Subgroups — C₂²×C₄×3₋¹⁺²⋊C₄ #13153561

Order 1728.

1612 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	C₂³×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂³×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×C₄×Dic₃	—	3	1	no	maximal
5	C₂²×C₄×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₂²×C₄×3₋¹⁺²	C₄	1	8	yes	complemented
8	C₂²×C₄×3₋¹⁺²	C₂²	1	6	yes	—
9	C₂²×3₋¹⁺²⋊C₄	C₄	1	16	yes	direct product factor
10	C₂²×3₋¹⁺²⋊C₄	C₂²	1	12	yes	—
11	C₄×3₋¹⁺²⋊C₄	C₂²	1	16	yes	direct product factor
12	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
13	C₂³×C₃×Dic₃	—	3	2	no	—
14	C₂×C₃×C₄×Dic₃	—	3	12	no	—
15	C₂³×C₄×C₉	—	2	1	no	—
16	C₂³×C₄×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂³×Dic₉	C₂×C₃	1	2	yes	—
18	C₂×C₄×Dic₉	C₂×C₃	1	12	yes	complemented
19	C₂³×3₋¹⁺²	C₂×C₄	1	8	yes	—
20	C₂³×3₋¹⁺²	C₂×C₄	1	4	yes	—
21	C₂³×3₋¹⁺²	C₂³	1	3	yes	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	24	yes	complemented
23	C₂×C₄×3₋¹⁺²	C₂³	1	4	yes	—
24	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	48	yes	direct product factor
25	C₂×3₋¹⁺²⋊C₄	C₂³	1	8	yes	—
26	C₂²×C₃×C₄²	—	9	1	no	—
27	C₂²×C₄×Dic₃	—	3	1	no	—
28	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃²×C₄	Dic₃	1	8	yes	—
30	C₂²×C₃²×C₄	C₂×D₃	1	6	yes	—
31	C₂²×C₃×Dic₃	—	3	16	no	—
32	C₂²×C₃×Dic₃	—	3	12	no	—
33	C₂⁴×C₉	—	2	1	no	—
34	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
35	C₃×C₄×Dic₃	—	3	16	no	—
36	C₂²×C₄×C₉	—	2	8	no	—
37	C₂²×C₄×C₉	—	2	6	no	—
38	C₂²×C₄×C₉	C₃×C₄	1	8	yes	complemented
39	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
40	C₂²×Dic₉	C₃×C₄	1	16	yes	complemented
41	C₂²×Dic₉	C₂²×C₃	1	12	yes	—
42	C₄×Dic₉	C₂²×C₃	1	16	yes	complemented
43	C₂²×3₋¹⁺²	C₄²	1	16	yes	complemented
44	C₂²×3₋¹⁺²	C₂²×C₄	1	12	yes	—
45	C₂²×3₋¹⁺²	C₂²×C₄	1	6	yes	—
46	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
47	C₄×3₋¹⁺²	C₂²×C₄	1	8	yes	complemented
48	3₋¹⁺²⋊C₄	C₂²×C₄	1	16	yes	direct product factor
49	C₂³×C₃×C₄	—	3	1	no	—
50	C₂³×C₃×C₄	—	9	2	no	—
51	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
52	C₂³×Dic₃	—	3	2	no	—
53	C₂×C₃×C₄²	—	9	12	no	—
54	C₂×C₄×Dic₃	—	3	12	no	—
55	C₂³×C₃²	C₄×D₃	1	8	yes	—
56	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
57	C₂³×C₃²	C₂²×D₃	1	3	yes	—
58	C₂×C₃²×C₄	C₂×Dic₃	1	24	yes	—
59	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
60	C₂×C₃×Dic₃	—	3	48	no	—
61	C₂×C₃×Dic₃	—	3	8	no	—
62	C₂³×C₉	—	2	8	no	—
63	C₂³×C₉	—	2	4	no	—
64	C₂³×C₉	—	2	3	no	—
65	C₂³×C₉	C₂×C₃×C₄	1	8	yes	—
66	C₂³×C₉	C₂×C₃×C₄	1	4	yes	—
67	C₂³×C₉	C₂³×C₃	1	3	yes	—
68	C₂×C₄×C₉	—	2	24	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	C₂×C₃×C₄	1	24	yes	complemented
71	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
72	C₂×Dic₉	C₂×C₃×C₄	1	48	yes	complemented
73	C₂×Dic₉	C₂³×C₃	1	8	yes	—
74	C₂²×C₄²	—	9	1	no	Sylow
75	C₂×3₋¹⁺²	C₂×C₄²	1	12	yes	complemented
76	C₂×3₋¹⁺²	C₂³×C₄	1	2	yes	—
77	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
78	C₂⁴×C₃	—	3	1	no	—
79	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	3	8	no	—
81	C₂²×C₃×C₄	—	3	6	no	—
82	C₂²×C₃×C₄	—	9	12	no	—
83	C₂²×C₃×C₄	—	9	16	no	—
84	C₂²×C₃×C₄	C₃×Dic₃	1	8	yes	—
85	C₂²×C₃×C₄	C₂×C₃×D₃	1	6	yes	—
86	C₂²×Dic₃	—	3	16	no	—
87	C₂²×Dic₃	—	3	12	no	—
88	C₃×C₄²	—	9	16	no	—
89	C₄×Dic₃	—	3	16	no	—
90	C₂²×C₃²	C₄×Dic₃	1	16	yes	—
91	C₂²×C₃²	C₂×C₄×D₃	1	12	yes	—
92	C₂²×C₃²	C₂²×Dic₃	1	6	yes	—
93	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
94	C₃²×C₄	C₂²×Dic₃	1	8	yes	—
95	C₃×Dic₃	—	3	16	no	—
96	C₂²×C₉	—	2	1	no	—
97	C₂²×C₉	—	2	16	no	—
98	C₂²×C₉	—	2	12	no	—
99	C₂²×C₉	—	2	6	no	—
100	C₂²×C₉	C₃×C₄²	1	16	yes	complemented
101	C₂²×C₉	C₂²×C₃×C₄	1	12	yes	—
102	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
103	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
104	C₄×C₉	—	2	8	no	—
105	C₄×C₉	C₂²×C₃×C₄	1	8	yes	complemented
106	Dic₉	C₂²×C₃×C₄	1	16	yes	complemented
107	C₂³×C₄	—	9	2	no	—
108	C₂³×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
109	C₂×C₄²	—	9	12	no	—
110	3₋¹⁺²	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₃	—	3	8	no	—
112	C₂³×C₃	—	3	4	no	—
113	C₂³×C₃	—	3	3	no	—
114	C₂³×C₃	C₃×C₄×D₃	1	8	yes	—
115	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
116	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
117	C₂×C₃×C₄	—	3	24	no	—
118	C₂×C₃×C₄	—	3	4	no	—
119	C₂×C₃×C₄	—	9	8	no	—
120	C₂×C₃×C₄	—	9	48	no	—
121	C₂×C₃×C₄	C₂×C₃×Dic₃	1	24	yes	—
122	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
123	C₂×Dic₃	—	3	8	no	—
124	C₂×Dic₃	—	3	48	no	—
125	C₂×C₃²	C₂×C₄×Dic₃	1	12	yes	—
126	C₂×C₃²	C₂²×C₄×D₃	1	2	yes	—
127	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
128	C₂×C₉	—	2	12	no	—
129	C₂×C₉	—	2	2	no	—
130	C₂×C₉	—	2	1	no	—
131	C₂×C₉	C₂×C₃×C₄²	1	12	yes	complemented
132	C₂×C₉	C₂³×C₃×C₄	1	2	yes	—
133	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
134	C₂⁴	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
135	C₂²×C₄	—	9	16	no	—
136	C₂²×C₄	—	9	12	no	—
137	C₂²×C₄	3₋¹⁺²⋊C₄	1	8	yes	direct product factor
138	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	6	yes	—
139	C₄²	—	9	16	no	—
140	C₂²×C₃	—	3	1	no	—
141	C₂²×C₃	—	3	12	no	—
142	C₂²×C₃	—	3	6	no	—
143	C₂²×C₃	—	3	16	no	—
144	C₂²×C₃	C₃×C₄×Dic₃	1	16	yes	—
145	C₂²×C₃	C₂×C₃×C₄×D₃	1	12	yes	—
146	C₂²×C₃	C₂²×C₃×Dic₃	1	6	yes	—
147	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
148	C₃×C₄	—	3	8	no	—
149	C₃×C₄	—	9	16	no	—
150	C₃×C₄	C₂²×C₃×Dic₃	1	8	yes	—
151	Dic₃	—	3	16	no	—
152	C₃²	C₂²×C₄×Dic₃	1	1	yes	characteristic
153	C₉	—	2	1	no	—
154	C₉	C₂²×C₃×C₄²	1	1	yes	characteristic, complemented
155	C₂³	C₄×AGL(1,ℤ/9)	1	8	yes	—
156	C₂³	C₂×3₋¹⁺²⋊C₄	1	4	yes	—
157	C₂³	C₂²×AGL(1,ℤ/9)	1	3	yes	—
158	C₂×C₄	—	9	8	no	—
159	C₂×C₄	—	9	48	no	—
160	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	24	yes	direct product factor
161	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	4	yes	—
162	C₂×C₃	—	3	2	no	—
163	C₂×C₃	—	3	12	no	—
164	C₂×C₃	—	3	1	no	—
165	C₂×C₃	C₂×C₃×C₄×Dic₃	1	12	yes	—
166	C₂×C₃	C₂²×C₃×C₄×D₃	1	2	yes	—
167	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
168	C₂²	C₄×3₋¹⁺²⋊C₄	1	16	yes	direct product factor
169	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	12	yes	—
170	C₂²	C₂²×3₋¹⁺²⋊C₄	1	6	yes	—
171	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
172	C₄	—	9	16	no	—
173	C₄	C₂²×3₋¹⁺²⋊C₄	1	8	yes	direct product factor
174	C₃	—	3	1	no	—
175	C₃	C₂²×C₃×C₄×Dic₃	1	1	yes	characteristic
176	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	12	yes	direct product factor
177	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	2	yes	—
178	C₂	C₂³×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂²×C₃×C₄²
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂³×C₄×3₋¹⁺²	C₂
socle	C₂⁴×C₃	C₂×C₃×D₃
prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂²×C₉	C₂⁴×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×3₋¹⁺²⋊C₄	C₂	direct (N × H)
5. C₂²×C₄×Dic₉	C₃	semidirect
7. C₂²×C₄×3₋¹⁺²	C₄	semidirect
9. C₂²×3₋¹⁺²⋊C₄	C₄	direct (N × H)
11. C₄×3₋¹⁺²⋊C₄	C₂²	direct (N × H)
18. C₂×C₄×Dic₉	C₂×C₃	semidirect
22. C₂×C₄×3₋¹⁺²	C₂×C₄	semidirect
24. C₂×3₋¹⁺²⋊C₄	C₂×C₄	direct (N × H)
38. C₂²×C₄×C₉	C₃×C₄	semidirect
40. C₂²×Dic₉	C₃×C₄	semidirect
42. C₄×Dic₉	C₂²×C₃	semidirect
43. C₂²×3₋¹⁺²	C₄²	semidirect
47. C₄×3₋¹⁺²	C₂²×C₄	semidirect
48. 3₋¹⁺²⋊C₄	C₂²×C₄	direct (N × H)
70. C₂×C₄×C₉	C₂×C₃×C₄	semidirect
72. C₂×Dic₉	C₂×C₃×C₄	semidirect
75. C₂×3₋¹⁺²	C₂×C₄²	semidirect
100. C₂²×C₉	C₃×C₄²	semidirect
105. C₄×C₉	C₂²×C₃×C₄	semidirect
106. Dic₉	C₂²×C₃×C₄	semidirect
110. 3₋¹⁺²	C₂²×C₄²	semidirect
131. C₂×C₉	C₂×C₃×C₄²	semidirect
137. C₂²×C₄	3₋¹⁺²⋊C₄	direct (N × H)
154. C₉	C₂²×C₃×C₄²	semidirect
160. C₂×C₄	C₂×3₋¹⁺²⋊C₄	direct (N × H)
168. C₂²	C₄×3₋¹⁺²⋊C₄	direct (N × H)
173. C₄	C₂²×3₋¹⁺²⋊C₄	direct (N × H)
176. C₂	C₂×C₄×3₋¹⁺²⋊C₄	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×C₄×3₋¹⁺²	2
C₂³×3₋¹⁺²⋊C₄	2
C₂×C₄×3₋¹⁺²⋊C₄	2
C₂²×C₃×C₄×Dic₃	3
C₂²×C₄×Dic₉	3

Subgroups — C22×C4×3−1+2⋊C4 #13153561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×3₋¹⁺²⋊C₄ #13153561