Subgroups — C₉×#5812 #13153639

Order 1728.

954 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₉×Dic₃	C₂	1	8	yes	maximal, complemented
3	C₉×#926	C₂	1	6	yes	maximal
4	C₃×#5812	C₃	1	1	yes	maximal, characteristic
5	C₉×#519	—	3	1	no	maximal
6	C₂×C₃×Q₈×C₉	C₄	1	8	yes	complemented
7	C₂×C₃×Q₈×C₉	C₂²	1	4	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
9	C₉×#217	C₂²	1	4	yes	—
10	C₉×C₄⋊Dic₃	C₂²	1	12	yes	—
11	C₄×C₉×Dic₃	C₂²	1	12	yes	—
12	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
13	C₃×Q₈×Dic₃	C₂×C₃	1	8	yes	—
14	C₃×#926	C₂×C₃	1	6	yes	—
15	C₂²×Q₈×C₉	—	2	1	no	—
16	C₂²×Q₈×C₉	D₃	1	1	yes	characteristic
17	C₄×Q₈×C₉	—	3	8	no	—
18	C₉×#117	—	3	6	no	—
19	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
20	C₃×Q₈×C₉	C₂×C₄	1	16	yes	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	12	yes	—
22	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
23	C₂×C₉×Dic₃	C₂³	1	8	yes	—
24	C₃×#519	—	3	1	no	—
25	#5812	C₉	1	1	yes	characteristic, direct product factor
26	C₂×C₃²×Q₈	C₃×C₄	1	8	yes	—
27	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
29	C₃×#217	C₂²×C₃	1	4	yes	—
30	C₃×C₄⋊Dic₃	C₂²×C₃	1	12	yes	—
31	C₃×C₄×Dic₃	C₂²×C₃	1	12	yes	—
32	C₂×Q₈×C₉	—	2	8	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	Dic₃	1	8	yes	complemented
35	C₂×Q₈×C₉	C₂×D₃	1	4	yes	—
36	C₂²×C₄×C₉	—	2	3	no	—
37	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
38	C₉×C₄⋊C₄	—	3	12	no	—
39	C₉×C₂²⋊C₄	—	3	4	no	—
40	C₄²×C₉	—	3	12	no	—
41	C₂²×C₃×C₉	—	2	2	no	—
42	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
43	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
44	C₃×C₄×C₉	C₂²×C₄	1	12	yes	—
45	C₉×Dic₃	—	2	8	no	—
46	C₂²×C₃×Q₈	—	2	1	no	—
47	C₂²×C₃×Q₈	C₂×C₉	1	1	yes	characteristic
48	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
49	C₃×C₄×Q₈	—	3	8	no	—
50	C₃×#117	—	3	6	no	—
51	Q₈×Dic₃	C₂×C₉	1	8	yes	complemented
52	#926	C₂×C₉	1	6	yes	—
53	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
54	C₃²×Q₈	C₂×C₃×C₄	1	16	yes	—
55	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
56	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
57	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
58	C₂³×C₉	—	2	1	no	—
59	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
60	Q₈×C₉	—	2	16	no	—
61	Q₈×C₉	C₂×Dic₃	1	16	yes	—
62	C₂×C₄×C₉	—	2	12	no	—
63	C₂×C₄×C₉	—	2	6	no	—
64	C₂×C₄×C₉	—	3	8	no	—
65	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
66	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
67	#519	—	3	1	no	Sylow
68	C₂×C₃×C₉	—	2	2	no	—
69	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₃×C₉	2₋¹⁺⁴	1	2	yes	—
71	C₂×C₃×Q₈	—	2	8	no	—
72	C₂×C₃×Q₈	—	2	4	no	—
73	C₂×C₃×Q₈	C₄×C₉	1	8	yes	complemented
74	C₂×C₃×Q₈	C₂²×C₉	1	4	yes	—
75	C₂×C₃×Q₈	C₃×Dic₃	1	8	yes	—
76	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
77	C₂²×C₃×C₄	—	2	3	no	—
78	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
79	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
80	C₃×C₄⋊C₄	—	3	12	no	—
81	C₃×C₂²⋊C₄	—	3	4	no	—
82	C₃×C₄²	—	3	12	no	—
83	#217	C₂²×C₉	1	4	yes	—
84	C₄⋊Dic₃	C₂²×C₉	1	12	yes	—
85	C₄×Dic₃	C₂²×C₉	1	12	yes	—
86	C₂²×C₃²	—	2	2	no	—
87	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
88	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
89	C₃²×C₄	C₂²×C₃×C₄	1	12	yes	—
90	C₃×Dic₃	—	2	8	no	—
91	C₂²×C₉	—	2	4	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	—	2	2	no	—
95	C₂²×C₉	C₂²×Dic₃	1	2	yes	—
96	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
97	C₄×C₉	—	2	12	no	—
98	C₄×C₉	—	6	8	no	—
99	C₄×C₉	C₂²×Dic₃	1	12	yes	—
100	C₂²×Q₈	D₃×C₉	1	1	yes	characteristic
101	C₄×Q₈	—	3	8	no	—
102	#117	—	3	6	no	—
103	C₃×C₉	#519	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	2	1	no	—
105	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
106	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
107	C₃×Q₈	—	2	16	no	—
108	C₃×Q₈	C₂×C₄×C₉	1	16	yes	—
109	C₃×Q₈	C₂×C₃×Dic₃	1	16	yes	—
110	C₂×C₃×C₄	—	2	12	no	—
111	C₂×C₃×C₄	—	2	6	no	—
112	C₂×C₃×C₄	—	3	8	no	—
113	C₂×C₃×C₄	C₂×C₄×C₉	1	12	yes	—
114	C₂×C₃×C₄	C₂³×C₉	1	6	yes	—
115	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
116	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
117	C₂×Dic₃	C₂³×C₉	1	8	yes	—
118	C₂×C₃²	—	2	2	no	—
119	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
120	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
121	C₂×C₉	—	2	4	no	—
122	C₂×C₉	—	2	2	no	—
123	C₂×C₉	—	2	2	no	—
124	C₂×C₉	—	2	1	no	—
125	C₂×C₉	#1007	1	2	yes	—
126	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
127	C₂×Q₈	C₉×Dic₃	1	8	yes	complemented
128	C₂×Q₈	C₂×D₃×C₉	1	4	yes	—
129	C₂²×C₄	C₂×D₃×C₉	1	3	yes	—
130	C₄⋊C₄	—	3	12	no	—
131	C₂²⋊C₄	—	3	4	no	—
132	C₄²	—	3	12	no	—
133	C₂²×C₃	—	2	4	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	—	2	1	no	—
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	C₂²×C₄×C₉	1	2	yes	—
139	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
140	C₂²×C₃	C₂²×C₃×Dic₃	1	2	yes	—
141	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
142	C₃×C₄	—	2	12	no	—
143	C₃×C₄	—	6	8	no	—
144	C₃×C₄	C₂²×C₄×C₉	1	12	yes	—
145	C₃×C₄	C₂²×C₃×Dic₃	1	12	yes	—
146	Dic₃	—	2	8	no	—
147	C₃²	C₃×#519	1	1	yes	characteristic
148	C₉	—	2	1	no	—
149	C₉	#5812	1	1	yes	characteristic, direct product factor
150	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
151	Q₈	C₂×C₉×Dic₃	1	16	yes	—
152	C₂×C₄	—	3	8	no	—
153	C₂×C₄	C₂×C₉×Dic₃	1	12	yes	—
154	C₂×C₄	C₂²×D₃×C₉	1	6	yes	—
155	C₂×C₃	—	2	4	no	—
156	C₂×C₃	—	2	2	no	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	2	2	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
161	C₂×C₃	C₉×2₋¹⁺⁴	1	2	yes	—
162	C₂×C₃	C₃×#1007	1	2	yes	—
163	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
164	C₂²	—	2	2	no	—
165	C₂²	C₂²×C₉×Dic₃	1	2	yes	—
166	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
167	C₄	—	6	8	no	—
168	C₄	C₂²×C₉×Dic₃	1	12	yes	—
169	C₃	—	2	1	no	—
170	C₃	C₉×#519	1	1	yes	characteristic, complemented
171	C₃	C₃×#5812	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	C₉×#1007	1	2	yes	—
174	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂×C₃	C₂³×C₄×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂³×C₃²	C₂³×C₃
hypercenter	C₂²×Q₈×C₉	D₃
hypocenter	C₃	C₉×#519
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×Q₈×C₉

Complements

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

N	H ≅ G/N	type
2. Q₈×C₉×Dic₃	C₂	semidirect
6. C₂×C₃×Q₈×C₉	C₄	semidirect
25. #5812	C₉	direct (N × H)
34. C₂×Q₈×C₉	Dic₃	semidirect
51. Q₈×Dic₃	C₂×C₉	semidirect
73. C₂×C₃×Q₈	C₄×C₉	semidirect
103. C₃×C₉	#519	semidirect
127. C₂×Q₈	C₉×Dic₃	semidirect
149. C₉	#5812	direct (N × H)
170. C₃	C₉×#519	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×Q₈×C₉	2
Q₈×C₉×Dic₃	2
C₉×#926	2
C₃×#5812	3
C₉×#519	3

Subgroups — C9×#5812 #13153639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5812 #13153639