Subgroups — C₃×D₄×D₈ #85143

Order 384.

620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₈	C₂	1	2	yes	maximal, complemented
2	C₃×D₄²	C₂	1	2	yes	maximal, complemented
3	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#466	C₂	1	2	yes	maximal, complemented
5	C₃×#459	C₂	1	2	yes	maximal, complemented
6	C₃×#447	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×D₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	—	2	2	no	—
11	C₂²×C₃×D₄	C₂²	1	4	yes	complemented
12	C₂×C₃×D₈	—	2	8	no	—
13	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₈	C₂²	1	4	yes	complemented
15	C₂×C₃×D₈	C₂²	1	2	yes	complemented
16	C₂×C₃×D₈	C₂²	1	2	yes	complemented
17	C₂²×C₃×C₈	C₂²	1	2	yes	complemented
18	C₃×#127	C₂²	1	2	yes	complemented
19	C₃×#121	—	2	2	no	—
20	C₃×#121	C₂²	1	4	yes	complemented
21	C₃×#120	—	2	4	no	—
22	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
23	C₃×C₄×D₄	C₂²	1	2	yes	complemented
24	C₃×#107	C₂²	1	1	yes	characteristic, complemented
25	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
26	C₃×#102	C₂²	1	2	yes	complemented
27	C₃×#102	C₂²	1	4	yes	complemented
28	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
30	C₂²×D₈	C₂×C₃	1	2	yes	complemented
31	D₄²	C₂×C₃	1	2	yes	complemented
32	#493	C₂×C₃	1	1	yes	characteristic, complemented
33	#466	C₂×C₃	1	2	yes	complemented
34	#459	C₂×C₃	1	2	yes	complemented
35	#447	C₂×C₃	1	4	yes	complemented
36	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂⁴×C₃	—	2	4	no	—
39	C₂×C₃×D₄	—	2	8	no	—
40	C₂×C₃×D₄	—	2	4	no	—
41	C₂×C₃×D₄	—	2	4	no	—
42	C₂×C₃×D₄	—	2	2	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	4	4	no	—
45	C₂×C₃×D₄	—	4	8	no	—
46	C₂×C₃×D₄	D₄	1	1	yes	characteristic
47	C₂×C₃×D₄	C₂³	1	4	yes	—
48	C₂×C₃×D₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	—	2	2	no	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₃×D₈	—	2	8	no	—
52	C₃×D₈	—	2	4	no	—
53	C₃×D₈	—	4	2	no	—
54	C₃×D₈	D₄	1	4	yes	direct product factor
55	C₂×C₃×C₈	—	2	4	no	—
56	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂³	1	2	yes	—
58	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
59	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₃×C₂²⋊C₄	—	2	2	no	—
62	C₃×C₂²⋊C₄	—	2	4	no	—
63	C₃×C₂²⋊C₄	D₄	1	2	yes	—
64	C₃×C₄²	C₂³	1	1	yes	characteristic
65	C₂²×D₄	—	2	2	no	—
66	C₂²×D₄	C₂²×C₃	1	4	yes	complemented
67	C₂×D₈	—	2	8	no	—
68	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×D₈	C₂²×C₃	1	4	yes	complemented
70	C₂×D₈	C₂²×C₃	1	2	yes	complemented
71	C₂×D₈	C₂²×C₃	1	2	yes	complemented
72	C₂²×C₈	C₂²×C₃	1	2	yes	complemented
73	#127	C₂²×C₃	1	2	yes	complemented
74	#121	—	2	2	no	—
75	#121	C₂²×C₃	1	4	yes	complemented
76	#120	—	2	4	no	—
77	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
78	C₄×D₄	C₂²×C₃	1	2	yes	complemented
79	#107	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
81	#102	C₂²×C₃	1	2	yes	complemented
82	#102	C₂²×C₃	1	4	yes	complemented
83	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
84	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
85	C₂³×C₃	—	2	2	no	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	—	4	8	no	—
88	C₂³×C₃	—	4	8	no	—
89	C₂³×C₃	—	8	4	no	—
90	C₂³×C₃	C₂×D₄	1	2	yes	—
91	C₃×D₄	—	2	4	no	—
92	C₃×D₄	—	2	2	no	—
93	C₃×D₄	—	2	4	no	—
94	C₃×D₄	—	4	8	no	—
95	C₃×D₄	—	4	8	no	—
96	C₃×D₄	—	4	4	no	—
97	C₃×D₄	—	8	2	no	—
98	C₃×D₄	D₈	1	4	yes	direct product factor
99	C₃×D₄	C₂×D₄	1	4	yes	complemented
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	4	4	no	—
104	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
108	C₃×C₈	—	2	2	no	—
109	C₃×C₈	—	2	1	no	—
110	C₃×C₈	C₂×D₄	1	2	yes	complemented
111	C₂⁴	—	2	4	no	—
112	C₂×D₄	—	2	8	no	—
113	C₂×D₄	—	2	4	no	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	4	4	no	—
118	C₂×D₄	—	4	8	no	—
119	C₂×D₄	C₃×D₄	1	1	yes	characteristic
120	C₂×D₄	C₂³×C₃	1	4	yes	—
121	C₂×D₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	C₂³×C₃	1	2	yes	—
124	D₈	—	2	8	no	—
125	D₈	—	2	4	no	—
126	D₈	—	4	2	no	—
127	D₈	C₃×D₄	1	4	yes	direct product factor
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₈	C₂³×C₃	1	2	yes	—
131	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
132	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
133	C₄⋊C₄	C₂³×C₃	1	2	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	C₃×D₄	1	2	yes	—
137	C₄²	C₂³×C₃	1	1	yes	characteristic
138	C₂²×C₃	—	2	2	no	—
139	C₂²×C₃	—	2	2	no	—
140	C₂²×C₃	—	2	4	no	—
141	C₂²×C₃	—	4	2	no	—
142	C₂²×C₃	—	4	2	no	—
143	C₂²×C₃	—	4	4	no	—
144	C₂²×C₃	—	4	4	no	—
145	C₂²×C₃	—	4	4	no	—
146	C₂²×C₃	—	8	16	no	—
147	C₂²×C₃	C₂×D₈	1	4	yes	complemented
148	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
149	C₃×C₄	—	2	2	no	—
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	C₂×D₈	1	2	yes	complemented
153	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂³	—	2	2	no	—
156	C₂³	—	2	4	no	—
157	C₂³	—	4	8	no	—
158	C₂³	—	4	8	no	—
159	C₂³	—	8	4	no	—
160	C₂³	C₂×C₃×D₄	1	2	yes	—
161	D₄	—	2	4	no	—
162	D₄	—	2	2	no	—
163	D₄	—	2	4	no	—
164	D₄	—	4	8	no	—
165	D₄	—	4	8	no	—
166	D₄	—	4	4	no	—
167	D₄	—	8	2	no	—
168	D₄	C₃×D₈	1	4	yes	direct product factor
169	D₄	C₂×C₃×D₄	1	4	yes	complemented
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	C₂×C₃×D₄	1	2	yes	complemented
181	C₂×C₃	—	2	4	no	—
182	C₂×C₃	—	4	4	no	—
183	C₂×C₃	—	8	4	no	—
184	C₂×C₃	D₄²	1	1	yes	characteristic
185	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
186	C₂×C₃	#576	1	1	yes	characteristic
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	—	8	16	no	—
196	C₂²	C₂×C₃×D₈	1	4	yes	complemented
197	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	2	no	—
201	C₄	C₂×C₃×D₈	1	2	yes	complemented
202	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
204	C₃	D₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
205	C₂	—	2	4	no	—
206	C₂	—	4	4	no	—
207	C₂	—	8	4	no	—
208	C₂	C₃×D₄²	1	1	yes	characteristic
209	C₂	C₂²×C₃×D₈	1	1	yes	characteristic
210	C₂	C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₈	C₂	semidirect
2. C₃×D₄²	C₂	semidirect
3. C₃×#493	C₂	semidirect
4. C₃×#466	C₂	semidirect
5. C₃×#459	C₂	semidirect
6. C₃×#447	C₂	semidirect
7. C₃×C₄×D₈	C₂	semidirect
8. C₃×C₈×D₄	C₂	semidirect
9. D₄×D₈	C₃	direct (N × H)
11. C₂²×C₃×D₄	C₂²	semidirect
14. C₂×C₃×D₈	C₂²	semidirect
15. C₂×C₃×D₈	C₂²	semidirect
16. C₂×C₃×D₈	C₂²	semidirect
17. C₂²×C₃×C₈	C₂²	semidirect
18. C₃×#127	C₂²	semidirect
20. C₃×#121	C₂²	semidirect
23. C₃×C₄×D₄	C₂²	semidirect
24. C₃×#107	C₂²	semidirect
25. C₃×C₄⋊C₈	C₂²	semidirect
26. C₃×#102	C₂²	semidirect
27. C₃×#102	C₂²	semidirect
28. C₃×C₂²⋊C₈	C₂²	semidirect
29. C₃×C₄×C₈	C₂²	semidirect
30. C₂²×D₈	C₂×C₃	semidirect
31. D₄²	C₂×C₃	semidirect
32. #493	C₂×C₃	semidirect
33. #466	C₂×C₃	semidirect
34. #459	C₂×C₃	semidirect
35. #447	C₂×C₃	semidirect
36. C₄×D₈	C₂×C₃	semidirect
37. C₈×D₄	C₂×C₃	semidirect
54. C₃×D₈	D₄	direct (N × H)
66. C₂²×D₄	C₂²×C₃	semidirect
69. C₂×D₈	C₂²×C₃	semidirect
70. C₂×D₈	C₂²×C₃	semidirect
71. C₂×D₈	C₂²×C₃	semidirect
72. C₂²×C₈	C₂²×C₃	semidirect
73. #127	C₂²×C₃	semidirect
75. #121	C₂²×C₃	semidirect
78. C₄×D₄	C₂²×C₃	semidirect
79. #107	C₂²×C₃	semidirect
80. C₄⋊C₈	C₂²×C₃	semidirect
81. #102	C₂²×C₃	semidirect
82. #102	C₂²×C₃	semidirect
83. C₂²⋊C₈	C₂²×C₃	semidirect
84. C₄×C₈	C₂²×C₃	semidirect
98. C₃×D₄	D₈	direct (N × H)
99. C₃×D₄	C₂×D₄	semidirect
110. C₃×C₈	C₂×D₄	semidirect
127. D₈	C₃×D₄	direct (N × H)
147. C₂²×C₃	C₂×D₈	semidirect
152. C₃×C₄	C₂×D₈	semidirect
168. D₄	C₃×D₈	direct (N × H)
169. D₄	C₂×C₃×D₄	semidirect
180. C₈	C₂×C₃×D₄	semidirect
196. C₂²	C₂×C₃×D₈	semidirect
201. C₄	C₂×C₃×D₈	semidirect
204. C₃	D₄×D₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₈	2
C₃×D₄²	2
C₃×#493	2
C₃×#466	2
C₃×#459	2
C₃×#447	2
C₃×C₄×D₈	2
C₃×C₈×D₄	2
D₄×D₈	3

Subgroups — C3×D4×D8 #85143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×D₈ #85143