Subgroups — C₄×D₄×C₃⋊D₃ #10596015

Order 576.

1278 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
3	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#638	C₂	1	2	yes	maximal, complemented
6	#64102	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#64090	C₂	1	2	yes	maximal, complemented
9	C₂²⋊C₄×C₃⋊D₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₃×D₄	—	3	4	no	maximal
13	C₂³×C₃⋊D₃	C₂²	1	2	yes	—
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	2	yes	complemented
16	C₂×#638	C₂²	1	2	yes	—
17	C₂²×C₃⋊Dic₃	C₂²	1	2	yes	complemented
18	D₄×C₃⋊D₃	C₄	1	8	yes	direct product factor
19	C₂×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₃⋊D₃	—	2	8	no	—
21	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	—
23	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	complemented
26	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	complemented
27	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₃²×C₂²⋊C₄	C₂²	1	2	yes	complemented
29	C₃²×C₄²	C₂²	1	1	yes	characteristic, complemented
30	#3757	C₂²	1	2	yes	complemented
31	#3752	C₂²	1	2	yes	complemented
32	#3752	C₂²	1	2	yes	complemented
33	C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic, complemented
34	#3750	C₂²	1	2	yes	complemented
35	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
37	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×D₃×D₄	—	3	4	no	—
39	C₂²×C₄×D₃	—	3	8	no	—
40	C₃×C₄×D₄	D₃	1	4	yes	complemented
41	D₄×Dic₃	—	3	4	no	—
42	C₄×#67	—	3	8	no	—
43	#893	—	3	4	no	—
44	D₃×C₄⋊C₄	—	3	4	no	—
45	#67⋊C₄	—	3	8	no	—
46	D₃×C₂²⋊C₄	—	3	8	no	—
47	C₄×D₁₂	—	3	4	no	—
48	C₄²×D₃	—	3	4	no	—
49	C₂³×C₃²	C₂³	1	2	yes	—
50	C₂²×C₃⋊D₃	—	2	4	no	—
51	C₂²×C₃⋊D₃	—	2	4	no	—
52	C₂²×C₃⋊D₃	C₂×C₄	1	8	yes	complemented
53	C₂²×C₃⋊D₃	C₂³	1	2	yes	—
54	C₂²×C₃⋊D₃	C₂³	1	1	yes	characteristic
55	C₃²×D₄	C₂×C₄	1	4	yes	complemented
56	C₂×C₃²×C₄	—	2	4	no	—
57	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂³	1	2	yes	—
59	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
61	#638	C₂×C₄	1	8	yes	complemented
62	C₂×C₃⋊Dic₃	—	2	4	no	—
63	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₃⋊Dic₃	C₂³	1	2	yes	—
65	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
66	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
67	C₄⋊(C₃⋊D₃)	C₂×C₄	1	4	yes	complemented
68	C₄×C₃⋊D₃	—	2	4	no	—
69	C₄×C₃⋊D₃	—	2	2	no	—
70	C₄×C₃⋊D₃	—	2	4	no	—
71	C₄×C₃⋊D₃	—	2	4	no	—
72	C₄×C₃⋊D₃	C₂×C₄	1	4	yes	complemented
73	C₄×C₃⋊D₃	D₄	1	4	yes	direct product factor
74	C₂×C₄×D₄	—	9	1	no	Sylow
75	C₂³×D₃	—	3	8	no	—
76	C₂×C₃×D₄	C₂×D₃	1	4	yes	—
77	C₂²×C₃×C₄	C₂×D₃	1	8	yes	complemented
78	C₂×#67	—	3	8	no	—
79	C₂²×Dic₃	—	3	8	no	—
80	D₃×D₄	—	3	32	no	—
81	C₂×D₁₂	—	3	4	no	—
82	C₂×C₄×D₃	—	3	4	no	—
83	C₂×C₄×D₃	—	3	4	no	—
84	C₂×C₄×D₃	—	3	4	no	—
85	C₂×C₄×D₃	—	3	8	no	—
86	C₂×C₄×D₃	—	3	8	no	—
87	C₂×C₄×D₃	—	3	8	no	—
88	C₂×C₄×D₃	—	6	32	no	—
89	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	complemented
90	C₃×C₂²⋊C₄	C₂×D₃	1	8	yes	complemented
91	C₃×C₄²	C₂×D₃	1	4	yes	complemented
92	#217	—	3	8	no	—
93	(C₂×D₃)⋊C₄	—	3	8	no	—
94	(C₂×D₃)⋊C₄	—	3	8	no	—
95	C₄⋊Dic₃	—	3	4	no	—
96	Dic₃⋊C₄	—	3	8	no	—
97	C₄×Dic₃	—	3	4	no	—
98	C₄×Dic₃	—	3	4	no	—
99	C₄×Dic₃	—	3	4	no	—
100	C₂²×C₃²	—	2	2	no	—
101	C₂²×C₃²	—	2	2	no	—
102	C₂²×C₃²	C₂²×C₄	1	4	yes	complemented
103	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
104	C₂×C₃⋊D₃	—	2	2	no	—
105	C₂×C₃⋊D₃	—	2	2	no	—
106	C₂×C₃⋊D₃	—	2	16	no	—
107	C₂×C₃⋊D₃	C₂²×C₄	1	2	yes	—
108	C₂×C₃⋊D₃	C₂²×C₄	1	4	yes	complemented
109	C₂×C₃⋊D₃	C₂×D₄	1	2	yes	—
110	C₂×C₃⋊D₃	#41	1	2	yes	—
111	C₃²×C₄	—	2	2	no	—
112	C₃²×C₄	—	2	1	no	—
113	C₃²×C₄	C₂²×C₄	1	2	yes	complemented
114	C₃²×C₄	C₂×D₄	1	2	yes	complemented
115	C₃⋊Dic₃	—	2	2	no	—
116	C₃⋊Dic₃	—	2	1	no	—
117	C₃⋊Dic₃	C₂²×C₄	1	2	yes	complemented
118	C₃⋊Dic₃	C₂×D₄	1	2	yes	complemented
119	C₂²×D₄	—	9	1	no	—
120	C₂³×C₄	—	9	2	no	—
121	C₄×D₄	—	9	1	no	—
122	C₄×D₄	—	9	2	no	—
123	C₄×D₄	—	9	1	no	—
124	C₄×D₄	—	9	2	no	—
125	C₄×D₄	—	9	1	no	—
126	C₄×D₄	C₃⋊D₃	1	1	yes	characteristic, direct product factor
127	C₂×C₄⋊C₄	—	9	1	no	—
128	C₂×C₂²⋊C₄	—	9	2	no	—
129	C₂×C₄²	—	9	1	no	—
130	C₂³×C₃	C₂²×D₃	1	8	yes	—
131	C₂²×D₃	—	3	4	no	—
132	C₂²×D₃	—	3	32	no	—
133	C₂²×D₃	—	3	8	no	—
134	C₂²×D₃	—	6	16	no	—
135	C₂²×D₃	—	6	16	no	—
136	C₃×D₄	C₄×D₃	1	16	yes	complemented
137	C₂×C₃×C₄	—	2	16	no	—
138	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
139	C₂×C₃×C₄	C₂²×D₃	1	8	yes	—
140	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
141	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
142	#67	—	3	32	no	—
143	C₂×Dic₃	—	3	4	no	—
144	C₂×Dic₃	—	3	4	no	—
145	C₂×Dic₃	—	3	4	no	—
146	C₂×Dic₃	—	3	8	no	—
147	C₂×Dic₃	—	6	16	no	—
148	D₁₂	—	3	16	no	—
149	C₄×D₃	—	3	16	no	—
150	C₄×D₃	—	3	16	no	—
151	C₄×D₃	—	6	16	no	—
152	C₄×D₃	—	6	8	no	—
153	C₄×D₃	—	6	16	no	—
154	C₄×D₃	—	6	16	no	—
155	C₂×C₃²	—	2	4	no	—
156	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₃²	C₂×#41	1	1	yes	characteristic
159	C₃⋊D₃	—	2	4	no	—
160	C₃⋊D₃	C₄×D₄	1	4	yes	direct product factor
161	C₂⁴	—	9	2	no	—
162	C₂×D₄	—	9	2	no	—
163	C₂×D₄	—	9	8	no	—
164	C₂×D₄	—	9	1	no	—
165	C₂×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
166	C₂²×C₄	—	9	1	no	—
167	C₂²×C₄	—	9	1	no	—
168	C₂²×C₄	—	9	2	no	—
169	C₂²×C₄	—	9	1	no	—
170	C₂²×C₄	—	9	2	no	—
171	C₂²×C₄	—	9	2	no	—
172	C₂²×C₄	—	9	2	no	—
173	C₂²×C₄	—	18	8	no	—
174	C₂²×C₄	C₂×C₃⋊D₃	1	2	yes	complemented
175	C₄⋊C₄	—	9	2	no	—
176	C₄⋊C₄	—	9	1	no	—
177	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
178	C₂²⋊C₄	—	9	2	no	—
179	C₂²⋊C₄	—	9	2	no	—
180	C₂²⋊C₄	—	9	2	no	—
181	C₂²⋊C₄	C₂×C₃⋊D₃	1	2	yes	complemented
182	C₄²	—	9	1	no	—
183	C₄²	—	9	1	no	—
184	C₄²	—	9	1	no	—
185	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
186	C₂²×C₃	—	2	8	no	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	C₂×C₄×D₃	1	16	yes	complemented
189	C₂²×C₃	C₂³×D₃	1	4	yes	—
190	C₂×D₃	—	3	8	no	—
191	C₂×D₃	—	3	8	no	—
192	C₂×D₃	—	3	8	no	—
193	C₂×D₃	—	3	16	no	—
194	C₂×D₃	—	6	64	no	—
195	C₂×D₃	—	6	8	no	—
196	C₂×D₃	—	6	8	no	—
197	C₃×C₄	—	2	8	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	C₂×C₄×D₃	1	8	yes	complemented
200	C₃×C₄	D₃×D₄	1	8	yes	complemented
201	Dic₃	—	3	8	no	—
202	Dic₃	—	3	8	no	—
203	Dic₃	—	6	8	no	—
204	Dic₃	—	6	4	no	—
205	C₃²	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	9	1	no	—
207	C₂³	—	9	2	no	—
208	C₂³	—	9	8	no	—
209	C₂³	—	18	4	no	—
210	C₂³	—	18	4	no	—
211	C₂³	C₂²×C₃⋊D₃	1	2	yes	—
212	D₄	—	9	8	no	—
213	D₄	—	9	4	no	—
214	D₄	C₄×C₃⋊D₃	1	4	yes	direct product factor
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	9	1	no	—
217	C₂×C₄	—	9	1	no	—
218	C₂×C₄	—	9	4	no	—
219	C₂×C₄	—	9	4	no	—
220	C₂×C₄	—	9	1	no	—
221	C₂×C₄	—	9	2	no	—
222	C₂×C₄	—	18	4	no	—
223	C₂×C₄	—	18	4	no	—
224	C₂×C₄	—	18	2	no	—
225	C₂×C₄	—	18	4	no	—
226	C₂×C₄	—	18	4	no	—
227	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃⋊D₃	1	2	yes	—
229	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
231	C₂×C₃	—	2	16	no	—
232	C₂×C₃	C₂²×C₄×D₃	1	4	yes	—
233	C₂×C₃	C₂×D₃×D₄	1	4	yes	—
234	C₂×C₃	D₃×#41	1	4	yes	—
235	D₃	—	3	16	no	—
236	D₃	—	6	16	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	9	2	no	—
240	C₂²	—	9	2	no	—
241	C₂²	—	9	2	no	—
242	C₂²	—	9	4	no	—
243	C₂²	—	18	16	no	—
244	C₂²	—	18	2	no	—
245	C₂²	—	18	2	no	—
246	C₂²	C₂×C₄×C₃⋊D₃	1	4	yes	complemented
247	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	9	2	no	—
251	C₄	—	9	2	no	—
252	C₄	—	18	2	no	—
253	C₄	—	18	1	no	—
254	C₄	C₂×C₄×C₃⋊D₃	1	2	yes	complemented
255	C₄	D₄×C₃⋊D₃	1	2	yes	direct product factor
256	C₃	C₄×D₃×D₄	1	4	yes	complemented
257	C₂	—	2	4	no	—
258	C₂	—	9	4	no	—
259	C₂	—	18	4	no	—
260	C₂	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
261	C₂	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
262	C₂	#41×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₄
Frattini	C₂²	C₂³×C₃⋊D₃
Fitting	C₃²×C₄×D₄	C₂
socle, cosocle	C₂²×C₃²	C₂⁴
prime radical	C₂×D₄×C₃⋊D₃	C₂
hypercenter	C₄×D₄	C₃⋊D₃
hypocenter	C₃²	C₂×C₄×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₃⋊D₃	C₂	semidirect
3. C₃²×C₄×D₄	C₂	semidirect
4. D₄×C₃⋊Dic₃	C₂	semidirect
5. C₄×#638	C₂	semidirect
6. #64102	C₂	semidirect
7. C₄⋊C₄×C₃⋊D₃	C₂	semidirect
8. #64090	C₂	semidirect
9. C₂²⋊C₄×C₃⋊D₃	C₂	semidirect
10. C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
11. C₄²×C₃⋊D₃	C₂	semidirect
15. C₂²×C₃²×C₄	C₂²	semidirect
17. C₂²×C₃⋊Dic₃	C₂²	semidirect
18. D₄×C₃⋊D₃	C₄	direct (N × H)
25. C₂×C₄×C₃⋊D₃	C₂²	semidirect
26. C₂×C₄×C₃⋊D₃	C₂²	semidirect
27. C₃²×C₄⋊C₄	C₂²	semidirect
28. C₃²×C₂²⋊C₄	C₂²	semidirect
29. C₃²×C₄²	C₂²	semidirect
30. #3757	C₂²	semidirect
31. #3752	C₂²	semidirect
32. #3752	C₂²	semidirect
33. C₄⋊(C₃⋊Dic₃)	C₂²	semidirect
34. #3750	C₂²	semidirect
35. C₄×C₃⋊Dic₃	C₂²	semidirect
36. C₄×C₃⋊Dic₃	C₂²	semidirect
37. C₄×C₃⋊Dic₃	C₂²	semidirect
40. C₃×C₄×D₄	D₃	semidirect
52. C₂²×C₃⋊D₃	C₂×C₄	semidirect
55. C₃²×D₄	C₂×C₄	semidirect
61. #638	C₂×C₄	semidirect
67. C₄⋊(C₃⋊D₃)	C₂×C₄	semidirect
72. C₄×C₃⋊D₃	C₂×C₄	semidirect
73. C₄×C₃⋊D₃	D₄	direct (N × H)
77. C₂²×C₃×C₄	C₂×D₃	semidirect
89. C₃×C₄⋊C₄	C₂×D₃	semidirect
90. C₃×C₂²⋊C₄	C₂×D₃	semidirect
91. C₃×C₄²	C₂×D₃	semidirect
102. C₂²×C₃²	C₂²×C₄	semidirect
108. C₂×C₃⋊D₃	C₂²×C₄	semidirect
113. C₃²×C₄	C₂²×C₄	semidirect
114. C₃²×C₄	C₂×D₄	semidirect
117. C₃⋊Dic₃	C₂²×C₄	semidirect
118. C₃⋊Dic₃	C₂×D₄	semidirect
126. C₄×D₄	C₃⋊D₃	direct (N × H)
136. C₃×D₄	C₄×D₃	semidirect
160. C₃⋊D₃	C₄×D₄	direct (N × H)
174. C₂²×C₄	C₂×C₃⋊D₃	semidirect
177. C₄⋊C₄	C₂×C₃⋊D₃	semidirect
181. C₂²⋊C₄	C₂×C₃⋊D₃	semidirect
185. C₄²	C₂×C₃⋊D₃	semidirect
188. C₂²×C₃	C₂×C₄×D₃	semidirect
199. C₃×C₄	C₂×C₄×D₃	semidirect
200. C₃×C₄	D₃×D₄	semidirect
205. C₃²	C₂×C₄×D₄	semidirect
214. D₄	C₄×C₃⋊D₃	direct (N × H)
246. C₂²	C₂×C₄×C₃⋊D₃	semidirect
254. C₄	C₂×C₄×C₃⋊D₃	semidirect
255. C₄	D₄×C₃⋊D₃	direct (N × H)
256. C₃	C₄×D₃×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₃⋊D₃	2
C₂²×C₄×C₃⋊D₃	2
C₃²×C₄×D₄	2
D₄×C₃⋊Dic₃	2
C₄×#638	2
#64102	2
C₄⋊C₄×C₃⋊D₃	2
#64090	2
C₂²⋊C₄×C₃⋊D₃	2
C₄×C₄⋊(C₃⋊D₃)	2
C₄²×C₃⋊D₃	2
C₄×D₃×D₄	3

Subgroups — C4×D4×C3⋊D3 #10596015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄×C₃⋊D₃ #10596015