Subgroups — #13143741

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6226	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	#11720538	C₂	1	2	yes	maximal
4	#11720537	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	#10592378	—	3	1	no	maximal
7	#10588359	—	3	3	no	maximal
8	C₂×#6226	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₃⋊Dic₉)	C₄	1	1	yes	characteristic
13	#88534	C₄	1	2	yes	—
14	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
16	C₄×#634	—	3	1	no	—
17	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
18	#63642	—	3	2	no	—
19	#63641	—	3	2	no	—
20	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
21	C₄×Dic₁₈	—	3	3	no	—
22	C₄×C₈×C₉	D₃	1	3	yes	—
23	#63375	—	3	6	no	—
24	Dic₉⋊C₈	—	3	6	no	—
25	C₄⋊(C₉⋊C₈)	—	3	3	no	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
30	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
31	#6226	—	2	2	no	—
32	C₃×C₈×C₉	—	2	1	no	—
33	C₃×C₈×C₉	Q₈	1	2	yes	complemented
34	C₃⋊(C₉⋊C₈)	—	2	2	no	—
35	#4679	—	9	3	no	—
36	#4679	—	9	1	no	—
37	C₂×#634	—	3	1	no	—
38	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
42	#3750	—	3	2	no	—
43	C₄×C₃⋊Dic₃	—	3	2	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
45	C₂×Dic₁₈	—	3	3	no	—
46	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
47	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
48	C₄²×C₉	C₂×D₃	1	3	yes	—
49	C₄⋊Dic₉	—	3	3	no	—
50	Dic₉⋊C₄	—	3	6	no	—
51	C₄×Dic₉	—	3	6	no	—
52	C₂×C₉⋊C₈	—	3	6	no	—
53	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₄×C₉	—	2	1	no	—
55	C₃×C₄×C₉	M₁₆	1	2	yes	—
56	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
57	C₃×C₄×C₉	#41	1	1	yes	characteristic
58	C₃⋊Dic₉	—	2	2	no	—
59	C₃⋊Dic₉	—	2	2	no	—
60	C₄×Dic₆	—	9	3	no	—
61	C₄×Dic₆	—	9	1	no	—
62	C₃×C₄×C₈	D₉	1	3	yes	—
63	C₃×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
64	#815	—	9	6	no	—
65	#815	—	9	2	no	—
66	Dic₃⋊C₈	—	9	6	no	—
67	Dic₃⋊C₈	—	9	2	no	—
68	C₄⋊(C₃⋊C₈)	—	9	3	no	—
69	C₄⋊(C₃⋊C₈)	—	9	1	no	—
70	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₂×C₃⋊Dic₃	—	3	2	no	—
74	C₂×C₃⋊Dic₃	—	3	2	no	—
75	#634	—	6	2	no	—
76	C₃²×C₈	—	2	1	no	—
77	C₃²×C₈	Dic₆	1	2	yes	—
78	C₃⋊(C₃⋊C₈)	—	6	2	no	—
79	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
80	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
81	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
82	C₂×Dic₉	—	3	6	no	—
83	C₂×Dic₉	—	3	6	no	—
84	Dic₁₈	—	6	6	no	—
85	C₈×C₉	—	2	3	no	—
86	C₈×C₉	Dic₆	1	6	yes	complemented
87	C₉⋊C₈	—	6	6	no	—
88	#446	—	27	1	no	Sylow
89	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₃×C₉	#131	1	1	yes	characteristic
92	C₂×Dic₆	—	9	3	no	—
93	C₂×Dic₆	—	9	1	no	—
94	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
95	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
96	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₃×C₄²	C₂×D₉	1	3	yes	—
99	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₄⋊Dic₃	—	9	3	no	—
101	C₄⋊Dic₃	—	9	1	no	—
102	Dic₃⋊C₄	—	9	6	no	—
103	Dic₃⋊C₄	—	9	2	no	—
104	C₄×Dic₃	—	9	6	no	—
105	C₄×Dic₃	—	9	2	no	—
106	C₂×C₃⋊C₈	—	9	6	no	—
107	C₂×C₃⋊C₈	—	9	2	no	—
108	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
109	C₃²×C₄	—	2	1	no	—
110	C₃²×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
111	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
112	C₃²×C₄	#235	1	1	yes	characteristic
113	C₃⋊Dic₃	—	6	2	no	—
114	C₃⋊Dic₃	—	6	2	no	—
115	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
116	C₄×C₉	—	2	3	no	—
117	C₄×C₉	ΓL(1,𝔽₂₅)	1	6	yes	—
118	C₄×C₉	C₂×Dic₆	1	3	yes	—
119	C₄×C₉	#235	1	3	yes	—
120	Dic₉	—	6	6	no	—
121	Dic₉	—	6	6	no	—
122	C₄×Q₈	—	27	1	no	—
123	C₄⋊C₈	—	27	2	no	—
124	C₄⋊C₈	—	27	1	no	—
125	#97	—	27	2	no	—
126	C₄×C₈	C₃⋊D₉	1	1	yes	characteristic
127	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
129	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
130	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
131	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
134	C₂×Dic₃	—	9	6	no	—
135	C₂×Dic₃	—	9	6	no	—
136	C₂×Dic₃	—	9	2	no	—
137	C₂×Dic₃	—	9	2	no	—
138	Dic₆	—	18	6	no	—
139	Dic₆	—	18	2	no	—
140	C₃×C₈	—	2	3	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	Dic₁₈	1	6	yes	complemented
143	C₃×C₈	#634	1	2	yes	—
144	C₃⋊C₈	—	18	6	no	—
145	C₃⋊C₈	—	18	2	no	—
146	C₂×C₃²	C₄×Dic₆	1	1	yes	characteristic
147	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
148	C₂×C₃²	#901	1	1	yes	characteristic
149	C₂×C₉	C₄×Dic₆	1	3	yes	—
150	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	3	yes	—
151	C₂×C₉	#901	1	3	yes	—
152	C₂×Q₈	—	27	1	no	—
153	C₂×C₈	—	27	2	no	—
154	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
156	C₄⋊C₄	—	27	2	no	—
157	C₄⋊C₄	—	27	1	no	—
158	C₄²	—	27	2	no	—
159	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
160	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
161	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	3	no	—
164	C₃×C₄	#3663	1	6	yes	—
165	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
166	C₃×C₄	#3697	1	3	yes	—
167	C₃×C₄	#3743	1	2	yes	—
168	C₃×C₄	C₂×#634	1	1	yes	characteristic
169	C₃×C₄	#3828	1	1	yes	characteristic
170	Dic₃	—	18	2	no	—
171	Dic₃	—	18	2	no	—
172	Dic₃	—	18	6	no	—
173	Dic₃	—	18	6	no	—
174	C₃²	#4679	1	1	yes	characteristic
175	C₉	#4679	1	3	yes	complemented
176	Q₈	—	54	2	no	—
177	C₂×C₄	—	27	2	no	—
178	C₂×C₄	—	27	2	no	—
179	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
180	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	54	2	no	—
184	C₈	#6226	1	2	yes	complemented
185	C₂×C₃	C₄×Dic₁₈	1	3	yes	—
186	C₂×C₃	C₂×#3663	1	3	yes	—
187	C₂×C₃	#63464	1	3	yes	—
188	C₂×C₃	C₄×#634	1	1	yes	characteristic
189	C₂×C₃	C₂×#3743	1	1	yes	characteristic
190	C₂×C₃	#64110	1	1	yes	characteristic
191	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	54	2	no	—
194	C₄	—	54	2	no	—
195	C₄	#88523	1	2	yes	—
196	C₄	C₂×#6226	1	1	yes	characteristic
197	C₄	#88736	1	1	yes	characteristic
198	C₃	#10588359	1	3	yes	complemented
199	C₃	#10592378	1	1	yes	characteristic
200	C₂	C₄×#6226	1	1	yes	characteristic
201	C₂	C₂×#88523	1	1	yes	characteristic
202	C₂	#11721770	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×C₈	C₃⋊D₉
hypocenter	C₃×C₉	#446
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
33. C₃×C₈×C₉	Q₈	semidirect
86. C₈×C₉	Dic₆	semidirect
127. C₃×C₉	#446	semidirect
142. C₃×C₈	Dic₁₈	semidirect
175. C₉	#4679	semidirect
184. C₈	#6226	semidirect
198. C₃	#10588359	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6226	2
C₃×C₄×C₈×C₉	2
#11720538	2
#11720537	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592378	3
#10588359	3