Subgroups — C₃²×C₈×C₁₆² #72196205

Order 18432.

13008 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈²×C₁₆	C₂	1	3	yes	maximal
2	C₃²×C₄×C₁₆²	C₂	1	4	yes	maximal
3	C₃×C₈×C₁₆²	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₈³	C₂²	1	1	yes	characteristic
5	C₃²×C₄×C₈×C₁₆	C₄	1	12	yes	—
6	C₃²×C₄×C₈×C₁₆	C₂²	1	6	yes	—
7	C₂×C₃²×C₁₆²	C₄	1	16	yes	—
8	C₃×C₈²×C₁₆	C₂×C₃	1	12	yes	—
9	C₃×C₄×C₁₆²	C₂×C₃	1	16	yes	—
10	C₃²×C₄×C₈²	C₂×C₄	1	6	yes	—
11	C₃²×C₄×C₈²	C₂³	1	1	yes	characteristic
12	C₃²×C₄²×C₁₆	C₂×C₄	1	12	yes	—
13	C₂×C₃²×C₈×C₁₆	C₈	1	48	yes	—
14	C₂×C₃²×C₈×C₁₆	C₂×C₄	1	24	yes	—
15	C₃²×C₁₆²	C₈	1	64	yes	direct product factor
16	C₈×C₁₆²	C₃²	1	1	yes	characteristic, direct product factor, Sylow
17	C₃×C₈³	C₂²×C₃	1	4	yes	—
18	C₃×C₄×C₈×C₁₆	C₃×C₄	1	48	yes	—
19	C₃×C₄×C₈×C₁₆	C₂²×C₃	1	24	yes	—
20	C₂×C₃×C₁₆²	C₃×C₄	1	64	yes	—
21	C₈²×C₁₆	C₂×C₃²	1	3	yes	—
22	C₄×C₁₆²	C₂×C₃²	1	4	yes	—
23	C₂×C₃²×C₄×C₁₆	C₄²	1	24	yes	—
24	C₂×C₃²×C₄×C₁₆	C₂×C₈	1	48	yes	—
25	C₃²×C₄²×C₈	C₄²	1	4	yes	—
26	C₃²×C₄²×C₈	C₂²×C₄	1	3	yes	—
27	C₂×C₃²×C₈²	C₂×C₈	1	24	yes	—
28	C₂×C₃²×C₈²	C₂²×C₄	1	4	yes	—
29	C₃²×C₈×C₁₆	C₁₆	1	192	yes	direct product factor
30	C₃²×C₈×C₁₆	C₂×C₈	1	96	yes	—
31	C₃×C₄²×C₁₆	C₂×C₃×C₄	1	48	yes	—
32	C₂×C₃×C₈×C₁₆	C₃×C₈	1	192	yes	—
33	C₂×C₃×C₈×C₁₆	C₂×C₃×C₄	1	96	yes	—
34	C₃×C₄×C₈²	C₂×C₃×C₄	1	24	yes	—
35	C₃×C₄×C₈²	C₂³×C₃	1	4	yes	—
36	C₃×C₁₆²	C₃×C₈	1	256	yes	direct product factor
37	C₂²×C₃²×C₁₆	C₄×C₈	1	48	yes	—
38	C₂×C₃²×C₄×C₈	C₄×C₈	1	24	yes	—
39	C₂×C₃²×C₄×C₈	C₂×C₄²	1	6	yes	—
40	C₂×C₃²×C₄×C₈	C₂²×C₈	1	12	yes	—
41	C₃²×C₄³	C₂×C₄²	1	1	yes	characteristic
42	C₃²×C₄×C₁₆	C₄×C₈	1	96	yes	—
43	C₃²×C₄×C₁₆	C₂×C₁₆	1	192	yes	—
44	C₃²×C₈²	C₂×C₁₆	1	96	yes	—
45	C₃²×C₈²	C₂²×C₈	1	16	yes	—
46	C₄×C₈×C₁₆	C₃²×C₄	1	12	yes	—
47	C₄×C₈×C₁₆	C₂²×C₃²	1	6	yes	—
48	C₂×C₁₆²	C₃²×C₄	1	16	yes	—
49	C₈³	C₂²×C₃²	1	1	yes	characteristic
50	C₂×C₃×C₄×C₁₆	C₃×C₄²	1	96	yes	—
51	C₂×C₃×C₄×C₁₆	C₂×C₃×C₈	1	192	yes	—
52	C₃×C₄²×C₈	C₃×C₄²	1	16	yes	—
53	C₃×C₄²×C₈	C₂²×C₃×C₄	1	12	yes	—
54	C₂×C₃×C₈²	C₂×C₃×C₈	1	96	yes	—
55	C₂×C₃×C₈²	C₂²×C₃×C₄	1	16	yes	—
56	C₃×C₈×C₁₆	C₃×C₁₆	1	768	yes	direct product factor
57	C₃×C₈×C₁₆	C₂×C₃×C₈	1	384	yes	—
58	C₂²×C₃²×C₈	C₈²	1	16	yes	—
59	C₂²×C₃²×C₈	C₂×C₄×C₈	1	12	yes	—
60	C₂×C₃²×C₄²	C₄³	1	1	yes	characteristic
61	C₂×C₃²×C₄²	C₂×C₄×C₈	1	6	yes	—
62	C₂×C₃²×C₁₆	C₈²	1	96	yes	—
63	C₂×C₃²×C₁₆	C₄×C₁₆	1	192	yes	—
64	C₃²×C₄×C₈	C₄×C₁₆	1	96	yes	—
65	C₃²×C₄×C₈	C₂×C₄×C₈	1	24	yes	—
66	C₃²×C₄×C₈	C₂²×C₁₆	1	48	yes	—
67	C₄²×C₁₆	C₂×C₃²×C₄	1	12	yes	—
68	C₂×C₈×C₁₆	C₃²×C₈	1	48	yes	—
69	C₂×C₈×C₁₆	C₂×C₃²×C₄	1	24	yes	—
70	C₄×C₈²	C₂×C₃²×C₄	1	6	yes	—
71	C₄×C₈²	C₂³×C₃²	1	1	yes	characteristic
72	C₁₆²	C₃²×C₈	1	64	yes	direct product factor
73	C₂²×C₃×C₁₆	C₃×C₄×C₈	1	192	yes	—
74	C₂×C₃×C₄×C₈	C₃×C₄×C₈	1	96	yes	—
75	C₂×C₃×C₄×C₈	C₂×C₃×C₄²	1	24	yes	—
76	C₂×C₃×C₄×C₈	C₂²×C₃×C₈	1	48	yes	—
77	C₃×C₄³	C₂×C₃×C₄²	1	4	yes	—
78	C₃×C₄×C₁₆	C₃×C₄×C₈	1	384	yes	—
79	C₃×C₄×C₁₆	C₂×C₃×C₁₆	1	768	yes	—
80	C₃×C₈²	C₂×C₃×C₁₆	1	384	yes	—
81	C₃×C₈²	C₂²×C₃×C₈	1	64	yes	—
82	C₂²×C₃²×C₄	C₂×C₈²	1	4	yes	—
83	C₂²×C₃²×C₄	C₄²×C₈	1	3	yes	—
84	C₂×C₃²×C₈	C₈×C₁₆	1	96	yes	—
85	C₂×C₃²×C₈	C₂×C₈²	1	24	yes	—
86	C₂×C₃²×C₈	C₂×C₄×C₁₆	1	48	yes	—
87	C₃²×C₄²	C₄²×C₈	1	4	yes	—
88	C₃²×C₄²	C₂×C₄×C₁₆	1	24	yes	—
89	C₃²×C₁₆	C₈×C₁₆	1	192	yes	direct product factor
90	C₂×C₄×C₁₆	C₃²×C₄²	1	24	yes	—
91	C₂×C₄×C₁₆	C₂×C₃²×C₈	1	48	yes	—
92	C₄²×C₈	C₃²×C₄²	1	4	yes	—
93	C₄²×C₈	C₂²×C₃²×C₄	1	3	yes	—
94	C₂×C₈²	C₂×C₃²×C₈	1	24	yes	—
95	C₂×C₈²	C₂²×C₃²×C₄	1	4	yes	—
96	C₈×C₁₆	C₃²×C₁₆	1	192	yes	direct product factor
97	C₈×C₁₆	C₂×C₃²×C₈	1	96	yes	—
98	C₂²×C₃×C₈	C₃×C₈²	1	64	yes	—
99	C₂²×C₃×C₈	C₂×C₃×C₄×C₈	1	48	yes	—
100	C₂×C₃×C₄²	C₃×C₄³	1	4	yes	—
101	C₂×C₃×C₄²	C₂×C₃×C₄×C₈	1	24	yes	—
102	C₂×C₃×C₁₆	C₃×C₈²	1	384	yes	—
103	C₂×C₃×C₁₆	C₃×C₄×C₁₆	1	768	yes	—
104	C₃×C₄×C₈	C₃×C₄×C₁₆	1	384	yes	—
105	C₃×C₄×C₈	C₂×C₃×C₄×C₈	1	96	yes	—
106	C₃×C₄×C₈	C₂²×C₃×C₁₆	1	192	yes	—
107	C₂³×C₃²	C₄×C₈²	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₄×C₈²	1	6	yes	—
109	C₂×C₃²×C₄	C₂×C₈×C₁₆	1	24	yes	—
110	C₂×C₃²×C₄	C₄²×C₁₆	1	12	yes	—
111	C₃²×C₈	C₁₆²	1	64	yes	direct product factor
112	C₃²×C₈	C₂×C₈×C₁₆	1	48	yes	—
113	C₂²×C₁₆	C₃²×C₄×C₈	1	48	yes	—
114	C₂×C₄×C₈	C₃²×C₄×C₈	1	24	yes	—
115	C₂×C₄×C₈	C₂×C₃²×C₄²	1	6	yes	—
116	C₂×C₄×C₈	C₂²×C₃²×C₈	1	12	yes	—
117	C₄³	C₂×C₃²×C₄²	1	1	yes	characteristic
118	C₄×C₁₆	C₃²×C₄×C₈	1	96	yes	—
119	C₄×C₁₆	C₂×C₃²×C₁₆	1	192	yes	—
120	C₈²	C₂×C₃²×C₁₆	1	96	yes	—
121	C₈²	C₂²×C₃²×C₈	1	16	yes	—
122	C₂²×C₃×C₄	C₂×C₃×C₈²	1	16	yes	—
123	C₂²×C₃×C₄	C₃×C₄²×C₈	1	12	yes	—
124	C₂×C₃×C₈	C₃×C₈×C₁₆	1	384	yes	—
125	C₂×C₃×C₈	C₂×C₃×C₈²	1	96	yes	—
126	C₂×C₃×C₈	C₂×C₃×C₄×C₁₆	1	192	yes	—
127	C₃×C₄²	C₃×C₄²×C₈	1	16	yes	—
128	C₃×C₄²	C₂×C₃×C₄×C₁₆	1	96	yes	—
129	C₃×C₁₆	C₃×C₈×C₁₆	1	768	yes	direct product factor
130	C₂²×C₃²	C₈³	1	1	yes	characteristic
131	C₂²×C₃²	C₄×C₈×C₁₆	1	6	yes	—
132	C₃²×C₄	C₂×C₁₆²	1	16	yes	—
133	C₃²×C₄	C₄×C₈×C₁₆	1	12	yes	—
134	C₂²×C₈	C₃²×C₈²	1	16	yes	—
135	C₂²×C₈	C₂×C₃²×C₄×C₈	1	12	yes	—
136	C₂×C₄²	C₃²×C₄³	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₃²×C₄×C₈	1	6	yes	—
138	C₂×C₁₆	C₃²×C₈²	1	96	yes	—
139	C₂×C₁₆	C₃²×C₄×C₁₆	1	192	yes	—
140	C₄×C₈	C₃²×C₄×C₁₆	1	96	yes	—
141	C₄×C₈	C₂×C₃²×C₄×C₈	1	24	yes	—
142	C₄×C₈	C₂²×C₃²×C₁₆	1	48	yes	—
143	C₂³×C₃	C₃×C₄×C₈²	1	4	yes	—
144	C₂×C₃×C₄	C₃×C₄×C₈²	1	24	yes	—
145	C₂×C₃×C₄	C₂×C₃×C₈×C₁₆	1	96	yes	—
146	C₂×C₃×C₄	C₃×C₄²×C₁₆	1	48	yes	—
147	C₃×C₈	C₃×C₁₆²	1	256	yes	direct product factor
148	C₃×C₈	C₂×C₃×C₈×C₁₆	1	192	yes	—
149	C₂×C₃²	C₄×C₁₆²	1	4	yes	—
150	C₂×C₃²	C₈²×C₁₆	1	3	yes	—
151	C₂²×C₄	C₂×C₃²×C₈²	1	4	yes	—
152	C₂²×C₄	C₃²×C₄²×C₈	1	3	yes	—
153	C₂×C₈	C₃²×C₈×C₁₆	1	96	yes	—
154	C₂×C₈	C₂×C₃²×C₈²	1	24	yes	—
155	C₂×C₈	C₂×C₃²×C₄×C₁₆	1	48	yes	—
156	C₄²	C₃²×C₄²×C₈	1	4	yes	—
157	C₄²	C₂×C₃²×C₄×C₁₆	1	24	yes	—
158	C₁₆	C₃²×C₈×C₁₆	1	192	yes	direct product factor
159	C₂²×C₃	C₃×C₄×C₈×C₁₆	1	24	yes	—
160	C₂²×C₃	C₃×C₈³	1	4	yes	—
161	C₃×C₄	C₂×C₃×C₁₆²	1	64	yes	—
162	C₃×C₄	C₃×C₄×C₈×C₁₆	1	48	yes	—
163	C₃²	C₈×C₁₆²	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	C₃²×C₄×C₈²	1	1	yes	characteristic
165	C₂×C₄	C₂×C₃²×C₈×C₁₆	1	24	yes	—
166	C₂×C₄	C₃²×C₄²×C₁₆	1	12	yes	—
167	C₂×C₄	C₃²×C₄×C₈²	1	6	yes	—
168	C₈	C₃²×C₁₆²	1	64	yes	direct product factor
169	C₈	C₂×C₃²×C₈×C₁₆	1	48	yes	—
170	C₂×C₃	C₃×C₄×C₁₆²	1	16	yes	—
171	C₂×C₃	C₃×C₈²×C₁₆	1	12	yes	—
172	C₂²	C₃²×C₄×C₈×C₁₆	1	6	yes	—
173	C₂²	C₃²×C₈³	1	1	yes	characteristic
174	C₄	C₂×C₃²×C₁₆²	1	16	yes	—
175	C₄	C₃²×C₄×C₈×C₁₆	1	12	yes	—
176	C₃	C₃×C₈×C₁₆²	1	4	yes	direct product factor
177	C₂	C₃²×C₄×C₁₆²	1	4	yes	—
178	C₂	C₃²×C₈²×C₁₆	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈²	C₂³×C₃²
socle, prime radical	C₂³×C₃²	C₄×C₈²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₈×C₁₆²	C₃	direct (N × H)
15. C₃²×C₁₆²	C₈	direct (N × H)
16. C₈×C₁₆²	C₃²	direct (N × H)
29. C₃²×C₈×C₁₆	C₁₆	direct (N × H)
36. C₃×C₁₆²	C₃×C₈	direct (N × H)
56. C₃×C₈×C₁₆	C₃×C₁₆	direct (N × H)
72. C₁₆²	C₃²×C₈	direct (N × H)
89. C₃²×C₁₆	C₈×C₁₆	direct (N × H)
96. C₈×C₁₆	C₃²×C₁₆	direct (N × H)
111. C₃²×C₈	C₁₆²	direct (N × H)
129. C₃×C₁₆	C₃×C₈×C₁₆	direct (N × H)
147. C₃×C₈	C₃×C₁₆²	direct (N × H)
158. C₁₆	C₃²×C₈×C₁₆	direct (N × H)
163. C₃²	C₈×C₁₆²	direct (N × H)
168. C₈	C₃²×C₁₆²	direct (N × H)
176. C₃	C₃×C₈×C₁₆²	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×C₈²×C₁₆	2
C₃²×C₄×C₁₆²	2
C₃×C₈×C₁₆²	3

Subgroups — C32×C8×C162 #72196205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₈×C₁₆² #72196205