Subgroups — C₂³×C₃³×C₆₄ #72192002

Order 13824.

9016 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃³×C₆₄	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃²×C₆₄	C₃	1	13	yes	maximal, direct product factor
4	C₂³×C₃³×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₃³×C₃₂	C₄	1	7	yes	—
6	C₂²×C₃³×C₃₂	C₂²	1	7	yes	—
7	C₂×C₃³×C₆₄	C₂²	1	28	yes	direct product factor
8	C₂³×C₃²×C₃₂	C₂×C₃	1	13	yes	—
9	C₂²×C₃²×C₆₄	C₂×C₃	1	182	yes	direct product factor
10	C₂³×C₃×C₆₄	C₃²	1	13	yes	direct product factor
11	C₂³×C₃³×C₈	C₈	1	1	yes	characteristic
12	C₂²×C₃³×C₁₆	C₈	1	7	yes	—
13	C₂²×C₃³×C₁₆	C₂×C₄	1	7	yes	—
14	C₂×C₃³×C₃₂	C₂×C₄	1	21	yes	—
15	C₂×C₃³×C₃₂	C₂³	1	7	yes	—
16	C₃³×C₆₄	C₂³	1	8	yes	direct product factor
17	C₂³×C₃²×C₁₆	C₃×C₄	1	13	yes	—
18	C₂²×C₃²×C₃₂	C₃×C₄	1	91	yes	—
19	C₂²×C₃²×C₃₂	C₂²×C₃	1	91	yes	—
20	C₂×C₃²×C₆₄	C₂²×C₃	1	364	yes	direct product factor
21	C₂³×C₃³×C₄	C₁₆	1	1	yes	characteristic
22	C₂²×C₃³×C₈	C₁₆	1	7	yes	—
23	C₂²×C₃³×C₈	C₂×C₈	1	7	yes	—
24	C₂×C₃³×C₁₆	C₂×C₈	1	21	yes	—
25	C₂×C₃³×C₁₆	C₂²×C₄	1	7	yes	—
26	C₃³×C₃₂	C₂²×C₄	1	7	yes	—
27	C₃³×C₃₂	C₂⁴	1	1	yes	characteristic
28	C₂³×C₃×C₃₂	C₂×C₃²	1	13	yes	—
29	C₂²×C₃×C₆₄	C₂×C₃²	1	182	yes	direct product factor
30	C₂³×C₃²×C₈	C₃×C₈	1	13	yes	—
31	C₂²×C₃²×C₁₆	C₃×C₈	1	91	yes	—
32	C₂²×C₃²×C₁₆	C₂×C₃×C₄	1	91	yes	—
33	C₂×C₃²×C₃₂	C₂×C₃×C₄	1	273	yes	—
34	C₂×C₃²×C₃₂	C₂³×C₃	1	91	yes	—
35	C₃²×C₆₄	C₂³×C₃	1	104	yes	direct product factor
36	C₂³×C₆₄	C₃³	1	1	yes	characteristic, direct product factor, Sylow
37	C₂⁴×C₃³	C₃₂	1	1	yes	characteristic
38	C₂²×C₃³×C₄	C₃₂	1	7	yes	—
39	C₂²×C₃³×C₄	C₂×C₁₆	1	7	yes	—
40	C₂×C₃³×C₈	C₂×C₁₆	1	21	yes	—
41	C₂×C₃³×C₈	C₂²×C₈	1	7	yes	—
42	C₃³×C₁₆	C₂²×C₈	1	7	yes	—
43	C₃³×C₁₆	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₃×C₁₆	C₃²×C₄	1	13	yes	—
45	C₂²×C₃×C₃₂	C₃²×C₄	1	91	yes	—
46	C₂²×C₃×C₃₂	C₂²×C₃²	1	91	yes	—
47	C₂×C₃×C₆₄	C₂²×C₃²	1	364	yes	direct product factor
48	C₂³×C₃²×C₄	C₃×C₁₆	1	13	yes	—
49	C₂²×C₃²×C₈	C₃×C₁₆	1	91	yes	—
50	C₂²×C₃²×C₈	C₂×C₃×C₈	1	91	yes	—
51	C₂×C₃²×C₁₆	C₂×C₃×C₈	1	273	yes	—
52	C₂×C₃²×C₁₆	C₂²×C₃×C₄	1	91	yes	—
53	C₃²×C₃₂	C₂²×C₃×C₄	1	91	yes	—
54	C₃²×C₃₂	C₂⁴×C₃	1	13	yes	—
55	C₂³×C₃₂	C₂×C₃³	1	1	yes	characteristic
56	C₂²×C₆₄	C₂×C₃³	1	14	yes	direct product factor
57	C₂³×C₃³	C₆₄	1	8	yes	direct product factor
58	C₂³×C₃³	C₂×C₃₂	1	7	yes	—
59	C₂×C₃³×C₄	C₂×C₃₂	1	21	yes	—
60	C₂×C₃³×C₄	C₂²×C₁₆	1	7	yes	—
61	C₃³×C₈	C₂²×C₁₆	1	7	yes	—
62	C₃³×C₈	C₂³×C₈	1	1	yes	characteristic
63	C₂³×C₃×C₈	C₃²×C₈	1	13	yes	—
64	C₂²×C₃×C₁₆	C₃²×C₈	1	91	yes	—
65	C₂²×C₃×C₁₆	C₂×C₃²×C₄	1	91	yes	—
66	C₂×C₃×C₃₂	C₂×C₃²×C₄	1	273	yes	—
67	C₂×C₃×C₃₂	C₂³×C₃²	1	91	yes	—
68	C₃×C₆₄	C₂³×C₃²	1	104	yes	direct product factor
69	C₂⁴×C₃²	C₃×C₃₂	1	13	yes	—
70	C₂²×C₃²×C₄	C₃×C₃₂	1	91	yes	—
71	C₂²×C₃²×C₄	C₂×C₃×C₁₆	1	91	yes	—
72	C₂×C₃²×C₈	C₂×C₃×C₁₆	1	273	yes	—
73	C₂×C₃²×C₈	C₂²×C₃×C₈	1	91	yes	—
74	C₃²×C₁₆	C₂²×C₃×C₈	1	91	yes	—
75	C₃²×C₁₆	C₂³×C₃×C₄	1	13	yes	—
76	C₂³×C₁₆	C₃³×C₄	1	1	yes	characteristic
77	C₂²×C₃₂	C₃³×C₄	1	7	yes	—
78	C₂²×C₃₂	C₂²×C₃³	1	7	yes	—
79	C₂×C₆₄	C₂²×C₃³	1	28	yes	direct product factor
80	C₂²×C₃³	C₂×C₆₄	1	28	yes	direct product factor
81	C₂²×C₃³	C₂²×C₃₂	1	7	yes	—
82	C₃³×C₄	C₂²×C₃₂	1	7	yes	—
83	C₃³×C₄	C₂³×C₁₆	1	1	yes	characteristic
84	C₂³×C₃×C₄	C₃²×C₁₆	1	13	yes	—
85	C₂²×C₃×C₈	C₃²×C₁₆	1	91	yes	—
86	C₂²×C₃×C₈	C₂×C₃²×C₈	1	91	yes	—
87	C₂×C₃×C₁₆	C₂×C₃²×C₈	1	273	yes	—
88	C₂×C₃×C₁₆	C₂²×C₃²×C₄	1	91	yes	—
89	C₃×C₃₂	C₂²×C₃²×C₄	1	91	yes	—
90	C₃×C₃₂	C₂⁴×C₃²	1	13	yes	—
91	C₂³×C₃²	C₃×C₆₄	1	104	yes	direct product factor
92	C₂³×C₃²	C₂×C₃×C₃₂	1	91	yes	—
93	C₂×C₃²×C₄	C₂×C₃×C₃₂	1	273	yes	—
94	C₂×C₃²×C₄	C₂²×C₃×C₁₆	1	91	yes	—
95	C₃²×C₈	C₂²×C₃×C₁₆	1	91	yes	—
96	C₃²×C₈	C₂³×C₃×C₈	1	13	yes	—
97	C₂³×C₈	C₃³×C₈	1	1	yes	characteristic
98	C₂²×C₁₆	C₃³×C₈	1	7	yes	—
99	C₂²×C₁₆	C₂×C₃³×C₄	1	7	yes	—
100	C₂×C₃₂	C₂×C₃³×C₄	1	21	yes	—
101	C₂×C₃₂	C₂³×C₃³	1	7	yes	—
102	C₆₄	C₂³×C₃³	1	8	yes	direct product factor
103	C₂×C₃³	C₂²×C₆₄	1	14	yes	direct product factor
104	C₂×C₃³	C₂³×C₃₂	1	1	yes	characteristic
105	C₂⁴×C₃	C₃²×C₃₂	1	13	yes	—
106	C₂²×C₃×C₄	C₃²×C₃₂	1	91	yes	—
107	C₂²×C₃×C₄	C₂×C₃²×C₁₆	1	91	yes	—
108	C₂×C₃×C₈	C₂×C₃²×C₁₆	1	273	yes	—
109	C₂×C₃×C₈	C₂²×C₃²×C₈	1	91	yes	—
110	C₃×C₁₆	C₂²×C₃²×C₈	1	91	yes	—
111	C₃×C₁₆	C₂³×C₃²×C₄	1	13	yes	—
112	C₂²×C₃²	C₂×C₃×C₆₄	1	364	yes	direct product factor
113	C₂²×C₃²	C₂²×C₃×C₃₂	1	91	yes	—
114	C₃²×C₄	C₂²×C₃×C₃₂	1	91	yes	—
115	C₃²×C₄	C₂³×C₃×C₁₆	1	13	yes	—
116	C₂³×C₄	C₃³×C₁₆	1	1	yes	characteristic
117	C₂²×C₈	C₃³×C₁₆	1	7	yes	—
118	C₂²×C₈	C₂×C₃³×C₈	1	7	yes	—
119	C₂×C₁₆	C₂×C₃³×C₈	1	21	yes	—
120	C₂×C₁₆	C₂²×C₃³×C₄	1	7	yes	—
121	C₃₂	C₂²×C₃³×C₄	1	7	yes	—
122	C₃₂	C₂⁴×C₃³	1	1	yes	characteristic
123	C₃³	C₂³×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₃	C₃²×C₆₄	1	104	yes	direct product factor
125	C₂³×C₃	C₂×C₃²×C₃₂	1	91	yes	—
126	C₂×C₃×C₄	C₂×C₃²×C₃₂	1	273	yes	—
127	C₂×C₃×C₄	C₂²×C₃²×C₁₆	1	91	yes	—
128	C₃×C₈	C₂²×C₃²×C₁₆	1	91	yes	—
129	C₃×C₈	C₂³×C₃²×C₈	1	13	yes	—
130	C₂×C₃²	C₂²×C₃×C₆₄	1	182	yes	direct product factor
131	C₂×C₃²	C₂³×C₃×C₃₂	1	13	yes	—
132	C₂⁴	C₃³×C₃₂	1	1	yes	characteristic
133	C₂²×C₄	C₃³×C₃₂	1	7	yes	—
134	C₂²×C₄	C₂×C₃³×C₁₆	1	7	yes	—
135	C₂×C₈	C₂×C₃³×C₁₆	1	21	yes	—
136	C₂×C₈	C₂²×C₃³×C₈	1	7	yes	—
137	C₁₆	C₂²×C₃³×C₈	1	7	yes	—
138	C₁₆	C₂³×C₃³×C₄	1	1	yes	characteristic
139	C₂²×C₃	C₂×C₃²×C₆₄	1	364	yes	direct product factor
140	C₂²×C₃	C₂²×C₃²×C₃₂	1	91	yes	—
141	C₃×C₄	C₂²×C₃²×C₃₂	1	91	yes	—
142	C₃×C₄	C₂³×C₃²×C₁₆	1	13	yes	—
143	C₃²	C₂³×C₃×C₆₄	1	13	yes	direct product factor
144	C₂³	C₃³×C₆₄	1	8	yes	direct product factor
145	C₂³	C₂×C₃³×C₃₂	1	7	yes	—
146	C₂×C₄	C₂×C₃³×C₃₂	1	21	yes	—
147	C₂×C₄	C₂²×C₃³×C₁₆	1	7	yes	—
148	C₈	C₂²×C₃³×C₁₆	1	7	yes	—
149	C₈	C₂³×C₃³×C₈	1	1	yes	characteristic
150	C₂×C₃	C₂²×C₃²×C₆₄	1	182	yes	direct product factor
151	C₂×C₃	C₂³×C₃²×C₃₂	1	13	yes	—
152	C₂²	C₂×C₃³×C₆₄	1	28	yes	direct product factor
153	C₂²	C₂²×C₃³×C₃₂	1	7	yes	—
154	C₄	C₂²×C₃³×C₃₂	1	7	yes	—
155	C₄	C₂³×C₃³×C₁₆	1	1	yes	characteristic
156	C₃	C₂³×C₃²×C₆₄	1	13	yes	direct product factor
157	C₂	C₂²×C₃³×C₆₄	1	14	yes	direct product factor
158	C₂	C₂³×C₃³×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂⁴×C₃³
socle, prime radical	C₂⁴×C₃³	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃³×C₆₄	C₂	direct (N × H)
3. C₂³×C₃²×C₆₄	C₃	direct (N × H)
7. C₂×C₃³×C₆₄	C₂²	direct (N × H)
9. C₂²×C₃²×C₆₄	C₂×C₃	direct (N × H)
10. C₂³×C₃×C₆₄	C₃²	direct (N × H)
16. C₃³×C₆₄	C₂³	direct (N × H)
20. C₂×C₃²×C₆₄	C₂²×C₃	direct (N × H)
29. C₂²×C₃×C₆₄	C₂×C₃²	direct (N × H)
35. C₃²×C₆₄	C₂³×C₃	direct (N × H)
36. C₂³×C₆₄	C₃³	direct (N × H)
47. C₂×C₃×C₆₄	C₂²×C₃²	direct (N × H)
56. C₂²×C₆₄	C₂×C₃³	direct (N × H)
57. C₂³×C₃³	C₆₄	direct (N × H)
68. C₃×C₆₄	C₂³×C₃²	direct (N × H)
79. C₂×C₆₄	C₂²×C₃³	direct (N × H)
80. C₂²×C₃³	C₂×C₆₄	direct (N × H)
91. C₂³×C₃²	C₃×C₆₄	direct (N × H)
102. C₆₄	C₂³×C₃³	direct (N × H)
103. C₂×C₃³	C₂²×C₆₄	direct (N × H)
112. C₂²×C₃²	C₂×C₃×C₆₄	direct (N × H)
123. C₃³	C₂³×C₆₄	direct (N × H)
124. C₂³×C₃	C₃²×C₆₄	direct (N × H)
130. C₂×C₃²	C₂²×C₃×C₆₄	direct (N × H)
139. C₂²×C₃	C₂×C₃²×C₆₄	direct (N × H)
143. C₃²	C₂³×C₃×C₆₄	direct (N × H)
144. C₂³	C₃³×C₆₄	direct (N × H)
150. C₂×C₃	C₂²×C₃²×C₆₄	direct (N × H)
152. C₂²	C₂×C₃³×C₆₄	direct (N × H)
156. C₃	C₂³×C₃²×C₆₄	direct (N × H)
157. C₂	C₂²×C₃³×C₆₄	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×C₃³×C₃₂	2
C₂²×C₃³×C₆₄	2
C₂³×C₃²×C₆₄	3

Subgroups — C23×C33×C64 #72192002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃³×C₆₄ #72192002