Subgroups — C₂²×C₃×C₄⋊C₁₆ #10652001

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₁₆	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊C₁₆	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₁₆	—	2	8	no	—
9	C₂²×C₃×C₁₆	C₂²	1	12	yes	—
10	C₂×C₃×C₄×C₈	C₄	1	6	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	6	yes	—
12	C₃×C₄⋊C₁₆	C₂²	1	16	yes	direct product factor
13	C₂³×C₁₆	C₂×C₃	1	2	yes	—
14	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂×C₄⋊C₁₆	C₂×C₃	1	12	yes	direct product factor
16	C₂³×C₃×C₄	C₈	1	1	yes	characteristic
17	C₂³×C₃×C₄	C₈	1	1	yes	characteristic
18	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₈	—	2	4	no	—
20	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
21	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
22	C₂²×C₃×C₈	D₄	1	4	yes	—
23	C₂²×C₃×C₈	Q₈	1	4	yes	—
24	C₂²×C₃×C₈	C₂³	1	3	yes	—
25	C₂×C₃×C₄²	C₈	1	6	yes	—
26	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
27	C₂×C₃×C₁₆	—	2	24	no	—
28	C₂×C₃×C₁₆	C₂³	1	8	yes	—
29	C₃×C₄×C₈	C₂×C₄	1	12	yes	—
30	C₃×C₄×C₈	C₂³	1	4	yes	—
31	C₂³×C₈	C₃×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
33	C₂²×C₄²	C₃×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	—	2	8	no	—
35	C₂²×C₁₆	C₂²×C₃	1	12	yes	—
36	C₂×C₄×C₈	C₃×C₄	1	6	yes	—
37	C₂×C₄×C₈	C₂²×C₃	1	6	yes	—
38	C₄⋊C₁₆	C₂²×C₃	1	16	yes	direct product factor
39	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₃×C₄	—	2	4	no	—
41	C₂²×C₃×C₄	C₁₆	1	8	yes	complemented
42	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
43	C₂²×C₃×C₄	C₂×C₈	1	6	yes	—
44	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
45	C₂²×C₃×C₄	C₂×C₈	1	6	yes	—
46	C₂²×C₃×C₄	M₁₆	1	4	yes	—
47	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
48	C₂×C₃×C₈	—	2	12	no	—
49	C₂×C₃×C₈	C₄⋊C₄	1	12	yes	—
50	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
51	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
52	C₂×C₃×C₈	C₂×D₄	1	6	yes	—
53	C₂×C₃×C₈	C₂×Q₈	1	6	yes	—
54	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
55	C₃×C₄²	C₂×C₈	1	12	yes	—
56	C₃×C₄²	C₂²×C₄	1	4	yes	—
57	C₃×C₁₆	—	2	8	no	—
58	C₂³×C₄	C₃×C₈	1	1	yes	characteristic
59	C₂³×C₄	C₃×C₈	1	1	yes	characteristic
60	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂²×C₈	—	2	4	no	—
62	C₂²×C₈	C₂×C₃×C₄	1	6	yes	—
63	C₂²×C₈	C₂×C₃×C₄	1	3	yes	—
64	C₂²×C₈	C₃×D₄	1	4	yes	—
65	C₂²×C₈	C₃×Q₈	1	4	yes	—
66	C₂²×C₈	C₂³×C₃	1	3	yes	—
67	C₂×C₄²	C₃×C₈	1	6	yes	—
68	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
69	C₂×C₁₆	—	2	24	no	—
70	C₂×C₁₆	C₂³×C₃	1	8	yes	—
71	C₄×C₈	C₂×C₃×C₄	1	12	yes	—
72	C₄×C₈	C₂³×C₃	1	4	yes	—
73	C₂³×C₃	C₄⋊C₈	1	4	yes	—
74	C₂³×C₃	C₂×C₁₆	1	4	yes	—
75	C₂³×C₃	M₃₂	1	4	yes	—
76	C₂³×C₃	C₂²×C₈	1	3	yes	—
77	C₂×C₃×C₄	—	2	12	no	—
78	C₂×C₃×C₄	C₄⋊C₈	1	12	yes	—
79	C₂×C₃×C₄	C₂×C₁₆	1	24	yes	complemented
80	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
81	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
82	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
83	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
84	C₂×C₃×C₄	C₂×M₁₆	1	6	yes	—
85	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
86	C₃×C₈	—	2	4	no	—
87	C₃×C₈	C₂×C₄⋊C₄	1	6	yes	—
88	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
89	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
90	C₂⁴	C₂×C₃×C₈	1	1	yes	characteristic
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	C₃×C₁₆	1	8	yes	complemented
93	C₂²×C₄	C₃×C₄⋊C₄	1	4	yes	—
94	C₂²×C₄	C₂×C₃×C₈	1	6	yes	—
95	C₂²×C₄	C₂×C₃×C₈	1	3	yes	—
96	C₂²×C₄	C₂×C₃×C₈	1	6	yes	—
97	C₂²×C₄	C₃×M₁₆	1	4	yes	—
98	C₂²×C₄	C₂²×C₃×C₄	1	3	yes	—
99	C₂×C₈	—	2	12	no	—
100	C₂×C₈	C₃×C₄⋊C₄	1	12	yes	—
101	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
102	C₂×C₈	C₂²×C₃×C₄	1	3	yes	—
103	C₂×C₈	C₂×C₃×D₄	1	6	yes	—
104	C₂×C₈	C₂×C₃×Q₈	1	6	yes	—
105	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
106	C₄²	C₂×C₃×C₈	1	12	yes	—
107	C₄²	C₂²×C₃×C₄	1	4	yes	—
108	C₁₆	—	2	8	no	—
109	C₂²×C₃	C₄⋊C₁₆	1	16	yes	direct product factor
110	C₂²×C₃	C₂×C₄⋊C₈	1	6	yes	—
111	C₂²×C₃	C₂²×C₁₆	1	6	yes	—
112	C₂²×C₃	C₂×M₃₂	1	6	yes	—
113	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
114	C₃×C₄	—	2	4	no	—
115	C₃×C₄	C₂×C₄⋊C₈	1	6	yes	—
116	C₃×C₄	C₂²×C₁₆	1	8	yes	complemented
117	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
119	C₂³	C₃×C₄⋊C₈	1	4	yes	—
120	C₂³	C₂×C₃×C₁₆	1	4	yes	—
121	C₂³	C₃×M₃₂	1	4	yes	—
122	C₂³	C₂²×C₃×C₈	1	3	yes	—
123	C₂×C₄	—	2	12	no	—
124	C₂×C₄	C₃×C₄⋊C₈	1	12	yes	—
125	C₂×C₄	C₂×C₃×C₁₆	1	24	yes	complemented
126	C₂×C₄	C₂×C₃×C₄⋊C₄	1	6	yes	—
127	C₂×C₄	C₂²×C₃×C₈	1	4	yes	—
128	C₂×C₄	C₂²×C₃×C₈	1	4	yes	—
129	C₂×C₄	C₂²×C₃×C₈	1	3	yes	—
130	C₂×C₄	C₂×C₃×M₁₆	1	6	yes	—
131	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	C₂×C₃×C₄⋊C₄	1	6	yes	—
134	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
135	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
136	C₂×C₃	C₂×C₄⋊C₁₆	1	12	yes	direct product factor
137	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
138	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
139	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
140	C₂²	C₃×C₄⋊C₁₆	1	16	yes	direct product factor
141	C₂²	C₂×C₃×C₄⋊C₈	1	6	yes	—
142	C₂²	C₂²×C₃×C₁₆	1	6	yes	—
143	C₂²	C₂×C₃×M₃₂	1	6	yes	—
144	C₂²	C₂³×C₃×C₈	1	1	yes	characteristic
145	C₄	—	2	4	no	—
146	C₄	C₂×C₃×C₄⋊C₈	1	6	yes	—
147	C₄	C₂²×C₃×C₁₆	1	8	yes	complemented
148	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
149	C₄	C₂²×C₃×M₁₆	1	1	yes	characteristic
150	C₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
151	C₂	C₂×C₃×C₄⋊C₁₆	1	12	yes	direct product factor
152	C₂	C₂²×C₃×C₄⋊C₈	1	1	yes	characteristic
153	C₂	C₂³×C₃×C₁₆	1	1	yes	characteristic
154	C₂	C₂²×C₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₈	C₂²
commutator	C₂	C₂³×C₃×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄⋊C₁₆	C₂	direct (N × H)
4. C₂²×C₄⋊C₁₆	C₃	direct (N × H)
12. C₃×C₄⋊C₁₆	C₂²	direct (N × H)
15. C₂×C₄⋊C₁₆	C₂×C₃	direct (N × H)
38. C₄⋊C₁₆	C₂²×C₃	direct (N × H)
41. C₂²×C₃×C₄	C₁₆	semidirect
79. C₂×C₃×C₄	C₂×C₁₆	semidirect
92. C₂²×C₄	C₃×C₁₆	semidirect
109. C₂²×C₃	C₄⋊C₁₆	direct (N × H)
116. C₃×C₄	C₂²×C₁₆	semidirect
125. C₂×C₄	C₂×C₃×C₁₆	semidirect
136. C₂×C₃	C₂×C₄⋊C₁₆	direct (N × H)
140. C₂²	C₃×C₄⋊C₁₆	direct (N × H)
147. C₄	C₂²×C₃×C₁₆	semidirect
150. C₃	C₂²×C₄⋊C₁₆	direct (N × H)
151. C₂	C₂×C₃×C₄⋊C₁₆	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×C₃×C₁₆	2
C₂²×C₃×C₄×C₈	2
C₂×C₃×C₄⋊C₁₆	2
C₂²×C₄⋊C₁₆	3

Subgroups — C22×C3×C4⋊C16 #10652001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄⋊C₁₆ #10652001