Subgroups — #13138455

Order 1728.

440 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88523	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6195	C₂	1	1	yes	maximal, characteristic
4	C₃×#63375	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₃)⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720282	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₉⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590726	—	3	1	no	maximal
9	#10588792	—	3	1	no	maximal
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×#6195	C₄	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#88523	—	2	2	no	—
16	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	D₃×C₈×C₉	—	2	2	no	—
20	(C₂×D₃)⋊Dic₉	C₄	1	1	yes	characteristic
21	#88445	C₄	1	1	yes	characteristic
22	Dic₉⋊Dic₃	C₄	1	1	yes	characteristic
23	C₂×#3743	—	3	1	no	—
24	C₂×C₃×D₃×C₈	D₃	1	1	yes	characteristic
25	C₄×#626	—	3	1	no	—
26	C₃×#815	—	3	1	no	—
27	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
28	#63557	—	3	1	no	—
29	(C₂×D₃)⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂²×C₈×C₉	—	3	1	no	—
31	C₄×#611	—	3	1	no	—
32	C₂×#3663	—	3	1	no	—
33	C₂×#3663	—	6	1	no	—
34	#63389	—	3	1	no	—
35	(C₂×D₉)⋊C₈	—	3	1	no	—
36	#63375	D₃	1	1	yes	characteristic, complemented
37	Dic₉⋊C₈	—	3	1	no	—
38	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₃⋊D₉	—	2	1	no	—
43	C₄×C₃⋊D₉	—	2	1	no	—
44	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
47	C₄×D₃×C₉	—	2	1	no	—
48	C₄×D₃×C₉	—	2	1	no	—
49	#6195	—	2	2	no	—
50	C₃×C₈×C₉	D₄	1	2	yes	complemented
51	C₃⋊(C₉⋊C₈)	—	2	1	no	—
52	C₉×C₃⋊C₈	—	2	1	no	—
53	C₃×C₉⋊C₈	—	2	1	no	—
54	#5111	—	9	1	no	—
55	#4706	—	9	1	no	—
56	C₂×C₄×C₃⋊D₃	—	3	1	no	—
57	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
58	C₂×#626	—	3	1	no	—
59	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
61	#3743	—	6	2	no	—
62	C₃×C₄×Dic₃	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×D₃×C₈	—	2	2	no	—
66	Dic₃⋊Dic₃	—	3	1	no	—
67	#3722	—	3	1	no	—
68	(C₂×D₃)⋊Dic₃	—	3	1	no	—
69	C₂²×C₄×C₉	—	3	1	no	—
70	C₂×#611	—	3	1	no	—
71	C₂×C₄×D₉	—	3	1	no	—
72	C₂×C₄×D₉	—	6	1	no	—
73	C₂×C₈×C₉	—	2	1	no	—
74	C₂×C₈×C₉	—	3	1	no	—
75	C₂×C₈×C₉	—	6	2	no	—
76	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
77	#3676	—	3	1	no	—
78	(C₂×D₉)⋊C₄	—	3	1	no	—
79	Dic₉⋊C₄	—	3	1	no	—
80	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
81	C₂×C₉⋊C₈	—	3	1	no	—
82	C₂×C₉⋊C₈	—	6	1	no	—
83	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
84	#3663	—	6	4	no	—
85	#3663	—	6	2	no	—
86	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃⋊D₉	—	2	1	no	—
88	C₂×C₃⋊D₉	—	2	1	no	—
89	C₂×C₃⋊D₉	—	2	1	no	—
90	C₂×D₃×C₉	—	2	1	no	—
91	C₂×D₃×C₉	—	2	1	no	—
92	C₂×D₃×C₉	M₁₆	1	2	yes	complemented
93	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄×C₉	#41	1	1	yes	characteristic
95	C₃⋊Dic₉	—	2	1	no	—
96	C₉×Dic₃	—	2	1	no	—
97	C₃×Dic₉	—	2	1	no	—
98	C₃×Dic₉	—	2	1	no	—
99	C₂²×C₃×C₈	—	3	1	no	—
100	C₄×#67	—	9	1	no	—
101	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
102	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
103	C₂×ΓL(1,𝔽₂₅)	—	18	1	no	—
104	C₂×D₃×C₈	D₉	1	1	yes	characteristic, complemented
105	C₄×D₁₂	—	9	1	no	—
106	C₃×#97	—	9	1	no	—
107	#830	—	9	1	no	—
108	(C₂×D₃)⋊C₈	—	9	1	no	—
109	(C₂×D₃)⋊C₈	—	9	1	no	—
110	(C₂×D₃)⋊C₈	—	9	1	no	—
111	#815	—	3	1	no	—
112	Dic₃⋊C₈	—	9	1	no	—
113	C₄⋊(C₃⋊C₈)	—	9	1	no	—
114	C₂²×C₃⋊D₃	—	3	1	no	—
115	C₂²×C₃×D₃	C₄×D₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₃⋊Dic₃	—	3	1	no	—
118	C₄×C₃⋊D₃	—	6	1	no	—
119	C₄×C₃⋊D₃	—	6	1	no	—
120	C₂×C₃×Dic₃	—	3	1	no	—
121	C₂×C₃×Dic₃	—	3	1	no	—
122	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
123	C₃×C₄×D₃	—	2	1	no	—
124	C₃×C₄×D₃	—	2	1	no	—
125	#626	—	6	2	no	—
126	C₂³×C₉	—	3	1	no	—
127	C₂²×D₉	—	3	1	no	—
128	C₂²×D₉	—	6	1	no	—
129	C₃²×C₈	#67	1	2	yes	—
130	C₃⋊(C₃⋊C₈)	—	6	1	no	—
131	C₃×C₃⋊C₈	—	2	1	no	—
132	C₃×C₃⋊C₈	—	6	1	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	3	1	no	—
135	C₂×C₄×C₉	—	6	1	no	—
136	C₂×C₄×C₉	—	6	1	no	—
137	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
138	#611	—	6	2	no	—
139	C₂×Dic₉	—	3	1	no	—
140	C₂×Dic₉	—	6	1	no	—
141	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
142	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
143	C₄×D₉	—	6	2	no	—
144	C₄×D₉	—	6	2	no	—
145	C₄×D₉	—	6	1	no	—
146	C₄×D₉	—	6	1	no	—
147	C₈×C₉	—	2	2	no	—
148	C₈×C₉	—	6	1	no	—
149	C₈×C₉	D₁₂	1	2	yes	complemented
150	C₉⋊C₈	—	2	1	no	—
151	C₉⋊C₈	—	6	2	no	—
152	C₉⋊C₈	—	6	1	no	—
153	#435	—	27	1	no	Sylow
154	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
155	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
156	C₂×C₃×C₉	#131	1	1	yes	characteristic
157	C₃⋊D₉	—	4	1	no	—
158	D₃×C₉	—	2	2	no	—
159	C₂²×C₃×C₄	—	3	1	no	—
160	C₂×#67	—	9	1	no	—
161	C₂×D₁₂	—	9	1	no	—
162	C₂×C₄×D₃	—	9	1	no	—
163	C₂×C₄×D₃	—	9	1	no	—
164	C₂×C₄×D₃	—	18	1	no	—
165	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
166	C₂×C₃×C₈	—	2	1	no	—
167	C₂×C₃×C₈	—	3	1	no	—
168	C₂×C₃×C₈	—	6	2	no	—
169	C₂×C₃×C₈	—	9	1	no	—
170	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
171	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
172	C₃×C₄²	—	9	1	no	—
173	#217	—	9	1	no	—
174	(C₂×D₃)⋊C₄	—	9	1	no	—
175	(C₂×D₃)⋊C₄	—	9	1	no	—
176	(C₂×D₃)⋊C₄	—	9	1	no	—
177	C₄⋊Dic₃	—	9	1	no	—
178	Dic₃⋊C₄	—	9	1	no	—
179	C₄×Dic₃	—	3	1	no	—
180	C₂×C₃⋊C₈	—	3	1	no	—
181	C₂×C₃⋊C₈	—	9	1	no	—
182	C₂×C₃⋊C₈	—	9	1	no	—
183	C₂×C₃⋊C₈	—	18	1	no	—
184	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
185	ΓL(1,𝔽₂₅)	—	18	4	no	—
186	ΓL(1,𝔽₂₅)	—	18	2	no	—
187	ΓL(1,𝔽₂₅)	—	18	2	no	—
188	D₃×C₈	—	2	2	no	—
189	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₃⋊D₃	—	6	1	no	—
191	C₂×C₃⋊D₃	—	6	1	no	—
192	C₂×C₃⋊D₃	—	6	1	no	—
193	C₂×C₃×D₃	—	2	1	no	—
194	C₂×C₃×D₃	—	2	1	no	—
195	C₂×C₃×D₃	ΓL(1,𝔽₂₅)	1	2	yes	—
196	C₃²×C₄	#235	1	1	yes	characteristic
197	C₃²×C₄	C₂×#67	1	1	yes	characteristic
198	C₃⋊Dic₃	—	6	1	no	—
199	C₃×Dic₃	—	2	1	no	—
200	C₃×Dic₃	—	6	1	no	—
201	C₃×Dic₃	—	6	1	no	—
202	C₂²×C₉	—	2	1	no	—
203	C₂²×C₉	—	3	2	no	—
204	C₂²×C₉	—	6	1	no	—
205	C₂²×C₉	—	6	1	no	—
206	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
207	C₂×D₉	—	6	2	no	—
208	C₂×D₉	—	6	1	no	—
209	C₂×D₉	—	6	1	no	—
210	C₂×D₉	—	6	1	no	—
211	C₂×D₉	—	12	1	no	—
212	C₂×D₉	—	12	1	no	—
213	C₄×C₉	—	2	1	no	—
214	C₄×C₉	—	2	1	no	—
215	C₄×C₉	—	6	1	no	—
216	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
217	C₄×C₉	#235	1	1	yes	characteristic
218	Dic₉	—	2	1	no	—
219	Dic₉	—	2	1	no	—
220	Dic₉	—	6	2	no	—
221	Dic₉	—	6	1	no	—
222	C₂×M₁₆	—	27	1	no	—
223	C₂²×C₈	—	3	1	no	—
224	C₄×D₄	—	27	1	no	—
225	C₄⋊C₈	—	27	1	no	—
226	C₂²⋊C₈	—	27	1	no	—
227	C₂²⋊C₈	—	27	1	no	—
228	#97	—	9	1	no	—
229	C₃×C₉	#435	1	1	yes	characteristic, complemented, Sylow
230	C₂³×C₃	—	3	1	no	—
231	C₂²×D₃	—	9	1	no	—
232	C₂²×D₃	—	9	1	no	—
233	C₂²×D₃	—	18	1	no	—
234	C₂²×D₃	C₄×D₉	1	1	yes	characteristic
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	3	1	no	—
237	C₂×C₃×C₄	—	6	1	no	—
238	C₂×C₃×C₄	—	6	1	no	—
239	C₂×C₃×C₄	—	9	1	no	—
240	C₂×C₃×C₄	—	9	1	no	—
241	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
243	#67	—	18	2	no	—
244	C₂×Dic₃	—	3	1	no	—
245	C₂×Dic₃	—	3	1	no	—
246	C₂×Dic₃	—	9	1	no	—
247	C₂×Dic₃	—	9	1	no	—
248	C₂×Dic₃	—	18	1	no	—
249	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
250	D₁₂	—	18	2	no	—
251	C₄×D₃	—	2	1	no	—
252	C₄×D₃	—	2	1	no	—
253	C₄×D₃	—	18	2	no	—
254	C₄×D₃	—	18	2	no	—
255	C₄×D₃	—	18	1	no	—
256	C₄×D₃	—	18	1	no	—
257	C₄×D₃	—	18	1	no	—
258	C₄×D₃	—	18	1	no	—
259	C₃×C₈	—	2	2	no	—
260	C₃×C₈	—	6	1	no	—
261	C₃×C₈	—	18	1	no	—
262	C₃×C₈	#611	1	2	yes	complemented
263	C₃×C₈	#626	1	2	yes	—
264	C₃⋊C₈	—	2	1	no	—
265	C₃⋊C₈	—	6	1	no	—
266	C₃⋊C₈	—	18	2	no	—
267	C₃⋊C₈	—	18	1	no	—
268	C₃⋊C₈	—	18	1	no	—
269	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
270	C₂×C₃²	#901	1	1	yes	characteristic
271	C₂×C₃²	C₄×#67	1	1	yes	characteristic
272	C₃⋊D₃	—	12	1	no	—
273	C₃×D₃	—	2	2	no	—
274	C₂×C₉	—	2	1	no	—
275	C₂×C₉	—	2	1	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	—	6	2	no	—
278	C₂×C₉	C₄×D₁₂	1	1	yes	characteristic
279	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
280	C₂×C₉	#907	1	1	yes	characteristic
281	D₉	—	12	2	no	—
282	D₉	—	12	1	no	—
283	C₂×D₄	—	27	1	no	—
284	C₂²×C₄	—	3	1	no	—
285	C₂²×C₄	—	27	1	no	—
286	M₁₆	—	54	2	no	—
287	C₂×C₈	—	3	1	no	—
288	C₂×C₈	—	6	2	no	—
289	C₂×C₈	—	9	1	no	—
290	C₂×C₈	—	27	1	no	—
291	C₂×C₈	D₃×D₉	1	1	yes	characteristic
292	C₄⋊C₄	—	27	1	no	—
293	C₂²⋊C₄	—	27	1	no	—
294	C₂²⋊C₄	—	27	1	no	—
295	C₄²	—	9	1	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	—	3	2	no	—
298	C₂²×C₃	—	6	1	no	—
299	C₂²×C₃	—	6	1	no	—
300	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
301	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
302	C₂×D₃	—	2	1	no	—
303	C₂×D₃	—	2	1	no	—
304	C₂×D₃	—	18	2	no	—
305	C₂×D₃	—	18	1	no	—
306	C₂×D₃	—	18	1	no	—
307	C₂×D₃	—	18	1	no	—
308	C₂×D₃	—	18	1	no	—
309	C₂×D₃	—	18	1	no	—
310	C₂×D₃	—	18	1	no	—
311	C₂×D₃	—	36	1	no	—
312	C₂×D₃	—	36	1	no	—
313	C₂×D₃	#3663	1	2	yes	complemented
314	C₃×C₄	—	2	1	no	—
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	—	6	1	no	—
317	C₃×C₄	—	18	1	no	—
318	C₃×C₄	—	18	1	no	—
319	C₃×C₄	#3697	1	1	yes	characteristic
320	C₃×C₄	C₂×#611	1	1	yes	characteristic
321	C₃×C₄	#3798	1	1	yes	characteristic
322	C₃×C₄	C₂×#626	1	1	yes	characteristic
323	Dic₃	—	2	1	no	—
324	Dic₃	—	6	1	no	—
325	Dic₃	—	6	1	no	—
326	Dic₃	—	18	2	no	—
327	Dic₃	—	18	1	no	—
328	Dic₃	—	18	1	no	—
329	C₃²	#5111	1	1	yes	characteristic
330	C₉	—	2	1	no	—
331	C₉	#4706	1	1	yes	characteristic, complemented
332	C₂³	—	3	1	no	—
333	C₂³	—	27	1	no	—
334	D₄	—	54	2	no	—
335	C₂×C₄	—	3	1	no	—
336	C₂×C₄	—	6	1	no	—
337	C₂×C₄	—	6	1	no	—
338	C₂×C₄	—	9	1	no	—
339	C₂×C₄	—	9	1	no	—
340	C₂×C₄	—	27	1	no	—
341	C₂×C₄	—	54	1	no	—
342	C₂×C₄	—	54	1	no	—
343	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
344	C₈	—	6	1	no	—
345	C₈	—	18	1	no	—
346	C₈	—	54	1	no	—
347	C₈	#6195	1	2	yes	complemented
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	2	1	no	—
350	C₂×C₃	—	2	1	no	—
351	C₂×C₃	—	6	2	no	—
352	C₂×C₃	C₂×#3663	1	1	yes	characteristic
353	C₂×C₃	#63464	1	1	yes	characteristic
354	C₂×C₃	C₄×#611	1	1	yes	characteristic
355	C₂×C₃	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
356	C₂×C₃	#63804	1	1	yes	characteristic
357	C₂×C₃	C₄×#626	1	1	yes	characteristic
358	D₃	—	2	2	no	—
359	D₃	—	36	2	no	—
360	D₃	—	36	1	no	—
361	D₃	—	36	1	no	—
362	C₂²	—	3	2	no	—
363	C₂²	—	6	1	no	—
364	C₂²	—	6	1	no	—
365	C₂²	—	54	1	no	—
366	C₂²	—	54	1	no	—
367	C₂²	—	54	1	no	—
368	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
369	C₄	—	6	1	no	—
370	C₄	—	18	1	no	—
371	C₄	—	18	1	no	—
372	C₄	—	54	1	no	—
373	C₄	#88640	1	1	yes	characteristic
374	C₄	C₂×#6195	1	1	yes	characteristic
375	C₃	—	2	1	no	—
376	C₃	#10588792	1	1	yes	characteristic, complemented
377	C₃	#10590726	1	1	yes	characteristic
378	C₂	—	6	2	no	—
379	C₂	—	108	1	no	—
380	C₂	D₃×#3663	1	1	yes	characteristic
381	C₂	#11720808	1	1	yes	characteristic
382	C₂	C₄×#6195	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂×#6195	C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	#435
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#88523	C₂	semidirect
2. C₂×D₃×C₈×C₉	C₂	semidirect
4. C₃×#63375	C₂	semidirect
5. (C₂×D₃)⋊(C₉⋊C₈)	C₂	semidirect
6. #11720282	C₂	semidirect
7. Dic₉⋊(C₃⋊C₈)	C₂	semidirect
36. #63375	D₃	semidirect
50. C₃×C₈×C₉	D₄	semidirect
92. C₂×D₃×C₉	M₁₆	semidirect
104. C₂×D₃×C₈	D₉	semidirect
149. C₈×C₉	D₁₂	semidirect
229. C₃×C₉	#435	semidirect
262. C₃×C₈	#611	semidirect
313. C₂×D₃	#3663	semidirect
331. C₉	#4706	semidirect
347. C₈	#6195	semidirect
376. C₃	#10588792	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#88523	2
C₂×D₃×C₈×C₉	2
C₄×#6195	2
C₃×#63375	2
(C₂×D₃)⋊(C₉⋊C₈)	2
#11720282	2
Dic₉⋊(C₃⋊C₈)	2
#10590726	3
#10588792	3