Subgroups — C₂²×C₃×C₃⋊(C₃⋊C₁₆) #13163134

Order 1728.

892 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊(C₃⋊C₁₆)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊(C₃⋊C₁₆)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×C₃⋊C₁₆	—	3	4	no	maximal
5	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
6	C₂×C₃³×C₈	C₄	1	3	yes	—
7	C₂×C₃³×C₈	C₂²	1	3	yes	—
8	C₃×C₃⋊(C₃⋊C₁₆)	C₂²	1	4	yes	direct product factor
9	C₂²×C₃²×C₈	—	2	4	no	—
10	C₂²×C₃²×C₈	D₃	1	4	yes	—
11	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₃	1	6	yes	direct product factor
13	C₂×C₃×C₃⋊C₁₆	—	3	24	no	—
14	C₂³×C₃³	C₈	1	1	yes	characteristic
15	C₂×C₃³×C₄	C₈	1	3	yes	—
16	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
17	C₃³×C₈	C₂×C₄	1	3	yes	—
18	C₃³×C₈	C₂³	1	1	yes	characteristic
19	C₂²×C₃×C₁₆	—	9	1	no	—
20	C₂²×C₃⋊C₁₆	—	3	4	no	—
21	C₂²×C₃²×C₄	—	2	4	no	—
22	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
23	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	—	2	12	no	—
25	C₂×C₃²×C₈	—	2	12	no	—
26	C₂×C₃²×C₈	Dic₃	1	12	yes	—
27	C₂×C₃²×C₈	C₃×C₄	1	3	yes	—
28	C₂×C₃²×C₈	C₂×D₃	1	12	yes	—
29	C₂×C₃²×C₈	C₂²×C₃	1	3	yes	—
30	C₃⋊(C₃⋊C₁₆)	C₂²×C₃	1	4	yes	direct product factor
31	C₃×C₃⋊C₁₆	—	3	16	no	—
32	C₂²×C₃³	C₁₆	1	4	yes	complemented
33	C₂²×C₃³	C₂×C₈	1	3	yes	—
34	C₃³×C₄	C₂×C₈	1	3	yes	—
35	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	—	2	4	no	—
37	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
38	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
39	C₂×C₃×C₁₆	—	9	6	no	—
40	C₂×C₃⋊C₁₆	—	3	24	no	—
41	C₂³×C₃²	—	2	4	no	—
42	C₂³×C₃²	C₃⋊C₈	1	4	yes	—
43	C₂³×C₃²	C₃×C₈	1	1	yes	characteristic
44	C₂×C₃²×C₄	—	2	12	no	—
45	C₂×C₃²×C₄	—	2	12	no	—
46	C₂×C₃²×C₄	C₃⋊C₈	1	12	yes	—
47	C₂×C₃²×C₄	C₃×C₈	1	3	yes	—
48	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
49	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
50	C₃²×C₈	—	2	12	no	—
51	C₃²×C₈	—	2	4	no	—
52	C₃²×C₈	C₂×Dic₃	1	12	yes	—
53	C₃²×C₈	C₂×C₃×C₄	1	3	yes	—
54	C₃²×C₈	C₂²×D₃	1	4	yes	—
55	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
56	C₂²×C₁₆	—	9	1	no	Sylow
57	C₂×C₃³	C₂×C₁₆	1	6	yes	complemented
58	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
59	C₂²×C₃×C₄	—	2	4	no	—
60	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
61	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
62	C₂×C₃×C₈	—	2	12	no	—
63	C₂×C₃×C₈	—	2	12	no	—
64	C₂×C₃×C₈	C₃×Dic₃	1	12	yes	—
65	C₂×C₃×C₈	C₃⋊Dic₃	1	3	yes	—
66	C₂×C₃×C₈	C₂×C₃×D₃	1	12	yes	—
67	C₂×C₃×C₈	C₂×C₃⋊D₃	1	3	yes	—
68	C₃×C₁₆	—	9	4	no	—
69	C₃⋊C₁₆	—	3	16	no	—
70	C₂²×C₃²	—	2	16	no	—
71	C₂²×C₃²	—	2	12	no	—
72	C₂²×C₃²	C₃⋊C₁₆	1	16	yes	complemented
73	C₂²×C₃²	C₃×C₁₆	1	4	yes	complemented
74	C₂²×C₃²	C₂×C₃⋊C₈	1	12	yes	—
75	C₂²×C₃²	C₂×C₃×C₈	1	3	yes	—
76	C₃²×C₄	—	2	12	no	—
77	C₃²×C₄	—	2	4	no	—
78	C₃²×C₄	C₂×C₃⋊C₈	1	12	yes	—
79	C₃²×C₄	C₂×C₃×C₈	1	3	yes	—
80	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
81	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
83	C₂×C₁₆	—	9	6	no	—
84	C₃³	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₃	—	2	4	no	—
86	C₂³×C₃	C₃×C₃⋊C₈	1	4	yes	—
87	C₂³×C₃	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	12	no	—
89	C₂×C₃×C₄	—	2	12	no	—
90	C₂×C₃×C₄	C₃×C₃⋊C₈	1	12	yes	—
91	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	3	yes	—
92	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
93	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
94	C₃×C₈	—	2	12	no	—
95	C₃×C₈	—	2	4	no	—
96	C₃×C₈	C₂×C₃×Dic₃	1	12	yes	—
97	C₃×C₈	C₂×C₃⋊Dic₃	1	3	yes	—
98	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
99	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
100	C₂×C₃²	—	2	4	no	—
101	C₂×C₃²	—	2	24	no	—
102	C₂×C₃²	C₂×C₃⋊C₁₆	1	24	yes	complemented
103	C₂×C₃²	C₂×C₃×C₁₆	1	6	yes	complemented
104	C₂×C₃²	C₂²×C₃⋊C₈	1	4	yes	—
105	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
106	C₂²×C₄	C₃×C₃⋊Dic₃	1	1	yes	characteristic
107	C₂×C₈	C₃×C₃⋊Dic₃	1	3	yes	—
108	C₂×C₈	C₂×C₃×C₃⋊D₃	1	3	yes	—
109	C₁₆	—	9	4	no	—
110	C₂²×C₃	—	2	12	no	—
111	C₂²×C₃	—	2	16	no	—
112	C₂²×C₃	C₃×C₃⋊C₁₆	1	16	yes	complemented
113	C₂²×C₃	C₃⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
114	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	12	yes	—
115	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
116	C₃×C₄	—	2	12	no	—
117	C₃×C₄	—	2	4	no	—
118	C₃×C₄	C₂×C₃×C₃⋊C₈	1	12	yes	—
119	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
120	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
121	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
122	C₃²	—	2	4	no	—
123	C₃²	C₂²×C₃⋊C₁₆	1	4	yes	complemented
124	C₃²	C₂²×C₃×C₁₆	1	1	yes	characteristic, complemented
125	C₂³	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
126	C₂×C₄	C₃×C₃⋊(C₃⋊C₈)	1	3	yes	—
127	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	3	yes	—
128	C₈	C₂×C₃×C₃⋊Dic₃	1	3	yes	—
129	C₈	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
130	C₂×C₃	—	2	4	no	—
131	C₂×C₃	—	2	24	no	—
132	C₂×C₃	C₂×C₃×C₃⋊C₁₆	1	24	yes	complemented
133	C₂×C₃	C₂×C₃⋊(C₃⋊C₁₆)	1	6	yes	direct product factor
134	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	4	yes	—
135	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
136	C₂²	C₃×C₃⋊(C₃⋊C₁₆)	1	4	yes	direct product factor
137	C₂²	C₂×C₃×C₃⋊(C₃⋊C₈)	1	3	yes	—
138	C₄	C₂×C₃×C₃⋊(C₃⋊C₈)	1	3	yes	—
139	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
140	C₃	—	2	4	no	—
141	C₃	C₂²×C₃×C₃⋊C₁₆	1	4	yes	complemented
142	C₃	C₂²×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic, direct product factor
143	C₂	C₂×C₃×C₃⋊(C₃⋊C₁₆)	1	6	yes	direct product factor
144	C₂	C₂²×C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₂²×C₃×C₁₆
Frattini	C₈	C₂²×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	C₈
cosocle	C₃²×C₈	C₂³×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊(C₃⋊C₁₆)	C₂	direct (N × H)
3. C₂²×C₃⋊(C₃⋊C₁₆)	C₃	direct (N × H)
8. C₃×C₃⋊(C₃⋊C₁₆)	C₂²	direct (N × H)
12. C₂×C₃⋊(C₃⋊C₁₆)	C₂×C₃	direct (N × H)
30. C₃⋊(C₃⋊C₁₆)	C₂²×C₃	direct (N × H)
32. C₂²×C₃³	C₁₆	semidirect
57. C₂×C₃³	C₂×C₁₆	semidirect
72. C₂²×C₃²	C₃⋊C₁₆	semidirect
73. C₂²×C₃²	C₃×C₁₆	semidirect
84. C₃³	C₂²×C₁₆	semidirect
102. C₂×C₃²	C₂×C₃⋊C₁₆	semidirect
103. C₂×C₃²	C₂×C₃×C₁₆	semidirect
112. C₂²×C₃	C₃×C₃⋊C₁₆	semidirect
113. C₂²×C₃	C₃⋊(C₃⋊C₁₆)	direct (N × H)
123. C₃²	C₂²×C₃⋊C₁₆	semidirect
124. C₃²	C₂²×C₃×C₁₆	semidirect
132. C₂×C₃	C₂×C₃×C₃⋊C₁₆	semidirect
133. C₂×C₃	C₂×C₃⋊(C₃⋊C₁₆)	direct (N × H)
136. C₂²	C₃×C₃⋊(C₃⋊C₁₆)	direct (N × H)
141. C₃	C₂²×C₃×C₃⋊C₁₆	semidirect
142. C₃	C₂²×C₃⋊(C₃⋊C₁₆)	direct (N × H)
143. C₂	C₂×C₃×C₃⋊(C₃⋊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
C₂²×C₃³×C₈	2
C₂×C₃×C₃⋊(C₃⋊C₁₆)	2
C₂²×C₃⋊(C₃⋊C₁₆)	3
C₂²×C₃×C₃⋊C₁₆	3

Subgroups — C22×C3×C3⋊(C3⋊C16) #13163134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₃⋊(C₃⋊C₁₆) #13163134