Subgroups — D₃×#576 #87202

Order 384.

1512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
3	#5835	C₂	1	2	yes	maximal, complemented
4	#5774	C₂	1	6	yes	maximal, complemented
5	D₃×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
6	#5769	C₂	1	3	yes	maximal, complemented
7	D₃×#135	C₂	1	3	yes	maximal, complemented
8	#5757	C₂	1	3	yes	maximal, complemented
9	C₂×D₃×D₈	C₂	1	3	yes	maximal, complemented
10	#5752	C₂	1	1	yes	maximal, characteristic, complemented
11	D₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#576	—	3	1	no	maximal, Sylow
13	C₃×H(5,𝔽₂)	C₂²	1	2	yes	complemented
14	#1009	—	2	2	no	—
15	#1009	C₂²	1	6	yes	complemented
16	#1009	C₂²	1	2	yes	complemented
17	D₃×#41	—	2	2	no	—
18	D₃×#41	C₂²	1	1	yes	characteristic
19	D₃×#41	C₂²	1	6	yes	complemented
20	#1004	—	2	6	no	—
21	#1004	C₂²	1	6	yes	complemented
22	C₂×D₃×D₄	—	2	6	no	—
23	C₂×D₃×D₄	C₂²	1	6	yes	complemented
24	C₃×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
25	C₃×#135	C₂²	1	3	yes	complemented
26	C₂×C₃×D₈	C₂²	1	3	yes	complemented
27	C₃×#131	C₂²	1	1	yes	characteristic, complemented
28	#950	C₂²	1	6	yes	complemented
29	#949	C₂²	1	6	yes	complemented
30	#948	C₂²	1	1	yes	characteristic, complemented
31	#932	C₂²	1	6	yes	complemented
32	C₂×#213	C₂²	1	6	yes	complemented
33	#919	C₂²	1	3	yes	complemented
34	D₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
35	#916	C₂²	1	6	yes	complemented
36	#914	C₂²	1	12	yes	complemented
37	D₃×SD₁₆	C₂²	1	6	yes	complemented
38	#912	C₂²	1	3	yes	complemented
39	#911	C₂²	1	12	yes	complemented
40	D₃×D₈	C₂²	1	3	yes	complemented
41	D₃×D₈	C₂²	1	6	yes	complemented
42	D₃×D₈	C₂²	1	6	yes	complemented
43	D₃×D₈	C₂²	1	6	yes	complemented
44	#908	C₂²	1	6	yes	complemented
45	#907	C₂²	1	3	yes	complemented
46	D₃×M₁₆	C₂²	1	3	yes	complemented
47	#904	C₂²	1	3	yes	complemented
48	C₂×D₂₄	C₂²	1	3	yes	complemented
49	#901	C₂²	1	3	yes	complemented
50	C₂×D₃×C₈	C₂²	1	3	yes	complemented
51	C₂×H(5,𝔽₂)	—	3	2	no	—
52	#576	—	3	2	no	—
53	#576	—	3	6	no	—
54	#576	—	3	3	no	—
55	#576	—	3	3	no	—
56	#576	—	3	1	no	—
57	#576	D₃	1	1	yes	characteristic, direct product factor
58	C₂×AGL(1,ℤ/8)	—	3	6	no	—
59	C₂×#135	—	3	3	no	—
60	C₂²×D₈	—	3	3	no	—
61	C₂×#131	—	3	1	no	—
62	C₂³×D₃	—	2	6	no	—
63	C₃×#41	—	2	2	no	—
64	C₃×#41	C₂³	1	1	yes	characteristic
65	C₃×#41	C₂³	1	6	yes	complemented
66	C₂×C₃×D₄	—	2	6	no	—
67	C₂×C₃×D₄	C₂³	1	6	yes	complemented
68	C₂×#67	—	2	12	no	—
69	C₂×#67	—	2	6	no	—
70	#239	—	2	2	no	—
71	#239	D₄	1	1	yes	characteristic
72	#239	C₂³	1	6	yes	complemented
73	D₃×Q₈	D₄	1	1	yes	characteristic
74	D₃×Q₈	C₂³	1	2	yes	—
75	#237	—	2	6	no	—
76	#237	—	2	6	no	—
77	#237	D₄	1	3	yes	—
78	#237	C₂³	1	6	yes	complemented
79	D₃×D₄	—	2	6	no	—
80	D₃×D₄	—	2	6	no	—
81	D₃×D₄	—	2	12	no	—
82	D₃×D₄	—	2	6	no	—
83	D₃×D₄	—	2	6	no	—
84	D₃×D₄	—	2	6	no	—
85	D₃×D₄	D₄	1	3	yes	—
86	D₃×D₄	C₂³	1	6	yes	—
87	D₃×D₄	C₂³	1	12	yes	complemented
88	#235	—	2	6	no	—
89	#235	—	2	6	no	—
90	#235	C₂³	1	3	yes	—
91	#235	C₂³	1	6	yes	complemented
92	C₂×D₁₂	—	2	6	no	—
93	C₂×D₁₂	C₂³	1	6	yes	complemented
94	C₂×C₄×D₃	—	2	6	no	—
95	C₂×C₄×D₃	C₂³	1	3	yes	—
96	C₃×Q₁₆	C₂³	1	1	yes	characteristic, complemented
97	C₃×SD₁₆	C₂³	1	6	yes	complemented
98	C₃×D₈	C₂³	1	3	yes	complemented
99	C₃×D₈	C₂³	1	6	yes	complemented
100	C₃×M₁₆	C₂³	1	3	yes	complemented
101	C₂×C₃×C₈	C₂³	1	3	yes	complemented
102	#216	C₂³	1	2	yes	complemented
103	#215	C₂³	1	6	yes	complemented
104	#214	C₂³	1	6	yes	complemented
105	#213	C₂³	1	6	yes	complemented
106	#213	C₂³	1	12	yes	complemented
107	#208	C₂³	1	3	yes	complemented
108	C₂×C₃⋊C₈	C₂³	1	3	yes	complemented
109	Dic₁₂	C₂³	1	1	yes	characteristic, complemented
110	D₂₄	C₂³	1	3	yes	complemented
111	D₂₄	C₂³	1	6	yes	complemented
112	#204	C₂³	1	6	yes	complemented
113	ΓL(1,𝔽₂₅)	C₂³	1	6	yes	complemented
114	D₃×C₈	C₂³	1	1	yes	characteristic
115	D₃×C₈	C₂³	1	3	yes	—
116	D₃×C₈	C₂³	1	3	yes	complemented
117	D₃×C₈	C₂³	1	3	yes	complemented
118	H(5,𝔽₂)	—	3	2	no	—
119	H(5,𝔽₂)	—	3	6	no	—
120	H(5,𝔽₂)	—	3	6	no	—
121	H(5,𝔽₂)	—	6	6	no	—
122	H(5,𝔽₂)	—	6	2	no	—
123	H(5,𝔽₂)	C₂×D₃	1	2	yes	complemented
124	C₂×#41	—	3	1	no	—
125	C₂×#41	—	3	6	no	—
126	C₂×#41	—	6	2	no	—
127	C₂²×D₄	—	3	6	no	—
128	C₂²×D₄	—	6	6	no	—
129	AGL(1,ℤ/8)	—	3	6	no	—
130	AGL(1,ℤ/8)	—	3	6	no	—
131	AGL(1,ℤ/8)	—	3	12	no	—
132	AGL(1,ℤ/8)	—	3	12	no	—
133	AGL(1,ℤ/8)	—	3	6	no	—
134	AGL(1,ℤ/8)	C₂×D₃	1	6	yes	complemented
135	#135	—	3	6	no	—
136	#135	—	3	3	no	—
137	#135	—	3	6	no	—
138	#135	—	3	3	no	—
139	#135	—	3	3	no	—
140	#135	C₂×D₃	1	3	yes	complemented
141	C₂×Q₁₆	—	3	1	no	—
142	C₂×SD₁₆	—	3	6	no	—
143	C₂×D₈	—	3	6	no	—
144	C₂×D₈	—	3	3	no	—
145	C₂×D₈	—	3	6	no	—
146	C₂×D₈	—	3	6	no	—
147	C₂×D₈	—	3	6	no	—
148	C₂×D₈	—	3	3	no	—
149	C₂×D₈	C₂×D₃	1	3	yes	complemented
150	#131	—	3	1	no	—
151	#131	—	3	3	no	—
152	#131	—	3	3	no	—
153	#131	C₂×D₃	1	1	yes	characteristic, complemented
154	C₂×M₁₆	—	3	3	no	—
155	C₂²×C₈	—	3	3	no	—
156	C₂³×C₃	—	2	6	no	—
157	C₂²×D₃	—	2	6	no	—
158	C₂²×D₃	—	2	6	no	—
159	C₂²×D₃	—	2	12	no	—
160	C₂²×D₃	—	8	12	no	—
161	C₂²×D₃	C₂×D₄	1	3	yes	—
162	C₃×Q₈	C₂×D₄	1	1	yes	characteristic
163	C₃×Q₈	C₂⁴	1	2	yes	—
164	C₃×D₄	—	2	6	no	—
165	C₃×D₄	—	2	6	no	—
166	C₃×D₄	C₂×D₄	1	3	yes	—
167	C₃×D₄	C₂⁴	1	6	yes	—
168	C₂×C₃×C₄	—	2	6	no	—
169	C₂×C₃×C₄	C₂⁴	1	3	yes	—
170	#67	—	2	6	no	—
171	#67	—	2	6	no	—
172	#67	—	2	6	no	—
173	#67	—	2	12	no	—
174	#67	C₂×D₄	1	6	yes	—
175	C₂×Dic₃	—	2	6	no	—
176	C₂×Dic₃	C₂×D₄	1	3	yes	—
177	D₁₂	—	2	6	no	—
178	D₁₂	—	2	6	no	—
179	D₁₂	C₂×D₄	1	3	yes	—
180	D₁₂	C₂⁴	1	6	yes	—
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	—	2	6	no	—
183	C₄×D₃	—	2	6	no	—
184	C₄×D₃	C₂×D₄	1	3	yes	—
185	C₄×D₃	C₂×D₄	1	3	yes	—
186	C₄×D₃	C₂⁴	1	1	yes	characteristic
187	C₄×D₃	C₂⁴	1	3	yes	—
188	Dic₆	—	2	2	no	—
189	Dic₆	C₂×D₄	1	3	yes	—
190	Dic₆	C₂⁴	1	2	yes	—
191	C₃×C₈	C₂⁴	1	1	yes	characteristic
192	C₃×C₈	C₂⁴	1	3	yes	—
193	C₃⋊C₈	C₂⁴	1	1	yes	characteristic
194	C₃⋊C₈	C₂⁴	1	3	yes	—
195	C₂⁴	—	6	6	no	—
196	#41	—	2	2	no	—
197	#41	—	3	1	no	—
198	#41	—	3	3	no	—
199	#41	—	3	3	no	—
200	#41	—	3	6	no	—
201	#41	—	3	6	no	—
202	#41	—	3	6	no	—
203	#41	—	6	6	no	—
204	#41	—	6	6	no	—
205	#41	—	6	6	no	—
206	#41	—	6	6	no	—
207	#41	—	6	2	no	—
208	#41	C₂²×D₃	1	1	yes	characteristic
209	#41	C₂²×D₃	1	6	yes	complemented
210	C₂×Q₈	—	3	1	no	—
211	C₂×Q₈	—	3	2	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	—	3	3	no	—
214	C₂×D₄	—	3	6	no	—
215	C₂×D₄	—	3	12	no	—
216	C₂×D₄	—	3	6	no	—
217	C₂×D₄	—	6	6	no	—
218	C₂×D₄	—	6	6	no	—
219	C₂×D₄	—	6	6	no	—
220	C₂×D₄	—	6	6	no	—
221	C₂×D₄	—	6	12	no	—
222	C₂×D₄	—	6	6	no	—
223	C₂×D₄	—	6	6	no	—
224	C₂×D₄	—	6	12	no	—
225	C₂×D₄	—	6	6	no	—
226	C₂×D₄	C₂²×D₃	1	6	yes	complemented
227	C₂²×C₄	—	3	3	no	—
228	C₂²×C₄	—	6	6	no	—
229	Q₁₆	—	3	2	no	—
230	Q₁₆	—	3	1	no	—
231	Q₁₆	C₂²×D₃	1	1	yes	characteristic, complemented
232	SD₁₆	—	3	6	no	—
233	SD₁₆	—	3	6	no	—
234	SD₁₆	—	3	6	no	—
235	SD₁₆	C₂²×D₃	1	6	yes	complemented
236	D₈	—	3	6	no	—
237	D₈	—	3	12	no	—
238	D₈	—	3	3	no	—
239	D₈	—	3	6	no	—
240	D₈	C₂²×D₃	1	3	yes	complemented
241	D₈	C₂²×D₃	1	6	yes	complemented
242	M₁₆	—	3	3	no	—
243	M₁₆	—	3	6	no	—
244	M₁₆	C₂²×D₃	1	3	yes	complemented
245	C₂×C₈	—	3	3	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	—	3	3	no	—
248	C₂×C₈	—	3	3	no	—
249	C₂×C₈	—	3	3	no	—
250	C₂×C₈	C₂²×D₃	1	3	yes	complemented
251	C₂²×C₃	—	2	6	no	—
252	C₂²×C₃	—	8	6	no	—
253	C₂²×C₃	C₂²×D₄	1	3	yes	—
254	C₂×D₃	—	2	6	no	—
255	C₂×D₃	—	2	6	no	—
256	C₂×D₃	—	4	12	no	—
257	C₂×D₃	—	8	12	no	—
258	C₂×D₃	—	8	6	no	—
259	C₂×D₃	C₂²×D₄	1	3	yes	—
260	C₂×D₃	C₂²×D₄	1	1	yes	characteristic
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	C₂²×D₄	1	3	yes	—
263	C₃×C₄	C₂⁵	1	1	yes	characteristic
264	Dic₃	—	2	2	no	—
265	Dic₃	C₂²×D₄	1	1	yes	characteristic
266	Dic₃	C₂²×D₄	1	3	yes	—
267	C₂³	—	2	6	no	—
268	C₂³	—	3	3	no	—
269	C₂³	—	6	6	no	—
270	C₂³	—	6	12	no	—
271	C₂³	—	6	6	no	—
272	C₂³	—	24	12	no	—
273	Q₈	—	3	3	no	—
274	Q₈	—	3	2	no	—
275	Q₈	—	6	2	no	—
276	Q₈	D₃×D₄	1	1	yes	characteristic
277	Q₈	C₂³×D₃	1	2	yes	—
278	D₄	—	2	6	no	—
279	D₄	—	2	6	no	—
280	D₄	—	3	6	no	—
281	D₄	—	3	3	no	—
282	D₄	—	3	6	no	—
283	D₄	—	6	6	no	—
284	D₄	—	6	6	no	—
285	D₄	—	6	12	no	—
286	D₄	—	6	6	no	—
287	D₄	—	6	6	no	—
288	D₄	—	6	6	no	—
289	D₄	D₃×D₄	1	3	yes	—
290	D₄	C₂³×D₃	1	6	yes	—
291	C₂×C₄	—	2	6	no	—
292	C₂×C₄	—	3	3	no	—
293	C₂×C₄	—	3	3	no	—
294	C₂×C₄	—	3	3	no	—
295	C₂×C₄	—	3	1	no	—
296	C₂×C₄	—	3	3	no	—
297	C₂×C₄	—	6	6	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	6	no	—
300	C₂×C₄	—	6	6	no	—
301	C₂×C₄	C₂³×D₃	1	3	yes	—
302	C₈	—	3	1	no	—
303	C₈	—	3	3	no	—
304	C₈	C₂³×D₃	1	1	yes	characteristic
305	C₈	C₂³×D₃	1	3	yes	—
306	C₂×C₃	—	2	3	no	—
307	C₂×C₃	—	4	6	no	—
308	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
309	D₃	—	2	3	no	—
310	D₃	—	4	6	no	—
311	D₃	#576	1	2	yes	direct product factor
312	C₂²	—	2	6	no	—
313	C₂²	—	3	1	no	—
314	C₂²	—	3	3	no	—
315	C₂²	—	6	6	no	—
316	C₂²	—	6	6	no	—
317	C₂²	—	8	6	no	—
318	C₂²	—	12	12	no	—
319	C₂²	—	24	6	no	—
320	C₂²	—	24	12	no	—
321	C₂²	C₂×D₃×D₄	1	3	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	3	1	no	—
324	C₄	—	3	3	no	—
325	C₄	—	6	2	no	—
326	C₄	C₂×D₃×D₄	1	3	yes	—
327	C₄	C₂⁴×D₃	1	1	yes	characteristic
328	C₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	3	no	—
330	C₂	—	3	2	no	—
331	C₂	—	4	6	no	—
332	C₂	—	6	3	no	—
333	C₂	—	12	6	no	—
334	C₂	C₂²×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₃×#576	C₂
socle	C₂×C₃	C₂³×D₄
hypercenter	#576	D₃
hypocenter	C₃	C₂×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. D₃×H(5,𝔽₂)	C₂	semidirect
2. C₃×#576	C₂	semidirect
3. #5835	C₂	semidirect
4. #5774	C₂	semidirect
5. D₃×AGL(1,ℤ/8)	C₂	semidirect
6. #5769	C₂	semidirect
7. D₃×#135	C₂	semidirect
8. #5757	C₂	semidirect
9. C₂×D₃×D₈	C₂	semidirect
10. #5752	C₂	semidirect
11. D₃×#131	C₂	semidirect
13. C₃×H(5,𝔽₂)	C₂²	semidirect
15. #1009	C₂²	semidirect
16. #1009	C₂²	semidirect
19. D₃×#41	C₂²	semidirect
21. #1004	C₂²	semidirect
23. C₂×D₃×D₄	C₂²	semidirect
24. C₃×AGL(1,ℤ/8)	C₂²	semidirect
25. C₃×#135	C₂²	semidirect
26. C₂×C₃×D₈	C₂²	semidirect
27. C₃×#131	C₂²	semidirect
28. #950	C₂²	semidirect
29. #949	C₂²	semidirect
30. #948	C₂²	semidirect
31. #932	C₂²	semidirect
32. C₂×#213	C₂²	semidirect
33. #919	C₂²	semidirect
34. D₃×Q₁₆	C₂²	semidirect
35. #916	C₂²	semidirect
36. #914	C₂²	semidirect
37. D₃×SD₁₆	C₂²	semidirect
38. #912	C₂²	semidirect
39. #911	C₂²	semidirect
40. D₃×D₈	C₂²	semidirect
41. D₃×D₈	C₂²	semidirect
42. D₃×D₈	C₂²	semidirect
43. D₃×D₈	C₂²	semidirect
44. #908	C₂²	semidirect
45. #907	C₂²	semidirect
46. D₃×M₁₆	C₂²	semidirect
47. #904	C₂²	semidirect
48. C₂×D₂₄	C₂²	semidirect
49. #901	C₂²	semidirect
50. C₂×D₃×C₈	C₂²	semidirect
57. #576	D₃	direct (N × H)
65. C₃×#41	C₂³	semidirect
67. C₂×C₃×D₄	C₂³	semidirect
72. #239	C₂³	semidirect
78. #237	C₂³	semidirect
87. D₃×D₄	C₂³	semidirect
91. #235	C₂³	semidirect
93. C₂×D₁₂	C₂³	semidirect
96. C₃×Q₁₆	C₂³	semidirect
97. C₃×SD₁₆	C₂³	semidirect
98. C₃×D₈	C₂³	semidirect
99. C₃×D₈	C₂³	semidirect
100. C₃×M₁₆	C₂³	semidirect
101. C₂×C₃×C₈	C₂³	semidirect
102. #216	C₂³	semidirect
103. #215	C₂³	semidirect
104. #214	C₂³	semidirect
105. #213	C₂³	semidirect
106. #213	C₂³	semidirect
107. #208	C₂³	semidirect
108. C₂×C₃⋊C₈	C₂³	semidirect
109. Dic₁₂	C₂³	semidirect
110. D₂₄	C₂³	semidirect
111. D₂₄	C₂³	semidirect
112. #204	C₂³	semidirect
113. ΓL(1,𝔽₂₅)	C₂³	semidirect
116. D₃×C₈	C₂³	semidirect
117. D₃×C₈	C₂³	semidirect
123. H(5,𝔽₂)	C₂×D₃	semidirect
134. AGL(1,ℤ/8)	C₂×D₃	semidirect
140. #135	C₂×D₃	semidirect
149. C₂×D₈	C₂×D₃	semidirect
153. #131	C₂×D₃	semidirect
209. #41	C₂²×D₃	semidirect
226. C₂×D₄	C₂²×D₃	semidirect
231. Q₁₆	C₂²×D₃	semidirect
235. SD₁₆	C₂²×D₃	semidirect
240. D₈	C₂²×D₃	semidirect
241. D₈	C₂²×D₃	semidirect
244. M₁₆	C₂²×D₃	semidirect
250. C₂×C₈	C₂²×D₃	semidirect
311. D₃	#576	direct (N × H)
328. C₃	C₂×#576	semidirect

Maximal subgroups

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

subgroup	index
D₃×H(5,𝔽₂)	2
C₃×#576	2
#5835	2
#5774	2
D₃×AGL(1,ℤ/8)	2
#5769	2
D₃×#135	2
#5757	2
C₂×D₃×D₈	2
#5752	2
D₃×#131	2
C₂×#576	3

Subgroups — D3×#576 #87202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#576 #87202