Subgroups — C₃×#10588229 #13135648

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Dic₉	C₂	1	2	yes	maximal
3	C₃×#4551	—	3	1	no	maximal
4	#10588229	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₃×Dic₉	—	2	2	no	—
7	C₂×C₃×C₈×C₉	—	2	2	no	—
8	C₂×C₃×C₈×C₉	C₄	1	2	yes	complemented
9	C₃×C₄⋊Dic₉	—	2	2	no	—
10	C₃×C₄⋊Dic₉	C₄	1	4	yes	complemented
11	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
13	C₂²×C₈×C₉	—	2	1	no	—
14	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
15	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
16	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
20	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
21	C₂×C₃×Dic₉	—	2	4	no	—
22	C₂×C₃×Dic₉	—	4	2	no	—
23	C₂×C₃×Dic₉	—	4	2	no	—
24	C₃×C₈×C₉	—	2	2	no	—
25	C₃×#340	—	9	1	no	—
26	#4551	—	3	1	no	—
27	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	—	6	2	no	—
29	C₂×C₃²×C₈	—	2	2	no	—
30	C₂×C₃²×C₈	Dic₃	1	2	yes	—
31	C₃×C₄⋊Dic₃	—	3	4	no	—
32	C₃×C₄⋊Dic₃	—	6	2	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
35	C₂²×Dic₉	—	2	2	no	—
36	C₂×C₈×C₉	—	2	4	no	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	C₃×C₄	1	2	yes	complemented
40	C₄⋊Dic₉	—	2	2	no	—
41	C₄⋊Dic₉	C₃×C₄	1	4	yes	complemented
42	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃×C₉	D₈	1	1	yes	characteristic
45	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
46	C₂²×C₃×C₉	Q₁₆	1	1	yes	characteristic
47	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
48	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
50	C₃×Dic₉	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	9	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂³×C₃²	D₁₂	1	1	yes	characteristic
57	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
59	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
60	C₂×C₃²×C₄	#67	1	2	yes	—
61	C₂×C₃×Dic₃	—	6	4	no	—
62	C₂×C₃×Dic₃	—	12	2	no	—
63	C₂×C₃×Dic₃	—	12	2	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
66	C₃²×C₈	—	2	2	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
74	C₂×C₄×C₉	C₃×Q₈	1	1	yes	characteristic
75	C₂×Dic₉	—	2	4	no	—
76	C₂×Dic₉	—	4	2	no	—
77	C₂×Dic₉	—	4	2	no	—
78	C₈×C₉	—	2	4	no	—
79	C₈×C₉	—	2	2	no	—
80	#340	—	9	1	no	Sylow
81	C₂×C₃×C₉	#95	1	1	yes	characteristic
82	C₂×C₃×C₉	#102	1	2	yes	—
83	C₂×C₃×C₉	#103	1	2	yes	—
84	C₂×C₃×C₉	#106	1	1	yes	characteristic
85	C₂×C₃×C₉	#107	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	18	2	no	—
88	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	6	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	Dic₉	1	2	yes	complemented
96	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
97	C₃×C₄⋊C₄	—	9	4	no	—
98	C₃×C₄⋊C₄	—	18	2	no	—
99	C₄⋊Dic₃	—	3	4	no	—
100	C₄⋊Dic₃	—	6	2	no	—
101	C₂²×C₃²	#204	1	2	yes	—
102	C₂²×C₃²	D₂₄	1	1	yes	characteristic
103	C₂²×C₃²	Dic₁₂	1	1	yes	characteristic
104	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
105	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
106	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
107	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
108	C₃²×C₄	#217	1	1	yes	characteristic
109	C₃×Dic₃	—	12	4	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	2	no	—
115	C₂²×C₉	C₃×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₉	C₃×D₈	1	1	yes	characteristic
118	C₂²×C₉	C₃×SD₁₆	1	2	yes	—
119	C₂²×C₉	C₃×Q₁₆	1	1	yes	characteristic
120	C₄×C₉	—	2	1	no	—
121	C₄×C₉	—	2	2	no	—
122	C₄×C₉	—	2	1	no	—
123	C₄×C₉	C₃×C₄²	1	1	yes	characteristic
124	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
126	Dic₉	—	4	4	no	—
127	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	9	2	no	—
129	C₃×C₉	#340	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	D₃₆	1	1	yes	characteristic
132	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	18	4	no	—
138	C₂×C₃×C₄	—	36	2	no	—
139	C₂×C₃×C₄	—	36	2	no	—
140	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
142	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
143	C₂×C₃×C₄	#611	1	2	yes	—
144	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×#67	1	2	yes	—
148	C₂×Dic₃	—	6	4	no	—
149	C₂×Dic₃	—	12	2	no	—
150	C₂×Dic₃	—	12	2	no	—
151	C₃×C₈	—	2	4	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	—	2	2	no	—
154	C₂×C₃²	#816	1	2	yes	—
155	C₂×C₃²	#817	1	1	yes	characteristic
156	C₂×C₃²	#818	1	1	yes	characteristic
157	C₂×C₃²	#821	1	2	yes	—
158	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	2	1	no	—
164	C₂×C₉	C₃×#95	1	1	yes	characteristic
165	C₂×C₉	C₃×#102	1	2	yes	—
166	C₂×C₉	C₃×#103	1	2	yes	—
167	C₂×C₉	C₃×#106	1	1	yes	characteristic
168	C₂×C₉	C₃×#107	1	1	yes	characteristic
169	C₂²×C₄	—	18	2	no	—
170	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₃×Dic₉	1	2	yes	complemented
173	C₄⋊C₄	—	9	4	no	—
174	C₄⋊C₄	—	18	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	Dic₃₆	1	1	yes	characteristic
181	C₂²×C₃	#3664	1	2	yes	—
182	C₂²×C₃	D₇₂	1	1	yes	characteristic
183	C₂²×C₃	C₄⋊Dic₉	1	1	yes	characteristic
184	C₂²×C₃	(C₂×D₉)⋊C₄	1	2	yes	—
185	C₂²×C₃	C₃×#204	1	2	yes	—
186	C₂²×C₃	C₃×D₂₄	1	1	yes	characteristic
187	C₂²×C₃	C₃×Dic₁₂	1	1	yes	characteristic
188	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
189	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	36	4	no	—
194	C₃×C₄	C₄×Dic₉	1	1	yes	characteristic
195	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
196	C₃×C₄	#3676	1	1	yes	characteristic
197	C₃×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
198	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
199	C₃×C₄	C₃×#217	1	1	yes	characteristic
200	Dic₃	—	12	4	no	—
201	C₃²	#4551	1	1	yes	characteristic
202	C₉	—	2	1	no	—
203	C₉	C₃×#340	1	1	yes	characteristic, complemented
204	C₂³	C₃×D₃₆	1	1	yes	characteristic
205	C₂×C₄	—	18	4	no	—
206	C₂×C₄	—	36	2	no	—
207	C₂×C₄	—	36	2	no	—
208	C₂×C₄	C₃×Dic₁₈	1	1	yes	characteristic
209	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
210	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
211	C₂×C₄	C₃×#611	1	2	yes	—
212	C₈	—	2	2	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	#63376	1	2	yes	—
219	C₂×C₃	#63377	1	1	yes	characteristic
220	C₂×C₃	#63378	1	1	yes	characteristic
221	C₂×C₃	#63380	1	2	yes	—
222	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
223	C₂×C₃	C₃×#816	1	2	yes	—
224	C₂×C₃	C₃×#817	1	1	yes	characteristic
225	C₂×C₃	C₃×#818	1	1	yes	characteristic
226	C₂×C₃	C₃×#821	1	2	yes	—
227	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
228	C₂²	C₃×Dic₃₆	1	1	yes	characteristic
229	C₂²	C₃×#3664	1	2	yes	—
230	C₂²	C₃×D₇₂	1	1	yes	characteristic
231	C₂²	C₃×C₄⋊Dic₉	1	1	yes	characteristic
232	C₂²	C₃×(C₂×D₉)⋊C₄	1	2	yes	—
233	C₄	—	36	4	no	—
234	C₄	C₃×C₄×Dic₉	1	1	yes	characteristic
235	C₄	C₃×Dic₉⋊C₄	1	2	yes	—
236	C₄	C₃×#3676	1	1	yes	characteristic
237	C₃	—	2	1	no	—
238	C₃	#10588229	1	1	yes	characteristic, direct product factor
239	C₃	C₃×#4551	1	1	yes	characteristic
240	C₂	C₃×#63376	1	2	yes	—
241	C₂	C₃×#63377	1	1	yes	characteristic
242	C₂	C₃×#63378	1	1	yes	characteristic
243	C₂	C₃×#63380	1	2	yes	—
244	C₂	C₃×C₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃₆
commutator	C₄×C₉	C₃×C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	D₁₂
cosocle	C₂²×C₄×C₉	C₂²×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₃×#340
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. #10588229	C₃	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
10. C₃×C₄⋊Dic₉	C₄	semidirect
39. C₂×C₈×C₉	C₃×C₄	semidirect
41. C₄⋊Dic₉	C₃×C₄	semidirect
95. C₂×C₃×C₈	Dic₉	semidirect
129. C₃×C₉	#340	semidirect
172. C₂×C₈	C₃×Dic₉	semidirect
203. C₉	C₃×#340	semidirect
238. C₃	#10588229	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₂×C₃×C₄⋊Dic₉	2
C₃×#4551	3
#10588229	3

Subgroups — C3×#10588229 #13135648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588229 #13135648