Subgroups — C₂³×D₃×Q₁₆ #11708967

Order 768.

11166 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃×Q₈	C₂	1	2	yes	maximal
2	C₂³×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#216	C₂	1	2	yes	maximal, complemented
4	C₂²×D₃×Q₁₆	C₂	1	56	yes	maximal, direct product factor
5	C₂³×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Q₁₆	—	3	1	no	maximal, Sylow
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂²×D₃×Q₈	—	2	28	no	—
10	C₂²×D₃×Q₈	C₂²	1	56	yes	—
11	C₂³×C₄×D₃	—	2	2	no	—
12	C₂³×C₄×D₃	C₂²	1	1	yes	characteristic
13	C₂³×Dic₆	—	2	2	no	—
14	C₂³×Dic₆	C₂²	1	2	yes	—
15	C₂²×C₃×Q₁₆	C₂²	1	28	yes	complemented
16	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₂²×#216	C₂²	1	56	yes	complemented
18	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×D₃×Q₁₆	C₂²	1	448	yes	direct product factor
20	C₂²×Dic₁₂	C₂²	1	28	yes	complemented
21	C₂²×D₃×C₈	C₂²	1	28	yes	—
22	C₂⁴×Q₈	—	3	2	no	—
23	C₂³×Q₁₆	—	3	2	no	—
24	C₂³×Q₁₆	—	3	56	no	—
25	C₂³×Q₁₆	—	3	1	no	—
26	C₂³×Q₁₆	D₃	1	1	yes	characteristic, direct product factor
27	C₂⁴×C₈	—	3	1	no	—
28	C₂⁴×D₃	D₄	1	1	yes	characteristic
29	C₂²×C₃×Q₈	—	2	14	no	—
30	C₂²×C₃×Q₈	C₂³	1	28	yes	—
31	C₂³×C₃×C₄	—	2	2	no	—
32	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂³×Dic₃	—	2	2	no	—
34	C₂³×Dic₃	D₄	1	1	yes	characteristic
35	C₂×D₃×Q₈	—	2	336	no	—
36	C₂×D₃×Q₈	C₂³	1	224	yes	—
37	C₂²×C₄×D₃	—	2	56	no	—
38	C₂²×C₄×D₃	D₄	1	14	yes	—
39	C₂²×C₄×D₃	C₂³	1	14	yes	—
40	C₂²×Dic₆	—	2	14	no	—
41	C₂²×Dic₆	—	2	56	no	—
42	C₂²×Dic₆	C₂³	1	28	yes	—
43	C₂×C₃×Q₁₆	C₂³	1	112	yes	complemented
44	C₂²×C₃×C₈	C₂³	1	14	yes	—
45	C₂×#216	C₂³	1	224	yes	complemented
46	C₂²×C₃⋊C₈	C₂³	1	14	yes	—
47	D₃×Q₁₆	C₂³	1	512	yes	direct product factor
48	C₂×Dic₁₂	C₂³	1	112	yes	complemented
49	C₂×D₃×C₈	C₂³	1	112	yes	—
50	C₂³×Q₈	—	3	56	no	—
51	C₂³×Q₈	—	3	2	no	—
52	C₂³×Q₈	—	6	2	no	—
53	C₂³×Q₈	—	6	28	no	—
54	C₂³×Q₈	C₂×D₃	1	2	yes	—
55	C₂⁴×C₄	—	3	1	no	—
56	C₂⁴×C₄	—	6	2	no	—
57	C₂²×Q₁₆	—	3	56	no	—
58	C₂²×Q₁₆	—	3	448	no	—
59	C₂²×Q₁₆	—	3	28	no	—
60	C₂²×Q₁₆	C₂×D₃	1	28	yes	direct product factor
61	C₂³×C₈	—	3	1	no	—
62	C₂³×C₈	—	3	28	no	—
63	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
65	C₂³×D₃	Q₁₆	1	16	yes	direct product factor
66	C₂³×D₃	C₂×D₄	1	14	yes	—
67	C₂×C₃×Q₈	—	2	84	no	—
68	C₂×C₃×Q₈	C₂⁴	1	56	yes	—
69	C₂²×C₃×C₄	—	2	28	no	—
70	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
71	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
72	C₂²×Dic₃	—	2	28	no	—
73	C₂²×Dic₃	C₂×D₄	1	14	yes	—
74	D₃×Q₈	—	2	448	no	—
75	D₃×Q₈	C₂⁴	1	128	yes	—
76	C₂×C₄×D₃	—	2	224	no	—
77	C₂×C₄×D₃	C₂×D₄	1	84	yes	—
78	C₂×C₄×D₃	C₂⁴	1	28	yes	—
79	C₂×Dic₆	—	2	84	no	—
80	C₂×Dic₆	—	2	224	no	—
81	C₂×Dic₆	C₂⁴	1	56	yes	—
82	C₃×Q₁₆	C₂⁴	1	64	yes	complemented
83	C₂×C₃×C₈	C₂⁴	1	28	yes	—
84	#216	C₂⁴	1	128	yes	complemented
85	C₂×C₃⋊C₈	C₂⁴	1	28	yes	—
86	Dic₁₂	C₂⁴	1	64	yes	complemented
87	D₃×C₈	C₂⁴	1	64	yes	—
88	C₂⁵	—	3	1	no	—
89	C₂²×Q₈	—	2	14	no	—
90	C₂²×Q₈	—	3	224	no	—
91	C₂²×Q₈	—	3	28	no	—
92	C₂²×Q₈	—	6	56	no	—
93	C₂²×Q₈	—	6	336	no	—
94	C₂²×Q₈	—	6	14	no	—
95	C₂²×Q₈	C₂²×D₃	1	28	yes	—
96	C₂³×C₄	—	2	2	no	—
97	C₂³×C₄	—	3	1	no	—
98	C₂³×C₄	—	3	14	no	—
99	C₂³×C₄	—	3	14	no	—
100	C₂³×C₄	—	6	56	no	—
101	C₂³×C₄	—	6	2	no	—
102	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂×Q₁₆	—	3	224	no	—
104	C₂×Q₁₆	—	3	512	no	—
105	C₂×Q₁₆	—	3	112	no	—
106	C₂×Q₁₆	C₂²×D₃	1	112	yes	direct product factor
107	C₂²×C₈	—	3	14	no	—
108	C₂²×C₈	—	3	112	no	—
109	C₂²×C₈	C₂²×D₃	1	14	yes	—
110	C₂³×C₃	C₂×Q₁₆	1	8	yes	complemented
111	C₂³×C₃	C₂²×D₄	1	7	yes	—
112	C₂²×D₃	C₂×Q₁₆	1	112	yes	direct product factor
113	C₂²×D₃	C₂²×D₄	1	28	yes	—
114	C₃×Q₈	—	2	56	no	—
115	C₃×Q₈	C₂⁵	1	16	yes	—
116	C₂×C₃×C₄	—	2	56	no	—
117	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
118	C₂×C₃×C₄	C₂⁵	1	7	yes	—
119	C₂×Dic₃	—	2	56	no	—
120	C₂×Dic₃	C₂²×D₄	1	28	yes	—
121	C₄×D₃	—	2	128	no	—
122	C₄×D₃	C₂²×D₄	1	56	yes	—
123	C₄×D₃	C₂⁵	1	8	yes	—
124	Dic₆	—	2	56	no	—
125	Dic₆	—	2	128	no	—
126	Dic₆	C₂⁵	1	16	yes	—
127	C₃×C₈	C₂⁵	1	8	yes	—
128	C₃⋊C₈	C₂⁵	1	8	yes	—
129	C₂⁴	—	3	16	no	—
130	C₂⁴	—	3	14	no	—
131	C₂⁴	D₃×D₄	1	1	yes	characteristic
132	C₂×Q₈	—	2	84	no	—
133	C₂×Q₈	—	3	128	no	—
134	C₂×Q₈	—	3	56	no	—
135	C₂×Q₈	—	6	224	no	—
136	C₂×Q₈	—	6	448	no	—
137	C₂×Q₈	—	6	84	no	—
138	C₂×Q₈	C₂³×D₃	1	56	yes	—
139	C₂²×C₄	—	2	28	no	—
140	C₂²×C₄	—	3	14	no	—
141	C₂²×C₄	—	3	84	no	—
142	C₂²×C₄	—	3	28	no	—
143	C₂²×C₄	—	6	224	no	—
144	C₂²×C₄	—	6	28	no	—
145	C₂²×C₄	D₃×D₄	1	7	yes	—
146	C₂²×C₄	C₂³×D₃	1	7	yes	—
147	Q₁₆	—	3	128	no	—
148	Q₁₆	—	3	64	no	—
149	Q₁₆	C₂³×D₃	1	64	yes	direct product factor
150	C₂×C₈	—	3	28	no	—
151	C₂×C₈	—	3	64	no	—
152	C₂×C₈	C₂³×D₃	1	28	yes	—
153	C₂²×C₃	C₂²×Q₁₆	1	28	yes	complemented
154	C₂²×C₃	C₂³×D₄	1	7	yes	—
155	C₂×D₃	C₂²×Q₁₆	1	112	yes	direct product factor
156	C₂×D₃	C₂³×D₄	1	8	yes	—
157	C₃×C₄	—	2	16	no	—
158	C₃×C₄	C₂³×D₄	1	7	yes	—
159	C₃×C₄	C₂⁶	1	1	yes	characteristic
160	Dic₃	—	2	16	no	—
161	Dic₃	C₂³×D₄	1	8	yes	—
162	C₂³	—	3	28	no	—
163	C₂³	—	3	112	no	—
164	C₂³	D₃×Q₁₆	1	8	yes	direct product factor
165	C₂³	C₂×D₃×D₄	1	7	yes	—
166	Q₈	—	2	56	no	—
167	Q₈	—	3	16	no	—
168	Q₈	—	6	128	no	—
169	Q₈	—	6	56	no	—
170	Q₈	C₂⁴×D₃	1	16	yes	—
171	C₂×C₄	—	2	56	no	—
172	C₂×C₄	—	3	28	no	—
173	C₂×C₄	—	3	56	no	—
174	C₂×C₄	—	3	8	no	—
175	C₂×C₄	—	6	128	no	—
176	C₂×C₄	—	6	56	no	—
177	C₂×C₄	C₂×D₃×D₄	1	21	yes	—
178	C₂×C₄	C₂⁴×D₃	1	7	yes	—
179	C₈	—	3	8	no	—
180	C₈	C₂⁴×D₃	1	8	yes	—
181	C₂×C₃	C₂³×Q₁₆	1	14	yes	complemented
182	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
183	D₃	C₂³×Q₁₆	1	16	yes	direct product factor
184	C₂²	—	3	8	no	—
185	C₂²	—	3	112	no	—
186	C₂²	C₂×D₃×Q₁₆	1	28	yes	direct product factor
187	C₂²	C₂²×D₃×D₄	1	7	yes	—
188	C₄	—	2	16	no	—
189	C₄	—	3	8	no	—
190	C₄	—	6	16	no	—
191	C₄	C₂²×D₃×D₄	1	7	yes	—
192	C₄	C₂⁵×D₃	1	1	yes	characteristic
193	C₃	C₂⁴×Q₁₆	1	1	yes	characteristic, complemented, Sylow
194	C₂	—	3	16	no	—
195	C₂	C₂²×D₃×Q₁₆	1	14	yes	direct product factor
196	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁶
Frattini	C₄	C₂⁵×D₃
Fitting	C₂³×C₃×Q₁₆	C₂
socle	C₂⁴×C₃	C₂×D₄
prime radical	C₂⁴×D₃	D₄
hypercenter	C₂³×Q₁₆	D₃
hypocenter	C₃	C₂⁴×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Q₁₆	C₂	semidirect
3. C₂³×#216	C₂	semidirect
4. C₂²×D₃×Q₁₆	C₂	direct (N × H)
5. C₂³×Dic₁₂	C₂	semidirect
15. C₂²×C₃×Q₁₆	C₂²	semidirect
17. C₂²×#216	C₂²	semidirect
19. C₂×D₃×Q₁₆	C₂²	direct (N × H)
20. C₂²×Dic₁₂	C₂²	semidirect
26. C₂³×Q₁₆	D₃	direct (N × H)
43. C₂×C₃×Q₁₆	C₂³	semidirect
45. C₂×#216	C₂³	semidirect
47. D₃×Q₁₆	C₂³	direct (N × H)
48. C₂×Dic₁₂	C₂³	semidirect
60. C₂²×Q₁₆	C₂×D₃	direct (N × H)
65. C₂³×D₃	Q₁₆	direct (N × H)
82. C₃×Q₁₆	C₂⁴	semidirect
84. #216	C₂⁴	semidirect
86. Dic₁₂	C₂⁴	semidirect
106. C₂×Q₁₆	C₂²×D₃	direct (N × H)
110. C₂³×C₃	C₂×Q₁₆	semidirect
112. C₂²×D₃	C₂×Q₁₆	direct (N × H)
149. Q₁₆	C₂³×D₃	direct (N × H)
153. C₂²×C₃	C₂²×Q₁₆	semidirect
155. C₂×D₃	C₂²×Q₁₆	direct (N × H)
164. C₂³	D₃×Q₁₆	direct (N × H)
181. C₂×C₃	C₂³×Q₁₆	semidirect
183. D₃	C₂³×Q₁₆	direct (N × H)
186. C₂²	C₂×D₃×Q₁₆	direct (N × H)
193. C₃	C₂⁴×Q₁₆	semidirect
195. C₂	C₂²×D₃×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃×Q₈	2
C₂³×C₃×Q₁₆	2
C₂³×#216	2
C₂²×D₃×Q₁₆	2
C₂³×Dic₁₂	2
C₂³×D₃×C₈	2
C₂⁴×Q₁₆	3

Subgroups — C23×D3×Q16 #11708967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₃×Q₁₆ #11708967