Subgroups — #13144378

Order 1728.

480 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	#11720571	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	2	yes	maximal, complemented
4	#11720516	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592699	—	3	1	no	maximal
7	#10588678	—	3	3	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
15	C₃²×#117	D₃	1	1	yes	characteristic
16	#63657	—	3	2	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	2	no	—
18	#63630	—	3	1	no	—
19	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	C₉×#117	D₃	1	3	yes	—
21	#63389	—	3	6	no	—
22	C₄⋊(C₉⋊C₈)	—	3	6	no	—
23	#63362	—	3	3	no	—
24	C₄×C₉⋊C₈	—	3	3	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₃⋊(C₉⋊C₈)	—	2	2	no	—
32	C₃⋊(C₉⋊C₈)	—	2	2	no	—
33	#5000	—	9	3	no	—
34	#5000	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
37	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
38	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
39	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
42	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
43	C₉×C₄⋊C₄	Dic₃	1	6	yes	—
44	C₉×C₂²⋊C₄	Dic₃	1	6	yes	—
45	C₄²×C₉	C₂×D₃	1	3	yes	—
46	C₄²×C₉	C₂×D₃	1	3	yes	—
47	C₂×C₉⋊C₈	—	3	6	no	—
48	C₂×C₉⋊C₈	—	3	6	no	—
49	C₂²×C₃×C₉	—	2	1	no	—
50	C₂²×C₃×C₉	—	2	2	no	—
51	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₄×C₉	—	2	1	no	—
53	C₃×C₄×C₉	—	2	2	no	—
54	C₃×C₄×C₉	—	2	2	no	—
55	C₃×C₄×C₉	#41	1	2	yes	—
56	C₃×#117	D₉	1	3	yes	—
57	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
58	#830	—	9	6	no	—
59	#830	—	9	2	no	—
60	C₄⋊(C₃⋊C₈)	—	9	6	no	—
61	C₄⋊(C₃⋊C₈)	—	9	2	no	—
62	#803	—	9	3	no	—
63	#803	—	9	1	no	—
64	C₄×C₃⋊C₈	—	9	3	no	—
65	C₄×C₃⋊C₈	—	9	1	no	—
66	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
69	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
70	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂³×C₉	C₂×Dic₃	1	3	yes	—
73	C₃⋊(C₃⋊C₈)	—	6	2	no	—
74	C₃⋊(C₃⋊C₈)	—	6	2	no	—
75	C₂×C₄×C₉	—	2	6	no	—
76	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
77	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
78	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
79	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
80	C₉⋊C₈	—	6	6	no	—
81	C₉⋊C₈	—	6	6	no	—
82	#433	—	27	1	no	Sylow
83	C₂×C₃×C₉	—	4	1	no	—
84	C₂×C₃×C₉	#117	1	1	yes	characteristic
85	C₂×C₃×C₉	#131	1	2	yes	—
86	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
87	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
88	C₃×C₄⋊C₄	Dic₉	1	6	yes	—
89	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
90	C₃×C₂²⋊C₄	Dic₉	1	6	yes	—
91	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
92	C₃×C₄²	C₂×D₉	1	3	yes	—
93	C₃×C₄²	C₂×D₉	1	3	yes	—
94	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
96	C₂×C₃⋊C₈	—	9	6	no	—
97	C₂×C₃⋊C₈	—	9	6	no	—
98	C₂×C₃⋊C₈	—	9	2	no	—
99	C₂×C₃⋊C₈	—	9	2	no	—
100	C₂²×C₃²	—	2	1	no	—
101	C₂²×C₃²	—	2	2	no	—
102	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
103	C₃²×C₄	—	2	1	no	—
104	C₃²×C₄	—	2	2	no	—
105	C₃²×C₄	—	2	2	no	—
106	C₃²×C₄	#235	1	2	yes	—
107	C₂²×C₉	—	2	3	no	—
108	C₂²×C₉	—	2	6	no	—
109	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
110	C₄×C₉	—	2	3	no	—
111	C₄×C₉	—	2	6	no	—
112	C₄×C₉	—	2	6	no	—
113	C₄×C₉	#235	1	6	yes	—
114	#117	C₃⋊D₉	1	1	yes	characteristic
115	C₄⋊C₈	—	27	2	no	—
116	C₂²⋊C₈	—	27	2	no	—
117	#97	—	27	1	no	—
118	C₄×C₈	—	27	1	no	—
119	C₃×C₉	#433	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₃	C₂×Dic₉	1	3	yes	—
121	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	6	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
125	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
126	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
127	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
128	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
129	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
130	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
132	C₃⋊C₈	—	18	6	no	—
133	C₃⋊C₈	—	18	6	no	—
134	C₃⋊C₈	—	18	2	no	—
135	C₃⋊C₈	—	18	2	no	—
136	C₂×C₃²	—	4	1	no	—
137	C₂×C₃²	#926	1	1	yes	characteristic
138	C₂×C₃²	#948	1	2	yes	—
139	C₂×C₉	—	4	3	no	—
140	C₂×C₉	#926	1	3	yes	—
141	C₂×C₉	#948	1	6	yes	—
142	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
143	C₂×C₈	—	27	2	no	—
144	C₂×C₈	—	27	2	no	—
145	C₄⋊C₄	C₃⋊Dic₉	1	2	yes	—
146	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	—
147	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
148	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
149	C₂²×C₃	—	2	3	no	—
150	C₂²×C₃	—	2	6	no	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
154	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
155	C₃×C₄	—	2	3	no	—
156	C₃×C₄	—	2	6	no	—
157	C₃×C₄	—	2	6	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	#3697	1	6	yes	—
162	C₃×C₄	#3828	1	2	yes	—
163	C₃²	#5000	1	1	yes	characteristic
164	C₉	#5000	1	3	yes	complemented
165	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
168	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
169	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
171	C₈	—	54	2	no	—
172	C₈	—	54	2	no	—
173	C₂×C₃	—	4	3	no	—
174	C₂×C₃	—	4	1	no	—
175	C₂×C₃	#63488	1	3	yes	—
176	C₂×C₃	#63510	1	6	yes	—
177	C₂×C₃	#64135	1	1	yes	characteristic
178	C₂×C₃	#64157	1	2	yes	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	#88736	1	2	yes	—
186	C₃	#10588678	1	3	yes	complemented
187	C₃	#10592699	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	#11721815	1	1	yes	characteristic
190	C₂	#11721865	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#117	C₃⋊D₉
hypocenter	C₃×C₉	#433
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
4. #11720516	C₂	semidirect
5. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
119. C₃×C₉	#433	semidirect
164. C₉	#5000	semidirect
186. C₃	#10588678	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
#11720571	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#11720516	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592699	3
#10588678	3