Subgroups — C₂×#11720585 #13144851

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720585	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63664	—	3	1	no	maximal
6	C₂×#63396	—	3	3	no	maximal
7	C₃×C₉×#41	—	2	1	no	—
8	C₃×C₉×#41	—	2	1	no	—
9	C₃×C₉×#41	C₄	1	2	yes	complemented
10	C₃×C₉×#41	C₂²	1	2	yes	—
11	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃×D₄×C₉	—	2	1	no	—
13	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₄×C₉	—	2	1	no	—
15	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊Dic₉	—	2	1	no	—
17	C₄×C₃⋊Dic₉	—	2	1	no	—
18	C₄×C₃⋊Dic₉	C₂²	1	2	yes	complemented
19	#88532	—	2	1	no	—
20	#88532	C₂²	1	2	yes	complemented
21	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
22	C₂×C₃²×#41	D₃	1	1	yes	characteristic
23	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
24	C₂×#3748	—	3	1	no	—
25	C₂×C₉×#41	D₃	1	3	yes	—
26	#63664	—	3	4	no	—
27	C₂×C₄×Dic₉	—	3	3	no	—
28	C₂×#3667	—	3	3	no	—
29	#63396	—	3	12	no	—
30	C₂³×C₃×C₉	—	2	1	no	—
31	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
32	C₃×Q₈×C₉	—	2	1	no	—
33	C₃×Q₈×C₉	C₂×C₄	1	2	yes	complemented
34	C₃×D₄×C₉	—	2	2	no	—
35	C₃×D₄×C₉	—	2	2	no	—
36	C₃×D₄×C₉	—	2	1	no	—
37	C₃×D₄×C₉	C₂×C₄	1	2	yes	complemented
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	2	no	—
40	C₂×C₃×C₄×C₉	—	2	2	no	—
41	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
45	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
47	C₂×C₃⋊Dic₉	—	2	4	no	—
48	C₂×C₃⋊Dic₉	—	2	2	no	—
49	C₃⋊(C₉⋊C₈)	—	2	2	no	—
50	C₂×#837	—	9	3	no	—
51	C₂×#837	—	9	1	no	—
52	C₃²×#41	—	2	1	no	—
53	C₃²×#41	—	2	1	no	—
54	C₃²×#41	Dic₃	1	2	yes	—
55	C₃²×#41	C₂×D₃	1	2	yes	—
56	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
57	C₂×C₃²×D₄	—	2	1	no	—
58	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
59	C₂²×C₃²×C₄	—	2	1	no	—
60	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃⋊Dic₃	—	6	1	no	—
62	C₄×C₃⋊Dic₃	—	3	2	no	—
63	C₄×C₃⋊Dic₃	—	6	1	no	—
64	#3748	—	3	2	no	—
65	#3748	—	6	1	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
67	C₉×#41	—	2	3	no	—
68	C₉×#41	—	2	3	no	—
69	C₉×#41	Dic₃	1	6	yes	complemented
70	C₉×#41	C₂×D₃	1	6	yes	—
71	C₂×Q₈×C₉	Dic₃	1	3	yes	complemented
72	C₂×D₄×C₉	—	2	3	no	—
73	C₂×D₄×C₉	Dic₃	1	3	yes	complemented
74	C₂²×C₄×C₉	—	2	3	no	—
75	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
76	C₂²×Dic₉	—	6	3	no	—
77	C₄×Dic₉	—	3	6	no	—
78	C₄×Dic₉	—	6	3	no	—
79	#3667	—	3	6	no	—
80	#3667	—	6	3	no	—
81	C₂×C₉⋊C₈	—	6	3	no	—
82	C₂²×C₃×C₉	—	2	2	no	—
83	C₂²×C₃×C₉	—	2	2	no	—
84	C₂²×C₃×C₉	—	4	2	no	—
85	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
88	C₃×C₄×C₉	—	2	2	no	—
89	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₃⋊Dic₉	—	2	4	no	—
94	C₂×C₃×#41	D₉	1	3	yes	—
95	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
96	C₂×C₄×Dic₃	—	9	3	no	—
97	C₂×C₄×Dic₃	—	9	1	no	—
98	C₂×#208	—	9	3	no	—
99	C₂×#208	—	9	1	no	—
100	#837	—	9	12	no	—
101	#837	—	9	4	no	—
102	C₂³×C₃²	—	2	1	no	—
103	C₂³×C₃²	#67	1	1	yes	characteristic
104	C₃²×Q₈	—	2	1	no	—
105	C₃²×Q₈	C₂×Dic₃	1	2	yes	—
106	C₃²×D₄	—	2	2	no	—
107	C₃²×D₄	—	2	2	no	—
108	C₃²×D₄	—	2	1	no	—
109	C₃²×D₄	C₂×Dic₃	1	2	yes	—
110	C₂×C₃²×C₄	—	2	1	no	—
111	C₂×C₃²×C₄	—	2	2	no	—
112	C₂×C₃²×C₄	—	2	2	no	—
113	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
115	C₂×C₃²×C₄	#67	1	1	yes	characteristic
116	C₂×C₃²×C₄	#67	1	1	yes	characteristic
117	C₂×C₃²×C₄	#67	1	1	yes	characteristic
118	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃⋊Dic₃	—	6	4	no	—
120	C₂×C₃⋊Dic₃	—	6	2	no	—
121	C₂³×C₉	—	2	3	no	—
122	C₂³×C₉	#67	1	3	yes	—
123	C₃⋊(C₃⋊C₈)	—	6	2	no	—
124	Q₈×C₉	—	2	3	no	—
125	Q₈×C₉	C₂×Dic₃	1	6	yes	complemented
126	D₄×C₉	—	2	6	no	—
127	D₄×C₉	—	2	6	no	—
128	D₄×C₉	—	2	3	no	—
129	D₄×C₉	C₂×Dic₃	1	6	yes	complemented
130	C₂×C₄×C₉	—	2	3	no	—
131	C₂×C₄×C₉	—	2	6	no	—
132	C₂×C₄×C₉	—	2	6	no	—
133	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
134	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
135	C₂×C₄×C₉	#67	1	3	yes	—
136	C₂×C₄×C₉	#67	1	3	yes	—
137	C₂×C₄×C₉	#67	1	3	yes	—
138	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
139	C₂×Dic₉	—	6	12	no	—
140	C₂×Dic₉	—	6	6	no	—
141	C₉⋊C₈	—	6	6	no	—
142	C₂×#104	—	27	1	no	Sylow
143	C₂×C₃×C₉	—	2	1	no	—
144	C₂×C₃×C₉	—	2	1	no	—
145	C₂×C₃×C₉	—	4	2	no	—
146	C₂×C₃×C₉	#104	1	2	yes	complemented
147	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₃×#41	—	2	3	no	—
149	C₃×#41	—	2	3	no	—
150	C₃×#41	—	2	1	no	—
151	C₃×#41	—	2	1	no	—
152	C₃×#41	Dic₉	1	6	yes	complemented
153	C₃×#41	C₂×D₉	1	6	yes	—
154	C₃×#41	C₃⋊Dic₃	1	2	yes	—
155	C₃×#41	C₂×C₃⋊D₃	1	2	yes	—
156	C₂×C₃×Q₈	Dic₉	1	3	yes	complemented
157	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
158	C₂×C₃×D₄	—	2	3	no	—
159	C₂×C₃×D₄	—	2	1	no	—
160	C₂×C₃×D₄	Dic₉	1	3	yes	complemented
161	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
162	C₂²×C₃×C₄	—	2	3	no	—
163	C₂²×C₃×C₄	—	2	1	no	—
164	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
165	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
166	C₂²×Dic₃	—	18	3	no	—
167	C₂²×Dic₃	—	18	1	no	—
168	C₄×Dic₃	—	9	6	no	—
169	C₄×Dic₃	—	9	2	no	—
170	C₄×Dic₃	—	18	3	no	—
171	C₄×Dic₃	—	18	1	no	—
172	#208	—	9	6	no	—
173	#208	—	9	2	no	—
174	#208	—	18	3	no	—
175	#208	—	18	1	no	—
176	C₂×C₃⋊C₈	—	18	3	no	—
177	C₂×C₃⋊C₈	—	18	1	no	—
178	C₂²×C₃²	—	2	2	no	—
179	C₂²×C₃²	—	2	2	no	—
180	C₂²×C₃²	—	4	2	no	—
181	C₂²×C₃²	#217	1	1	yes	characteristic
182	C₂²×C₃²	#217	1	1	yes	characteristic
183	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
184	C₃²×C₄	—	2	2	no	—
185	C₃²×C₄	#217	1	1	yes	characteristic
186	C₃²×C₄	#217	1	1	yes	characteristic
187	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₃²×C₄	C₂×#67	1	1	yes	characteristic
189	C₃⋊Dic₃	—	6	4	no	—
190	C₂²×C₉	—	2	6	no	—
191	C₂²×C₉	—	2	6	no	—
192	C₂²×C₉	—	4	6	no	—
193	C₂²×C₉	#217	1	3	yes	—
194	C₂²×C₉	#217	1	3	yes	—
195	C₂²×C₉	C₂×#67	1	3	yes	—
196	C₄×C₉	—	2	6	no	—
197	C₄×C₉	#217	1	3	yes	—
198	C₄×C₉	#217	1	3	yes	—
199	C₄×C₉	C₂²×Dic₃	1	3	yes	—
200	C₄×C₉	C₂×#67	1	3	yes	—
201	Dic₉	—	6	12	no	—
202	C₂×#41	C₃⋊D₉	1	1	yes	characteristic
203	C₂×M₁₆	—	27	1	no	—
204	C₂×C₄²	—	27	1	no	—
205	#104	—	27	4	no	—
206	C₃×C₉	C₂×#104	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₃	—	2	3	no	—
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	#611	1	3	yes	—
210	C₂³×C₃	#638	1	1	yes	characteristic
211	C₃×Q₈	—	2	3	no	—
212	C₃×Q₈	—	2	1	no	—
213	C₃×Q₈	C₂×Dic₉	1	6	yes	complemented
214	C₃×Q₈	C₂×C₃⋊Dic₃	1	2	yes	—
215	C₃×D₄	—	2	6	no	—
216	C₃×D₄	—	2	6	no	—
217	C₃×D₄	—	2	2	no	—
218	C₃×D₄	—	2	2	no	—
219	C₃×D₄	—	2	3	no	—
220	C₃×D₄	—	2	1	no	—
221	C₃×D₄	C₂×Dic₉	1	6	yes	complemented
222	C₃×D₄	C₂×C₃⋊Dic₃	1	2	yes	—
223	C₂×C₃×C₄	—	2	3	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	6	no	—
226	C₂×C₃×C₄	—	2	6	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
230	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
231	C₂×C₃×C₄	#611	1	3	yes	—
232	C₂×C₃×C₄	#611	1	3	yes	—
233	C₂×C₃×C₄	#611	1	3	yes	—
234	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
235	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
237	C₂×C₃×C₄	#638	1	1	yes	characteristic
238	C₂×C₃×C₄	#638	1	1	yes	characteristic
239	C₂×C₃×C₄	#638	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
241	C₂×Dic₃	—	18	4	no	—
242	C₂×Dic₃	—	18	12	no	—
243	C₂×Dic₃	—	18	6	no	—
244	C₂×Dic₃	—	18	2	no	—
245	C₃⋊C₈	—	18	6	no	—
246	C₃⋊C₈	—	18	2	no	—
247	C₂×C₃²	—	2	1	no	—
248	C₂×C₃²	—	2	1	no	—
249	C₂×C₃²	—	4	2	no	—
250	C₂×C₃²	#837	1	2	yes	—
251	C₂×C₃²	C₂×#217	1	1	yes	characteristic
252	C₂×C₉	—	2	3	no	—
253	C₂×C₉	—	2	3	no	—
254	C₂×C₉	—	4	6	no	—
255	C₂×C₉	#837	1	6	yes	complemented
256	C₂×C₉	C₂×#217	1	3	yes	—
257	#41	—	2	1	no	—
258	#41	—	2	1	no	—
259	#41	C₃⋊Dic₉	1	2	yes	complemented
260	#41	C₂×C₃⋊D₉	1	2	yes	—
261	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic, complemented
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic, complemented
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	54	1	no	—
266	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
267	M₁₆	—	27	2	no	—
268	M₁₆	—	54	1	no	—
269	C₂×C₈	—	54	1	no	—
270	C₄²	—	27	2	no	—
271	C₄²	—	54	1	no	—
272	C₂²×C₃	—	2	6	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	6	no	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	4	6	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	#3676	1	3	yes	—
279	C₂²×C₃	#3676	1	3	yes	—
280	C₂²×C₃	C₂×#611	1	3	yes	—
281	C₂²×C₃	#3757	1	1	yes	characteristic
282	C₂²×C₃	#3757	1	1	yes	characteristic
283	C₂²×C₃	C₂×#638	1	1	yes	characteristic
284	C₃×C₄	—	2	6	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	#3676	1	3	yes	—
287	C₃×C₄	#3676	1	3	yes	—
288	C₃×C₄	C₂²×Dic₉	1	3	yes	—
289	C₃×C₄	C₂×#611	1	3	yes	—
290	C₃×C₄	#3757	1	1	yes	characteristic
291	C₃×C₄	#3757	1	1	yes	characteristic
292	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
293	C₃×C₄	C₂×#638	1	1	yes	characteristic
294	Dic₃	—	18	4	no	—
295	Dic₃	—	18	12	no	—
296	C₃²	C₂×#837	1	1	yes	characteristic
297	C₉	C₂×#837	1	3	yes	complemented
298	C₂³	—	2	1	no	—
299	C₂³	#6233	1	1	yes	characteristic
300	Q₈	—	2	1	no	—
301	Q₈	C₂×C₃⋊Dic₉	1	2	yes	complemented
302	D₄	—	2	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	2	1	no	—
305	D₄	C₂×C₃⋊Dic₉	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	54	4	no	—
310	C₂×C₄	—	54	2	no	—
311	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
313	C₂×C₄	#6233	1	1	yes	characteristic
314	C₂×C₄	#6233	1	1	yes	characteristic
315	C₂×C₄	#6233	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
317	C₈	—	54	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	3	no	—
321	C₂×C₃	—	2	3	no	—
322	C₂×C₃	—	4	2	no	—
323	C₂×C₃	—	4	6	no	—
324	C₂×C₃	#63396	1	6	yes	complemented
325	C₂×C₃	C₂×#3676	1	3	yes	—
326	C₂×C₃	#63664	1	2	yes	—
327	C₂×C₃	C₂×#3757	1	1	yes	characteristic
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	#88551	1	1	yes	characteristic
332	C₂²	#88551	1	1	yes	characteristic
333	C₂²	C₂×#6233	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	54	4	no	—
336	C₄	#88551	1	1	yes	characteristic
337	C₄	#88551	1	1	yes	characteristic
338	C₄	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
339	C₄	C₂×#6233	1	1	yes	characteristic
340	C₃	C₂×#63396	1	3	yes	complemented
341	C₃	C₂×#63664	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	4	2	no	—
345	C₂	#11720585	1	2	yes	direct product factor
346	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂²×C₃²	#217
prime radical	C₂×C₃²×#41	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×#41	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#104
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₃⋊Dic₉	C₂	semidirect
3. C₂×#88532	C₂	semidirect
4. #11720585	C₂	direct (N × H)
9. C₃×C₉×#41	C₄	semidirect
11. C₂×C₃×Q₈×C₉	C₄	semidirect
13. C₂×C₃×D₄×C₉	C₄	semidirect
18. C₄×C₃⋊Dic₉	C₂²	semidirect
20. #88532	C₂²	semidirect
33. C₃×Q₈×C₉	C₂×C₄	semidirect
37. C₃×D₄×C₉	C₂×C₄	semidirect
69. C₉×#41	Dic₃	semidirect
71. C₂×Q₈×C₉	Dic₃	semidirect
73. C₂×D₄×C₉	Dic₃	semidirect
125. Q₈×C₉	C₂×Dic₃	semidirect
129. D₄×C₉	C₂×Dic₃	semidirect
146. C₂×C₃×C₉	#104	semidirect
152. C₃×#41	Dic₉	semidirect
156. C₂×C₃×Q₈	Dic₉	semidirect
160. C₂×C₃×D₄	Dic₉	semidirect
206. C₃×C₉	C₂×#104	semidirect
213. C₃×Q₈	C₂×Dic₉	semidirect
221. C₃×D₄	C₂×Dic₉	semidirect
255. C₂×C₉	#837	semidirect
259. #41	C₃⋊Dic₉	semidirect
261. C₂×Q₈	C₃⋊Dic₉	semidirect
263. C₂×D₄	C₃⋊Dic₉	semidirect
297. C₉	C₂×#837	semidirect
301. Q₈	C₂×C₃⋊Dic₉	semidirect
305. D₄	C₂×C₃⋊Dic₉	semidirect
324. C₂×C₃	#63396	semidirect
340. C₃	C₂×#63396	semidirect
345. C₂	#11720585	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×C₃×C₉×#41	2
C₂×C₄×C₃⋊Dic₉	2
C₂×#88532	2
#11720585	2
C₂×#63664	3
C₂×#63396	3

Subgroups — C2×#11720585 #13144851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720585 #13144851