Subgroups — C₂×#11720571 #13144796

Order 1728.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
3	#11720571	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63657	—	3	1	no	maximal
5	C₂×#63389	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₉⋊C₈)	—	2	4	no	—
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
15	C₂²×C₃⋊(C₃⋊C₈)	—	3	2	no	—
16	C₂³×C₄×C₉	D₃	1	3	yes	—
17	#63657	—	3	4	no	—
18	C₂²×C₉⋊C₈	—	3	6	no	—
19	#63389	—	3	12	no	—
20	C₂³×C₃×C₉	—	2	2	no	—
21	C₂³×C₃×C₉	—	2	2	no	—
22	C₂³×C₃×C₉	C₈	1	4	yes	complemented
23	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₃⋊(C₉⋊C₈)	—	2	4	no	—
33	C₂×#830	—	9	3	no	—
34	C₂×#830	—	9	1	no	—
35	C₂⁴×C₃²	Dic₃	1	1	yes	characteristic
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
38	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂⁴×C₉	Dic₃	1	3	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
44	C₂²×C₄×C₉	—	2	12	no	—
45	C₂²×C₄×C₉	Dic₃	1	3	yes	—
46	C₂²×C₄×C₉	Dic₃	1	6	yes	—
47	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
48	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
49	C₂×C₉⋊C₈	—	3	12	no	—
50	C₂×C₉⋊C₈	—	6	12	no	—
51	C₂²×C₃×C₉	—	2	8	no	—
52	C₂²×C₃×C₉	—	2	2	no	—
53	C₂²×C₃×C₉	—	2	2	no	—
54	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
56	C₂²×C₃×C₉	C₂×C₈	1	4	yes	complemented
57	C₂²×C₃×C₉	M₁₆	1	2	yes	—
58	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	—	2	2	no	—
60	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
61	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
62	C₂³×C₃×C₄	D₉	1	3	yes	—
63	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	—	9	6	no	—
65	C₂²×C₃⋊C₈	—	9	2	no	—
66	#830	—	9	12	no	—
67	#830	—	9	4	no	—
68	C₂³×C₃²	—	2	2	no	—
69	C₂³×C₃²	—	2	2	no	—
70	C₂³×C₃²	C₃⋊C₈	1	4	yes	—
71	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
72	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	—	2	4	no	—
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	#67	1	4	yes	—
79	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂³×C₉	—	2	6	no	—
81	C₂³×C₉	—	2	6	no	—
82	C₂³×C₉	C₃⋊C₈	1	12	yes	complemented
83	C₂³×C₉	C₂×Dic₃	1	6	yes	—
84	C₂³×C₉	C₂×Dic₃	1	3	yes	—
85	C₃⋊(C₃⋊C₈)	—	6	4	no	—
86	C₂×C₄×C₉	—	2	6	no	—
87	C₂×C₄×C₉	—	2	12	no	—
88	C₂×C₄×C₉	—	2	12	no	—
89	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
90	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
91	C₂×C₄×C₉	#67	1	12	yes	—
92	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
93	C₉⋊C₈	—	6	12	no	—
94	C₂×C₂²⋊C₈	—	27	1	no	Sylow
95	C₂×C₃×C₉	—	2	4	no	—
96	C₂×C₃×C₉	C₂²⋊C₈	1	4	yes	complemented
97	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂⁴×C₃	Dic₉	1	3	yes	—
101	C₂⁴×C₃	C₃⋊Dic₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	12	no	—
103	C₂²×C₃×C₄	—	2	4	no	—
104	C₂²×C₃×C₄	Dic₉	1	3	yes	—
105	C₂²×C₃×C₄	Dic₉	1	6	yes	—
106	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
107	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
108	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
110	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
111	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₃⋊C₈	—	9	12	no	—
113	C₂×C₃⋊C₈	—	9	4	no	—
114	C₂×C₃⋊C₈	—	18	12	no	—
115	C₂×C₃⋊C₈	—	18	4	no	—
116	C₂²×C₃²	—	2	8	no	—
117	C₂²×C₃²	—	2	2	no	—
118	C₂²×C₃²	—	2	2	no	—
119	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
120	C₂²×C₃²	C₂×C₃⋊C₈	1	4	yes	—
121	C₂²×C₃²	#208	1	2	yes	—
122	C₂²×C₃²	#217	1	2	yes	—
123	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	#217	1	2	yes	—
126	C₃²×C₄	C₂×#67	1	2	yes	—
127	C₂²×C₉	—	2	24	no	—
128	C₂²×C₉	—	2	6	no	—
129	C₂²×C₉	—	2	6	no	—
130	C₂²×C₉	C₂×C₃⋊C₈	1	6	yes	—
131	C₂²×C₉	C₂×C₃⋊C₈	1	12	yes	complemented
132	C₂²×C₉	#208	1	6	yes	—
133	C₂²×C₉	#217	1	6	yes	—
134	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
135	C₄×C₉	—	2	6	no	—
136	C₄×C₉	#217	1	6	yes	—
137	C₄×C₉	C₂×#67	1	6	yes	—
138	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
139	C₂²×C₈	—	27	2	no	—
140	C₂²⋊C₈	—	27	4	no	—
141	C₃×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	6	no	—
143	C₂³×C₃	—	2	6	no	—
144	C₂³×C₃	—	2	2	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	C₉⋊C₈	1	12	yes	complemented
147	C₂³×C₃	C₂×Dic₉	1	6	yes	—
148	C₂³×C₃	C₂×Dic₉	1	3	yes	—
149	C₂³×C₃	C₃⋊(C₃⋊C₈)	1	4	yes	—
150	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
151	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	12	no	—
155	C₂×C₃×C₄	—	2	12	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	4	no	—
158	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
159	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
160	C₂×C₃×C₄	#611	1	12	yes	—
161	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
162	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
164	C₂×C₃×C₄	#638	1	4	yes	—
165	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
166	C₃⋊C₈	—	18	12	no	—
167	C₃⋊C₈	—	18	4	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	#830	1	4	yes	—
170	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
171	C₂×C₃²	C₂×#208	1	1	yes	characteristic
172	C₂×C₃²	C₂×#217	1	1	yes	characteristic
173	C₂×C₉	—	2	12	no	—
174	C₂×C₉	#830	1	12	yes	complemented
175	C₂×C₉	C₂²×C₃⋊C₈	1	3	yes	—
176	C₂×C₉	C₂×#208	1	3	yes	—
177	C₂×C₉	C₂×#217	1	3	yes	—
178	C₂⁴	C₃⋊Dic₉	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
181	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
182	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
183	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
184	C₂×C₈	—	27	4	no	—
185	C₂×C₈	—	54	4	no	—
186	C₂²×C₃	—	2	6	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	24	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	—	2	6	no	—
192	C₂²×C₃	C₂×C₉⋊C₈	1	6	yes	—
193	C₂²×C₃	C₂×C₉⋊C₈	1	12	yes	complemented
194	C₂²×C₃	#3667	1	6	yes	—
195	C₂²×C₃	#3676	1	6	yes	—
196	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
197	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
198	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
199	C₂²×C₃	#3748	1	2	yes	—
200	C₂²×C₃	#3757	1	2	yes	—
201	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
202	C₃×C₄	—	2	6	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	#3676	1	6	yes	—
205	C₃×C₄	C₂×#611	1	6	yes	—
206	C₃×C₄	#3757	1	2	yes	—
207	C₃×C₄	C₂×#638	1	2	yes	—
208	C₃²	C₂×#830	1	1	yes	characteristic
209	C₉	C₂×#830	1	3	yes	complemented
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	C₃⋊(C₉⋊C₈)	1	4	yes	complemented
213	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
214	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
219	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
220	C₂×C₄	#6233	1	4	yes	—
221	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
222	C₈	—	54	4	no	—
223	C₂×C₃	—	2	4	no	—
224	C₂×C₃	—	2	12	no	—
225	C₂×C₃	#63389	1	12	yes	complemented
226	C₂×C₃	C₂²×C₉⋊C₈	1	3	yes	—
227	C₂×C₃	C₂×#3667	1	3	yes	—
228	C₂×C₃	C₂×#3676	1	3	yes	—
229	C₂×C₃	#63657	1	4	yes	—
230	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
231	C₂×C₃	C₂×#3748	1	1	yes	characteristic
232	C₂×C₃	C₂×#3757	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	—
237	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	4	yes	complemented
238	C₂²	#88532	1	2	yes	—
239	C₂²	#88551	1	2	yes	—
240	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	#88551	1	2	yes	—
243	C₄	C₂×#6233	1	2	yes	—
244	C₃	C₂×#63389	1	3	yes	complemented
245	C₃	C₂×#63657	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	#11720571	1	4	yes	direct product factor
248	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
249	C₂	C₂×#88532	1	1	yes	characteristic
250	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂×Dic₃
prime radical	C₂⁴×C₃²	Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
3. #11720571	C₂	direct (N × H)
13. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
22. C₂³×C₃×C₉	C₈	semidirect
56. C₂²×C₃×C₉	C₂×C₈	semidirect
82. C₂³×C₉	C₃⋊C₈	semidirect
96. C₂×C₃×C₉	C₂²⋊C₈	semidirect
131. C₂²×C₉	C₂×C₃⋊C₈	semidirect
141. C₃×C₉	C₂×C₂²⋊C₈	semidirect
146. C₂³×C₃	C₉⋊C₈	semidirect
174. C₂×C₉	#830	semidirect
193. C₂²×C₃	C₂×C₉⋊C₈	semidirect
209. C₉	C₂×#830	semidirect
212. C₂³	C₃⋊(C₉⋊C₈)	semidirect
225. C₂×C₃	#63389	semidirect
237. C₂²	C₂×C₃⋊(C₉⋊C₈)	semidirect
244. C₃	C₂×#63389	semidirect
247. C₂	#11720571	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720571	2
C₂×#63657	3
C₂×#63389	3

Subgroups — C2×#11720571 #13144796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720571 #13144796