Subgroups — #13144774

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#135	C₂	1	1	yes	maximal, characteristic
2	#11721865	C₂	1	2	yes	maximal, complemented
3	#11720585	C₂	1	2	yes	maximal, complemented
4	#11720538	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720535	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592896	—	3	1	no	maximal
7	#10588877	—	3	3	no	maximal
8	C₃×C₉×#41	C₂²	1	2	yes	—
9	C₃×C₉×Q₁₆	C₄	1	1	yes	characteristic, complemented
10	C₃×C₉×SD₁₆	C₄	1	2	yes	complemented
11	C₃×C₉×D₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
14	#88532	—	2	2	no	—
15	#88532	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃²×#135	D₃	1	1	yes	characteristic
19	#64157	—	3	2	no	—
20	#63664	—	3	2	no	—
21	#63646	—	3	1	no	—
22	#63642	—	3	1	no	—
23	C₉×#135	D₃	1	3	yes	—
24	#63510	—	3	6	no	—
25	#63396	—	3	6	no	—
26	#63375	—	3	3	no	—
27	#63372	—	3	3	no	—
28	C₃×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₃×D₄×C₉	—	2	2	no	—
30	C₃×D₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃⋊Dic₉	—	2	1	no	—
34	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
35	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
36	C₃⋊(C₉⋊C₈)	—	2	2	no	—
37	C₃⋊(C₉⋊C₈)	D₄	1	2	yes	—
38	#5197	—	9	3	no	—
39	#5197	—	9	1	no	—
40	C₃²×#41	C₂×D₃	1	2	yes	—
41	C₃²×Q₁₆	Dic₃	1	1	yes	characteristic
42	C₃²×SD₁₆	Dic₃	1	2	yes	—
43	C₃²×D₈	Dic₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄×C₃⋊Dic₃	—	3	1	no	—
46	#3748	—	3	2	no	—
47	#3748	—	6	2	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
50	C₉×#41	C₂×D₃	1	6	yes	—
51	C₉×Q₁₆	Dic₃	1	3	yes	complemented
52	C₉×SD₁₆	Dic₃	1	6	yes	complemented
53	C₉×D₈	Dic₃	1	3	yes	complemented
54	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
55	C₄×Dic₉	—	3	3	no	—
56	#3667	—	3	6	no	—
57	#3667	—	6	6	no	—
58	C₂×C₉⋊C₈	—	3	3	no	—
59	C₂×C₉⋊C₈	—	6	6	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	#41	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃⋊Dic₉	—	4	1	no	—
66	C₃×#135	D₉	1	3	yes	—
67	C₃×#135	C₃⋊D₃	1	1	yes	characteristic
68	#948	—	9	6	no	—
69	#948	—	9	2	no	—
70	#837	—	9	6	no	—
71	#837	—	9	2	no	—
72	#819	—	9	3	no	—
73	#819	—	9	1	no	—
74	#815	—	9	3	no	—
75	#815	—	9	1	no	—
76	C₃²×Q₈	C₂×Dic₃	1	2	yes	—
77	C₃²×D₄	—	2	2	no	—
78	C₃²×D₄	C₂×Dic₃	1	2	yes	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃⋊Dic₃	—	6	1	no	—
82	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
83	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
84	C₃⋊(C₃⋊C₈)	—	3	2	no	—
85	C₃⋊(C₃⋊C₈)	—	6	2	no	—
86	Q₈×C₉	C₂×Dic₃	1	6	yes	—
87	D₄×C₉	—	2	6	no	—
88	D₄×C₉	C₂×Dic₃	1	6	yes	—
89	C₂×C₄×C₉	—	2	6	no	—
90	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
91	C₂×Dic₉	—	6	3	no	—
92	C₈×C₉	C₂×Dic₃	1	3	yes	—
93	C₈×C₉	C₂×Dic₃	1	3	yes	—
94	C₉⋊C₈	—	3	6	no	—
95	C₉⋊C₈	—	6	6	no	—
96	#444	—	27	1	no	Sylow
97	C₂×C₃×C₉	—	2	1	no	—
98	C₂×C₃×C₉	—	4	2	no	—
99	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
100	C₃×#41	C₂×D₉	1	6	yes	—
101	C₃×#41	C₂×C₃⋊D₃	1	2	yes	—
102	C₃×Q₁₆	Dic₉	1	3	yes	complemented
103	C₃×Q₁₆	C₃⋊Dic₃	1	1	yes	characteristic
104	C₃×SD₁₆	Dic₉	1	6	yes	complemented
105	C₃×SD₁₆	C₃⋊Dic₃	1	2	yes	—
106	C₃×D₈	Dic₉	1	3	yes	complemented
107	C₃×D₈	C₃⋊Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
109	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
110	C₄×Dic₃	—	9	3	no	—
111	C₄×Dic₃	—	9	1	no	—
112	#208	—	9	6	no	—
113	#208	—	9	2	no	—
114	#208	—	18	2	no	—
115	#208	—	18	6	no	—
116	C₂×C₃⋊C₈	—	9	3	no	—
117	C₂×C₃⋊C₈	—	9	1	no	—
118	C₂×C₃⋊C₈	—	18	2	no	—
119	C₂×C₃⋊C₈	—	18	6	no	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	#237	1	1	yes	characteristic
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
124	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
125	C₃⋊Dic₃	—	12	1	no	—
126	C₂²×C₉	—	2	6	no	—
127	C₂²×C₉	#237	1	3	yes	—
128	C₄×C₉	—	2	6	no	—
129	C₄×C₉	D₃×D₄	1	3	yes	—
130	C₄×C₉	C₂²×Dic₃	1	3	yes	—
131	Dic₉	—	12	3	no	—
132	#135	C₃⋊D₉	1	1	yes	characteristic
133	#131	—	27	2	no	—
134	#108	—	27	1	no	—
135	#104	—	27	2	no	—
136	#97	—	27	1	no	—
137	C₃×C₉	#444	1	1	yes	characteristic, complemented, Sylow
138	C₃×Q₈	C₂×Dic₉	1	6	yes	—
139	C₃×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
140	C₃×D₄	—	2	6	no	—
141	C₃×D₄	—	2	2	no	—
142	C₃×D₄	C₂×Dic₉	1	6	yes	—
143	C₃×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
144	C₂×C₃×C₄	—	2	6	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
147	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
148	C₂×Dic₃	—	18	3	no	—
149	C₂×Dic₃	—	18	1	no	—
150	C₃×C₈	C₂×Dic₉	1	3	yes	—
151	C₃×C₈	C₂×Dic₉	1	3	yes	—
152	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
153	C₃×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
154	C₃⋊C₈	—	9	6	no	—
155	C₃⋊C₈	—	9	2	no	—
156	C₃⋊C₈	—	18	6	no	—
157	C₃⋊C₈	—	18	2	no	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	—	4	2	no	—
160	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
161	C₂×C₉	—	2	3	no	—
162	C₂×C₉	—	4	6	no	—
163	C₂×C₉	D₄×Dic₃	1	3	yes	—
164	#41	C₂×C₃⋊D₉	1	2	yes	—
165	Q₁₆	C₃⋊Dic₉	1	1	yes	characteristic, complemented
166	SD₁₆	C₃⋊Dic₉	1	2	yes	complemented
167	D₈	C₃⋊Dic₉	1	1	yes	characteristic, complemented
168	M₁₆	—	27	2	no	—
169	M₁₆	—	54	2	no	—
170	C₂×C₈	—	27	1	no	—
171	C₂×C₈	—	54	2	no	—
172	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
173	C₄²	—	27	1	no	—
174	C₂²×C₃	—	2	6	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	#3699	1	3	yes	—
177	C₂²×C₃	#3830	1	1	yes	characteristic
178	C₃×C₄	—	2	6	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	D₄×D₉	1	3	yes	—
181	C₃×C₄	C₂²×Dic₉	1	3	yes	—
182	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
183	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
184	Dic₃	—	36	1	no	—
185	Dic₃	—	36	3	no	—
186	C₃²	#5197	1	1	yes	characteristic
187	C₉	#5197	1	3	yes	complemented
188	Q₈	C₂×C₃⋊Dic₉	1	2	yes	—
189	D₄	—	2	2	no	—
190	D₄	C₂×C₃⋊Dic₉	1	2	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	54	1	no	—
193	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
194	C₈	—	27	2	no	—
195	C₈	—	54	2	no	—
196	C₈	C₂×C₃⋊Dic₉	1	1	yes	characteristic
197	C₈	C₂×C₃⋊Dic₉	1	1	yes	characteristic
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	—	2	3	no	—
200	C₂×C₃	—	4	2	no	—
201	C₂×C₃	—	4	6	no	—
202	C₂×C₃	D₄×Dic₉	1	3	yes	—
203	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	#88742	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	108	1	no	—
208	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
209	C₄	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
210	C₃	#10588877	1	3	yes	complemented
211	C₃	#10592896	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	4	2	no	—
214	C₂	D₄×C₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₃⋊D₉
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#135	C₂
socle	C₂×C₃²	D₄×Dic₃
prime radical	C₃²×#135	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#135	C₃⋊D₉
hypocenter	C₃×C₉	#444
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #11721865	C₂	semidirect
3. #11720585	C₂	semidirect
4. #11720538	C₂	semidirect
5. #11720535	C₂	semidirect
9. C₃×C₉×Q₁₆	C₄	semidirect
10. C₃×C₉×SD₁₆	C₄	semidirect
11. C₃×C₉×D₈	C₄	semidirect
51. C₉×Q₁₆	Dic₃	semidirect
52. C₉×SD₁₆	Dic₃	semidirect
53. C₉×D₈	Dic₃	semidirect
102. C₃×Q₁₆	Dic₉	semidirect
104. C₃×SD₁₆	Dic₉	semidirect
106. C₃×D₈	Dic₉	semidirect
137. C₃×C₉	#444	semidirect
165. Q₁₆	C₃⋊Dic₉	semidirect
166. SD₁₆	C₃⋊Dic₉	semidirect
167. D₈	C₃⋊Dic₉	semidirect
187. C₉	#5197	semidirect
210. C₃	#10588877	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#135	2
#11721865	2
#11720585	2
#11720538	2
#11720535	2
#10592896	3
#10588877	3