Subgroups — C₂×C₃×C₈×Dic₃ #10592195

Order 576.

362 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×Dic₃	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₄×C₈	—	3	1	no	maximal
6	C₂×C₈×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₃	C₄	1	2	yes	—
9	C₂×C₃²×C₈	C₄	1	4	yes	complemented
10	C₂×C₃²×C₈	C₂²	1	2	yes	—
11	C₃×C₄×Dic₃	C₄	1	2	yes	—
12	C₃×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃×C₈	—	2	1	no	—
16	C₂²×C₃×C₈	—	3	1	no	—
17	C₂²×C₃×C₈	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄²	—	3	1	no	—
20	C₂×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×C₈	—	3	4	no	—
23	C₈×Dic₃	C₂×C₃	1	4	yes	direct product factor
24	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃×Dic₃	C₈	1	8	yes	direct product factor
30	C₂×C₃×Dic₃	C₂×C₄	1	4	yes	—
31	C₃²×C₈	C₂×C₄	1	4	yes	complemented
32	C₃×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₄×C₈	—	3	1	no	Sylow
34	C₂²×C₃×C₄	—	2	1	no	—
35	C₂²×C₃×C₄	—	3	2	no	—
36	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₂²×Dic₃	C₃×C₄	1	2	yes	—
39	C₂×C₃×C₈	—	2	4	no	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	—	3	4	no	—
42	C₂×C₃×C₈	—	3	2	no	—
43	C₂×C₃×C₈	Dic₃	1	4	yes	direct product factor
44	C₂×C₃×C₈	C₃×C₄	1	4	yes	complemented
45	C₂×C₃×C₈	C₂×D₃	1	2	yes	—
46	C₂×C₃×C₈	C₂²×C₃	1	2	yes	—
47	C₃×C₄²	—	3	2	no	—
48	C₃×C₄²	—	3	2	no	—
49	C₄×Dic₃	C₃×C₄	1	2	yes	—
50	C₄×Dic₃	C₂²×C₃	1	2	yes	—
51	C₂×C₃⋊C₈	C₃×C₄	1	4	yes	complemented
52	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	—
53	C₂²×C₃²	C₄²	1	2	yes	—
54	C₂²×C₃²	C₂×C₈	1	4	yes	—
55	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
56	C₃²×C₄	C₄²	1	2	yes	—
57	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃×Dic₃	C₂×C₈	1	8	yes	direct product factor
60	C₂²×C₈	—	3	1	no	—
61	C₂²×C₈	C₃×D₃	1	1	yes	characteristic
62	C₂×C₄²	—	3	1	no	—
63	C₄×C₈	—	3	4	no	—
64	C₂³×C₃	—	2	1	no	—
65	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
66	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄	—	2	1	no	—
68	C₂×C₃×C₄	—	2	2	no	—
69	C₂×C₃×C₄	—	2	2	no	—
70	C₂×C₃×C₄	—	2	1	no	—
71	C₂×C₃×C₄	—	3	8	no	—
72	C₂×C₃×C₄	—	3	4	no	—
73	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃×C₄	C₄×D₃	1	2	yes	—
75	C₂×C₃×C₄	C₂×Dic₃	1	2	yes	—
76	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₂×Dic₃	C₃×C₈	1	8	yes	direct product factor
82	C₂×Dic₃	C₂×C₃×C₄	1	4	yes	—
83	C₃×C₈	—	2	4	no	—
84	C₃×C₈	—	3	4	no	—
85	C₃×C₈	C₂×Dic₃	1	4	yes	direct product factor
86	C₃×C₈	C₂×C₃×C₄	1	4	yes	complemented
87	C₃⋊C₈	C₂×C₃×C₄	1	4	yes	complemented
88	C₂×C₃²	C₄×C₈	1	4	yes	complemented
89	C₂×C₃²	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₃²	C₂²×C₈	1	2	yes	—
91	C₂²×C₄	—	3	2	no	—
92	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
93	C₂×C₈	—	3	4	no	—
94	C₂×C₈	—	3	2	no	—
95	C₂×C₈	C₃×Dic₃	1	4	yes	direct product factor
96	C₂×C₈	C₂×C₃×D₃	1	2	yes	—
97	C₄²	—	3	2	no	—
98	C₄²	—	3	2	no	—
99	C₂²×C₃	—	2	4	no	—
100	C₂²×C₃	—	2	2	no	—
101	C₂²×C₃	—	2	1	no	—
102	C₂²×C₃	D₃×C₈	1	4	yes	—
103	C₂²×C₃	C₄×Dic₃	1	2	yes	—
104	C₂²×C₃	C₃×C₄²	1	2	yes	—
105	C₂²×C₃	C₂×C₃×C₈	1	4	yes	—
106	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
108	C₃×C₄	—	2	2	no	—
109	C₃×C₄	—	2	1	no	—
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	—	3	8	no	—
112	C₃×C₄	C₄×Dic₃	1	2	yes	—
113	C₃×C₄	C₃×C₄²	1	2	yes	—
114	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
115	C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
118	Dic₃	C₂×C₃×C₈	1	8	yes	direct product factor
119	C₃²	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
120	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
121	C₂×C₄	—	3	8	no	—
122	C₂×C₄	—	3	4	no	—
123	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
124	C₂×C₄	C₃×C₄×D₃	1	2	yes	—
125	C₂×C₄	C₂×C₃×Dic₃	1	2	yes	—
126	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
127	C₈	—	3	4	no	—
128	C₈	C₂×C₃×Dic₃	1	4	yes	direct product factor
129	C₂×C₃	—	2	2	no	—
130	C₂×C₃	—	2	1	no	—
131	C₂×C₃	—	2	4	no	—
132	C₂×C₃	C₈×Dic₃	1	4	yes	direct product factor
133	C₂×C₃	C₃×C₄×C₈	1	4	yes	complemented
134	C₂×C₃	C₂×D₃×C₈	1	2	yes	—
135	C₂×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
136	C₂×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
137	C₂×C₃	C₂²×C₃×C₈	1	2	yes	—
138	C₂²	C₃×D₃×C₈	1	4	yes	—
139	C₂²	C₃×C₄×Dic₃	1	2	yes	—
140	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
141	C₄	—	3	8	no	—
142	C₄	C₃×C₄×Dic₃	1	2	yes	—
143	C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
144	C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
145	C₃	—	2	1	no	—
146	C₃	C₂×C₈×Dic₃	1	1	yes	characteristic, direct product factor
147	C₃	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented
148	C₂	C₃×C₈×Dic₃	1	4	yes	direct product factor
149	C₂	C₂×C₃×D₃×C₈	1	2	yes	—
150	C₂	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₈	D₃
commutator, hypocenter	C₃	C₂×C₃×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₈×Dic₃	C₂	direct (N × H)
6. C₂×C₈×Dic₃	C₃	direct (N × H)
9. C₂×C₃²×C₈	C₄	semidirect
13. C₂×C₃×C₃⋊C₈	C₄	semidirect
23. C₈×Dic₃	C₂×C₃	direct (N × H)
29. C₂×C₃×Dic₃	C₈	direct (N × H)
31. C₃²×C₈	C₂×C₄	semidirect
32. C₃×C₃⋊C₈	C₂×C₄	semidirect
43. C₂×C₃×C₈	Dic₃	direct (N × H)
44. C₂×C₃×C₈	C₃×C₄	semidirect
51. C₂×C₃⋊C₈	C₃×C₄	semidirect
59. C₃×Dic₃	C₂×C₈	direct (N × H)
81. C₂×Dic₃	C₃×C₈	direct (N × H)
85. C₃×C₈	C₂×Dic₃	direct (N × H)
86. C₃×C₈	C₂×C₃×C₄	semidirect
87. C₃⋊C₈	C₂×C₃×C₄	semidirect
88. C₂×C₃²	C₄×C₈	semidirect
95. C₂×C₈	C₃×Dic₃	direct (N × H)
118. Dic₃	C₂×C₃×C₈	direct (N × H)
119. C₃²	C₂×C₄×C₈	semidirect
128. C₈	C₂×C₃×Dic₃	direct (N × H)
132. C₂×C₃	C₈×Dic₃	direct (N × H)
133. C₂×C₃	C₃×C₄×C₈	semidirect
146. C₃	C₂×C₈×Dic₃	direct (N × H)
147. C₃	C₂×C₃×C₄×C₈	semidirect
148. C₂	C₃×C₈×Dic₃	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×C₃²×C₈	2
C₂×C₃×C₄×Dic₃	2
C₂²×C₃×C₃⋊C₈	2
C₃×C₈×Dic₃	2
C₂×C₃×C₄×C₈	3
C₂×C₈×Dic₃	3

Subgroups — C2×C3×C8×Dic3 #10592195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×Dic₃ #10592195