Subgroups — C₂×#11720273 #13139360

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720273	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63572	—	3	1	no	maximal
6	C₂×Dic₉⋊C₈	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₄	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₄	1	1	yes	characteristic
10	C₄×C₃⋊Dic₉	C₄	1	2	yes	—
11	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
12	C₂×C₉×C₃⋊C₈	—	2	2	no	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₉⋊C₈	—	2	2	no	—
15	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
19	#63572	—	3	4	no	—
20	C₂²×C₈×C₉	—	3	1	no	—
21	C₂×C₄×Dic₉	—	3	1	no	—
22	C₂×C₄×Dic₉	—	6	1	no	—
23	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic
24	Dic₉⋊C₈	—	3	4	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	C₈	1	4	yes	complemented
32	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
33	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
34	C₉×C₃⋊C₈	—	2	2	no	—
35	C₃×C₉⋊C₈	—	2	2	no	—
36	C₂×Dic₃⋊C₈	—	9	1	no	—
37	C₂×Dic₃⋊C₈	—	9	1	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₃⋊Dic₃	—	3	1	no	—
40	C₂²×C₃⋊Dic₃	—	3	1	no	—
41	C₄×C₃⋊Dic₃	—	3	2	no	—
42	C₄×C₃⋊Dic₃	—	3	2	no	—
43	C₂×C₃×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×C₃⋊C₈	—	3	2	no	—
45	C₂×C₃×C₃⋊C₈	—	6	2	no	—
46	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
49	C₂²×Dic₉	—	3	1	no	—
50	C₂²×Dic₉	—	3	1	no	—
51	C₂²×Dic₉	—	6	1	no	—
52	C₂²×Dic₉	—	6	1	no	—
53	C₂×C₈×C₉	—	3	2	no	—
54	C₂×C₈×C₉	—	6	2	no	—
55	C₄×Dic₉	—	3	2	no	—
56	C₄×Dic₉	—	3	2	no	—
57	C₄×Dic₉	—	6	2	no	—
58	C₄×Dic₉	—	6	2	no	—
59	C₂×C₉⋊C₈	—	2	2	no	—
60	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
61	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
63	C₂²×C₃×C₉	M₁₆	1	2	yes	—
64	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
68	C₃⋊Dic₉	—	2	2	no	—
69	C₃⋊Dic₉	C₂×C₈	1	4	yes	complemented
70	C₂²×C₃×C₈	—	3	1	no	—
71	C₂²×C₃×C₈	—	9	1	no	—
72	C₂×C₄×Dic₃	—	9	1	no	—
73	C₂×C₄×Dic₃	—	9	1	no	—
74	C₂×C₄×Dic₃	—	18	1	no	—
75	C₂²×C₃⋊C₈	—	3	1	no	—
76	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
77	Dic₃⋊C₈	—	9	4	no	—
78	Dic₃⋊C₈	—	9	4	no	—
79	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	Dic₆	1	2	yes	—
81	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	#67	1	2	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	—	3	4	no	—
85	C₂×C₃⋊Dic₃	—	3	2	no	—
86	C₂×C₃⋊Dic₃	—	3	2	no	—
87	C₂×C₃⋊Dic₃	—	6	2	no	—
88	C₂³×C₉	—	2	1	no	—
89	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
90	C₃×C₃⋊C₈	—	2	2	no	—
91	C₃×C₃⋊C₈	—	6	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	Dic₆	1	2	yes	—
97	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
98	C₂×C₄×C₉	#67	1	2	yes	—
99	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
100	C₂×Dic₉	—	3	4	no	—
101	C₂×Dic₉	—	3	2	no	—
102	C₂×Dic₉	—	3	2	no	—
103	C₂×Dic₉	—	6	2	no	—
104	C₂×Dic₉	—	6	2	no	—
105	C₂×Dic₉	—	6	4	no	—
106	C₂×Dic₉	—	6	4	no	—
107	C₂×Dic₉	—	6	2	no	—
108	C₈×C₉	—	6	2	no	—
109	C₉⋊C₈	—	2	2	no	—
110	C₂×C₄⋊C₈	—	27	1	no	Sylow
111	C₂×C₃×C₉	C₄⋊C₈	1	4	yes	complemented
112	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
117	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
118	C₂²×Dic₃	—	9	1	no	—
119	C₂²×Dic₃	—	9	1	no	—
120	C₂²×Dic₃	—	9	1	no	—
121	C₂²×Dic₃	—	9	1	no	—
122	C₂²×Dic₃	—	18	1	no	—
123	C₂²×Dic₃	—	18	1	no	—
124	C₂×C₃×C₈	—	3	2	no	—
125	C₂×C₃×C₈	—	6	2	no	—
126	C₂×C₃×C₈	—	9	2	no	—
127	C₂×C₃×C₈	—	18	2	no	—
128	C₄×Dic₃	—	9	2	no	—
129	C₄×Dic₃	—	9	2	no	—
130	C₄×Dic₃	—	9	2	no	—
131	C₄×Dic₃	—	9	2	no	—
132	C₄×Dic₃	—	18	2	no	—
133	C₄×Dic₃	—	18	2	no	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
138	C₂²×C₃²	D₃×C₈	1	2	yes	—
139	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
141	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
142	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
143	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
144	C₃²×C₄	C₂×#67	1	1	yes	characteristic
145	C₃⋊Dic₃	—	3	4	no	—
146	C₃⋊Dic₃	—	6	2	no	—
147	C₂²×C₉	—	2	2	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	D₃×C₈	1	2	yes	—
152	C₂²×C₉	ΓL(1,𝔽₂₅)	1	2	yes	—
153	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
154	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
155	C₄×C₉	—	2	2	no	—
156	C₄×C₉	—	2	1	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
159	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
160	C₄×C₉	C₂×#67	1	1	yes	characteristic
161	Dic₉	—	3	4	no	—
162	Dic₉	—	6	4	no	—
163	Dic₉	—	6	4	no	—
164	Dic₉	—	6	2	no	—
165	C₂²×C₈	—	3	1	no	—
166	C₂²×C₈	—	9	1	no	—
167	C₂×C₄²	—	27	1	no	—
168	C₄⋊C₈	—	27	4	no	—
169	C₃×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
172	C₂³×C₃	#624	1	1	yes	characteristic
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
178	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
179	C₂×C₃×C₄	#611	1	2	yes	—
180	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
181	C₂×C₃×C₄	#624	1	1	yes	characteristic
182	C₂×C₃×C₄	#625	1	2	yes	—
183	C₂×C₃×C₄	#627	1	2	yes	—
184	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
185	C₂×Dic₃	—	9	4	no	—
186	C₂×Dic₃	—	9	2	no	—
187	C₂×Dic₃	—	9	4	no	—
188	C₂×Dic₃	—	9	2	no	—
189	C₂×Dic₃	—	9	2	no	—
190	C₂×Dic₃	—	9	2	no	—
191	C₂×Dic₃	—	18	2	no	—
192	C₂×Dic₃	—	18	2	no	—
193	C₂×Dic₃	—	18	4	no	—
194	C₂×Dic₃	—	18	4	no	—
195	C₂×Dic₃	—	18	2	no	—
196	C₂×Dic₃	—	18	2	no	—
197	C₃×C₈	—	6	2	no	—
198	C₃×C₈	—	18	2	no	—
199	C₃⋊C₈	—	2	2	no	—
200	C₃⋊C₈	—	6	2	no	—
201	C₂×C₃²	Dic₃⋊C₈	1	4	yes	—
202	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
203	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
204	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	—	2	4	no	—
208	C₂×C₉	—	2	1	no	—
209	C₂×C₉	Dic₃⋊C₈	1	4	yes	complemented
210	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
211	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
212	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	—	27	1	no	—
214	C₂²×C₄	—	27	1	no	—
215	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	6	2	no	—
218	C₂×C₈	—	9	2	no	—
219	C₂×C₈	—	18	2	no	—
220	C₄²	—	27	2	no	—
221	C₄²	—	27	2	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	C₈×D₉	1	2	yes	—
227	C₂²×C₃	#3663	1	2	yes	—
228	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
229	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
230	C₂²×C₃	#3710	1	2	yes	—
231	C₂²×C₃	#3712	1	2	yes	—
232	C₂²×C₃	#3724	1	2	yes	—
233	C₂²×C₃	C₂×#624	1	1	yes	characteristic
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
238	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
239	C₃×C₄	C₂×#611	1	1	yes	characteristic
240	C₃×C₄	#3724	1	2	yes	—
241	C₃×C₄	C₂×#625	1	1	yes	characteristic
242	C₃×C₄	C₂×#627	1	1	yes	characteristic
243	Dic₃	—	9	4	no	—
244	Dic₃	—	9	4	no	—
245	Dic₃	—	18	4	no	—
246	Dic₃	—	18	4	no	—
247	Dic₃	—	18	2	no	—
248	Dic₃	—	18	2	no	—
249	C₃²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
250	C₉	—	2	1	no	—
251	C₉	C₂×Dic₃⋊C₈	1	1	yes	characteristic, complemented
252	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	—	27	2	no	—
254	C₂×C₄	—	27	4	no	—
255	C₂×C₄	—	27	2	no	—
256	C₂×C₄	—	54	2	no	—
257	C₂×C₄	#6189	1	2	yes	—
258	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#6194	1	2	yes	—
260	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
261	C₈	—	6	2	no	—
262	C₈	—	18	2	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	4	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	Dic₉⋊C₈	1	4	yes	complemented
268	C₂×C₃	C₂×C₈×D₉	1	1	yes	characteristic
269	C₂×C₃	C₂×#3663	1	1	yes	characteristic
270	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
271	C₂×C₃	#63572	1	4	yes	—
272	C₂×C₃	C₂×#3710	1	1	yes	characteristic
273	C₂×C₃	C₂×#3712	1	1	yes	characteristic
274	C₂×C₃	C₂×#3724	1	1	yes	characteristic
275	C₂²	(C₃⋊D₉)⋊C₈	1	2	yes	—
276	C₂²	#88414	1	2	yes	—
277	C₂²	#88442	1	2	yes	—
278	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
279	C₄	—	27	4	no	—
280	C₄	—	54	2	no	—
281	C₄	#88442	1	2	yes	—
282	C₄	C₂×#6189	1	1	yes	characteristic
283	C₄	C₂×#6194	1	1	yes	characteristic
284	C₃	—	2	1	no	—
285	C₃	C₂×Dic₉⋊C₈	1	1	yes	characteristic, complemented
286	C₃	C₂×#63572	1	1	yes	characteristic
287	C₂	#11720273	1	4	yes	direct product factor
288	C₂	C₂×(C₃⋊D₉)⋊C₈	1	1	yes	characteristic
289	C₂	C₂×#88414	1	1	yes	characteristic
290	C₂	C₂×#88442	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypocenter	C₃×C₉	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720273	C₂	direct (N × H)
31. C₂×C₃⋊Dic₉	C₈	semidirect
69. C₃⋊Dic₉	C₂×C₈	semidirect
111. C₂×C₃×C₉	C₄⋊C₈	semidirect
169. C₃×C₉	C₂×C₄⋊C₈	semidirect
209. C₂×C₉	Dic₃⋊C₈	semidirect
251. C₉	C₂×Dic₃⋊C₈	semidirect
267. C₂×C₃	Dic₉⋊C₈	semidirect
285. C₃	C₂×Dic₉⋊C₈	semidirect
287. C₂	#11720273	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂²×C₉×C₃⋊C₈	2
C₂²×C₃×C₉⋊C₈	2
#11720273	2
C₂×#63572	3
C₂×Dic₉⋊C₈	3

Subgroups — C2×#11720273 #13139360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720273 #13139360