Subgroups — C₂×#11720289 #13139413

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#88532	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
4	#11720289	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63560	—	3	1	no	maximal
6	C₂×#63382	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₉	C₄	1	2	yes	—
9	C₂×C₃×Dic₁₈	—	2	1	no	—
10	C₂×C₃×Dic₁₈	—	2	1	no	—
11	C₂×C₃×Dic₁₈	—	2	1	no	—
12	C₂×C₃×Dic₁₈	—	2	1	no	—
13	C₂×C₃×Dic₁₈	C₄	1	2	yes	—
14	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
15	#88532	—	2	1	no	—
16	#88532	C₂²	1	2	yes	—
17	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
18	C₉×#208	—	2	1	no	—
19	C₉×#208	C₂²	1	2	yes	—
20	C₂×C₉×C₃⋊C₈	—	2	1	no	—
21	C₂²×C₃×Dic₆	—	3	1	no	—
22	C₂×#3748	—	3	1	no	—
23	C₂×C₃×#208	D₃	1	1	yes	characteristic
24	C₂²×Dic₁₈	D₃	1	1	yes	characteristic
25	#63560	—	3	4	no	—
26	C₂×C₉×M₁₆	—	3	1	no	—
27	C₂×#3667	—	3	1	no	—
28	C₂×#3667	—	6	1	no	—
29	#63382	—	3	4	no	—
30	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₂×C₃×Dic₉	—	2	2	no	—
38	C₂×C₃×Dic₉	—	2	2	no	—
39	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
40	C₃×Dic₁₈	—	2	2	no	—
41	C₃×Dic₁₈	—	2	2	no	—
42	C₃×Dic₁₈	—	2	2	no	—
43	C₃×Dic₁₈	—	2	2	no	—
44	C₃⋊(C₉⋊C₈)	—	2	2	no	—
45	C₉×C₃⋊C₈	—	2	2	no	—
46	C₂×#836	—	9	1	no	—
47	C₂×#824	—	9	1	no	—
48	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×Dic₃	—	3	2	no	—
50	C₂×C₃×Dic₆	—	3	2	no	—
51	C₂×C₃×Dic₆	—	3	2	no	—
52	C₂×C₃×Dic₆	—	6	1	no	—
53	C₂×C₃×Dic₆	—	6	1	no	—
54	C₂×C₃×Dic₆	—	6	1	no	—
55	C₂×C₃×Dic₆	—	6	1	no	—
56	#3748	—	3	2	no	—
57	#3748	—	6	1	no	—
58	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
59	C₃×#208	—	2	1	no	—
60	C₃×#208	C₂×D₃	1	2	yes	—
61	C₂×C₃×C₃⋊C₈	—	2	1	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
64	C₂²×Dic₉	Dic₃	1	2	yes	—
65	C₂×Dic₁₈	—	2	1	no	—
66	C₂×Dic₁₈	—	2	1	no	—
67	C₂×Dic₁₈	—	2	1	no	—
68	C₂×Dic₁₈	—	2	1	no	—
69	C₂×Dic₁₈	Dic₃	1	2	yes	—
70	C₂×Dic₁₈	C₂×D₃	1	2	yes	—
71	C₉×M₁₆	—	3	2	no	—
72	C₉×M₁₆	—	6	1	no	—
73	C₂×C₈×C₉	—	6	1	no	—
74	#3667	—	3	2	no	—
75	#3667	—	6	2	no	—
76	#3667	—	6	2	no	—
77	#3667	—	6	1	no	—
78	C₂×C₉⋊C₈	—	6	2	no	—
79	C₂×C₉⋊C₈	—	6	1	no	—
80	C₂²×C₃×C₉	—	2	2	no	—
81	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₃×Dic₉	—	2	4	no	—
89	C₂²×C₃×Q₈	—	9	1	no	—
90	C₂²×Dic₆	—	3	1	no	—
91	C₂×C₃×M₁₆	—	3	1	no	—
92	C₂×#208	—	9	1	no	—
93	C₂×#208	—	9	1	no	—
94	C₂×#208	—	18	1	no	—
95	C₂×#208	D₉	1	1	yes	characteristic
96	#836	—	9	4	no	—
97	#824	—	9	4	no	—
98	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
100	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
101	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
102	C₂×C₃²×C₄	#67	1	1	yes	characteristic
103	C₂×C₃²×C₄	#67	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	3	4	no	—
106	C₂×C₃×Dic₃	—	6	2	no	—
107	C₂×C₃×Dic₃	—	6	2	no	—
108	C₃×Dic₆	—	6	2	no	—
109	C₃×Dic₆	—	6	2	no	—
110	C₃×Dic₆	—	6	2	no	—
111	C₃×Dic₆	—	6	2	no	—
112	C₂³×C₉	—	2	1	no	—
113	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
114	C₃⋊(C₃⋊C₈)	—	6	2	no	—
115	C₃×C₃⋊C₈	—	2	2	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	2	1	no	—
122	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
123	C₂×C₄×C₉	#67	1	1	yes	characteristic
124	C₂×C₄×C₉	#67	1	1	yes	characteristic
125	C₂×C₄×C₉	#67	1	1	yes	characteristic
126	C₂×C₄×C₉	#67	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
128	C₂×Dic₉	—	2	2	no	—
129	C₂×Dic₉	—	2	2	no	—
130	C₂×Dic₉	C₂×Dic₃	1	4	yes	—
131	Dic₁₈	—	2	2	no	—
132	Dic₁₈	—	2	2	no	—
133	Dic₁₈	—	2	2	no	—
134	Dic₁₈	—	2	2	no	—
135	C₈×C₉	—	6	2	no	—
136	C₉⋊C₈	—	6	4	no	—
137	C₉⋊C₈	—	6	2	no	—
138	C₂×#101	—	27	1	no	Sylow
139	C₂×C₃×C₉	—	2	1	no	—
140	C₂×C₃×C₉	—	2	1	no	—
141	C₂×C₃×C₉	#101	1	2	yes	complemented
142	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₃×Q₈	—	9	2	no	—
144	C₂×C₃×Q₈	—	9	2	no	—
145	C₂×C₃×Q₈	—	18	1	no	—
146	C₂×C₃×Q₈	—	18	1	no	—
147	C₂×C₃×Q₈	—	18	1	no	—
148	C₂×C₃×Q₈	—	18	1	no	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	—	9	2	no	—
151	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
152	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
153	C₂²×Dic₃	—	3	2	no	—
154	C₂×Dic₆	—	3	2	no	—
155	C₂×Dic₆	—	3	2	no	—
156	C₂×Dic₆	—	6	1	no	—
157	C₂×Dic₆	—	6	1	no	—
158	C₂×Dic₆	—	6	1	no	—
159	C₂×Dic₆	—	6	1	no	—
160	C₃×M₁₆	—	3	2	no	—
161	C₃×M₁₆	—	6	1	no	—
162	C₂×C₃×C₈	—	6	1	no	—
163	#208	—	2	1	no	—
164	#208	—	9	2	no	—
165	#208	—	9	2	no	—
166	#208	—	18	2	no	—
167	#208	—	18	2	no	—
168	#208	—	18	1	no	—
169	#208	—	18	1	no	—
170	#208	C₂×D₉	1	2	yes	—
171	C₂×C₃⋊C₈	—	2	1	no	—
172	C₂×C₃⋊C₈	—	18	2	no	—
173	C₂×C₃⋊C₈	—	18	1	no	—
174	C₂×C₃⋊C₈	—	18	1	no	—
175	C₂²×C₃²	—	2	2	no	—
176	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
179	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
182	C₃²×C₄	C₂×#67	1	1	yes	characteristic
183	C₃×Dic₃	—	6	4	no	—
184	C₂²×C₉	—	2	2	no	—
185	C₂²×C₉	—	2	1	no	—
186	C₂²×C₉	—	2	1	no	—
187	C₂²×C₉	—	2	1	no	—
188	C₂²×C₉	—	4	2	no	—
189	C₂²×C₉	#217	1	1	yes	characteristic
190	C₂²×C₉	#217	1	1	yes	characteristic
191	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	2	1	no	—
195	C₄×C₉	—	2	1	no	—
196	C₄×C₉	#217	1	1	yes	characteristic
197	C₄×C₉	#217	1	1	yes	characteristic
198	C₄×C₉	C₂×#67	1	1	yes	characteristic
199	C₄×C₉	C₂×#67	1	1	yes	characteristic
200	Dic₉	—	2	4	no	—
201	C₂²×Q₈	—	9	1	no	—
202	C₂×M₁₆	—	3	1	no	—
203	C₂×M₁₆	—	27	1	no	—
204	#101	—	27	4	no	—
205	C₃×C₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	2	1	no	—
207	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
208	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
209	C₃×Q₈	—	18	2	no	—
210	C₃×Q₈	—	18	2	no	—
211	C₃×Q₈	—	18	2	no	—
212	C₃×Q₈	—	18	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	9	4	no	—
220	C₂×C₃×C₄	—	18	2	no	—
221	C₂×C₃×C₄	—	18	2	no	—
222	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
223	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
224	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
225	C₂×C₃×C₄	#611	1	1	yes	characteristic
226	C₂×C₃×C₄	#611	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
228	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
229	C₂×C₃×C₄	#625	1	1	yes	characteristic
230	C₂×C₃×C₄	#625	1	1	yes	characteristic
231	C₂×C₃×C₄	#626	1	1	yes	characteristic
232	C₂×C₃×C₄	#626	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
234	C₂×Dic₃	—	3	4	no	—
235	C₂×Dic₃	—	6	2	no	—
236	C₂×Dic₃	—	6	2	no	—
237	Dic₆	—	6	2	no	—
238	Dic₆	—	6	2	no	—
239	Dic₆	—	6	2	no	—
240	Dic₆	—	6	2	no	—
241	C₃×C₈	—	6	2	no	—
242	C₃⋊C₈	—	2	2	no	—
243	C₃⋊C₈	—	18	4	no	—
244	C₃⋊C₈	—	18	2	no	—
245	C₃⋊C₈	—	18	2	no	—
246	C₂×C₃²	—	2	1	no	—
247	C₂×C₃²	—	2	1	no	—
248	C₂×C₃²	#824	1	2	yes	—
249	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₂×C₉	—	2	1	no	—
251	C₂×C₉	—	2	1	no	—
252	C₂×C₉	—	2	2	no	—
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	4	1	no	—
255	C₂×C₉	—	4	1	no	—
256	C₂×C₉	#836	1	2	yes	complemented
257	C₂×C₉	C₂×#217	1	1	yes	characteristic
258	C₂×Q₈	—	9	2	no	—
259	C₂×Q₈	—	9	2	no	—
260	C₂×Q₈	—	18	1	no	—
261	C₂×Q₈	—	18	1	no	—
262	C₂×Q₈	—	18	1	no	—
263	C₂×Q₈	—	18	1	no	—
264	C₂²×C₄	—	9	2	no	—
265	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
266	M₁₆	—	3	2	no	—
267	M₁₆	—	6	1	no	—
268	M₁₆	—	27	2	no	—
269	M₁₆	—	54	1	no	—
270	C₂×C₈	—	6	1	no	—
271	C₂×C₈	—	54	1	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
279	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
280	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
281	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
282	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
283	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	18	4	no	—
289	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
290	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
291	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
292	C₃×C₄	C₂×#611	1	1	yes	characteristic
293	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
294	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
295	C₃×C₄	C₂×#625	1	1	yes	characteristic
296	C₃×C₄	C₂×#626	1	1	yes	characteristic
297	Dic₃	—	6	4	no	—
298	C₃²	C₂×#824	1	1	yes	characteristic
299	C₉	—	2	1	no	—
300	C₉	C₂×#836	1	1	yes	characteristic, complemented
301	C₂³	Dic₃×D₉	1	1	yes	characteristic
302	Q₈	—	18	2	no	—
303	Q₈	—	18	2	no	—
304	Q₈	—	18	2	no	—
305	Q₈	—	18	2	no	—
306	C₂×C₄	—	9	4	no	—
307	C₂×C₄	—	18	2	no	—
308	C₂×C₄	—	18	2	no	—
309	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
310	C₂×C₄	#6192	1	1	yes	characteristic
311	C₂×C₄	#6192	1	1	yes	characteristic
312	C₂×C₄	#6194	1	1	yes	characteristic
313	C₂×C₄	#6194	1	1	yes	characteristic
314	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
315	C₈	—	6	2	no	—
316	C₈	—	54	2	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	2	no	—
323	C₂×C₃	—	4	1	no	—
324	C₂×C₃	—	4	1	no	—
325	C₂×C₃	#63382	1	2	yes	complemented
326	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
327	C₂×C₃	#63560	1	2	yes	—
328	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
329	C₂²	—	2	2	no	—
330	C₂²	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
331	C₂²	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
332	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
333	C₄	—	18	4	no	—
334	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
335	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
336	C₄	C₂×#6192	1	1	yes	characteristic
337	C₄	C₂×#6194	1	1	yes	characteristic
338	C₃	—	2	1	no	—
339	C₃	C₂×#63382	1	1	yes	characteristic, complemented
340	C₃	C₂×#63560	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	#11720289	1	2	yes	direct product factor
344	C₂	C₂×(C₂×D₉)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₉)⋊Dic₃
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#101
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720289	C₂	direct (N × H)
141. C₂×C₃×C₉	#101	semidirect
205. C₃×C₉	C₂×#101	semidirect
256. C₂×C₉	#836	semidirect
300. C₉	C₂×#836	semidirect
325. C₂×C₃	#63382	semidirect
339. C₃	C₂×#63382	semidirect
343. C₂	#11720289	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×#88532	2
C₂×C₉×#208	2
#11720289	2
C₂×#63560	3
C₂×#63382	3

Subgroups — C2×#11720289 #13139413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720289 #13139413