Subgroups — C₂×#11720572 #13144798

Order 1728.

1650 conjugacy classes of subgroups.

The subgroup lattice has 264 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	C₂³×C₃⋊Dic₉	C₂	1	2	yes	maximal
3	#11720572	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63658	—	3	1	no	maximal
5	C₂×C₉⋊#95	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃⋊Dic₉	—	2	4	no	—
11	C₂²×C₃⋊Dic₉	—	2	4	no	—
12	C₂²×C₃⋊Dic₉	C₄	1	8	yes	complemented
13	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
14	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
15	C₂³×C₃⋊Dic₃	—	3	2	no	—
16	C₂³×C₄×C₉	D₃	1	3	yes	—
17	C₂³×Dic₉	—	3	6	no	—
18	#63658	—	3	4	no	—
19	C₉⋊#95	—	3	12	no	—
20	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
21	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
22	C₂³×C₃×C₉	D₄	1	2	yes	—
23	C₂³×C₃×C₉	D₄	1	4	yes	—
24	C₂³×C₃×C₉	Q₈	1	2	yes	—
25	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	8	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	complemented
29	C₂×C₃⋊Dic₉	—	2	16	no	—
30	C₂×C₃⋊Dic₉	—	2	4	no	—
31	C₂×C₃⋊Dic₉	—	2	4	no	—
32	C₂×C₃⋊Dic₉	C₂×C₄	1	8	yes	complemented
33	C₂×C₃⋊#95	—	9	3	no	—
34	C₂×C₃⋊#95	—	9	1	no	—
35	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
38	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
39	C₂²×C₃⋊Dic₃	—	3	8	no	—
40	C₂²×C₃⋊Dic₃	—	3	4	no	—
41	C₂²×C₃⋊Dic₃	—	6	4	no	—
42	C₂²×C₃⋊Dic₃	—	6	4	no	—
43	C₂⁴×C₉	C₂×D₃	1	3	yes	—
44	C₂²×C₄×C₉	—	2	12	no	—
45	C₂²×C₄×C₉	Dic₃	1	12	yes	complemented
46	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
47	C₂²×Dic₉	—	3	24	no	—
48	C₂²×Dic₉	—	3	12	no	—
49	C₂²×Dic₉	—	6	12	no	—
50	C₂²×Dic₉	—	6	12	no	—
51	C₂²×C₃×C₉	C₄²	1	4	yes	—
52	C₂²×C₃×C₉	C₂²⋊C₄	1	8	yes	—
53	C₂²×C₃×C₉	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₃×C₉	C₄⋊C₄	1	8	yes	—
55	C₂²×C₃×C₉	C₄⋊C₄	1	4	yes	—
56	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
60	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₄×C₉	—	2	4	no	—
62	C₃⋊Dic₉	—	2	8	no	—
63	C₂³×C₃×C₄	D₉	1	3	yes	—
64	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
65	C₂³×Dic₃	—	9	6	no	—
66	C₂³×Dic₃	—	9	2	no	—
67	C₃⋊#95	—	9	12	no	—
68	C₃⋊#95	—	9	4	no	—
69	C₂³×C₃²	Dic₆	1	2	yes	—
70	C₂³×C₃²	C₄×D₃	1	4	yes	—
71	C₂³×C₃²	D₁₂	1	2	yes	—
72	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
73	C₂³×C₃²	#67	1	4	yes	—
74	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
78	C₂×C₃⋊Dic₃	—	3	8	no	—
79	C₂×C₃⋊Dic₃	—	6	16	no	—
80	C₂×C₃⋊Dic₃	—	6	4	no	—
81	C₂×C₃⋊Dic₃	—	6	4	no	—
82	C₂³×C₉	Dic₆	1	6	yes	—
83	C₂³×C₉	C₄×D₃	1	12	yes	—
84	C₂³×C₉	D₁₂	1	6	yes	—
85	C₂³×C₉	C₂×Dic₃	1	6	yes	—
86	C₂³×C₉	#67	1	12	yes	—
87	C₂³×C₉	C₂²×D₃	1	3	yes	—
88	C₂×C₄×C₉	—	2	24	no	—
89	C₂×C₄×C₉	—	2	12	no	—
90	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	complemented
91	C₂×Dic₉	—	3	24	no	—
92	C₂×Dic₉	—	6	48	no	—
93	C₂×Dic₉	—	6	12	no	—
94	C₂×Dic₉	—	6	12	no	—
95	C₂×#95	—	27	1	no	Sylow
96	C₂×C₃×C₉	#95	1	8	yes	complemented
97	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂⁴×C₃	C₂×D₉	1	3	yes	—
103	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	—	2	12	no	—
105	C₂²×C₃×C₄	—	2	4	no	—
106	C₂²×C₃×C₄	Dic₉	1	12	yes	complemented
107	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
108	C₂²×C₃×C₄	C₃⋊Dic₃	1	4	yes	—
109	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
110	C₂²×Dic₃	—	9	24	no	—
111	C₂²×Dic₃	—	9	12	no	—
112	C₂²×Dic₃	—	9	8	no	—
113	C₂²×Dic₃	—	9	4	no	—
114	C₂²×Dic₃	—	18	12	no	—
115	C₂²×Dic₃	—	18	12	no	—
116	C₂²×Dic₃	—	18	4	no	—
117	C₂²×Dic₃	—	18	4	no	—
118	C₂²×C₃²	C₄×Dic₃	1	4	yes	—
119	C₂²×C₃²	Dic₃⋊C₄	1	8	yes	—
120	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
121	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
122	C₂²×C₃²	#217	1	4	yes	—
123	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
124	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
125	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
126	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
127	C₂²×C₃²	C₂×#67	1	2	yes	—
128	C₃²×C₄	—	2	4	no	—
129	C₃⋊Dic₃	—	6	8	no	—
130	C₂²×C₉	C₄×Dic₃	1	12	yes	—
131	C₂²×C₉	Dic₃⋊C₄	1	24	yes	—
132	C₂²×C₉	C₄⋊Dic₃	1	12	yes	—
133	C₂²×C₉	(C₂×D₃)⋊C₄	1	24	yes	—
134	C₂²×C₉	#217	1	12	yes	—
135	C₂²×C₉	C₂×Dic₆	1	3	yes	—
136	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
137	C₂²×C₉	C₂×D₁₂	1	3	yes	—
138	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
139	C₂²×C₉	C₂×#67	1	6	yes	—
140	C₄×C₉	—	2	12	no	—
141	Dic₉	—	6	24	no	—
142	C₂³×C₄	—	27	2	no	—
143	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
144	#95	—	27	4	no	—
145	C₃×C₉	C₂×#95	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₃	Dic₁₈	1	6	yes	—
147	C₂³×C₃	C₄×D₉	1	12	yes	—
148	C₂³×C₃	D₃₆	1	6	yes	—
149	C₂³×C₃	C₂×Dic₉	1	6	yes	—
150	C₂³×C₃	#611	1	12	yes	—
151	C₂³×C₃	C₂²×D₉	1	3	yes	—
152	C₂³×C₃	#634	1	2	yes	—
153	C₂³×C₃	C₄×C₃⋊D₃	1	4	yes	—
154	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	2	yes	—
155	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
156	C₂³×C₃	#638	1	4	yes	—
157	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	24	no	—
159	C₂×C₃×C₄	—	2	12	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	complemented
163	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
164	C₂×Dic₃	—	9	24	no	—
165	C₂×Dic₃	—	9	8	no	—
166	C₂×Dic₃	—	18	48	no	—
167	C₂×Dic₃	—	18	12	no	—
168	C₂×Dic₃	—	18	12	no	—
169	C₂×Dic₃	—	18	16	no	—
170	C₂×Dic₃	—	18	4	no	—
171	C₂×Dic₃	—	18	4	no	—
172	C₂×C₃²	C₃⋊#95	1	8	yes	—
173	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
174	C₂×C₃²	C₂×Dic₃⋊C₄	1	2	yes	—
175	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
176	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	2	yes	—
177	C₂×C₃²	C₂×#217	1	1	yes	characteristic
178	C₂×C₉	C₃⋊#95	1	24	yes	complemented
179	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
180	C₂×C₉	C₂×Dic₃⋊C₄	1	6	yes	—
181	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
182	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
183	C₂×C₉	C₂×#217	1	3	yes	—
184	C₂⁴	C₂×C₃⋊D₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	27	8	no	—
187	C₂²×C₄	—	27	4	no	—
188	C₂²×C₄	—	54	4	no	—
189	C₂²×C₄	—	54	4	no	—
190	C₂²×C₄	C₃⋊Dic₉	1	4	yes	complemented
191	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
192	C₂²×C₃	C₄×Dic₉	1	12	yes	—
193	C₂²×C₃	Dic₉⋊C₄	1	24	yes	—
194	C₂²×C₃	C₄⋊Dic₉	1	12	yes	—
195	C₂²×C₃	(C₂×D₉)⋊C₄	1	24	yes	—
196	C₂²×C₃	#3676	1	12	yes	—
197	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
198	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
199	C₂²×C₃	C₂×D₃₆	1	3	yes	—
200	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
201	C₂²×C₃	C₂×#611	1	6	yes	—
202	C₂²×C₃	C₄×C₃⋊Dic₃	1	4	yes	—
203	C₂²×C₃	#3750	1	8	yes	—
204	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
205	C₂²×C₃	#3752	1	8	yes	—
206	C₂²×C₃	#3757	1	4	yes	—
207	C₂²×C₃	C₂×#634	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	2	yes	—
209	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
210	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
211	C₂²×C₃	C₂×#638	1	2	yes	—
212	C₃×C₄	—	2	12	no	—
213	C₃×C₄	—	2	4	no	—
214	Dic₃	—	18	24	no	—
215	Dic₃	—	18	8	no	—
216	C₃²	C₂×C₃⋊#95	1	1	yes	characteristic
217	C₉	C₂×C₃⋊#95	1	3	yes	complemented
218	C₂³	#6226	1	2	yes	—
219	C₂³	C₄×C₃⋊D₉	1	4	yes	—
220	C₂³	C₄⋊(C₃⋊D₉)	1	2	yes	—
221	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
222	C₂³	#6233	1	4	yes	—
223	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	27	8	no	—
227	C₂×C₄	—	54	4	no	—
228	C₂×C₄	—	54	4	no	—
229	C₂×C₄	—	54	16	no	—
230	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	complemented
231	C₂×C₃	C₉⋊#95	1	24	yes	complemented
232	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
233	C₂×C₃	C₂×Dic₉⋊C₄	1	6	yes	—
234	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
235	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	6	yes	—
236	C₂×C₃	C₂×#3676	1	3	yes	—
237	C₂×C₃	#63658	1	8	yes	—
238	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
239	C₂×C₃	C₂×#3750	1	2	yes	—
240	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
241	C₂×C₃	C₂×#3752	1	2	yes	—
242	C₂×C₃	C₂×#3757	1	1	yes	characteristic
243	C₂²	C₄×C₃⋊Dic₉	1	4	yes	—
244	C₂²	#88534	1	8	yes	—
245	C₂²	C₄⋊(C₃⋊Dic₉)	1	4	yes	—
246	C₂²	#88536	1	8	yes	—
247	C₂²	#88551	1	4	yes	—
248	C₂²	C₂×#6226	1	1	yes	characteristic
249	C₂²	C₂×C₄×C₃⋊D₉	1	2	yes	—
250	C₂²	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
251	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
252	C₂²	C₂×#6233	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	54	8	no	—
255	C₃	C₂×C₉⋊#95	1	3	yes	complemented
256	C₃	C₂×#63658	1	1	yes	characteristic
257	C₂	#11720572	1	8	yes	direct product factor
258	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic
259	C₂	C₂×#88534	1	2	yes	—
260	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
261	C₂	C₂×#88536	1	2	yes	—
262	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#95
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
3. #11720572	C₂	direct (N × H)
8. C₂²×C₃×C₄×C₉	C₄	semidirect
12. C₂²×C₃⋊Dic₉	C₄	semidirect
28. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
32. C₂×C₃⋊Dic₉	C₂×C₄	semidirect
45. C₂²×C₄×C₉	Dic₃	semidirect
90. C₂×C₄×C₉	C₂×Dic₃	semidirect
96. C₂×C₃×C₉	#95	semidirect
106. C₂²×C₃×C₄	Dic₉	semidirect
145. C₃×C₉	C₂×#95	semidirect
162. C₂×C₃×C₄	C₂×Dic₉	semidirect
178. C₂×C₉	C₃⋊#95	semidirect
190. C₂²×C₄	C₃⋊Dic₉	semidirect
217. C₉	C₂×C₃⋊#95	semidirect
230. C₂×C₄	C₂×C₃⋊Dic₉	semidirect
231. C₂×C₃	C₉⋊#95	semidirect
255. C₃	C₂×C₉⋊#95	semidirect
257. C₂	#11720572	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
C₂³×C₃⋊Dic₉	2
#11720572	2
C₂×#63658	3
C₂×C₉⋊#95	3

Subgroups — C2×#11720572 #13144798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720572 #13144798