Subgroups — C₉×#4682 #13141406

Order 1728.

338 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#869	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#818	C₂	1	4	yes	maximal
4	C₃×#4682	C₃	1	1	yes	maximal, characteristic
5	C₉×#500	—	3	1	no	maximal
6	C₂×C₉×Dic₆	—	2	2	no	—
7	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
8	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₉×C₄⋊Dic₃	—	2	2	no	—
10	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₃×#869	C₂×C₃	1	2	yes	—
12	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
13	C₃×#818	C₂×C₃	1	4	yes	—
14	C₉×C₄⋊Q₈	—	3	2	no	—
15	C₉×#107	—	3	4	no	—
16	C₄×C₈×C₉	—	2	1	no	—
17	C₄×C₈×C₉	D₃	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
21	C₂×C₉×Dic₃	—	2	4	no	—
22	C₉×Dic₆	—	4	4	no	—
23	C₃×C₈×C₉	Q₈	1	4	yes	complemented
24	C₃×#500	—	3	1	no	—
25	#4682	C₉	1	1	yes	characteristic, direct product factor
26	C₂×C₃×Dic₆	—	2	2	no	—
27	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
28	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
29	C₃×C₄⋊Dic₃	—	2	2	no	—
30	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
31	C₂×Q₈×C₉	—	6	2	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
34	C₉×C₄⋊C₄	—	3	4	no	—
35	C₉×C₄⋊C₄	—	6	2	no	—
36	C₄²×C₉	—	2	1	no	—
37	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
39	C₃×C₄×C₉	D₈	1	2	yes	—
40	C₃×C₄×C₉	Q₁₆	1	2	yes	—
41	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
42	C₉×Dic₃	—	4	4	no	—
43	C₃×C₄⋊Q₈	—	3	2	no	—
44	#869	C₂×C₉	1	2	yes	—
45	C₃×#107	—	3	4	no	—
46	C₃×C₄×C₈	—	2	1	no	—
47	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic
48	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
49	#818	C₂×C₉	1	4	yes	—
50	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
53	C₂×C₃×Dic₃	—	2	4	no	—
54	C₃×Dic₆	—	4	4	no	—
55	C₃²×C₈	C₃×Q₈	1	4	yes	—
56	Q₈×C₉	—	12	4	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	—	2	1	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	6	4	no	—
61	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
62	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
64	C₈×C₉	—	2	4	no	—
65	C₈×C₉	Dic₆	1	4	yes	complemented
66	#500	—	3	1	no	Sylow
67	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₃×Q₈	—	6	2	no	—
71	C₂×Dic₆	—	2	2	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
74	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
75	C₃×C₄⋊C₄	—	3	4	no	—
76	C₃×C₄⋊C₄	—	6	2	no	—
77	C₃×C₄²	—	2	1	no	—
78	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
79	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
80	C₄⋊Dic₃	—	2	2	no	—
81	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
82	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
83	C₃²×C₄	C₃×D₈	1	2	yes	—
84	C₃²×C₄	C₃×Q₁₆	1	2	yes	—
85	C₃²×C₄	C₂×C₃×Q₈	1	2	yes	—
86	C₃×Dic₃	—	4	4	no	—
87	C₂²×C₉	—	2	1	no	—
88	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
89	C₄×C₉	—	2	2	no	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	—	2	2	no	—
92	C₄×C₉	—	12	4	no	—
93	C₄×C₉	D₂₄	1	2	yes	—
94	C₄×C₉	Dic₁₂	1	2	yes	—
95	C₄×C₉	C₂×Dic₆	1	2	yes	—
96	C₄⋊Q₈	—	3	2	no	—
97	#107	—	3	4	no	—
98	C₄×C₈	D₃×C₉	1	1	yes	characteristic
99	C₃×C₉	#500	1	1	yes	characteristic, complemented, Sylow
100	C₃×Q₈	—	12	4	no	—
101	C₂×C₃×C₄	—	2	1	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	6	4	no	—
105	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
106	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
108	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
109	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
111	C₂×Dic₃	—	2	4	no	—
112	Dic₆	—	4	4	no	—
113	C₃×C₈	—	2	4	no	—
114	C₃×C₈	Q₈×C₉	1	4	yes	complemented
115	C₃×C₈	C₃×Dic₆	1	4	yes	—
116	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
118	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
119	C₂×C₉	—	2	1	no	—
120	C₂×C₉	—	2	1	no	—
121	C₂×C₉	—	2	1	no	—
122	C₂×C₉	#869	1	1	yes	characteristic
123	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
124	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
125	C₂×Q₈	—	6	2	no	—
126	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
127	C₄⋊C₄	—	3	4	no	—
128	C₄⋊C₄	—	6	2	no	—
129	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
130	C₂²×C₃	—	2	1	no	—
131	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
132	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
133	C₃×C₄	—	2	2	no	—
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	12	4	no	—
137	C₃×C₄	C₉×D₈	1	2	yes	—
138	C₃×C₄	C₉×Q₁₆	1	2	yes	—
139	C₃×C₄	C₂×Q₈×C₉	1	2	yes	—
140	C₃×C₄	C₃×D₂₄	1	2	yes	—
141	C₃×C₄	C₃×Dic₁₂	1	2	yes	—
142	C₃×C₄	C₂×C₃×Dic₆	1	2	yes	—
143	Dic₃	—	4	4	no	—
144	C₃²	C₃×#500	1	1	yes	characteristic
145	C₉	—	2	1	no	—
146	C₉	#4682	1	1	yes	characteristic, direct product factor
147	Q₈	—	12	4	no	—
148	C₂×C₄	—	6	4	no	—
149	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
150	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
152	C₈	C₉×Dic₆	1	4	yes	complemented
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	—	2	1	no	—
155	C₂×C₃	—	2	1	no	—
156	C₂×C₃	C₉×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
158	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
159	C₂×C₃	C₃×#869	1	1	yes	characteristic
160	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
161	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
162	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
163	C₄	—	12	4	no	—
164	C₄	C₉×D₂₄	1	2	yes	—
165	C₄	C₉×Dic₁₂	1	2	yes	—
166	C₄	C₂×C₉×Dic₆	1	2	yes	—
167	C₃	—	2	1	no	—
168	C₃	C₉×#500	1	1	yes	characteristic, complemented
169	C₃	C₃×#4682	1	1	yes	characteristic
170	C₂	C₉×#869	1	1	yes	characteristic
171	C₂	C₂×C₉×D₂₄	1	1	yes	characteristic
172	C₂	C₂×C₉×Dic₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₉×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
23. C₃×C₈×C₉	Q₈	semidirect
25. #4682	C₉	direct (N × H)
65. C₈×C₉	Dic₆	semidirect
99. C₃×C₉	#500	semidirect
114. C₃×C₈	Q₈×C₉	semidirect
146. C₉	#4682	direct (N × H)
152. C₈	C₉×Dic₆	semidirect
168. C₃	C₉×#500	semidirect

Maximal subgroups

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

subgroup	index
C₉×#869	2
C₃×C₄×C₈×C₉	2
C₉×#818	2
C₃×#4682	3
C₉×#500	3

Subgroups — C9×#4682 #13141406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4682 #13141406