Subgroups — C₂×#11720295 #13139446

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720295	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63563	—	3	1	no	maximal
6	C₂×#63392	—	3	1	no	maximal
7	C₂³×D₃×C₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	—	2	1	no	—
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	—	2	2	no	—
12	C₂×C₉×D₁₂	C₄	1	4	yes	complemented
13	C₂×C₉×D₁₂	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	complemented
16	C₂×C₃×C₉⋊C₈	—	2	2	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	complemented
18	C₂²×C₃×D₁₂	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
20	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
21	C₂²×D₄×C₉	—	3	1	no	—
22	#63563	—	3	4	no	—
23	C₂×C₄⋊Dic₉	—	3	1	no	—
24	C₂×C₄⋊Dic₉	—	6	1	no	—
25	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
26	#63392	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂²×D₃×C₉	—	2	4	no	—
29	C₂²×D₃×C₉	—	2	2	no	—
30	C₂²×D₃×C₉	—	4	2	no	—
31	C₂²×D₃×C₉	—	4	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×C₃⋊Dic₉	—	2	2	no	—
37	C₂×C₃⋊Dic₉	—	4	2	no	—
38	C₉×D₁₂	—	2	4	no	—
39	C₉×D₁₂	—	2	2	no	—
40	C₉×D₁₂	C₂×C₄	1	4	yes	complemented
41	C₃×C₉⋊C₈	—	2	2	no	—
42	C₂×#832	—	9	1	no	—
43	C₂×#821	—	9	1	no	—
44	C₂³×C₃×D₃	—	2	1	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃⋊Dic₃	—	6	1	no	—
47	C₂×C₃×D₁₂	—	2	1	no	—
48	C₂×C₃×D₁₂	—	2	2	no	—
49	C₂×C₃×D₁₂	Dic₃	1	4	yes	—
50	C₂×C₃×D₁₂	C₂×D₃	1	2	yes	—
51	C₂⁴×C₉	—	6	1	no	—
52	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
53	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
54	C₂×C₃×C₃⋊C₈	—	3	2	no	—
55	C₂×C₃×C₃⋊C₈	—	6	2	no	—
56	C₂×D₄×C₉	—	3	4	no	—
57	C₂×D₄×C₉	—	3	2	no	—
58	C₂×D₄×C₉	—	6	1	no	—
59	C₂×D₄×C₉	—	6	2	no	—
60	C₂²×C₄×C₉	—	2	1	no	—
61	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂²×Dic₉	—	6	2	no	—
63	C₂²×Dic₉	—	6	1	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	C₂×C₉⋊C₈	—	2	2	no	—
69	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	complemented
70	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₃×C₉	D₈	1	2	yes	—
72	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
73	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×D₃×C₉	—	2	4	no	—
75	C₂×D₃×C₉	—	4	8	no	—
76	C₂×D₃×C₉	—	4	2	no	—
77	C₂×D₃×C₉	—	4	2	no	—
78	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
79	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
81	C₃⋊Dic₉	—	4	2	no	—
82	C₂²×C₃×D₄	—	3	1	no	—
83	C₂²×D₁₂	D₉	1	1	yes	characteristic
84	C₂²×C₃×C₈	—	9	1	no	—
85	C₂×C₄⋊Dic₃	—	9	1	no	—
86	C₂×C₄⋊Dic₃	—	9	1	no	—
87	C₂×C₄⋊Dic₃	—	18	1	no	—
88	C₂²×C₃⋊C₈	—	3	1	no	—
89	#832	—	9	4	no	—
90	#821	—	9	4	no	—
91	C₂³×C₃²	#67	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	4	no	—
93	C₂²×C₃×D₃	—	2	2	no	—
94	C₂²×C₃×D₃	—	4	2	no	—
95	C₂²×C₃×D₃	—	4	2	no	—
96	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
97	C₂×C₃²×C₄	#67	1	2	yes	—
98	C₂×C₃²×C₄	#67	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₃⋊Dic₃	—	6	2	no	—
101	C₂×C₃⋊Dic₃	—	12	2	no	—
102	C₃×D₁₂	—	2	4	no	—
103	C₃×D₁₂	—	2	2	no	—
104	C₃×D₁₂	C₂×Dic₃	1	4	yes	—
105	C₂³×C₉	—	2	1	no	—
106	C₂³×C₉	—	6	4	no	—
107	C₂³×C₉	—	6	2	no	—
108	C₂³×C₉	—	12	2	no	—
109	C₂³×C₉	—	12	2	no	—
110	C₂³×C₉	D₁₂	1	1	yes	characteristic
111	C₃×C₃⋊C₈	—	6	2	no	—
112	D₄×C₉	—	3	4	no	—
113	D₄×C₉	—	6	4	no	—
114	D₄×C₉	—	6	2	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
120	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
121	C₂×C₄×C₉	#67	1	2	yes	—
122	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
123	C₂×Dic₉	—	6	4	no	—
124	C₂×Dic₉	—	6	2	no	—
125	C₂×Dic₉	—	12	4	no	—
126	C₂×Dic₉	—	12	2	no	—
127	C₉⋊C₈	—	2	2	no	—
128	C₂×#102	—	27	1	no	Sylow
129	C₂×C₃×C₉	#102	1	4	yes	complemented
130	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
133	D₃×C₉	—	4	4	no	—
134	C₂⁴×C₃	—	6	1	no	—
135	C₂³×D₃	—	2	1	no	—
136	C₂×C₃×D₄	—	3	4	no	—
137	C₂×C₃×D₄	—	3	2	no	—
138	C₂×C₃×D₄	—	6	2	no	—
139	C₂×C₃×D₄	—	6	1	no	—
140	C₂²×C₃×C₄	—	2	1	no	—
141	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
143	C₂²×Dic₃	—	18	2	no	—
144	C₂²×Dic₃	—	18	1	no	—
145	C₂²×Dic₃	—	18	1	no	—
146	C₂×D₁₂	—	2	2	no	—
147	C₂×D₁₂	—	2	1	no	—
148	C₂×D₁₂	Dic₉	1	4	yes	complemented
149	C₂×D₁₂	C₂×D₉	1	2	yes	—
150	C₂×C₃×C₈	—	9	2	no	—
151	C₂×C₃×C₈	—	18	2	no	—
152	C₄⋊Dic₃	—	9	2	no	—
153	C₄⋊Dic₃	—	9	2	no	—
154	C₄⋊Dic₃	—	18	2	no	—
155	C₄⋊Dic₃	—	18	2	no	—
156	C₄⋊Dic₃	—	18	1	no	—
157	C₄⋊Dic₃	—	18	1	no	—
158	C₂×C₃⋊C₈	—	3	2	no	—
159	C₂×C₃⋊C₈	—	6	2	no	—
160	C₂²×C₃²	#213	1	2	yes	—
161	C₂²×C₃²	#214	1	2	yes	—
162	C₂²×C₃²	#217	1	2	yes	—
163	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
164	C₂×C₃×D₃	—	2	4	no	—
165	C₂×C₃×D₃	—	4	8	no	—
166	C₂×C₃×D₃	—	4	2	no	—
167	C₂×C₃×D₃	—	4	2	no	—
168	C₃²×C₄	#217	1	2	yes	—
169	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
170	C₃²×C₄	C₂×#67	1	1	yes	characteristic
171	C₃⋊Dic₃	—	12	2	no	—
172	C₂²×C₉	—	2	2	no	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	2	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	6	4	no	—
177	C₂²×C₉	—	12	8	no	—
178	C₂²×C₉	—	12	2	no	—
179	C₂²×C₉	—	12	2	no	—
180	C₂²×C₉	#204	1	2	yes	—
181	C₂²×C₉	D₂₄	1	2	yes	—
182	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
183	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	2	1	no	—
186	C₄×C₉	—	2	1	no	—
187	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
188	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
189	C₄×C₉	C₂×#67	1	1	yes	characteristic
190	Dic₉	—	12	4	no	—
191	Dic₉	—	12	2	no	—
192	C₂²×D₄	—	3	1	no	—
193	C₂²×C₈	—	9	1	no	—
194	C₂×C₄⋊C₄	—	27	1	no	—
195	#102	—	27	4	no	—
196	C₃×C₉	C₂×#102	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₃	—	2	1	no	—
198	C₂³×C₃	—	6	4	no	—
199	C₂³×C₃	—	6	2	no	—
200	C₂³×C₃	—	12	2	no	—
201	C₂³×C₃	—	12	2	no	—
202	C₂³×C₃	#611	1	1	yes	characteristic
203	C₂³×C₃	#626	1	1	yes	characteristic
204	C₂²×D₃	—	2	4	no	—
205	C₂²×D₃	—	2	2	no	—
206	C₂²×D₃	—	4	2	no	—
207	C₂²×D₃	—	4	2	no	—
208	C₃×D₄	—	3	4	no	—
209	C₃×D₄	—	6	4	no	—
210	C₃×D₄	—	6	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
216	C₂×C₃×C₄	#611	1	2	yes	—
217	C₂×C₃×C₄	#611	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
219	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
220	C₂×C₃×C₄	#625	1	2	yes	—
221	C₂×C₃×C₄	#626	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
223	C₂×Dic₃	—	18	4	no	—
224	C₂×Dic₃	—	18	2	no	—
225	C₂×Dic₃	—	18	2	no	—
226	C₂×Dic₃	—	36	4	no	—
227	C₂×Dic₃	—	36	2	no	—
228	C₂×Dic₃	—	36	2	no	—
229	D₁₂	—	2	4	no	—
230	D₁₂	—	2	2	no	—
231	D₁₂	C₂×Dic₉	1	4	yes	complemented
232	C₃×C₈	—	18	2	no	—
233	C₃⋊C₈	—	6	2	no	—
234	C₂×C₃²	#832	1	4	yes	—
235	C₂×C₃²	C₂×#213	1	1	yes	characteristic
236	C₂×C₃²	C₂×#214	1	1	yes	characteristic
237	C₂×C₃²	C₂×#217	1	1	yes	characteristic
238	C₃×D₃	—	4	4	no	—
239	C₂×C₉	—	2	4	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	12	4	no	—
244	C₂×C₉	#821	1	4	yes	complemented
245	C₂×C₉	C₂×#204	1	1	yes	characteristic
246	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
247	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
248	C₂⁴	—	6	1	no	—
249	C₂×D₄	—	3	4	no	—
250	C₂×D₄	—	3	2	no	—
251	C₂×D₄	—	6	2	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂²×C₄	—	54	1	no	—
254	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
255	C₂×C₈	—	9	2	no	—
256	C₂×C₈	—	18	2	no	—
257	C₄⋊C₄	—	27	2	no	—
258	C₄⋊C₄	—	54	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	2	no	—
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	6	4	no	—
264	C₂²×C₃	—	12	8	no	—
265	C₂²×C₃	—	12	2	no	—
266	C₂²×C₃	—	12	2	no	—
267	C₂²×C₃	#3672	1	2	yes	—
268	C₂²×C₃	#3673	1	2	yes	—
269	C₂²×C₃	#3676	1	2	yes	—
270	C₂²×C₃	C₂×#611	1	1	yes	characteristic
271	C₂²×C₃	#3714	1	2	yes	—
272	C₂²×C₃	#3716	1	2	yes	—
273	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
274	C₂²×C₃	C₂×#626	1	1	yes	characteristic
275	C₂×D₃	—	2	4	no	—
276	C₂×D₃	—	4	2	no	—
277	C₂×D₃	—	4	2	no	—
278	C₂×D₃	—	4	8	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	#3676	1	2	yes	—
283	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
284	C₃×C₄	C₂×#611	1	1	yes	characteristic
285	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
286	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
287	C₃×C₄	C₂×#625	1	1	yes	characteristic
288	Dic₃	—	36	4	no	—
289	Dic₃	—	36	2	no	—
290	Dic₃	—	36	2	no	—
291	C₃²	C₂×#832	1	1	yes	characteristic
292	C₉	—	2	1	no	—
293	C₉	C₂×#821	1	1	yes	characteristic, complemented
294	C₂³	—	6	2	no	—
295	C₂³	—	6	4	no	—
296	C₂³	—	12	2	no	—
297	C₂³	—	12	2	no	—
298	C₂³	#6195	1	1	yes	characteristic
299	D₄	—	3	4	no	—
300	D₄	—	6	2	no	—
301	D₄	—	6	4	no	—
302	C₂×C₄	—	54	2	no	—
303	C₂×C₄	—	108	2	no	—
304	C₂×C₄	D₃×Dic₉	1	2	yes	—
305	C₂×C₄	#6194	1	2	yes	—
306	C₂×C₄	#6195	1	1	yes	characteristic
307	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
308	C₈	—	18	2	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	12	4	no	—
314	C₂×C₃	#63392	1	4	yes	complemented
315	C₂×C₃	C₂×#3672	1	1	yes	characteristic
316	C₂×C₃	C₂×#3673	1	1	yes	characteristic
317	C₂×C₃	C₂×#3676	1	1	yes	characteristic
318	C₂×C₃	#63563	1	4	yes	—
319	C₂×C₃	C₂×#3714	1	1	yes	characteristic
320	C₂×C₃	C₂×#3716	1	1	yes	characteristic
321	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
322	D₃	—	4	4	no	—
323	C₂²	—	6	4	no	—
324	C₂²	—	12	8	no	—
325	C₂²	—	12	2	no	—
326	C₂²	—	12	2	no	—
327	C₂²	#88422	1	2	yes	—
328	C₂²	#88424	1	2	yes	—
329	C₂²	(C₂×D₃)⋊Dic₉	1	2	yes	—
330	C₂²	C₂×#6195	1	1	yes	characteristic
331	C₄	—	108	2	no	—
332	C₄	(C₂×D₃)⋊Dic₉	1	2	yes	—
333	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
334	C₄	C₂×#6194	1	1	yes	characteristic
335	C₃	—	2	1	no	—
336	C₃	C₂×#63392	1	1	yes	characteristic, complemented
337	C₃	C₂×#63563	1	1	yes	characteristic
338	C₂	—	12	4	no	—
339	C₂	#11720295	1	4	yes	direct product factor
340	C₂	C₂×#88422	1	1	yes	characteristic
341	C₂	C₂×#88424	1	1	yes	characteristic
342	C₂	C₂×(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6195
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂³×C₃²	#67
prime radical	C₂²×C₃×D₁₂	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#102
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
2. C₂×C₄⋊(C₃⋊Dic₉)	C₂	semidirect
3. C₂²×C₃×C₉⋊C₈	C₂	semidirect
4. #11720295	C₂	direct (N × H)
12. C₂×C₉×D₁₂	C₄	semidirect
15. C₄⋊(C₃⋊Dic₉)	C₂²	semidirect
17. C₂×C₃×C₉⋊C₈	C₂²	semidirect
25. C₂²×C₉⋊C₈	D₃	semidirect
40. C₉×D₁₂	C₂×C₄	semidirect
69. C₂×C₉⋊C₈	C₂×D₃	semidirect
129. C₂×C₃×C₉	#102	semidirect
148. C₂×D₁₂	Dic₉	semidirect
196. C₃×C₉	C₂×#102	semidirect
231. D₁₂	C₂×Dic₉	semidirect
244. C₂×C₉	#821	semidirect
293. C₉	C₂×#821	semidirect
314. C₂×C₃	#63392	semidirect
336. C₃	C₂×#63392	semidirect
339. C₂	#11720295	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×C₉×D₁₂	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂²×C₃×C₉⋊C₈	2
#11720295	2
C₂×#63563	3
C₂×#63392	3

Subgroups — C2×#11720295 #13139446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720295 #13139446