Subgroups — C₂×#11720518 #13144231

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×#88736	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720518	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63632	—	3	1	no	maximal
6	C₂×#63364	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	—	2	1	no	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×#6233	—	2	1	no	—
10	#88736	—	2	1	no	—
11	#88736	—	2	1	no	—
12	#88736	C₄	1	2	yes	complemented
13	#88736	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₃⋊D₉)	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄×C₃⋊D₉	—	2	1	no	—
16	C₂×#6226	C₄	1	1	yes	characteristic, complemented
17	C₃×C₄²×C₉	—	2	1	no	—
18	C₃×C₄²×C₉	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×#3828	—	3	1	no	—
23	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
24	C₂×#3748	—	3	1	no	—
25	C₂×#3697	—	3	3	no	—
26	#63632	—	3	4	no	—
27	C₂×C₄²×C₉	D₃	1	3	yes	complemented
28	C₂×#3667	—	3	3	no	—
29	#63364	—	3	12	no	—
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₂²×C₃⋊D₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	4	no	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
40	#6233	—	2	2	no	—
41	#6233	—	2	2	no	—
42	C₂×C₃⋊Dic₉	—	2	1	no	—
43	C₄⋊(C₃⋊D₉)	—	2	1	no	—
44	C₄⋊(C₃⋊D₉)	C₂×C₄	1	2	yes	complemented
45	C₄×C₃⋊D₉	—	2	2	no	—
46	C₄×C₃⋊D₉	—	2	2	no	—
47	#6226	—	2	1	no	—
48	#6226	C₂×C₄	1	2	yes	complemented
49	C₃⋊(C₉⋊C₈)	—	2	2	no	—
50	C₂×#805	—	9	3	no	—
51	C₂×#805	—	9	1	no	—
52	C₂²×C₃²×C₄	—	2	1	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
54	C₂×#638	—	6	1	no	—
55	#3828	—	3	2	no	—
56	#3828	—	3	2	no	—
57	#3828	—	6	1	no	—
58	#3828	—	6	1	no	—
59	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
60	C₂×C₄×C₃⋊D₃	—	6	1	no	—
61	C₂×#634	—	3	1	no	—
62	C₃²×C₄²	—	2	1	no	—
63	C₃²×C₄²	C₂×D₃	1	2	yes	—
64	#3748	—	3	2	no	—
65	#3748	—	6	1	no	—
66	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
67	C₂²×C₄×C₉	—	2	3	no	—
68	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
69	C₂×#611	—	6	3	no	—
70	#3697	—	3	6	no	—
71	#3697	—	3	6	no	—
72	#3697	—	6	3	no	—
73	#3697	—	6	3	no	—
74	C₂×D₃₆	—	3	3	no	—
75	C₂×C₄×D₉	—	6	3	no	—
76	C₂×Dic₁₈	—	3	3	no	—
77	C₄²×C₉	—	2	3	no	—
78	C₄²×C₉	C₂×D₃	1	6	yes	complemented
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	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃⋊D₉	—	2	2	no	—
87	C₂×C₃⋊D₉	—	4	2	no	—
88	C₃×C₄×C₉	—	2	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	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₃⋊Dic₉	—	2	2	no	—
94	C₂×#235	—	9	3	no	—
95	C₂×#235	—	9	1	no	—
96	C₂×C₃×C₄²	D₉	1	3	yes	complemented
97	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
98	C₂×#208	—	9	3	no	—
99	C₂×#208	—	9	1	no	—
100	#805	—	9	12	no	—
101	#805	—	9	4	no	—
102	C₂³×C₃²	#67	1	1	yes	characteristic
103	C₂²×C₃⋊D₃	—	6	1	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
109	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
110	C₂×C₃²×C₄	#67	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	#638	—	6	2	no	—
113	#638	—	6	2	no	—
114	C₂×C₃⋊Dic₃	—	6	1	no	—
115	C₄⋊(C₃⋊D₃)	—	3	2	no	—
116	C₄⋊(C₃⋊D₃)	—	6	1	no	—
117	C₄×C₃⋊D₃	—	6	2	no	—
118	C₄×C₃⋊D₃	—	6	2	no	—
119	#634	—	3	2	no	—
120	#634	—	6	1	no	—
121	C₂³×C₉	#67	1	3	yes	—
122	C₂²×D₉	—	6	3	no	—
123	C₃⋊(C₃⋊C₈)	—	6	2	no	—
124	C₂×C₄×C₉	—	2	12	no	—
125	C₂×C₄×C₉	—	2	6	no	—
126	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
127	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
128	C₂×C₄×C₉	D₁₂	1	3	yes	—
129	C₂×C₄×C₉	D₁₂	1	3	yes	—
130	C₂×C₄×C₉	#67	1	3	yes	—
131	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
132	#611	—	6	6	no	—
133	#611	—	6	6	no	—
134	C₂×Dic₉	—	6	3	no	—
135	D₃₆	—	3	6	no	—
136	D₃₆	—	6	3	no	—
137	C₄×D₉	—	6	6	no	—
138	C₄×D₉	—	6	6	no	—
139	Dic₁₈	—	3	6	no	—
140	Dic₁₈	—	6	3	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	#104	1	2	yes	complemented
146	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₃⋊D₉	—	4	2	no	—
148	C₂²×C₃×C₄	—	2	3	no	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
151	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
152	C₂×#67	—	18	3	no	—
153	C₂×#67	—	18	1	no	—
154	#235	—	9	6	no	—
155	#235	—	9	6	no	—
156	#235	—	9	2	no	—
157	#235	—	9	2	no	—
158	#235	—	18	3	no	—
159	#235	—	18	3	no	—
160	#235	—	18	1	no	—
161	#235	—	18	1	no	—
162	C₂×D₁₂	—	9	3	no	—
163	C₂×D₁₂	—	9	1	no	—
164	C₂×C₄×D₃	—	18	3	no	—
165	C₂×C₄×D₃	—	18	1	no	—
166	C₂×Dic₆	—	9	3	no	—
167	C₂×Dic₆	—	9	1	no	—
168	C₃×C₄²	—	2	3	no	—
169	C₃×C₄²	—	2	1	no	—
170	C₃×C₄²	C₂×D₉	1	6	yes	complemented
171	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
182	C₂×C₃⋊D₃	—	6	2	no	—
183	C₂×C₃⋊D₃	—	12	2	no	—
184	C₃²×C₄	—	2	4	no	—
185	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
188	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
189	C₃⋊Dic₃	—	6	2	no	—
190	C₂²×C₉	—	2	6	no	—
191	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
192	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
193	C₂²×C₉	C₂×#67	1	3	yes	—
194	C₂×D₉	—	6	6	no	—
195	C₂×D₉	—	12	6	no	—
196	C₄×C₉	—	2	12	no	—
197	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
198	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
199	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
200	C₄×C₉	C₂×D₁₂	1	3	yes	—
201	Dic₉	—	6	6	no	—
202	C₂×#41	—	27	1	no	—
203	C₂×M₁₆	—	27	1	no	—
204	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic, complemented
205	#104	—	27	4	no	—
206	C₃×C₉	C₂×#104	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₃	#611	1	3	yes	—
208	C₂³×C₃	#638	1	1	yes	characteristic
209	C₂²×D₃	—	18	3	no	—
210	C₂²×D₃	—	18	1	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	2	12	no	—
213	C₂×C₃×C₄	—	2	6	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
216	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
217	C₂×C₃×C₄	D₃₆	1	3	yes	—
218	C₂×C₃×C₄	D₃₆	1	3	yes	—
219	C₂×C₃×C₄	#611	1	3	yes	—
220	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
221	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
222	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
223	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
224	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
225	C₂×C₃×C₄	#638	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
227	#67	—	18	6	no	—
228	#67	—	18	6	no	—
229	#67	—	18	2	no	—
230	#67	—	18	2	no	—
231	C₂×Dic₃	—	18	3	no	—
232	C₂×Dic₃	—	18	1	no	—
233	D₁₂	—	9	6	no	—
234	D₁₂	—	9	2	no	—
235	D₁₂	—	18	3	no	—
236	D₁₂	—	18	1	no	—
237	C₄×D₃	—	18	6	no	—
238	C₄×D₃	—	18	6	no	—
239	C₄×D₃	—	18	2	no	—
240	C₄×D₃	—	18	2	no	—
241	Dic₆	—	9	6	no	—
242	Dic₆	—	9	2	no	—
243	Dic₆	—	18	3	no	—
244	Dic₆	—	18	1	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	#805	1	2	yes	—
250	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
251	C₃⋊D₃	—	12	2	no	—
252	C₂×C₉	—	2	3	no	—
253	C₂×C₉	—	2	3	no	—
254	C₂×C₉	#805	1	6	yes	complemented
255	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	3	yes	—
256	D₉	—	12	6	no	—
257	#41	—	27	2	no	—
258	#41	—	27	2	no	—
259	#41	—	54	1	no	—
260	#41	—	54	1	no	—
261	C₂×Q₈	—	27	1	no	—
262	C₂×D₄	—	27	1	no	—
263	C₂×D₄	—	54	1	no	—
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	—	2	1	no	—
271	C₄²	C₂×C₃⋊D₉	1	2	yes	complemented
272	C₂²×C₃	—	2	6	no	—
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
275	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
276	C₂²×C₃	C₂×#611	1	3	yes	—
277	C₂²×C₃	#3752	1	1	yes	characteristic
278	C₂²×C₃	#3752	1	1	yes	characteristic
279	C₂²×C₃	C₂×#638	1	1	yes	characteristic
280	C₂×D₃	—	18	6	no	—
281	C₂×D₃	—	18	2	no	—
282	C₂×D₃	—	36	6	no	—
283	C₂×D₃	—	36	2	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	12	no	—
286	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
287	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
288	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
289	C₃×C₄	C₂×D₃₆	1	3	yes	—
290	C₃×C₄	#3752	1	1	yes	characteristic
291	C₃×C₄	#3752	1	1	yes	characteristic
292	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
293	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
294	Dic₃	—	18	6	no	—
295	Dic₃	—	18	2	no	—
296	C₃²	C₂×#805	1	1	yes	characteristic
297	C₉	C₂×#805	1	3	yes	complemented
298	C₂³	—	54	1	no	—
299	C₂³	#6233	1	1	yes	characteristic
300	Q₈	—	27	2	no	—
301	Q₈	—	54	1	no	—
302	D₄	—	27	2	no	—
303	D₄	—	54	2	no	—
304	D₄	—	54	2	no	—
305	D₄	—	54	1	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	54	1	no	—
309	C₂×C₄	—	54	2	no	—
310	C₂×C₄	—	54	2	no	—
311	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
312	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
313	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
314	C₂×C₄	C₄⋊(C₃⋊D₉)	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	#63364	1	6	yes	complemented
323	C₂×C₃	C₂×(C₂×D₉)⋊C₄	1	3	yes	—
324	C₂×C₃	#63632	1	2	yes	—
325	C₂×C₃	C₂×#3752	1	1	yes	characteristic
326	D₃	—	36	2	no	—
327	D₃	—	36	6	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	54	2	no	—
330	C₂²	—	108	2	no	—
331	C₂²	#88536	1	1	yes	characteristic
332	C₂²	#88536	1	1	yes	characteristic
333	C₂²	C₂×#6233	1	1	yes	characteristic
334	C₄	—	2	4	no	—
335	C₄	—	54	2	no	—
336	C₄	#88536	1	1	yes	characteristic
337	C₄	#88536	1	1	yes	characteristic
338	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
339	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
340	C₃	C₂×#63364	1	3	yes	complemented
341	C₃	C₂×#63632	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	108	2	no	—
345	C₂	#11720518	1	2	yes	direct product factor
346	C₂	C₂×#88536	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊(C₃⋊D₉)
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂×#88736	C₂
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	C₂×#104
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₉	C₂	semidirect
3. C₂×#88532	C₂	semidirect
4. #11720518	C₂	direct (N × H)
12. #88736	C₄	semidirect
14. C₂×C₄⋊(C₃⋊D₉)	C₄	semidirect
16. C₂×#6226	C₄	semidirect
18. C₃×C₄²×C₉	C₂²	semidirect
20. #88532	C₂²	semidirect
27. C₂×C₄²×C₉	D₃	semidirect
44. C₄⋊(C₃⋊D₉)	C₂×C₄	semidirect
48. #6226	C₂×C₄	semidirect
78. C₄²×C₉	C₂×D₃	semidirect
96. C₂×C₃×C₄²	D₉	semidirect
145. C₂×C₃×C₉	#104	semidirect
170. C₃×C₄²	C₂×D₉	semidirect
204. C₂×C₄²	C₃⋊D₉	semidirect
206. C₃×C₉	C₂×#104	semidirect
254. C₂×C₉	#805	semidirect
271. C₄²	C₂×C₃⋊D₉	semidirect
297. C₉	C₂×#805	semidirect
322. C₂×C₃	#63364	semidirect
340. C₃	C₂×#63364	semidirect
345. C₂	#11720518	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×#88736	2
C₂×C₃×C₄²×C₉	2
C₂×#88532	2
#11720518	2
C₂×#63632	3
C₂×#63364	3

Subgroups — C2×#11720518 #13144231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720518 #13144231