Subgroups — C₂×C₈×Dic₆ #78443

Order 384.

440 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₈×Dic₃	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₆	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×Q₈	—	3	1	no	maximal, Sylow
8	C₂²×Dic₆	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₂×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₃	C₂²	1	2	yes	—
15	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
16	C₄×Dic₆	C₄	1	4	yes	—
17	C₄×Dic₆	C₂²	1	4	yes	—
18	C₃×C₄×C₈	C₂²	1	4	yes	—
19	Dic₃⋊C₈	C₂²	1	8	yes	—
20	C₈×Dic₃	C₂²	1	8	yes	—
21	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
22	C₂×C₄×Q₈	—	3	1	no	—
23	C₈×Q₈	—	3	8	no	—
24	C₂×C₄⋊C₈	—	3	2	no	—
25	C₂×C₄⋊C₈	—	3	1	no	—
26	C₂×C₄×C₈	—	3	2	no	—
27	C₂×C₄×C₈	D₃	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×Dic₃	C₂×C₄	1	2	yes	—
33	C₂×Dic₆	C₈	1	8	yes	direct product factor
34	C₂×Dic₆	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₈	—	2	2	no	—
36	C₂×C₃×C₈	Q₈	1	4	yes	complemented
37	C₂×C₃×C₈	C₂³	1	2	yes	—
38	C₂×C₃×C₈	C₂³	1	2	yes	—
39	C₃×C₄²	C₂×C₄	1	2	yes	—
40	C₃×C₄²	C₂³	1	2	yes	—
41	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
42	Dic₃⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₃	C₂×C₄	1	4	yes	—
44	C₄×Dic₃	C₂³	1	4	yes	—
45	C₂×C₃⋊C₈	—	2	4	no	—
46	C₂×C₃⋊C₈	C₂³	1	4	yes	—
47	C₂²×Q₈	—	3	1	no	—
48	C₂²×C₈	—	3	2	no	—
49	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₄×Q₈	—	3	4	no	—
52	C₄×Q₈	—	3	4	no	—
53	C₂×C₄⋊C₄	—	3	2	no	—
54	C₂×C₄⋊C₄	—	3	1	no	—
55	C₂×C₄²	—	3	2	no	—
56	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₄⋊C₈	—	3	8	no	—
58	C₄⋊C₈	—	3	4	no	—
59	C₄×C₈	—	3	8	no	—
60	C₄×C₈	C₂×D₃	1	4	yes	—
61	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
65	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
68	C₂×C₃×C₄	#41	1	2	yes	—
69	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₃	—	2	4	no	—
71	C₂×Dic₃	C₂×C₈	1	8	yes	—
72	C₂×Dic₃	C₂²×C₄	1	4	yes	—
73	C₂×Dic₃	C₂²×C₄	1	4	yes	—
74	Dic₆	C₂×C₈	1	16	yes	direct product factor
75	C₃×C₈	—	2	2	no	—
76	C₃×C₈	C₂×Q₈	1	4	yes	complemented
77	C₃⋊C₈	—	2	4	no	—
78	C₂×Q₈	—	3	8	no	—
79	C₂×Q₈	—	3	4	no	—
80	C₂²×C₄	—	3	2	no	—
81	C₂²×C₄	—	3	2	no	—
82	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	3	4	no	—
87	C₂×C₈	—	6	4	no	—
88	C₂×C₈	Dic₆	1	4	yes	direct product factor
89	C₂×C₈	C₂²×D₃	1	2	yes	—
90	C₂×C₈	C₂²×D₃	1	2	yes	—
91	C₄⋊C₄	—	3	8	no	—
92	C₄⋊C₄	—	3	4	no	—
93	C₄²	—	3	4	no	—
94	C₄²	—	3	4	no	—
95	C₄²	C₄×D₃	1	2	yes	—
96	C₄²	C₂²×D₃	1	2	yes	—
97	C₂²×C₃	C₄×Q₈	1	2	yes	—
98	C₂²×C₃	C₂²×C₈	1	2	yes	—
99	C₂²×C₃	#131	1	2	yes	—
100	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
101	C₃×C₄	—	2	2	no	—
102	C₃×C₄	C₄×Q₈	1	2	yes	—
103	C₃×C₄	C₂²×C₈	1	4	yes	—
104	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
105	C₃×C₄	C₂×#41	1	1	yes	characteristic
106	Dic₃	—	2	4	no	—
107	Dic₃	C₂²×C₈	1	8	yes	—
108	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
109	Q₈	—	3	16	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	3	8	no	—
112	C₂×C₄	—	3	4	no	—
113	C₂×C₄	—	3	4	no	—
114	C₂×C₄	—	6	4	no	—
115	C₂×C₄	D₃×C₈	1	4	yes	—
116	C₂×C₄	C₂×Dic₆	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
120	C₂×C₄	#235	1	2	yes	—
121	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	6	4	no	—
124	C₈	C₂×Dic₆	1	4	yes	direct product factor
125	C₂×C₃	C₈×Q₈	1	4	yes	complemented
126	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
128	C₂×C₃	C₂×#131	1	1	yes	characteristic
129	C₂²	C₄×Dic₆	1	2	yes	—
130	C₂²	C₂×D₃×C₈	1	2	yes	—
131	C₂²	#901	1	2	yes	—
132	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	3	8	no	—
135	C₄	—	6	4	no	—
136	C₄	C₄×Dic₆	1	2	yes	—
137	C₄	C₂×D₃×C₈	1	4	yes	—
138	C₄	C₂²×Dic₆	1	1	yes	characteristic
139	C₄	C₂×#235	1	1	yes	characteristic
140	C₃	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
141	C₂	C₈×Dic₆	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
143	C₂	C₂²×D₃×C₈	1	1	yes	characteristic
144	C₂	C₂×#901	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₃
commutator	C₂×C₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×Dic₆	C₂	direct (N × H)
33. C₂×Dic₆	C₈	direct (N × H)
36. C₂×C₃×C₈	Q₈	semidirect
74. Dic₆	C₂×C₈	direct (N × H)
76. C₃×C₈	C₂×Q₈	semidirect
88. C₂×C₈	Dic₆	direct (N × H)
124. C₈	C₂×Dic₆	direct (N × H)
125. C₂×C₃	C₈×Q₈	semidirect
140. C₃	C₂×C₈×Q₈	semidirect
141. C₂	C₈×Dic₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆	2
C₂×C₃×C₄×C₈	2
C₂×Dic₃⋊C₈	2
C₂×C₈×Dic₃	2
C₂×C₄⋊(C₃⋊C₈)	2
C₈×Dic₆	2
C₂×C₈×Q₈	3

Subgroups — C2×C8×Dic6 #78443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₆ #78443