Subgroups — C₄×D₃×D₈ #79584

Order 384.

616 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#213	C₂	1	2	yes	maximal, complemented
6	D₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	#4872	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×#102	C₂	1	2	yes	maximal, complemented
9	#4756	C₂	1	2	yes	maximal, complemented
10	C₄×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₈	—	3	1	no	maximal, Sylow
13	C₂×D₃×D₄	C₂²	1	2	yes	—
14	C₂²×C₄×D₃	—	2	2	no	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×D₄	C₂²	1	2	yes	complemented
17	D₄×Dic₃	C₂²	1	2	yes	complemented
18	C₂×#213	C₂²	1	2	yes	—
19	C₄×#67	—	2	2	no	—
20	D₃×D₈	C₄	1	8	yes	direct product factor
21	C₂×D₂₄	C₂²	1	1	yes	characteristic
22	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
23	C₂×D₃×C₈	C₂²	1	1	yes	characteristic
24	#893	C₂²	1	2	yes	complemented
25	D₃×C₄⋊C₄	C₂²	1	2	yes	—
26	#67⋊C₄	—	2	2	no	—
27	D₃×C₂²⋊C₄	—	2	2	no	—
28	C₄×D₁₂	C₂²	1	2	yes	complemented
29	C₄²×D₃	C₂²	1	1	yes	characteristic
30	C₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₃×#102	C₂²	1	2	yes	complemented
32	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
33	#832	C₂²	1	2	yes	complemented
34	#821	C₂²	1	2	yes	complemented
35	#818	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
37	C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
38	#809	C₂²	1	2	yes	complemented
39	#807	C₂²	1	2	yes	complemented
40	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₈	—	3	1	no	—
42	C₂×C₄×D₄	—	3	2	no	—
43	C₄×D₈	—	3	1	no	—
44	C₄×D₈	—	3	2	no	—
45	C₄×D₈	—	3	1	no	—
46	C₄×D₈	—	3	2	no	—
47	C₄×D₈	—	3	1	no	—
48	C₄×D₈	D₃	1	1	yes	characteristic, direct product factor
49	C₂×#107	—	3	1	no	—
50	C₂×#102	—	3	2	no	—
51	C₂×C₄×C₈	—	3	1	no	—
52	C₂³×D₃	—	2	2	no	—
53	C₂×C₃×D₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	—	2	2	no	—
55	C₂×#67	—	2	2	no	—
56	C₂²×Dic₃	—	2	2	no	—
57	D₃×D₄	—	2	4	no	—
58	D₃×D₄	C₂×C₄	1	8	yes	complemented
59	C₂×D₁₂	C₂³	1	2	yes	—
60	C₂×C₄×D₃	—	2	2	no	—
61	C₂×C₄×D₃	—	2	2	no	—
62	C₂×C₄×D₃	—	2	2	no	—
63	C₂×C₄×D₃	—	4	8	no	—
64	C₂×C₄×D₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₃	C₂³	1	1	yes	characteristic
67	C₃×D₈	C₂×C₄	1	4	yes	complemented
68	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
70	C₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₂²⋊C₄	—	2	2	no	—
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	#217	—	2	2	no	—
74	#213	C₂×C₄	1	8	yes	complemented
75	(C₂×D₃)⋊C₄	—	2	2	no	—
76	(C₂×D₃)⋊C₄	—	2	2	no	—
77	C₄⋊Dic₃	C₂³	1	2	yes	—
78	Dic₃⋊C₄	—	2	2	no	—
79	C₄×Dic₃	D₄	1	1	yes	characteristic
80	C₄×Dic₃	D₄	1	1	yes	characteristic
81	C₄×Dic₃	C₂³	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
84	D₂₄	C₂×C₄	1	4	yes	complemented
85	D₃×C₈	—	2	2	no	—
86	D₃×C₈	C₂×C₄	1	4	yes	complemented
87	C₂²×D₄	—	3	2	no	—
88	C₂³×C₄	—	6	2	no	—
89	C₂×D₈	—	3	2	no	—
90	C₂×D₈	—	3	8	no	—
91	C₂×D₈	—	3	1	no	—
92	C₂×D₈	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₈	—	3	1	no	—
94	C₂²×C₈	—	3	1	no	—
95	C₄×D₄	—	3	2	no	—
96	C₄×D₄	—	3	2	no	—
97	C₄×D₄	—	3	2	no	—
98	C₄×D₄	—	6	2	no	—
99	C₄×D₄	—	6	2	no	—
100	C₄×D₄	C₂×D₃	1	2	yes	complemented
101	C₂×C₄⋊C₄	—	3	2	no	—
102	C₂×C₂²⋊C₄	—	6	2	no	—
103	C₂×C₄²	—	3	1	no	—
104	#107	—	3	1	no	—
105	#107	—	3	2	no	—
106	#107	C₂×D₃	1	1	yes	characteristic, complemented
107	#102	—	3	2	no	—
108	#102	—	3	2	no	—
109	#102	—	3	2	no	—
110	#102	C₂×D₃	1	2	yes	complemented
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	—	3	1	no	—
113	C₄×C₈	—	3	1	no	—
114	C₄×C₈	C₂×D₃	1	1	yes	characteristic, complemented
115	C₂³×C₃	—	2	2	no	—
116	C₂²×D₃	—	2	2	no	—
117	C₂²×D₃	—	2	8	no	—
118	C₂²×D₃	—	4	4	no	—
119	C₂²×D₃	—	4	4	no	—
120	C₂²×D₃	C₂×D₄	1	1	yes	characteristic
121	C₃×D₄	—	2	2	no	—
122	C₃×D₄	C₂²×C₄	1	4	yes	complemented
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	4	4	no	—
125	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
128	#67	—	2	8	no	—
129	C₂×Dic₃	—	2	2	no	—
130	C₂×Dic₃	—	4	4	no	—
131	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₃	C₂×D₄	1	1	yes	characteristic
134	D₁₂	—	2	2	no	—
135	D₁₂	C₂²×C₄	1	4	yes	complemented
136	C₄×D₃	—	2	2	no	—
137	C₄×D₃	—	4	4	no	—
138	C₄×D₃	—	4	4	no	—
139	C₄×D₃	—	4	4	no	—
140	C₄×D₃	D₈	1	4	yes	direct product factor
141	C₄×D₃	C₂²×C₄	1	2	yes	—
142	C₄×D₃	#41	1	2	yes	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	C₂²×C₄	1	2	yes	complemented
145	C₃⋊C₈	—	2	1	no	—
146	C₃⋊C₈	C₂²×C₄	1	2	yes	complemented
147	C₂⁴	—	6	2	no	—
148	C₂×D₄	—	3	8	no	—
149	C₂×D₄	—	3	2	no	—
150	C₂×D₄	—	6	4	no	—
151	C₂×D₄	—	6	2	no	—
152	C₂×D₄	C₂²×D₃	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	3	1	no	—
155	C₂²×C₄	—	3	1	no	—
156	C₂²×C₄	—	3	1	no	—
157	C₂²×C₄	—	6	2	no	—
158	C₂²×C₄	—	6	2	no	—
159	C₂²×C₄	—	6	2	no	—
160	C₂²×C₄	—	6	2	no	—
161	C₂²×C₄	—	12	8	no	—
162	D₈	—	3	8	no	—
163	D₈	—	3	4	no	—
164	D₈	C₄×D₃	1	4	yes	direct product factor
165	C₂×C₈	—	3	1	no	—
166	C₂×C₈	—	3	4	no	—
167	C₂×C₈	—	3	1	no	—
168	C₂×C₈	—	6	2	no	—
169	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
171	C₄⋊C₄	—	3	2	no	—
172	C₄⋊C₄	—	6	2	no	—
173	C₄⋊C₄	C₂²×D₃	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	6	2	no	—
176	C₂²⋊C₄	—	6	2	no	—
177	C₂²⋊C₄	—	6	2	no	—
178	C₄²	—	3	1	no	—
179	C₄²	—	3	1	no	—
180	C₄²	—	3	1	no	—
181	C₄²	C₂²×D₃	1	1	yes	characteristic
182	C₂²×C₃	—	2	4	no	—
183	C₂²×C₃	—	4	2	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×D₃	—	2	4	no	—
187	C₂×D₃	—	4	2	no	—
188	C₂×D₃	—	4	2	no	—
189	C₂×D₃	—	4	16	no	—
190	C₂×D₃	C₄×D₄	1	2	yes	—
191	C₂×D₃	C₂×D₈	1	2	yes	—
192	C₂×D₃	#135	1	2	yes	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	C₂×D₈	1	2	yes	complemented
196	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
197	C₃×C₄	C₂×#41	1	1	yes	characteristic
198	Dic₃	—	2	1	no	—
199	Dic₃	—	4	2	no	—
200	Dic₃	C₄×D₄	1	2	yes	—
201	Dic₃	C₂×D₈	1	2	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	3	1	no	—
204	C₂³	—	6	8	no	—
205	C₂³	—	6	2	no	—
206	C₂³	—	12	4	no	—
207	C₂³	—	12	4	no	—
208	D₄	—	2	2	no	—
209	D₄	—	3	4	no	—
210	D₄	—	6	8	no	—
211	D₄	—	6	2	no	—
212	D₄	C₂×C₄×D₃	1	4	yes	complemented
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	3	1	no	—
215	C₂×C₄	—	3	1	no	—
216	C₂×C₄	—	3	4	no	—
217	C₂×C₄	—	3	2	no	—
218	C₂×C₄	—	3	2	no	—
219	C₂×C₄	—	3	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	6	2	no	—
222	C₂×C₄	—	6	2	no	—
223	C₂×C₄	—	12	4	no	—
224	C₂×C₄	—	12	4	no	—
225	C₂×C₄	—	12	4	no	—
226	C₂×C₄	—	12	4	no	—
227	C₂×C₄	D₃×D₄	1	1	yes	characteristic
228	C₂×C₄	D₃×D₄	1	1	yes	characteristic
229	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	3	2	no	—
232	C₈	—	6	1	no	—
233	C₈	C₂×C₄×D₃	1	2	yes	complemented
234	C₂×C₃	—	4	4	no	—
235	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
237	C₂×C₃	C₂×#135	1	1	yes	characteristic
238	D₃	—	4	4	no	—
239	D₃	C₄×D₈	1	4	yes	direct product factor
240	C₂²	—	2	4	no	—
241	C₂²	—	3	2	no	—
242	C₂²	—	3	2	no	—
243	C₂²	—	3	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	6	4	no	—
247	C₂²	—	12	16	no	—
248	C₂²	—	12	2	no	—
249	C₂²	—	12	2	no	—
250	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	3	2	no	—
253	C₄	—	3	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	6	1	no	—
256	C₄	—	12	2	no	—
257	C₄	D₃×D₈	1	2	yes	direct product factor
258	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
259	C₄	D₃×#41	1	1	yes	characteristic
260	C₃	C₂×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
261	C₂	—	3	4	no	—
262	C₂	—	4	4	no	—
263	C₂	—	12	4	no	—
264	C₂	C₄×D₃×D₄	1	1	yes	characteristic
265	C₂	C₂×D₃×D₈	1	1	yes	characteristic
266	C₂	D₃×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₄
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×D₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×D₃×D₈	C₂
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×D₈	D₃
hypocenter	C₃	C₂×C₄×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₃×D₄	C₂	semidirect
3. C₃×C₄×D₈	C₂	semidirect
4. Dic₃×D₈	C₂	semidirect
5. C₄×#213	C₂	semidirect
6. D₃×#107	C₂	semidirect
7. #4872	C₂	semidirect
8. D₃×#102	C₂	semidirect
9. #4756	C₂	semidirect
10. C₄×D₂₄	C₂	semidirect
11. C₄×D₃×C₈	C₂	semidirect
16. C₃×C₄×D₄	C₂²	semidirect
17. D₄×Dic₃	C₂²	semidirect
20. D₃×D₈	C₄	direct (N × H)
24. #893	C₂²	semidirect
28. C₄×D₁₂	C₂²	semidirect
30. C₃×#107	C₂²	semidirect
31. C₃×#102	C₂²	semidirect
32. C₃×C₄×C₈	C₂²	semidirect
33. #832	C₂²	semidirect
34. #821	C₂²	semidirect
35. #818	C₂²	semidirect
36. C₈×Dic₃	C₂²	semidirect
37. C₈×Dic₃	C₂²	semidirect
38. #809	C₂²	semidirect
39. #807	C₂²	semidirect
40. C₄×C₃⋊C₈	C₂²	semidirect
48. C₄×D₈	D₃	direct (N × H)
58. D₃×D₄	C₂×C₄	semidirect
67. C₃×D₈	C₂×C₄	semidirect
74. #213	C₂×C₄	semidirect
84. D₂₄	C₂×C₄	semidirect
86. D₃×C₈	C₂×C₄	semidirect
100. C₄×D₄	C₂×D₃	semidirect
106. #107	C₂×D₃	semidirect
110. #102	C₂×D₃	semidirect
114. C₄×C₈	C₂×D₃	semidirect
122. C₃×D₄	C₂²×C₄	semidirect
135. D₁₂	C₂²×C₄	semidirect
140. C₄×D₃	D₈	direct (N × H)
144. C₃×C₈	C₂²×C₄	semidirect
146. C₃⋊C₈	C₂²×C₄	semidirect
164. D₈	C₄×D₃	direct (N × H)
195. C₃×C₄	C₂×D₈	semidirect
201. Dic₃	C₂×D₈	semidirect
212. D₄	C₂×C₄×D₃	semidirect
233. C₈	C₂×C₄×D₃	semidirect
239. D₃	C₄×D₈	direct (N × H)
257. C₄	D₃×D₈	direct (N × H)
260. C₃	C₂×C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₈	2
C₄×D₃×D₄	2
C₃×C₄×D₈	2
Dic₃×D₈	2
C₄×#213	2
D₃×#107	2
#4872	2
D₃×#102	2
#4756	2
C₄×D₂₄	2
C₄×D₃×C₈	2
C₂×C₄×D₈	3

Subgroups — C4×D3×D8 #79584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₃×D₈ #79584