Subgroups — C₂×#11720266 #13139325

Order 1728.

682 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720266	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63562	—	3	1	no	maximal
6	C₂×#63369	—	3	1	no	maximal
7	C₂³×C₃×D₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₉×Dic₃	—	2	1	no	—
10	C₂×C₃×D₃₆	—	2	1	no	—
11	C₂×C₃×D₃₆	—	2	2	no	—
12	C₂×C₃×D₃₆	C₄	1	4	yes	complemented
13	C₂×C₃×D₃₆	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	complemented
16	C₉×C₄⋊Dic₃	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	2	yes	complemented
18	C₂²×C₃×D₁₂	—	3	1	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂×C₃×C₄⋊Dic₃	D₃	1	1	yes	characteristic
21	C₂²×D₃₆	D₃	1	1	yes	characteristic
22	#63562	—	3	4	no	—
23	C₂×C₉×C₄⋊C₄	—	3	1	no	—
24	C₂²×C₉⋊C₈	—	3	1	no	—
25	C₂²×C₉⋊C₈	—	6	1	no	—
26	#63369	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₃×D₉	—	2	4	no	—
29	C₂²×C₃×D₉	—	2	2	no	—
30	C₂²×C₃×D₉	—	4	2	no	—
31	C₂²×C₃×D₉	—	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×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	—	4	2	no	—
38	C₃×D₃₆	—	2	4	no	—
39	C₃×D₃₆	—	2	2	no	—
40	C₃×D₃₆	C₂×C₄	1	4	yes	complemented
41	C₃⋊(C₉⋊C₈)	—	2	2	no	—
42	C₂×#832	—	9	1	no	—
43	C₂×#809	—	9	1	no	—
44	C₂³×C₃×D₃	—	6	1	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×Dic₃	—	2	1	no	—
47	C₂×C₃×D₁₂	—	3	4	no	—
48	C₂×C₃×D₁₂	—	3	2	no	—
49	C₂×C₃×D₁₂	—	6	2	no	—
50	C₂×C₃×D₁₂	—	6	1	no	—
51	C₂³×D₉	—	2	1	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
53	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
54	C₃×C₄⋊Dic₃	—	2	1	no	—
55	C₃×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	6	1	no	—
58	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
59	C₂×D₃₆	—	2	1	no	—
60	C₂×D₃₆	—	2	2	no	—
61	C₂×D₃₆	Dic₃	1	4	yes	complemented
62	C₂×D₃₆	C₂×D₃	1	2	yes	—
63	C₉×C₄⋊C₄	—	3	2	no	—
64	C₉×C₄⋊C₄	—	6	1	no	—
65	C₂×C₉⋊C₈	—	3	2	no	—
66	C₂×C₉⋊C₈	—	6	2	no	—
67	C₂×C₉⋊C₈	—	6	4	no	—
68	C₂×C₉⋊C₈	—	6	2	no	—
69	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃×C₉	D₈	1	2	yes	—
71	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
72	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×D₉	—	2	4	no	—
74	C₂×C₃×D₉	—	4	8	no	—
75	C₂×C₃×D₉	—	4	2	no	—
76	C₂×C₃×D₉	—	4	2	no	—
77	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
78	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
80	C₉×Dic₃	—	4	2	no	—
81	C₂²×C₃×D₄	—	9	1	no	—
82	C₂²×D₁₂	—	3	1	no	—
83	C₂×C₃×C₄⋊C₄	—	3	1	no	—
84	C₂×C₄⋊Dic₃	D₉	1	1	yes	characteristic, complemented
85	C₂²×C₃⋊C₈	—	9	1	no	—
86	C₂²×C₃⋊C₈	—	9	1	no	—
87	C₂²×C₃⋊C₈	—	18	1	no	—
88	#832	—	9	4	no	—
89	#809	—	9	4	no	—
90	C₂³×C₃²	#67	1	1	yes	characteristic
91	C₂²×C₃×D₃	—	6	4	no	—
92	C₂²×C₃×D₃	—	6	2	no	—
93	C₂²×C₃×D₃	—	12	2	no	—
94	C₂²×C₃×D₃	—	12	2	no	—
95	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
96	C₂×C₃²×C₄	D₁₂	1	2	yes	—
97	C₂×C₃²×C₄	#67	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₂×C₃×Dic₃	—	4	2	no	—
101	C₃×D₁₂	—	3	4	no	—
102	C₃×D₁₂	—	6	4	no	—
103	C₃×D₁₂	—	6	2	no	—
104	C₂³×C₉	—	2	1	no	—
105	C₂³×C₉	#67	1	1	yes	characteristic
106	C₂²×D₉	—	2	4	no	—
107	C₂²×D₉	—	2	2	no	—
108	C₂²×D₉	—	4	2	no	—
109	C₂²×D₉	—	4	2	no	—
110	C₃⋊(C₃⋊C₈)	—	6	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	1	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	1	no	—
115	C₂×C₄×C₉	—	6	2	no	—
116	C₂×C₄×C₉	—	12	2	no	—
117	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
118	C₂×C₄×C₉	#67	1	1	yes	characteristic
119	C₂×C₄×C₉	#67	1	2	yes	—
120	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
121	D₃₆	—	2	4	no	—
122	D₃₆	—	2	2	no	—
123	D₃₆	C₂×Dic₃	1	4	yes	complemented
124	C₉⋊C₈	—	6	4	no	—
125	C₉⋊C₈	—	6	2	no	—
126	C₂×#102	—	27	1	no	Sylow
127	C₂×C₃×C₉	#102	1	4	yes	complemented
128	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₃×D₉	—	4	4	no	—
132	C₂⁴×C₃	—	18	1	no	—
133	C₂³×D₃	—	6	1	no	—
134	C₂×C₃×D₄	—	9	4	no	—
135	C₂×C₃×D₄	—	9	2	no	—
136	C₂×C₃×D₄	—	18	2	no	—
137	C₂×C₃×D₄	—	18	1	no	—
138	C₂²×C₃×C₄	—	2	1	no	—
139	C₂²×C₃×C₄	—	6	1	no	—
140	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
141	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
142	C₂²×Dic₃	—	2	1	no	—
143	C₂×D₁₂	—	3	4	no	—
144	C₂×D₁₂	—	3	2	no	—
145	C₂×D₁₂	—	6	1	no	—
146	C₂×D₁₂	—	6	2	no	—
147	C₃×C₄⋊C₄	—	3	2	no	—
148	C₃×C₄⋊C₄	—	6	1	no	—
149	C₄⋊Dic₃	—	2	1	no	—
150	C₄⋊Dic₃	C₂×D₉	1	2	yes	complemented
151	C₂×C₃⋊C₈	—	9	2	no	—
152	C₂×C₃⋊C₈	—	9	2	no	—
153	C₂×C₃⋊C₈	—	18	2	no	—
154	C₂×C₃⋊C₈	—	18	4	no	—
155	C₂×C₃⋊C₈	—	18	2	no	—
156	C₂×C₃⋊C₈	—	18	2	no	—
157	C₂²×C₃²	(C₂×D₃)⋊C₄	1	2	yes	—
158	C₂²×C₃²	#213	1	2	yes	—
159	C₂²×C₃²	#215	1	2	yes	—
160	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
161	C₂×C₃×D₃	—	6	4	no	—
162	C₂×C₃×D₃	—	12	8	no	—
163	C₂×C₃×D₃	—	12	2	no	—
164	C₂×C₃×D₃	—	12	2	no	—
165	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
167	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
168	C₃×Dic₃	—	4	2	no	—
169	C₂²×C₉	—	2	2	no	—
170	C₂²×C₉	—	2	2	no	—
171	C₂²×C₉	—	2	2	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	#213	1	2	yes	—
174	C₂²×C₉	#214	1	2	yes	—
175	C₂²×C₉	#217	1	2	yes	—
176	C₂²×C₉	C₂×#67	1	1	yes	characteristic
177	C₂×D₉	—	2	4	no	—
178	C₂×D₉	—	4	8	no	—
179	C₂×D₉	—	4	2	no	—
180	C₂×D₉	—	4	2	no	—
181	C₄×C₉	—	2	2	no	—
182	C₄×C₉	—	2	1	no	—
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	12	2	no	—
185	C₄×C₉	#217	1	2	yes	—
186	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
187	C₄×C₉	C₂×#67	1	1	yes	characteristic
188	C₂²×D₄	—	9	1	no	—
189	C₂²×C₈	—	27	1	no	—
190	C₂×C₄⋊C₄	—	3	1	no	—
191	#102	—	27	4	no	—
192	C₃×C₉	C₂×#102	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	18	4	no	—
195	C₂³×C₃	—	18	2	no	—
196	C₂³×C₃	—	36	2	no	—
197	C₂³×C₃	—	36	2	no	—
198	C₂³×C₃	#611	1	1	yes	characteristic
199	C₂³×C₃	#625	1	1	yes	characteristic
200	C₂²×D₃	—	6	4	no	—
201	C₂²×D₃	—	6	2	no	—
202	C₂²×D₃	—	12	2	no	—
203	C₂²×D₃	—	12	2	no	—
204	C₃×D₄	—	9	4	no	—
205	C₃×D₄	—	18	4	no	—
206	C₃×D₄	—	18	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	6	2	no	—
212	C₂×C₃×C₄	—	12	2	no	—
213	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
214	C₂×C₃×C₄	D₃₆	1	2	yes	—
215	C₂×C₃×C₄	#611	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
217	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
218	C₂×C₃×C₄	#625	1	1	yes	characteristic
219	C₂×C₃×C₄	#626	1	2	yes	—
220	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
221	C₂×Dic₃	—	2	2	no	—
222	C₂×Dic₃	—	4	2	no	—
223	D₁₂	—	3	4	no	—
224	D₁₂	—	6	4	no	—
225	D₁₂	—	6	2	no	—
226	C₃⋊C₈	—	18	4	no	—
227	C₃⋊C₈	—	18	2	no	—
228	C₃⋊C₈	—	18	2	no	—
229	C₂×C₃²	#809	1	4	yes	—
230	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₃²	C₂×#213	1	1	yes	characteristic
232	C₂×C₃²	C₂×#215	1	1	yes	characteristic
233	C₃×D₃	—	12	4	no	—
234	C₂×C₉	—	2	4	no	—
235	C₂×C₉	—	2	1	no	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	#832	1	4	yes	complemented
239	C₂×C₉	C₂×#213	1	1	yes	characteristic
240	C₂×C₉	C₂×#214	1	1	yes	characteristic
241	C₂×C₉	C₂×#217	1	1	yes	characteristic
242	D₉	—	4	4	no	—
243	C₂⁴	—	18	1	no	—
244	C₂×D₄	—	9	4	no	—
245	C₂×D₄	—	9	2	no	—
246	C₂×D₄	—	18	2	no	—
247	C₂×D₄	—	18	1	no	—
248	C₂²×C₄	—	6	1	no	—
249	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
250	C₂×C₈	—	27	2	no	—
251	C₂×C₈	—	54	2	no	—
252	C₄⋊C₄	—	3	2	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	18	4	no	—
259	C₂²×C₃	—	36	2	no	—
260	C₂²×C₃	—	36	2	no	—
261	C₂²×C₃	—	36	8	no	—
262	C₂²×C₃	(C₂×D₉)⋊C₄	1	2	yes	—
263	C₂²×C₃	#3673	1	2	yes	—
264	C₂²×C₃	#3675	1	2	yes	—
265	C₂²×C₃	C₂×#611	1	1	yes	characteristic
266	C₂²×C₃	#3713	1	2	yes	—
267	C₂²×C₃	#3715	1	2	yes	—
268	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
269	C₂²×C₃	C₂×#625	1	1	yes	characteristic
270	C₂×D₃	—	6	4	no	—
271	C₂×D₃	—	12	8	no	—
272	C₂×D₃	—	12	2	no	—
273	C₂×D₃	—	12	2	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	12	2	no	—
278	C₃×C₄	(C₂×D₉)⋊C₄	1	2	yes	—
279	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
280	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
281	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
282	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
283	C₃×C₄	C₂×#626	1	1	yes	characteristic
284	Dic₃	—	4	2	no	—
285	C₃²	C₂×#809	1	1	yes	characteristic
286	C₉	—	2	1	no	—
287	C₉	C₂×#832	1	1	yes	characteristic, complemented
288	C₂³	—	18	2	no	—
289	C₂³	—	18	4	no	—
290	C₂³	—	36	2	no	—
291	C₂³	—	36	2	no	—
292	C₂³	#6194	1	1	yes	characteristic
293	D₄	—	9	4	no	—
294	D₄	—	18	2	no	—
295	D₄	—	18	4	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	12	2	no	—
298	C₂×C₄	Dic₃×D₉	1	2	yes	—
299	C₂×C₄	#6192	1	2	yes	—
300	C₂×C₄	#6194	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
302	C₈	—	54	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	36	4	no	—
308	C₂×C₃	#63369	1	4	yes	complemented
309	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
310	C₂×C₃	C₂×#3673	1	1	yes	characteristic
311	C₂×C₃	C₂×#3675	1	1	yes	characteristic
312	C₂×C₃	#63562	1	4	yes	—
313	C₂×C₃	C₂×#3713	1	1	yes	characteristic
314	C₂×C₃	C₂×#3715	1	1	yes	characteristic
315	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
316	D₃	—	12	4	no	—
317	C₂²	—	18	4	no	—
318	C₂²	—	36	8	no	—
319	C₂²	—	36	2	no	—
320	C₂²	—	36	2	no	—
321	C₂²	#88421	1	2	yes	—
322	C₂²	#88425	1	2	yes	—
323	C₂²	(C₂×D₉)⋊Dic₃	1	2	yes	—
324	C₂²	C₂×#6194	1	1	yes	characteristic
325	C₄	—	12	2	no	—
326	C₄	(C₂×D₉)⋊Dic₃	1	2	yes	—
327	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
328	C₄	C₂×#6192	1	1	yes	characteristic
329	C₃	—	2	1	no	—
330	C₃	C₂×#63369	1	1	yes	characteristic, complemented
331	C₃	C₂×#63562	1	1	yes	characteristic
332	C₂	—	36	4	no	—
333	C₂	#11720266	1	4	yes	direct product factor
334	C₂	C₂×#88421	1	1	yes	characteristic
335	C₂	C₂×#88425	1	1	yes	characteristic
336	C₂	C₂×(C₂×D₉)⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6194
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×D₃₆	C₂
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. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
3. C₂×C₉×C₄⋊Dic₃	C₂	semidirect
4. #11720266	C₂	direct (N × H)
12. C₂×C₃×D₃₆	C₄	semidirect
15. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
17. C₉×C₄⋊Dic₃	C₂²	semidirect
40. C₃×D₃₆	C₂×C₄	semidirect
61. C₂×D₃₆	Dic₃	semidirect
84. C₂×C₄⋊Dic₃	D₉	semidirect
123. D₃₆	C₂×Dic₃	semidirect
127. C₂×C₃×C₉	#102	semidirect
150. C₄⋊Dic₃	C₂×D₉	semidirect
192. C₃×C₉	C₂×#102	semidirect
238. C₂×C₉	#832	semidirect
287. C₉	C₂×#832	semidirect
308. C₂×C₃	#63369	semidirect
330. C₃	C₂×#63369	semidirect
333. C₂	#11720266	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×D₃₆	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₂×C₉×C₄⋊Dic₃	2
#11720266	2
C₂×#63562	3
C₂×#63369	3

Subgroups — C2×#11720266 #13139325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720266 #13139325