Subgroups — C₂×C₈×D₁₂ #78456

Order 384.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 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, complemented
2	C₂×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₁₂	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×D₄	—	3	1	no	maximal, Sylow
8	C₂²×D₁₂	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃	C₂²	1	2	yes	—
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
15	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂×D₃×C₈	—	2	8	no	—
17	C₂×D₃×C₈	C₂²	1	8	yes	complemented
18	C₄×D₁₂	C₄	1	4	yes	—
19	C₄×D₁₂	C₂²	1	4	yes	—
20	C₃×C₄×C₈	C₂²	1	4	yes	complemented
21	(C₂×D₃)⋊C₈	C₂²	1	8	yes	complemented
22	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	complemented
23	C₂³×C₈	—	3	2	no	—
24	C₂×C₄×D₄	—	3	1	no	—
25	C₈×D₄	—	3	8	no	—
26	C₂×C₄⋊C₈	—	3	1	no	—
27	C₂×C₂²⋊C₈	—	3	2	no	—
28	C₂×C₄×C₈	D₃	1	1	yes	characteristic, complemented
29	C₂³×D₃	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	C₂×C₄	1	2	yes	—
34	C₂×D₁₂	C₈	1	8	yes	direct product factor
35	C₂×D₁₂	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₃	—	2	4	no	—
37	C₂×C₄×D₃	—	2	4	no	—
38	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₃	C₂³	1	4	yes	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	D₄	1	4	yes	complemented
42	C₂×C₃×C₈	C₂³	1	2	yes	—
43	C₂×C₃×C₈	C₂³	1	2	yes	—
44	C₃×C₄²	C₂×C₄	1	2	yes	—
45	C₃×C₄²	C₂³	1	2	yes	—
46	(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
47	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
48	C₂×C₃⋊C₈	—	2	4	no	—
49	C₂×C₃⋊C₈	C₂³	1	4	yes	—
50	D₃×C₈	—	2	16	no	—
51	C₂²×D₄	—	3	1	no	—
52	C₂³×C₄	—	3	2	no	—
53	C₂²×C₈	—	3	2	no	—
54	C₂²×C₈	—	3	8	no	—
55	C₂²×C₈	—	6	8	no	—
56	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₄×D₄	—	3	4	no	—
59	C₄×D₄	—	3	4	no	—
60	C₂×C₄⋊C₄	—	3	1	no	—
61	C₂×C₂²⋊C₄	—	3	2	no	—
62	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₄⋊C₈	—	3	4	no	—
64	C₂²⋊C₈	—	3	8	no	—
65	C₄×C₈	C₂×D₃	1	4	yes	complemented
66	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₃	—	2	4	no	—
68	C₂²×D₃	—	2	4	no	—
69	C₂²×D₃	C₂×C₈	1	8	yes	complemented
70	C₂²×D₃	C₂²×C₄	1	4	yes	—
71	C₂×C₃×C₄	—	2	2	no	—
72	C₂×C₃×C₄	C₂×C₈	1	4	yes	complemented
73	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
74	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄	#41	1	2	yes	—
78	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₃	—	2	4	no	—
80	C₂×Dic₃	C₂²×C₄	1	4	yes	—
81	D₁₂	C₂×C₈	1	16	yes	direct product factor
82	C₄×D₃	—	2	8	no	—
83	C₄×D₃	—	2	4	no	—
84	C₄×D₃	—	2	4	no	—
85	C₃×C₈	—	2	2	no	—
86	C₃×C₈	C₂×D₄	1	4	yes	complemented
87	C₃⋊C₈	—	2	4	no	—
88	C₂⁴	—	3	2	no	—
89	C₂×D₄	—	3	8	no	—
90	C₂×D₄	—	3	4	no	—
91	C₂²×C₄	—	3	2	no	—
92	C₂²×C₄	—	3	4	no	—
93	C₂²×C₄	—	3	4	no	—
94	C₂²×C₄	—	6	4	no	—
95	C₂²×C₄	—	6	4	no	—
96	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	3	4	no	—
101	C₂×C₈	—	6	4	no	—
102	C₂×C₈	—	6	16	no	—
103	C₂×C₈	D₁₂	1	4	yes	direct product factor
104	C₂×C₈	C₂²×D₃	1	2	yes	—
105	C₂×C₈	C₂²×D₃	1	2	yes	—
106	C₄⋊C₄	—	3	4	no	—
107	C₂²⋊C₄	—	3	8	no	—
108	C₄²	C₄×D₃	1	2	yes	—
109	C₄²	C₂²×D₃	1	2	yes	—
110	C₂²×C₃	C₄×D₄	1	2	yes	—
111	C₂²×C₃	C₂²×C₈	1	2	yes	—
112	C₂²×C₃	#131	1	2	yes	—
113	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₃	—	2	4	no	—
115	C₂×D₃	—	2	4	no	—
116	C₂×D₃	—	2	16	no	—
117	C₂×D₃	C₂²×C₈	1	8	yes	complemented
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	C₄×D₄	1	2	yes	—
120	C₃×C₄	C₂²×C₈	1	4	yes	complemented
121	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₃×C₄	C₂×#41	1	1	yes	characteristic
123	Dic₃	—	2	4	no	—
124	C₂³	—	3	8	no	—
125	C₂³	—	3	4	no	—
126	C₂³	—	6	4	no	—
127	C₂³	—	6	4	no	—
128	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
129	D₄	—	3	16	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	3	4	no	—
132	C₂×C₄	—	6	4	no	—
133	C₂×C₄	—	6	8	no	—
134	C₂×C₄	—	6	4	no	—
135	C₂×C₄	—	6	4	no	—
136	C₂×C₄	D₃×C₈	1	4	yes	complemented
137	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
138	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
140	C₂×C₄	C₂×D₁₂	1	2	yes	—
141	C₂×C₄	#235	1	2	yes	—
142	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	6	4	no	—
145	C₈	C₂×D₁₂	1	4	yes	direct product factor
146	C₂×C₃	C₈×D₄	1	4	yes	complemented
147	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
149	C₂×C₃	C₂×#131	1	1	yes	characteristic
150	D₃	—	2	8	no	—
151	C₂²	—	3	8	no	—
152	C₂²	—	6	16	no	—
153	C₂²	—	6	4	no	—
154	C₂²	—	6	4	no	—
155	C₂²	C₄×D₁₂	1	2	yes	—
156	C₂²	C₂×D₃×C₈	1	2	yes	—
157	C₂²	#901	1	2	yes	—
158	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	6	4	no	—
161	C₄	C₄×D₁₂	1	2	yes	—
162	C₄	C₂×D₃×C₈	1	4	yes	complemented
163	C₄	C₂²×D₁₂	1	1	yes	characteristic
164	C₄	C₂×#235	1	1	yes	characteristic
165	C₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
166	C₂	—	6	8	no	—
167	C₂	C₈×D₁₂	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₁₂	1	1	yes	characteristic
169	C₂	C₂²×D₃×C₈	1	1	yes	characteristic
170	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	C₂³×C₃	C₂²×C₄
prime radical	C₂²×D₁₂	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×C₈×D₄
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
1. C₂²×D₃×C₈	C₂	semidirect
3. C₂×C₃×C₄×C₈	C₂	semidirect
4. C₂×(C₂×D₃)⋊C₈	C₂	semidirect
5. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
6. C₈×D₁₂	C₂	direct (N × H)
17. C₂×D₃×C₈	C₂²	semidirect
20. C₃×C₄×C₈	C₂²	semidirect
21. (C₂×D₃)⋊C₈	C₂²	semidirect
22. C₄⋊(C₃⋊C₈)	C₂²	semidirect
28. C₂×C₄×C₈	D₃	semidirect
34. C₂×D₁₂	C₈	direct (N × H)
41. C₂×C₃×C₈	D₄	semidirect
65. C₄×C₈	C₂×D₃	semidirect
69. C₂²×D₃	C₂×C₈	semidirect
72. C₂×C₃×C₄	C₂×C₈	semidirect
81. D₁₂	C₂×C₈	direct (N × H)
86. C₃×C₈	C₂×D₄	semidirect
103. C₂×C₈	D₁₂	direct (N × H)
117. C₂×D₃	C₂²×C₈	semidirect
120. C₃×C₄	C₂²×C₈	semidirect
136. C₂×C₄	D₃×C₈	semidirect
145. C₈	C₂×D₁₂	direct (N × H)
146. C₂×C₃	C₈×D₄	semidirect
162. C₄	C₂×D₃×C₈	semidirect
165. C₃	C₂×C₈×D₄	semidirect
167. C₂	C₈×D₁₂	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×C₄×D₁₂	2
C₂×C₃×C₄×C₈	2
C₂×(C₂×D₃)⋊C₈	2
C₂×C₄⋊(C₃⋊C₈)	2
C₈×D₁₂	2
C₂×C₈×D₄	3

Subgroups — C2×C8×D12 #78456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₁₂ #78456