Subgroups — C₃×C₄²×Q₁₆ #10639106

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₁₆	C₂	1	3	yes	maximal
2	C₃×C₄²×Q₈	C₂	1	2	yes	maximal
3	C₃×C₄×#107	C₂	1	3	yes	maximal
4	C₃×C₄×#103	C₂	1	6	yes	maximal
5	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
6	C₄²×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Q₈	C₂²	1	6	yes	—
9	C₃×C₄×Q₁₆	C₄	1	24	yes	direct product factor
10	C₂×C₃×#107	C₂²	1	3	yes	—
11	C₂×C₃×#103	C₂²	1	6	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	3	yes	—
14	C₃×C₄×C₄⋊C₄	—	2	6	no	—
15	C₃×C₄×C₄⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄³	—	2	2	no	—
17	C₃×C₄³	C₂²	1	1	yes	characteristic
18	C₃×#340	C₂²	1	6	yes	—
19	C₂×C₄×Q₁₆	C₂×C₃	1	3	yes	—
20	C₄²×Q₈	C₂×C₃	1	2	yes	—
21	C₄×#107	C₂×C₃	1	3	yes	—
22	C₄×#103	C₂×C₃	1	6	yes	—
23	C₄²×C₈	C₂×C₃	1	1	yes	characteristic
24	C₂²×C₃×Q₈	C₂³	1	2	yes	—
25	C₂×C₃×Q₁₆	C₂×C₄	1	12	yes	—
26	C₂²×C₃×C₈	C₂³	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄×Q₈	—	2	12	no	—
29	C₃×C₄×Q₈	C₂×C₄	1	24	yes	—
30	C₂×C₃×C₄⋊C₄	—	2	6	no	—
31	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
32	C₂×C₃×C₄²	—	2	6	no	—
33	C₂×C₃×C₄²	—	2	6	no	—
34	C₂×C₃×C₄²	D₄	1	3	yes	—
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄²	C₂³	1	3	yes	—
37	C₃×#107	C₂×C₄	1	12	yes	—
38	C₃×#103	C₂×C₄	1	24	yes	—
39	C₃×C₄×C₈	—	2	6	no	—
40	C₃×C₄×C₈	C₂×C₄	1	12	yes	—
41	C₃×#95	—	2	6	no	—
42	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₄×Q₈	C₂²×C₃	1	6	yes	—
44	C₄×Q₁₆	C₃×C₄	1	24	yes	direct product factor
45	C₂×#107	C₂²×C₃	1	3	yes	—
46	C₂×#103	C₂²×C₃	1	6	yes	—
47	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
48	C₂×C₄×C₈	C₂²×C₃	1	3	yes	—
49	C₄×C₄⋊C₄	—	2	6	no	—
50	C₄×C₄⋊C₄	C₂²×C₃	1	6	yes	—
51	C₄³	—	2	2	no	—
52	C₄³	C₂²×C₃	1	1	yes	characteristic
53	#340	C₂²×C₃	1	6	yes	—
54	C₂×C₃×Q₈	—	2	6	no	—
55	C₂×C₃×Q₈	C₂²×C₄	1	12	yes	—
56	C₂²×C₃×C₄	—	2	2	no	—
57	C₂²×C₃×C₄	—	2	6	no	—
58	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
59	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
60	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
61	C₃×Q₁₆	C₄²	1	16	yes	direct product factor
62	C₂×C₃×C₈	—	2	6	no	—
63	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
64	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
65	C₃×C₄⋊C₄	—	2	24	no	—
66	C₃×C₄⋊C₄	—	2	6	no	—
67	C₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
68	C₃×C₄²	—	2	24	no	—
69	C₃×C₄²	—	2	6	no	—
70	C₃×C₄²	—	4	12	no	—
71	C₃×C₄²	Q₁₆	1	4	yes	direct product factor
72	C₃×C₄²	C₂²×C₄	1	6	yes	—
73	C₃×C₄²	#41	1	6	yes	—
74	C₂²×Q₈	C₂³×C₃	1	2	yes	—
75	C₂×Q₁₆	C₂×C₃×C₄	1	12	yes	—
76	C₂²×C₈	C₂³×C₃	1	3	yes	—
77	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
78	C₄×Q₈	—	2	12	no	—
79	C₄×Q₈	C₂×C₃×C₄	1	24	yes	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
82	C₂×C₄²	—	2	6	no	—
83	C₂×C₄²	—	2	6	no	—
84	C₂×C₄²	C₃×D₄	1	3	yes	—
85	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
86	C₂×C₄²	C₂³×C₃	1	3	yes	—
87	#107	C₂×C₃×C₄	1	12	yes	—
88	#103	C₂×C₃×C₄	1	24	yes	—
89	C₄×C₈	—	2	6	no	—
90	C₄×C₈	C₂×C₃×C₄	1	12	yes	—
91	#95	—	2	6	no	—
92	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
93	C₃×Q₈	—	2	12	no	—
94	C₃×Q₈	C₂×C₄²	1	8	yes	—
95	C₂×C₃×C₄	—	2	12	no	—
96	C₂×C₃×C₄	—	2	6	no	—
97	C₂×C₃×C₄	—	2	12	no	—
98	C₂×C₃×C₄	—	4	12	no	—
99	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
100	C₂×C₃×C₄	C₄×D₄	1	6	yes	—
101	C₂×C₃×C₄	C₂×Q₁₆	1	6	yes	—
102	C₂×C₃×C₄	#135	1	6	yes	—
103	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
104	C₂×C₃×C₄	C₂×#41	1	3	yes	—
105	C₃×C₈	—	2	6	no	—
106	C₃×C₈	C₂×C₄²	1	4	yes	—
107	C₂×Q₈	—	2	6	no	—
108	C₂×Q₈	C₂²×C₃×C₄	1	12	yes	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	6	no	—
111	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
112	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
113	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
114	Q₁₆	C₃×C₄²	1	16	yes	direct product factor
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
117	C₂×C₈	C₂²×C₃×C₄	1	6	yes	—
118	C₄⋊C₄	—	2	24	no	—
119	C₄⋊C₄	—	2	6	no	—
120	C₄⋊C₄	C₂²×C₃×C₄	1	12	yes	—
121	C₄²	—	2	24	no	—
122	C₄²	—	2	6	no	—
123	C₄²	—	4	12	no	—
124	C₄²	C₃×Q₁₆	1	4	yes	direct product factor
125	C₄²	C₂²×C₃×C₄	1	6	yes	—
126	C₄²	C₃×#41	1	6	yes	—
127	C₂²×C₃	C₂×C₄×D₄	1	3	yes	—
128	C₂²×C₃	C₂²×Q₁₆	1	1	yes	characteristic
129	C₂²×C₃	C₂×#135	1	3	yes	—
130	C₃×C₄	—	2	6	no	—
131	C₃×C₄	—	2	8	no	—
132	C₃×C₄	—	4	12	no	—
133	C₃×C₄	C₄×Q₁₆	1	12	yes	direct product factor
134	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
135	C₃×C₄	C₄×#41	1	3	yes	—
136	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
137	Q₈	—	2	12	no	—
138	Q₈	C₂×C₃×C₄²	1	8	yes	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	4	12	no	—
143	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
144	C₂×C₄	C₃×C₄×D₄	1	6	yes	—
145	C₂×C₄	C₂×C₃×Q₁₆	1	6	yes	—
146	C₂×C₄	C₃×#135	1	6	yes	—
147	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
148	C₂×C₄	C₂×C₃×#41	1	3	yes	—
149	C₈	—	2	6	no	—
150	C₈	C₂×C₃×C₄²	1	4	yes	—
151	C₂×C₃	C₄²×D₄	1	1	yes	characteristic
152	C₂×C₃	C₂×C₄×Q₁₆	1	3	yes	—
153	C₂×C₃	C₄×#135	1	3	yes	—
154	C₂²	C₂×C₃×C₄×D₄	1	3	yes	—
155	C₂²	C₂²×C₃×Q₁₆	1	1	yes	characteristic
156	C₂²	C₂×C₃×#135	1	3	yes	—
157	C₄	—	2	6	no	—
158	C₄	—	2	8	no	—
159	C₄	—	4	12	no	—
160	C₄	C₃×C₄×Q₁₆	1	12	yes	direct product factor
161	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
162	C₄	C₃×C₄×#41	1	3	yes	—
163	C₃	C₄²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	C₃×C₄²×D₄	1	1	yes	characteristic
165	C₂	C₂×C₃×C₄×Q₁₆	1	3	yes	—
166	C₂	C₃×C₄×#135	1	3	yes	—

	subgroup	G/H
center	C₂×C₃×C₄²	D₄
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄² ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄²×Q₁₆	C₃	direct (N × H)
9. C₃×C₄×Q₁₆	C₄	direct (N × H)
44. C₄×Q₁₆	C₃×C₄	direct (N × H)
61. C₃×Q₁₆	C₄²	direct (N × H)
71. C₃×C₄²	Q₁₆	direct (N × H)
114. Q₁₆	C₃×C₄²	direct (N × H)
124. C₄²	C₃×Q₁₆	direct (N × H)
133. C₃×C₄	C₄×Q₁₆	direct (N × H)
160. C₄	C₃×C₄×Q₁₆	direct (N × H)
163. C₃	C₄²×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
C₂×C₃×C₄×Q₁₆	2
C₃×C₄²×Q₈	2
C₃×C₄×#107	2
C₃×C₄×#103	2
C₃×C₄²×C₈	2
C₄²×Q₁₆	3

Subgroups — C3×C42×Q16 #10639106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄²×Q₁₆ #10639106