Subgroups — C₃²×C₈×Q₁₆ #11768626

Order 1152.

492 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	2	yes	maximal
2	C₃²×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃²×#335	C₂	1	1	yes	maximal, characteristic
4	C₃²×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₃×C₈×Q₁₆	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×Q₁₆	C₄	1	1	yes	characteristic
8	C₃²×C₄×Q₈	C₂²	1	2	yes	—
9	C₃²×#107	C₄	1	1	yes	characteristic
10	C₃²×C₄⋊C₈	—	2	2	no	—
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃²×#103	C₄	1	2	yes	—
13	C₃²×C₄×C₈	—	2	2	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₈×Q₈	C₂×C₃	1	8	yes	—
18	C₃×C₄×Q₁₆	C₂×C₃	1	4	yes	—
19	C₃×#335	C₂×C₃	1	4	yes	—
20	C₃×Q₈⋊C₈	C₂×C₃	1	8	yes	—
21	C₃×C₈²	C₂×C₃	1	4	yes	—
22	C₂×C₃²×Q₈	C₂×C₄	1	2	yes	—
23	C₃²×Q₁₆	C₈	1	4	yes	direct product factor
24	C₂×C₃²×C₈	—	2	1	no	—
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
30	C₃²×C₄⋊C₄	—	2	2	no	—
31	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₃²×C₄²	—	2	2	no	—
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	C₈×Q₁₆	C₃²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×Q₁₆	C₃×C₄	1	4	yes	—
36	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	—
37	C₃×#107	C₃×C₄	1	4	yes	—
38	C₃×C₄⋊C₈	—	2	8	no	—
39	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
40	C₃×#103	C₃×C₄	1	8	yes	—
41	C₃×C₄×C₈	—	2	8	no	—
42	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
45	C₃²×Q₈	—	2	2	no	—
46	C₃²×Q₈	C₂×C₈	1	4	yes	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	—	2	2	no	—
49	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	#41	1	1	yes	characteristic
52	C₃²×C₈	—	2	2	no	—
53	C₃²×C₈	—	2	1	no	—
54	C₃²×C₈	—	2	1	no	—
55	C₃²×C₈	—	4	2	no	—
56	C₃²×C₈	C₂×C₈	1	2	yes	—
57	C₃²×C₈	Q₁₆	1	2	yes	direct product factor
58	C₈×Q₈	C₂×C₃²	1	2	yes	—
59	C₄×Q₁₆	C₂×C₃²	1	1	yes	characteristic
60	#335	C₂×C₃²	1	1	yes	characteristic
61	Q₈⋊C₈	C₂×C₃²	1	2	yes	—
62	C₈²	C₂×C₃²	1	1	yes	characteristic
63	C₂×C₃×Q₈	C₂×C₃×C₄	1	8	yes	—
64	C₃×Q₁₆	C₃×C₈	1	16	yes	direct product factor
65	C₂×C₃×C₈	—	2	4	no	—
66	C₂×C₃×C₈	—	2	8	no	—
67	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
69	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
70	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
71	C₃×C₄⋊C₄	—	2	8	no	—
72	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
73	C₃×C₄²	—	2	8	no	—
74	C₃×C₄²	C₂³×C₃	1	4	yes	—
75	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
76	C₃²×C₄	—	2	4	no	—
77	C₃²×C₄	—	2	1	no	—
78	C₃²×C₄	—	4	2	no	—
79	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
80	C₃²×C₄	#131	1	1	yes	characteristic
81	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
82	C₃²×C₄	#135	1	1	yes	characteristic
83	C₂×Q₁₆	C₃²×C₄	1	1	yes	characteristic
84	C₄×Q₈	C₂²×C₃²	1	2	yes	—
85	#107	C₃²×C₄	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	C₂²×C₃²	1	2	yes	—
88	#103	C₃²×C₄	1	2	yes	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
91	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
92	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
93	C₃×Q₈	—	2	8	no	—
94	C₃×Q₈	C₂×C₃×C₈	1	16	yes	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	—	2	8	no	—
97	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
98	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
99	C₂×C₃×C₄	C₃×#41	1	4	yes	—
100	C₃×C₈	—	2	8	no	—
101	C₃×C₈	—	2	4	no	—
102	C₃×C₈	—	2	4	no	—
103	C₃×C₈	—	4	8	no	—
104	C₃×C₈	C₂×C₃×C₈	1	8	yes	—
105	C₃×C₈	C₃×Q₁₆	1	8	yes	direct product factor
106	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
107	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
108	C₂×C₃²	#443	1	1	yes	characteristic
109	C₂×Q₈	C₂×C₃²×C₄	1	2	yes	—
110	Q₁₆	C₃²×C₈	1	4	yes	direct product factor
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
114	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
115	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
116	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₃²×C₄	1	2	yes	—
119	C₄²	—	2	2	no	—
120	C₄²	C₂³×C₃²	1	1	yes	characteristic
121	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
122	C₃×C₄	—	2	16	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	4	8	no	—
125	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
126	C₃×C₄	C₃×#131	1	4	yes	—
127	C₃×C₄	C₂×C₃×Q₁₆	1	4	yes	—
128	C₃×C₄	C₃×#135	1	4	yes	—
129	C₃²	C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
130	Q₈	—	2	2	no	—
131	Q₈	C₂×C₃²×C₈	1	4	yes	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
136	C₂×C₄	C₃²×#41	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	C₂×C₃²×C₈	1	2	yes	—
142	C₈	C₃²×Q₁₆	1	2	yes	direct product factor
143	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
144	C₂×C₃	C₃×C₄×Q₁₆	1	4	yes	—
145	C₂×C₃	C₃×#443	1	4	yes	—
146	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	2	no	—
150	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
151	C₄	C₃²×#131	1	1	yes	characteristic
152	C₄	C₂×C₃²×Q₁₆	1	1	yes	characteristic
153	C₄	C₃²×#135	1	1	yes	characteristic
154	C₃	C₃×C₈×Q₁₆	1	4	yes	direct product factor
155	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
156	C₂	C₃²×C₄×Q₁₆	1	1	yes	characteristic
157	C₂	C₃²×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃×C₈×Q₁₆	C₃	direct (N × H)
23. C₃²×Q₁₆	C₈	direct (N × H)
34. C₈×Q₁₆	C₃²	direct (N × H)
57. C₃²×C₈	Q₁₆	direct (N × H)
64. C₃×Q₁₆	C₃×C₈	direct (N × H)
105. C₃×C₈	C₃×Q₁₆	direct (N × H)
110. Q₁₆	C₃²×C₈	direct (N × H)
129. C₃²	C₈×Q₁₆	direct (N × H)
142. C₈	C₃²×Q₁₆	direct (N × H)
154. C₃	C₃×C₈×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
C₃²×C₈×Q₈	2
C₃²×C₄×Q₁₆	2
C₃²×#335	2
C₃²×Q₈⋊C₈	2
C₃²×C₈²	2
C₃×C₈×Q₁₆	3

Subgroups — C32×C8×Q16 #11768626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×Q₁₆ #11768626