Subgroups — M₁₆×Dic₆ #78509

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#5118	C₂	1	2	yes	maximal
4	Dic₃×M₁₆	C₂	1	2	yes	maximal
5	#4924	C₂	1	1	yes	maximal, characteristic
6	#4701	C₂	1	4	yes	maximal, complemented
7	#4679	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₆	C₂	1	2	yes	maximal, complemented
9	Q₈×M₁₆	—	3	1	no	maximal, Sylow
10	C₂²×Dic₆	C₄	1	1	yes	characteristic
11	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
15	C₂×Dic₃⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	2	yes	—
17	C₂×#208	C₂²	1	2	yes	—
18	C₄×Dic₆	C₄	1	1	yes	characteristic
19	C₄×Dic₆	C₄	1	2	yes	—
20	C₄×Dic₆	C₄	1	1	yes	characteristic
21	C₄×Dic₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₆	C₂²	1	1	yes	characteristic
23	C₄×Dic₆	C₂²	1	2	yes	—
24	C₃×#97	C₂²	1	2	yes	—
25	C₃×C₄×C₈	C₂²	1	2	yes	—
26	#815	C₂²	1	4	yes	—
27	Dic₃⋊C₈	C₂²	1	4	yes	—
28	Dic₃⋊C₈	C₂²	1	4	yes	—
29	C₈×Dic₃	C₂²	1	4	yes	—
30	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
31	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
32	C₂×C₄×Q₈	—	3	1	no	—
33	#446	—	3	4	no	—
34	#446	—	3	2	no	—
35	C₈×Q₈	—	3	2	no	—
36	#423	—	3	2	no	—
37	#423	—	3	1	no	—
38	C₄×M₁₆	—	3	2	no	—
39	C₄×M₁₆	D₃	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×Dic₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₃	C₂×C₄	1	2	yes	—
45	C₂×Dic₆	—	2	4	no	—
46	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₆	C₂×C₄	1	2	yes	—
48	C₂×Dic₆	C₂×C₄	1	1	yes	characteristic
49	C₃×M₁₆	—	2	2	no	—
50	C₃×M₁₆	Q₈	1	4	yes	complemented
51	C₂×C₃×C₈	C₂³	1	2	yes	—
52	C₂×C₃×C₈	C₂³	1	2	yes	—
53	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
54	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄²	C₂³	1	1	yes	characteristic
56	C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
60	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₃⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₃	C₂×C₄	1	2	yes	—
65	C₄×Dic₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₃	C₂³	1	2	yes	—
67	C₄×Dic₃	C₂³	1	2	yes	—
68	#208	—	2	4	no	—
69	C₂×C₃⋊C₈	C₂³	1	4	yes	—
70	C₂²×Q₈	—	3	1	no	—
71	C₂×M₁₆	—	3	2	no	—
72	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
74	C₄×Q₈	—	3	1	no	—
75	C₄×Q₈	—	3	2	no	—
76	C₄×Q₈	—	3	1	no	—
77	C₄×Q₈	—	3	1	no	—
78	C₄×Q₈	—	3	2	no	—
79	C₄×Q₈	—	3	1	no	—
80	C₂×C₄⋊C₄	—	3	2	no	—
81	C₂×C₄⋊C₄	—	3	1	no	—
82	C₂×C₄²	—	3	2	no	—
83	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
84	C₄⋊C₈	—	3	4	no	—
85	C₄⋊C₈	—	3	4	no	—
86	C₄⋊C₈	—	3	2	no	—
87	C₄⋊C₈	—	3	2	no	—
88	#97	—	3	4	no	—
89	#97	C₂×D₃	1	2	yes	—
90	C₄×C₈	—	3	4	no	—
91	C₄×C₈	C₂×D₃	1	2	yes	—
92	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	4	1	no	—
95	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
101	C₂×C₃×C₄	#41	1	2	yes	—
102	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
103	C₂×Dic₃	—	2	4	no	—
104	C₂×Dic₃	—	4	2	no	—
105	C₂×Dic₃	C₂²×C₄	1	2	yes	—
106	C₂×Dic₃	C₂²×C₄	1	2	yes	—
107	C₂×Dic₃	C₂²×C₄	1	2	yes	—
108	C₂×Dic₃	C₂²×C₄	1	2	yes	—
109	Dic₆	—	2	2	no	—
110	Dic₆	—	2	4	no	—
111	Dic₆	M₁₆	1	4	yes	direct product factor
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	C₂×Q₈	1	4	yes	complemented
114	C₃⋊C₈	—	2	4	no	—
115	C₂×Q₈	—	3	1	no	—
116	C₂×Q₈	—	3	2	no	—
117	C₂×Q₈	—	3	1	no	—
118	C₂×Q₈	—	6	4	no	—
119	C₂²×C₄	—	3	2	no	—
120	C₂²×C₄	—	3	2	no	—
121	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
122	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	6	4	no	—
126	M₁₆	Dic₆	1	4	yes	direct product factor
127	C₂×C₈	—	3	4	no	—
128	C₂×C₈	C₂²×D₃	1	2	yes	—
129	C₂×C₈	C₂²×D₃	1	2	yes	—
130	C₄⋊C₄	—	3	2	no	—
131	C₄⋊C₄	—	3	2	no	—
132	C₄⋊C₄	—	3	2	no	—
133	C₄⋊C₄	—	3	2	no	—
134	C₄⋊C₄	—	3	2	no	—
135	C₄⋊C₄	—	3	1	no	—
136	C₄⋊C₄	—	3	1	no	—
137	C₄²	—	3	2	no	—
138	C₄²	—	3	2	no	—
139	C₄²	—	3	2	no	—
140	C₄²	—	3	2	no	—
141	C₄²	C₄×D₃	1	1	yes	characteristic
142	C₄²	C₄×D₃	1	1	yes	characteristic
143	C₄²	C₂²×D₃	1	1	yes	characteristic
144	C₄²	C₂²×D₃	1	1	yes	characteristic
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	—	2	1	no	—
147	C₂²×C₃	C₄×Q₈	1	2	yes	—
148	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	2	1	no	—
152	C₃×C₄	C₄×Q₈	1	2	yes	—
153	C₃×C₄	C₂×M₁₆	1	2	yes	—
154	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
155	C₃×C₄	C₂×#41	1	1	yes	characteristic
156	Dic₃	—	2	2	no	—
157	Dic₃	—	2	2	no	—
158	Dic₃	—	2	2	no	—
159	Dic₃	C₂×M₁₆	1	4	yes	—
160	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
161	Q₈	—	3	4	no	—
162	Q₈	—	6	4	no	—
163	Q₈	—	6	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	3	2	no	—
166	C₂×C₄	—	3	2	no	—
167	C₂×C₄	—	3	2	no	—
168	C₂×C₄	—	3	2	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	6	4	no	—
171	C₂×C₄	—	12	2	no	—
172	C₂×C₄	C₂×Dic₆	1	2	yes	—
173	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄	#235	1	2	yes	—
179	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	6	4	no	—
182	C₈	C₂×Dic₆	1	4	yes	complemented
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₂×C₃	#568	1	1	yes	characteristic
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	C₄×Dic₆	1	2	yes	—
190	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	3	4	no	—
195	C₄	—	6	2	no	—
196	C₄	—	6	2	no	—
197	C₄	—	6	2	no	—
198	C₄	C₄×Dic₆	1	2	yes	—
199	C₄	D₃×M₁₆	1	2	yes	—
200	C₄	C₂²×Dic₆	1	1	yes	characteristic
201	C₄	C₂×#235	1	1	yes	characteristic
202	C₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	2	no	—
204	C₂	C₂×C₄×Dic₆	1	1	yes	characteristic
205	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic
206	C₂	#5745	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×C₄×M₁₆	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
6. #4701	C₂	semidirect
7. #4679	C₂	semidirect
8. C₈×Dic₆	C₂	semidirect
50. C₃×M₁₆	Q₈	semidirect
111. Dic₆	M₁₆	direct (N × H)
113. C₃×C₈	C₂×Q₈	semidirect
126. M₁₆	Dic₆	direct (N × H)
182. C₈	C₂×Dic₆	semidirect
202. C₃	Q₈×M₁₆	semidirect

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
C₃×C₄×M₁₆	2
#5118	2
Dic₃×M₁₆	2
#4924	2
#4701	2
#4679	2
C₈×Dic₆	2
Q₈×M₁₆	3

Subgroups — M16×Dic6 #78509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₆ #78509