Subgroups — C₂×#11720296 #13139459

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 376 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×#88532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720296	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63558	—	3	1	no	maximal
6	C₂×#63391	—	3	1	no	maximal
7	C₂³×D₃×C₉	C₄	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×D₁₂	—	2	1	no	—
10	C₂×C₉×D₁₂	—	2	1	no	—
11	C₂×C₉×D₁₂	—	2	1	no	—
12	C₂×C₉×D₁₂	—	2	1	no	—
13	C₂×C₉×D₁₂	C₄	1	2	yes	—
14	C₂×C₉×D₁₂	C₂²	1	2	yes	—
15	#88532	—	2	1	no	—
16	#88532	C₂²	1	2	yes	complemented
17	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
18	C₃×#3667	—	2	1	no	—
19	C₃×#3667	C₂²	1	2	yes	complemented
20	C₂×C₃×C₉⋊C₈	—	2	1	no	—
21	C₂²×C₃×D₁₂	D₃	1	1	yes	characteristic
22	C₂×#3748	—	3	1	no	—
23	C₂×C₃×#208	—	3	1	no	—
24	C₂²×D₄×C₉	—	3	1	no	—
25	#63558	—	3	4	no	—
26	C₂×#3667	—	3	1	no	—
27	C₂×#3667	—	6	1	no	—
28	C₂×#3667	D₃	1	1	yes	characteristic, complemented
29	#63391	—	3	4	no	—
30	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₃×C₉	—	2	2	no	—
32	C₂²×D₃×C₉	—	2	2	no	—
33	C₂²×D₃×C₉	—	4	4	no	—
34	C₂²×D₃×C₉	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₉×D₁₂	—	2	2	no	—
42	C₉×D₁₂	—	2	2	no	—
43	C₉×D₁₂	—	2	2	no	—
44	C₉×D₁₂	—	2	2	no	—
45	C₃⋊(C₉⋊C₈)	—	2	2	no	—
46	C₃×C₉⋊C₈	—	2	2	no	—
47	C₂×#833	—	9	1	no	—
48	C₂×#823	—	9	1	no	—
49	C₂³×C₃×D₃	Dic₃	1	2	yes	—
50	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×D₁₂	—	2	1	no	—
52	C₂×C₃×D₁₂	—	2	1	no	—
53	C₂×C₃×D₁₂	—	2	1	no	—
54	C₂×C₃×D₁₂	—	2	1	no	—
55	C₂×C₃×D₁₂	Dic₃	1	2	yes	—
56	C₂×C₃×D₁₂	C₂×D₃	1	2	yes	—
57	C₂⁴×C₉	—	3	2	no	—
58	#3748	—	3	2	no	—
59	#3748	—	6	1	no	—
60	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
61	C₃×#208	—	3	2	no	—
62	C₃×#208	—	6	1	no	—
63	C₂×C₃×C₃⋊C₈	—	6	1	no	—
64	C₂×D₄×C₉	—	3	2	no	—
65	C₂×D₄×C₉	—	3	2	no	—
66	C₂×D₄×C₉	—	6	1	no	—
67	C₂×D₄×C₉	—	6	1	no	—
68	C₂×D₄×C₉	—	6	1	no	—
69	C₂×D₄×C₉	—	6	1	no	—
70	C₂²×C₄×C₉	—	2	1	no	—
71	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
72	#3667	—	2	1	no	—
73	#3667	—	3	2	no	—
74	#3667	—	6	2	no	—
75	#3667	—	6	2	no	—
76	#3667	—	6	1	no	—
77	#3667	C₂×D₃	1	2	yes	complemented
78	C₂×C₉⋊C₈	—	2	1	no	—
79	C₂×C₉⋊C₈	—	6	2	no	—
80	C₂×C₉⋊C₈	—	6	1	no	—
81	C₂²×C₃×C₉	—	2	2	no	—
82	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×D₃×C₉	—	2	4	no	—
86	C₂×D₃×C₉	—	4	4	no	—
87	C₂×D₃×C₉	—	4	4	no	—
88	C₂×D₃×C₉	—	4	4	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	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₃×D₄	—	3	1	no	—
94	C₂²×D₁₂	D₉	1	1	yes	characteristic
95	C₂×C₃×M₁₆	—	9	1	no	—
96	C₂×#208	—	3	1	no	—
97	C₂×#208	—	9	1	no	—
98	C₂×#208	—	9	1	no	—
99	C₂×#208	—	18	1	no	—
100	#833	—	9	4	no	—
101	#823	—	9	4	no	—
102	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₃×D₃	—	2	2	no	—
104	C₂²×C₃×D₃	—	2	2	no	—
105	C₂²×C₃×D₃	—	4	4	no	—
106	C₂²×C₃×D₃	C₂×Dic₃	1	4	yes	—
107	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	#67	1	1	yes	characteristic
109	C₂×C₃²×C₄	#67	1	1	yes	characteristic
110	C₂×C₃²×C₄	#67	1	1	yes	characteristic
111	C₂×C₃²×C₄	#67	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₃×D₁₂	—	2	2	no	—
114	C₃×D₁₂	—	2	2	no	—
115	C₃×D₁₂	—	2	2	no	—
116	C₃×D₁₂	—	2	2	no	—
117	C₂³×C₉	—	2	1	no	—
118	C₂³×C₉	—	3	4	no	—
119	C₂³×C₉	—	6	2	no	—
120	C₂³×C₉	—	6	2	no	—
121	C₂³×C₉	—	12	4	no	—
122	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
123	C₃⋊(C₃⋊C₈)	—	6	2	no	—
124	C₃×C₃⋊C₈	—	6	2	no	—
125	D₄×C₉	—	6	2	no	—
126	D₄×C₉	—	6	2	no	—
127	D₄×C₉	—	6	2	no	—
128	D₄×C₉	—	6	2	no	—
129	C₂×C₄×C₉	—	2	1	no	—
130	C₂×C₄×C₉	—	2	1	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	2	1	no	—
135	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
136	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
137	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
138	C₂×C₄×C₉	#67	1	1	yes	characteristic
139	C₂×C₄×C₉	#67	1	1	yes	characteristic
140	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
141	C₉⋊C₈	—	2	2	no	—
142	C₉⋊C₈	—	6	4	no	—
143	C₉⋊C₈	—	6	2	no	—
144	C₂×#100	—	27	1	no	Sylow
145	C₂×C₃×C₉	—	2	1	no	—
146	C₂×C₃×C₉	—	2	1	no	—
147	C₂×C₃×C₉	#100	1	2	yes	complemented
148	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	D₃×C₉	—	4	4	no	—
150	C₂⁴×C₃	—	3	2	no	—
151	C₂³×D₃	Dic₉	1	2	yes	—
152	C₂×C₃×D₄	—	3	2	no	—
153	C₂×C₃×D₄	—	3	2	no	—
154	C₂×C₃×D₄	—	6	1	no	—
155	C₂×C₃×D₄	—	6	1	no	—
156	C₂×C₃×D₄	—	6	1	no	—
157	C₂×C₃×D₄	—	6	1	no	—
158	C₂²×C₃×C₄	—	2	1	no	—
159	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
160	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
161	C₂×D₁₂	—	2	1	no	—
162	C₂×D₁₂	—	2	1	no	—
163	C₂×D₁₂	—	2	1	no	—
164	C₂×D₁₂	—	2	1	no	—
165	C₂×D₁₂	Dic₉	1	2	yes	—
166	C₂×D₁₂	C₂×D₉	1	2	yes	—
167	C₃×M₁₆	—	9	2	no	—
168	C₃×M₁₆	—	18	1	no	—
169	C₂×C₃×C₈	—	18	1	no	—
170	#208	—	3	2	no	—
171	#208	—	6	1	no	—
172	#208	—	9	2	no	—
173	#208	—	9	2	no	—
174	#208	—	18	2	no	—
175	#208	—	18	2	no	—
176	#208	—	18	1	no	—
177	#208	—	18	1	no	—
178	C₂×C₃⋊C₈	—	6	1	no	—
179	C₂×C₃⋊C₈	—	18	2	no	—
180	C₂×C₃⋊C₈	—	18	1	no	—
181	C₂×C₃⋊C₈	—	18	1	no	—
182	C₂²×C₃²	—	2	2	no	—
183	C₂²×C₃²	#217	1	1	yes	characteristic
184	C₂²×C₃²	#217	1	1	yes	characteristic
185	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
186	C₂×C₃×D₃	—	2	4	no	—
187	C₂×C₃×D₃	—	4	4	no	—
188	C₂×C₃×D₃	—	4	4	no	—
189	C₂×C₃×D₃	—	4	4	no	—
190	C₃²×C₄	#217	1	1	yes	characteristic
191	C₃²×C₄	#217	1	1	yes	characteristic
192	C₃²×C₄	C₂×#67	1	1	yes	characteristic
193	C₃²×C₄	C₂×#67	1	1	yes	characteristic
194	C₂²×C₉	—	2	2	no	—
195	C₂²×C₉	—	2	1	no	—
196	C₂²×C₉	—	2	1	no	—
197	C₂²×C₉	—	2	1	no	—
198	C₂²×C₉	—	4	2	no	—
199	C₂²×C₉	—	6	4	no	—
200	C₂²×C₉	—	12	4	no	—
201	C₂²×C₉	—	12	4	no	—
202	C₂²×C₉	—	12	4	no	—
203	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
206	C₄×C₉	—	2	1	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	2	1	no	—
209	C₄×C₉	—	2	1	no	—
210	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
213	C₄×C₉	C₂×#67	1	1	yes	characteristic
214	C₂²×D₄	—	3	1	no	—
215	C₂×M₁₆	—	9	1	no	—
216	C₂×M₁₆	—	27	1	no	—
217	#100	—	27	4	no	—
218	C₃×C₉	C₂×#100	1	1	yes	characteristic, complemented, Sylow
219	C₂³×C₃	—	2	1	no	—
220	C₂³×C₃	—	3	4	no	—
221	C₂³×C₃	—	6	2	no	—
222	C₂³×C₃	—	6	2	no	—
223	C₂³×C₃	—	12	4	no	—
224	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
225	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
226	C₂²×D₃	—	2	2	no	—
227	C₂²×D₃	—	2	2	no	—
228	C₂²×D₃	—	4	4	no	—
229	C₂²×D₃	C₂×Dic₉	1	4	yes	—
230	C₃×D₄	—	6	2	no	—
231	C₃×D₄	—	6	2	no	—
232	C₃×D₄	—	6	2	no	—
233	C₃×D₄	—	6	2	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	2	1	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
241	C₂×C₃×C₄	#611	1	1	yes	characteristic
242	C₂×C₃×C₄	#611	1	1	yes	characteristic
243	C₂×C₃×C₄	#611	1	1	yes	characteristic
244	C₂×C₃×C₄	#611	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
246	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
247	C₂×C₃×C₄	#625	1	1	yes	characteristic
248	C₂×C₃×C₄	#625	1	1	yes	characteristic
249	C₂×C₃×C₄	#626	1	1	yes	characteristic
250	C₂×C₃×C₄	#626	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
252	D₁₂	—	2	2	no	—
253	D₁₂	—	2	2	no	—
254	D₁₂	—	2	2	no	—
255	D₁₂	—	2	2	no	—
256	C₃×C₈	—	18	2	no	—
257	C₃⋊C₈	—	6	2	no	—
258	C₃⋊C₈	—	18	4	no	—
259	C₃⋊C₈	—	18	2	no	—
260	C₃⋊C₈	—	18	2	no	—
261	C₂×C₃²	—	2	1	no	—
262	C₂×C₃²	—	2	1	no	—
263	C₂×C₃²	#833	1	2	yes	—
264	C₂×C₃²	C₂×#217	1	1	yes	characteristic
265	C₃×D₃	—	4	4	no	—
266	C₂×C₉	—	2	1	no	—
267	C₂×C₉	—	2	1	no	—
268	C₂×C₉	—	2	2	no	—
269	C₂×C₉	—	2	1	no	—
270	C₂×C₉	—	4	1	no	—
271	C₂×C₉	—	4	1	no	—
272	C₂×C₉	—	12	4	no	—
273	C₂×C₉	#823	1	2	yes	complemented
274	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
275	C₂⁴	—	3	2	no	—
276	C₂×D₄	—	3	2	no	—
277	C₂×D₄	—	3	2	no	—
278	C₂×D₄	—	6	1	no	—
279	C₂×D₄	—	6	1	no	—
280	C₂×D₄	—	6	1	no	—
281	C₂×D₄	—	6	1	no	—
282	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
283	M₁₆	—	9	2	no	—
284	M₁₆	—	18	1	no	—
285	M₁₆	—	27	2	no	—
286	M₁₆	—	54	1	no	—
287	C₂×C₈	—	18	1	no	—
288	C₂×C₈	—	54	1	no	—
289	C₂²×C₃	—	2	2	no	—
290	C₂²×C₃	—	2	2	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	2	1	no	—
294	C₂²×C₃	—	4	2	no	—
295	C₂²×C₃	—	6	4	no	—
296	C₂²×C₃	—	12	4	no	—
297	C₂²×C₃	—	12	4	no	—
298	C₂²×C₃	—	12	4	no	—
299	C₂²×C₃	#3676	1	1	yes	characteristic
300	C₂²×C₃	#3676	1	1	yes	characteristic
301	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
302	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
303	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
304	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
305	C₂×D₃	—	2	4	no	—
306	C₂×D₃	—	4	4	no	—
307	C₂×D₃	—	4	4	no	—
308	C₂×D₃	—	4	4	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	1	no	—
313	C₃×C₄	#3676	1	1	yes	characteristic
314	C₃×C₄	#3676	1	1	yes	characteristic
315	C₃×C₄	C₂×#611	1	1	yes	characteristic
316	C₃×C₄	C₂×#611	1	1	yes	characteristic
317	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
318	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
319	C₃×C₄	C₂×#625	1	1	yes	characteristic
320	C₃×C₄	C₂×#626	1	1	yes	characteristic
321	C₃²	C₂×#833	1	1	yes	characteristic
322	C₉	—	2	1	no	—
323	C₉	C₂×#823	1	1	yes	characteristic, complemented
324	C₂³	—	3	4	no	—
325	C₂³	—	6	2	no	—
326	C₂³	—	6	2	no	—
327	C₂³	—	12	4	no	—
328	C₂³	D₃×Dic₉	1	1	yes	characteristic
329	D₄	—	6	2	no	—
330	D₄	—	6	2	no	—
331	D₄	—	6	2	no	—
332	D₄	—	6	2	no	—
333	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
334	C₂×C₄	#6194	1	1	yes	characteristic
335	C₂×C₄	#6194	1	1	yes	characteristic
336	C₂×C₄	#6195	1	1	yes	characteristic
337	C₂×C₄	#6195	1	1	yes	characteristic
338	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
339	C₈	—	18	2	no	—
340	C₈	—	54	2	no	—
341	C₂×C₃	—	2	1	no	—
342	C₂×C₃	—	2	1	no	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	2	no	—
347	C₂×C₃	—	4	1	no	—
348	C₂×C₃	—	4	1	no	—
349	C₂×C₃	—	12	4	no	—
350	C₂×C₃	#63391	1	2	yes	complemented
351	C₂×C₃	C₂×#3676	1	1	yes	characteristic
352	C₂×C₃	#63558	1	2	yes	—
353	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
354	D₃	—	4	4	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	6	4	no	—
357	C₂²	—	12	4	no	—
358	C₂²	—	12	4	no	—
359	C₂²	—	12	4	no	—
360	C₂²	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
361	C₂²	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
362	C₂²	C₂×D₃×Dic₉	1	1	yes	characteristic
363	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
364	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
365	C₄	C₂×#6194	1	1	yes	characteristic
366	C₄	C₂×#6195	1	1	yes	characteristic
367	C₃	—	2	1	no	—
368	C₃	C₂×#63391	1	1	yes	characteristic, complemented
369	C₃	C₂×#63558	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	12	4	no	—
373	C₂	#11720296	1	2	yes	direct product factor
374	C₂	C₂×(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊Dic₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	#217
prime radical	C₂²×C₃×D₁₂	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#100
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
2. C₂×#88532	C₂	semidirect
3. C₂×C₃×#3667	C₂	semidirect
4. #11720296	C₂	direct (N × H)
16. #88532	C₂²	semidirect
19. C₃×#3667	C₂²	semidirect
28. C₂×#3667	D₃	semidirect
77. #3667	C₂×D₃	semidirect
147. C₂×C₃×C₉	#100	semidirect
218. C₃×C₉	C₂×#100	semidirect
273. C₂×C₉	#823	semidirect
323. C₉	C₂×#823	semidirect
350. C₂×C₃	#63391	semidirect
368. C₃	C₂×#63391	semidirect
373. C₂	#11720296	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×#88532	2
C₂×C₃×#3667	2
#11720296	2
C₂×#63558	3
C₂×#63391	3

Subgroups — C2×#11720296 #13139459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720296 #13139459