Subgroups — C₂×#11720276 #13139367

Order 1728.

578 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720276	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63566	—	3	1	no	maximal
6	C₂×#63376	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	—	2	1	no	—
9	C₂²×C₃×Dic₉	—	2	1	no	—
10	C₂×C₃×Dic₁₈	—	2	1	no	—
11	C₂×C₃×Dic₁₈	—	2	2	no	—
12	C₂×C₃×Dic₁₈	C₄	1	4	yes	complemented
13	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
16	C₂×C₉×C₃⋊C₈	—	2	2	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₆	—	3	1	no	—
19	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
20	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
21	C₂²×Dic₁₈	D₃	1	1	yes	characteristic
22	#63566	—	3	4	no	—
23	C₂²×C₈×C₉	—	3	1	no	—
24	C₂×C₄⋊Dic₉	—	3	1	no	—
25	C₂×C₄⋊Dic₉	—	6	1	no	—
26	#63376	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₃⋊Dic₉	—	4	2	no	—
34	C₂×C₃×Dic₉	—	2	4	no	—
35	C₂×C₃×Dic₉	—	2	2	no	—
36	C₃×Dic₁₈	—	2	4	no	—
37	C₃×Dic₁₈	—	2	2	no	—
38	C₃×Dic₁₈	C₂×C₄	1	4	yes	complemented
39	C₉×C₃⋊C₈	—	2	2	no	—
40	C₂×#835	—	9	1	no	—
41	C₂×#816	—	9	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	6	1	no	—
44	C₂²×C₃×Dic₃	—	6	1	no	—
45	C₂×C₃×Dic₆	—	3	4	no	—
46	C₂×C₃×Dic₆	—	3	2	no	—
47	C₂×C₃×Dic₆	—	6	1	no	—
48	C₂×C₃×Dic₆	—	6	2	no	—
49	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
50	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
51	C₂×C₃×C₃⋊C₈	—	2	2	no	—
52	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
53	C₂²×C₄×C₉	—	2	1	no	—
54	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
55	C₂²×Dic₉	—	2	1	no	—
56	C₂²×Dic₉	—	6	2	no	—
57	C₂²×Dic₉	—	6	1	no	—
58	C₂×Dic₁₈	—	2	1	no	—
59	C₂×Dic₁₈	—	2	2	no	—
60	C₂×Dic₁₈	Dic₃	1	4	yes	complemented
61	C₂×Dic₁₈	C₂×D₃	1	2	yes	—
62	C₂×C₈×C₉	—	3	2	no	—
63	C₂×C₈×C₉	—	6	2	no	—
64	C₄⋊Dic₉	—	3	2	no	—
65	C₄⋊Dic₉	—	6	2	no	—
66	C₄⋊Dic₉	—	6	2	no	—
67	C₄⋊Dic₉	—	6	1	no	—
68	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
70	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
71	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
73	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₃⋊Dic₉	—	4	2	no	—
76	C₃×Dic₉	—	2	4	no	—
77	C₂²×C₃×Q₈	—	9	1	no	—
78	C₂²×Dic₆	—	3	1	no	—
79	C₂²×C₃×C₈	—	3	1	no	—
80	C₂×C₄⋊Dic₃	—	9	1	no	—
81	C₂×C₄⋊Dic₃	—	9	1	no	—
82	C₂×C₄⋊Dic₃	—	18	1	no	—
83	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
84	#835	—	9	4	no	—
85	#816	—	9	4	no	—
86	C₂³×C₃²	D₁₂	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
88	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
89	C₂×C₃²×C₄	#67	1	2	yes	—
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃⋊Dic₃	—	6	2	no	—
92	C₂×C₃⋊Dic₃	—	12	2	no	—
93	C₂×C₃×Dic₃	—	6	4	no	—
94	C₂×C₃×Dic₃	—	6	2	no	—
95	C₃×Dic₆	—	3	4	no	—
96	C₃×Dic₆	—	6	4	no	—
97	C₃×Dic₆	—	6	2	no	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	#67	1	1	yes	characteristic
100	C₃×C₃⋊C₈	—	2	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
106	C₂×C₄×C₉	#67	1	2	yes	—
107	C₂×C₄×C₉	#67	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
109	C₂×Dic₉	—	2	4	no	—
110	C₂×Dic₉	—	2	2	no	—
111	C₂×Dic₉	—	6	4	no	—
112	C₂×Dic₉	—	6	2	no	—
113	C₂×Dic₉	—	12	4	no	—
114	C₂×Dic₉	—	12	2	no	—
115	Dic₁₈	—	2	4	no	—
116	Dic₁₈	—	2	2	no	—
117	Dic₁₈	C₂×Dic₃	1	4	yes	complemented
118	C₈×C₉	—	6	2	no	—
119	C₂×#103	—	27	1	no	Sylow
120	C₂×C₃×C₉	#103	1	4	yes	complemented
121	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₃×Q₈	—	9	4	no	—
125	C₂×C₃×Q₈	—	9	2	no	—
126	C₂×C₃×Q₈	—	18	1	no	—
127	C₂×C₃×Q₈	—	18	2	no	—
128	C₂²×C₃×C₄	—	2	1	no	—
129	C₂²×C₃×C₄	—	18	1	no	—
130	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
132	C₂²×Dic₃	—	6	1	no	—
133	C₂²×Dic₃	—	18	2	no	—
134	C₂²×Dic₃	—	18	1	no	—
135	C₂²×Dic₃	—	18	1	no	—
136	C₂×Dic₆	—	3	4	no	—
137	C₂×Dic₆	—	3	2	no	—
138	C₂×Dic₆	—	6	2	no	—
139	C₂×Dic₆	—	6	1	no	—
140	C₂×C₃×C₈	—	3	2	no	—
141	C₂×C₃×C₈	—	6	2	no	—
142	C₄⋊Dic₃	—	9	2	no	—
143	C₄⋊Dic₃	—	9	2	no	—
144	C₄⋊Dic₃	—	18	2	no	—
145	C₄⋊Dic₃	—	18	2	no	—
146	C₄⋊Dic₃	—	18	1	no	—
147	C₄⋊Dic₃	—	18	1	no	—
148	C₂×C₃⋊C₈	—	2	2	no	—
149	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
150	C₂²×C₃²	#204	1	2	yes	—
151	C₂²×C₃²	Dic₁₂	1	2	yes	—
152	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
153	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
154	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
156	C₃²×C₄	C₂×#67	1	1	yes	characteristic
157	C₃⋊Dic₃	—	12	2	no	—
158	C₃×Dic₃	—	6	4	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	#215	1	2	yes	—
164	C₂²×C₉	#216	1	2	yes	—
165	C₂²×C₉	#217	1	2	yes	—
166	C₂²×C₉	C₂×#67	1	1	yes	characteristic
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	#217	1	2	yes	—
171	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
172	C₄×C₉	C₂×#67	1	1	yes	characteristic
173	Dic₉	—	2	4	no	—
174	Dic₉	—	12	4	no	—
175	Dic₉	—	12	2	no	—
176	C₂²×Q₈	—	9	1	no	—
177	C₂²×C₈	—	3	1	no	—
178	C₂×C₄⋊C₄	—	27	1	no	—
179	#103	—	27	4	no	—
180	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₃	—	2	1	no	—
182	C₂³×C₃	D₃₆	1	1	yes	characteristic
183	C₂³×C₃	#626	1	1	yes	characteristic
184	C₃×Q₈	—	9	4	no	—
185	C₃×Q₈	—	18	4	no	—
186	C₃×Q₈	—	18	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	18	4	no	—
192	C₂×C₃×C₄	—	18	2	no	—
193	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
194	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
195	C₂×C₃×C₄	#611	1	2	yes	—
196	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
197	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
198	C₂×C₃×C₄	#625	1	2	yes	—
199	C₂×C₃×C₄	#626	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
201	C₂×Dic₃	—	6	4	no	—
202	C₂×Dic₃	—	6	2	no	—
203	C₂×Dic₃	—	18	4	no	—
204	C₂×Dic₃	—	18	2	no	—
205	C₂×Dic₃	—	18	2	no	—
206	C₂×Dic₃	—	36	4	no	—
207	C₂×Dic₃	—	36	2	no	—
208	C₂×Dic₃	—	36	2	no	—
209	Dic₆	—	3	4	no	—
210	Dic₆	—	6	4	no	—
211	Dic₆	—	6	2	no	—
212	C₃×C₈	—	6	2	no	—
213	C₃⋊C₈	—	2	2	no	—
214	C₂×C₃²	#816	1	4	yes	—
215	C₂×C₃²	C₂×#204	1	1	yes	characteristic
216	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
217	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₉	—	2	4	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	2	1	no	—
222	C₂×C₉	#835	1	4	yes	complemented
223	C₂×C₉	C₂×#215	1	1	yes	characteristic
224	C₂×C₉	C₂×#216	1	1	yes	characteristic
225	C₂×C₉	C₂×#217	1	1	yes	characteristic
226	C₂×Q₈	—	9	4	no	—
227	C₂×Q₈	—	9	2	no	—
228	C₂×Q₈	—	18	1	no	—
229	C₂×Q₈	—	18	2	no	—
230	C₂²×C₄	—	18	1	no	—
231	C₂²×C₄	—	54	1	no	—
232	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
233	C₂×C₈	—	3	2	no	—
234	C₂×C₈	—	6	2	no	—
235	C₄⋊C₄	—	27	2	no	—
236	C₄⋊C₄	—	54	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	Dic₃₆	1	2	yes	—
242	C₂²×C₃	#3664	1	2	yes	—
243	C₂²×C₃	(C₂×D₉)⋊C₄	1	2	yes	—
244	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
245	C₂²×C₃	#3717	1	2	yes	—
246	C₂²×C₃	#3719	1	2	yes	—
247	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
248	C₂²×C₃	C₂×#626	1	1	yes	characteristic
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	18	4	no	—
253	C₃×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
254	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
255	C₃×C₄	C₂×#611	1	1	yes	characteristic
256	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
257	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
258	C₃×C₄	C₂×#625	1	1	yes	characteristic
259	Dic₃	—	6	4	no	—
260	Dic₃	—	36	4	no	—
261	Dic₃	—	36	2	no	—
262	Dic₃	—	36	2	no	—
263	C₃²	C₂×#816	1	1	yes	characteristic
264	C₉	—	2	1	no	—
265	C₉	C₂×#835	1	1	yes	characteristic, complemented
266	C₂³	#6192	1	1	yes	characteristic
267	Q₈	—	9	4	no	—
268	Q₈	—	18	2	no	—
269	Q₈	—	18	4	no	—
270	C₂×C₄	—	18	2	no	—
271	C₂×C₄	—	18	4	no	—
272	C₂×C₄	—	54	2	no	—
273	C₂×C₄	—	108	2	no	—
274	C₂×C₄	Dic₃×D₉	1	2	yes	—
275	C₂×C₄	#6192	1	1	yes	characteristic
276	C₂×C₄	#6194	1	2	yes	—
277	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
278	C₈	—	6	2	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	4	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	#63376	1	4	yes	complemented
284	C₂×C₃	C₂×Dic₃₆	1	1	yes	characteristic
285	C₂×C₃	C₂×#3664	1	1	yes	characteristic
286	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
287	C₂×C₃	#63566	1	4	yes	—
288	C₂×C₃	C₂×#3717	1	1	yes	characteristic
289	C₂×C₃	C₂×#3719	1	1	yes	characteristic
290	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
291	C₂²	#88416	1	2	yes	—
292	C₂²	#88418	1	2	yes	—
293	C₂²	(C₂×D₉)⋊Dic₃	1	2	yes	—
294	C₂²	C₂×#6192	1	1	yes	characteristic
295	C₄	—	18	4	no	—
296	C₄	—	108	2	no	—
297	C₄	(C₂×D₉)⋊Dic₃	1	2	yes	—
298	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
299	C₄	C₂×#6194	1	1	yes	characteristic
300	C₃	—	2	1	no	—
301	C₃	C₂×#63376	1	1	yes	characteristic, complemented
302	C₃	C₂×#63566	1	1	yes	characteristic
303	C₂	#11720276	1	4	yes	direct product factor
304	C₂	C₂×#88416	1	1	yes	characteristic
305	C₂	C₂×#88418	1	1	yes	characteristic
306	C₂	C₂×(C₂×D₉)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6192
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	D₁₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#103
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720276	C₂	direct (N × H)
12. C₂×C₃×Dic₁₈	C₄	semidirect
38. C₃×Dic₁₈	C₂×C₄	semidirect
60. C₂×Dic₁₈	Dic₃	semidirect
117. Dic₁₈	C₂×Dic₃	semidirect
120. C₂×C₃×C₉	#103	semidirect
180. C₃×C₉	C₂×#103	semidirect
222. C₂×C₉	#835	semidirect
265. C₉	C₂×#835	semidirect
283. C₂×C₃	#63376	semidirect
301. C₃	C₂×#63376	semidirect
303. C₂	#11720276	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×Dic₁₈	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂²×C₉×C₃⋊C₈	2
#11720276	2
C₂×#63566	3
C₂×#63376	3

Subgroups — C2×#11720276 #13139367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720276 #13139367