1	1	yes	characteristic
296	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
297	C₂×C₃	C₂×#807	1	1	yes	characteristic
298	C₂×C₃	#4998	1	1	yes	characteristic
299	C₂×C₃	C₄×#213	1	2	yes	—
300	C₂×C₃	C₄×#216	1	2	yes	—
301	C₂×C₃	C₂×#818	1	1	yes	characteristic
302	C₂×C₃	#5106	1	1	yes	characteristic
303	C₂²	C₂×#3714	1	1	yes	characteristic
304	C₂²	#63829	1	2	yes	—
305	C₂²	C₂×#3719	1	1	yes	characteristic
306	C₂²	C₄×#626	1	2	yes	—
307	C₂²	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	12	4	no	—
310	C₄	—	36	4	no	—
311	C₄	#63575	1	4	yes	direct product factor
312	C₄	#63855	1	1	yes	characteristic
313	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
314	C₄	C₄×#627	1	2	yes	—
315	C₃	—	2	1	no	—
316	C₃	C₄×#807	1	1	yes	characteristic, complemented
317	C₃	C₄×#818	1	1	yes	characteristic, complemented
318	C₂	C₄×#3714	1	2	yes	—
319	C₂	C₄×#3719	1	2	yes	—
320	C₂	#10591122	1	1	yes	characteristic
321	C₂	C₂×#63575	1	1	yes	characteristic
322	C₂	C₄×Dic₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#626
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	C₄×#107
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
17. C₃×C₄×C₃⋊C₈	C₄	semidirect
18. #63575	C₄	direct (N × H)
73. C₄×C₃⋊C₈	Dic₃	semidirect
91. C₃×C₃⋊C₈	C₄²	semidirect
146. C₃²×C₄	#107	semidirect
211. C₃⋊C₈	C₄×Dic₃	semidirect
257. C₃×C₄	#807	semidirect
258. C₃×C₄	#818	semidirect
269. C₃²	C₄×#107	semidirect
311. C₄	#63575	direct (N × H)
316. C₃	C₄×#807	semidirect
317. C₃	C₄×#818	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⋊Dic₃)	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×C₄×C₃⋊C₈	2
C₂×#63575	2
#10589164	2
C₄×#818	3
C₄×#807	3

Subgroups — C4×#63575 #11784474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#63575 #11784474