Subgroups — C₂×C₄×D₃×C₈ #78449

Order 384.

760 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₃×C₈	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄×C₈	—	3	1	no	maximal, Sylow
8	C₂²×C₄×D₃	C₄	1	2	yes	—
9	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×D₃×C₈	C₄	1	16	yes	direct product factor
16	C₂×D₃×C₈	C₂²	1	8	yes	—
17	C₄²×D₃	C₄	1	4	yes	—
18	C₄²×D₃	C₂²	1	4	yes	—
19	C₃×C₄×C₈	C₂²	1	4	yes	complemented
20	C₈×Dic₃	C₂²	1	8	yes	complemented
21	C₄×C₃⋊C₈	C₂²	1	4	yes	complemented
22	C₂³×C₈	—	3	2	no	—
23	C₂²×C₄²	—	3	1	no	—
24	C₂×C₄×C₈	—	3	2	no	—
25	C₂×C₄×C₈	—	3	1	no	—
26	C₂×C₄×C₈	—	3	8	no	—
27	C₂×C₄×C₈	D₃	1	1	yes	characteristic, direct product factor
28	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃	C₈	1	16	yes	direct product factor
34	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₃	C₂×C₄	1	8	yes	—
36	C₂×C₄×D₃	C₂×C₄	1	8	yes	—
37	C₂×C₄×D₃	C₂³	1	2	yes	—
38	C₂×C₃×C₈	C₂×C₄	1	8	yes	complemented
39	C₂×C₃×C₈	C₂³	1	4	yes	—
40	C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₃×C₄²	C₂³	1	2	yes	—
42	C₄×Dic₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₃	C₂×C₄	1	8	yes	—
44	C₄×Dic₃	C₂³	1	2	yes	—
45	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
46	C₂×C₃⋊C₈	C₂³	1	4	yes	—
47	D₃×C₈	C₂×C₄	1	32	yes	direct product factor
48	C₂³×C₄	—	3	2	no	—
49	C₂³×C₄	—	3	1	no	—
50	C₂²×C₈	—	3	2	no	—
51	C₂²×C₈	—	3	16	no	—
52	C₂²×C₈	—	3	8	no	—
53	C₂²×C₈	C₂×D₃	1	2	yes	—
54	C₂×C₄²	—	3	2	no	—
55	C₂×C₄²	—	3	1	no	—
56	C₂×C₄²	—	3	4	no	—
57	C₂×C₄²	—	3	4	no	—
58	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₄×C₈	—	3	8	no	—
60	C₄×C₈	—	3	4	no	—
61	C₄×C₈	C₂×D₃	1	4	yes	direct product factor
62	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×D₃	C₄²	1	4	yes	—
64	C₂²×D₃	C₂×C₈	1	8	yes	—
65	C₂²×D₃	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄	C₂×C₈	1	8	yes	complemented
67	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
69	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
70	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₃	C₄²	1	4	yes	—
72	C₂×Dic₃	C₂×C₈	1	8	yes	complemented
73	C₂×Dic₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₃	C₂²×C₄	1	4	yes	—
75	C₄×D₃	C₄²	1	8	yes	—
76	C₄×D₃	C₂×C₈	1	32	yes	direct product factor
77	C₄×D₃	C₂²×C₄	1	8	yes	—
78	C₃×C₈	C₂²×C₄	1	8	yes	complemented
79	C₃⋊C₈	C₂²×C₄	1	8	yes	complemented
80	C₂⁴	—	3	1	no	—
81	C₂²×C₄	—	3	1	no	—
82	C₂²×C₄	—	3	2	no	—
83	C₂²×C₄	—	3	16	no	—
84	C₂²×C₄	—	3	2	no	—
85	C₂²×C₄	—	3	8	no	—
86	C₂²×C₄	—	3	8	no	—
87	C₂²×C₄	—	3	2	no	—
88	C₂²×C₄	C₄×D₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₈	—	3	8	no	—
91	C₂×C₈	—	3	4	no	—
92	C₂×C₈	—	3	32	no	—
93	C₂×C₈	C₄×D₃	1	8	yes	direct product factor
94	C₂×C₈	C₂²×D₃	1	4	yes	—
95	C₄²	—	3	2	no	—
96	C₄²	—	3	8	no	—
97	C₄²	—	3	2	no	—
98	C₄²	C₄×D₃	1	2	yes	—
99	C₄²	C₂²×D₃	1	2	yes	—
100	C₂²×C₃	C₂×C₄²	1	2	yes	—
101	C₂²×C₃	C₂²×C₈	1	4	yes	—
102	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₃	C₄×C₈	1	16	yes	direct product factor
104	C₂×D₃	C₂×C₄²	1	4	yes	—
105	C₂×D₃	C₂²×C₈	1	8	yes	—
106	C₃×C₄	C₂×C₄²	1	2	yes	—
107	C₃×C₄	C₂²×C₈	1	8	yes	complemented
108	C₃×C₄	C₂³×C₄	1	2	yes	—
109	Dic₃	C₂×C₄²	1	4	yes	—
110	Dic₃	C₂²×C₈	1	8	yes	complemented
111	C₂³	—	3	4	no	—
112	C₂³	—	3	8	no	—
113	C₂³	—	3	2	no	—
114	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
115	C₂×C₄	—	3	4	no	—
116	C₂×C₄	—	3	8	no	—
117	C₂×C₄	—	3	2	no	—
118	C₂×C₄	—	3	4	no	—
119	C₂×C₄	—	3	8	no	—
120	C₂×C₄	—	3	32	no	—
121	C₂×C₄	—	3	8	no	—
122	C₂×C₄	D₃×C₈	1	8	yes	direct product factor
123	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
126	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
127	C₈	—	3	8	no	—
128	C₈	C₂×C₄×D₃	1	8	yes	direct product factor
129	C₂×C₃	C₂×C₄×C₈	1	4	yes	complemented
130	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
131	C₂×C₃	C₂³×C₈	1	2	yes	—
132	D₃	C₂×C₄×C₈	1	8	yes	direct product factor
133	C₂²	—	3	16	no	—
134	C₂²	—	3	4	no	—
135	C₂²	—	3	8	no	—
136	C₂²	C₄²×D₃	1	2	yes	—
137	C₂²	C₂×D₃×C₈	1	4	yes	—
138	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
139	C₄	—	3	4	no	—
140	C₄	—	3	8	no	—
141	C₄	C₄²×D₃	1	2	yes	—
142	C₄	C₂×D₃×C₈	1	8	yes	direct product factor
143	C₄	C₂²×C₄×D₃	1	2	yes	—
144	C₃	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
145	C₂	—	3	8	no	—
146	C₂	C₄×D₃×C₈	1	4	yes	direct product factor
147	C₂	C₂×C₄²×D₃	1	1	yes	characteristic
148	C₂	C₂²×D₃×C₈	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₃
commutator, hypocenter	C₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄×C₈	C₂	semidirect
4. C₂×C₈×Dic₃	C₂	semidirect
5. C₂×C₄×C₃⋊C₈	C₂	semidirect
6. C₄×D₃×C₈	C₂	direct (N × H)
15. C₂×D₃×C₈	C₄	direct (N × H)
19. C₃×C₄×C₈	C₂²	semidirect
20. C₈×Dic₃	C₂²	semidirect
21. C₄×C₃⋊C₈	C₂²	semidirect
27. C₂×C₄×C₈	D₃	direct (N × H)
33. C₂×C₄×D₃	C₈	direct (N × H)
38. C₂×C₃×C₈	C₂×C₄	semidirect
45. C₂×C₃⋊C₈	C₂×C₄	semidirect
47. D₃×C₈	C₂×C₄	direct (N × H)
61. C₄×C₈	C₂×D₃	direct (N × H)
66. C₂×C₃×C₄	C₂×C₈	semidirect
72. C₂×Dic₃	C₂×C₈	semidirect
76. C₄×D₃	C₂×C₈	direct (N × H)
78. C₃×C₈	C₂²×C₄	semidirect
79. C₃⋊C₈	C₂²×C₄	semidirect
93. C₂×C₈	C₄×D₃	direct (N × H)
103. C₂×D₃	C₄×C₈	direct (N × H)
107. C₃×C₄	C₂²×C₈	semidirect
110. Dic₃	C₂²×C₈	semidirect
122. C₂×C₄	D₃×C₈	direct (N × H)
128. C₈	C₂×C₄×D₃	direct (N × H)
129. C₂×C₃	C₂×C₄×C₈	semidirect
132. D₃	C₂×C₄×C₈	direct (N × H)
142. C₄	C₂×D₃×C₈	direct (N × H)
144. C₃	C₂²×C₄×C₈	semidirect
146. C₂	C₄×D₃×C₈	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×D₃×C₈	2
C₂×C₄²×D₃	2
C₂×C₃×C₄×C₈	2
C₂×C₈×Dic₃	2
C₂×C₄×C₃⋊C₈	2
C₄×D₃×C₈	2
C₂²×C₄×C₈	3

Subgroups — C2×C4×D3×C8 #78449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₃×C₈ #78449