Subgroups — #13138173

Order 1728.

446 conjugacy classes of subgroups.

The subgroup lattice has 360 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
2	C₄×(C₃⋊D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#88414	C₂	1	1	yes	maximal, characteristic
4	C₉×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₉⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590666	—	3	1	no	maximal
9	#10588521	—	3	1	no	maximal
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×(C₃⋊D₉)⋊C₄	C₄	1	1	yes	characteristic
12	C₂×(C₃⋊D₉)⋊C₄	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	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
17	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
18	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	Dic₃×Dic₉	C₄	1	1	yes	characteristic
21	Dic₃×Dic₉	C₄	1	1	yes	characteristic
22	#88414	—	2	2	no	—
23	C₂×#3743	—	3	1	no	—
24	C₄×#624	—	3	1	no	—
25	C₂×#3712	—	3	1	no	—
26	C₃×Dic₃⋊C₈	—	3	1	no	—
27	C₃×Dic₃⋊C₈	D₃	1	1	yes	characteristic
28	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
29	Dic₃⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂×#3663	—	3	1	no	—
31	C₂×#3663	—	3	1	no	—
32	C₂×#3663	—	6	1	no	—
33	C₄²×D₉	—	3	1	no	—
34	C₉×C₄⋊C₈	—	3	1	no	—
35	Dic₉⋊C₈	—	3	1	no	—
36	Dic₉⋊C₈	D₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₉⋊C₈)	—	3	1	no	—
38	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₃⋊D₉	D₄	1	2	yes	—
42	C₄×C₃⋊D₉	Q₈	1	2	yes	—
43	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
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	(C₃⋊D₉)⋊C₄	—	2	2	no	—
48	(C₃⋊D₉)⋊C₄	—	2	2	no	—
49	C₃×C₈×C₉	—	2	1	no	—
50	C₃⋊(C₉⋊C₈)	—	2	1	no	—
51	C₉×C₃⋊C₈	—	2	1	no	—
52	C₃×C₉⋊C₈	—	2	1	no	—
53	#4837	—	9	1	no	—
54	#4837	—	9	1	no	—
55	C₂×C₄×C₃⋊D₃	—	3	1	no	—
56	C₂×#624	—	3	1	no	—
57	C₂×#624	—	3	1	no	—
58	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
60	#3743	—	6	2	no	—
61	C₃×C₄×Dic₃	—	3	1	no	—
62	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
65	Dic₃²	—	3	1	no	—
66	Dic₃²	—	3	1	no	—
67	#3712	—	6	2	no	—
68	C₂×C₄×D₉	—	3	1	no	—
69	C₂×C₄×D₉	—	3	1	no	—
70	C₂×C₄×D₉	—	3	1	no	—
71	C₂×C₄×D₉	—	6	1	no	—
72	C₂×C₈×C₉	—	2	1	no	—
73	C₂×C₈×C₉	—	3	1	no	—
74	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
75	C₄²×C₉	—	3	1	no	—
76	C₄×Dic₉	—	3	1	no	—
77	C₄×Dic₉	—	3	1	no	—
78	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
79	C₂×C₉⋊C₈	—	3	1	no	—
80	C₂×C₉⋊C₈	—	6	1	no	—
81	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
82	#3663	—	6	2	no	—
83	#3663	—	6	2	no	—
84	#3663	—	6	4	no	—
85	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃⋊D₉	—	2	1	no	—
87	C₂×C₃⋊D₉	—	2	1	no	—
88	C₂×C₃⋊D₉	C₄⋊C₄	1	2	yes	—
89	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
91	C₃⋊Dic₉	C₄⋊C₄	1	2	yes	—
92	C₉×Dic₃	—	2	1	no	—
93	C₉×Dic₃	M₁₆	1	2	yes	complemented
94	C₃×Dic₉	—	2	1	no	—
95	C₃×Dic₉	M₁₆	1	2	yes	complemented
96	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
97	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
98	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
99	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
100	C₂×ΓL(1,𝔽₂₅)	—	18	1	no	—
101	C₄²×D₃	—	9	1	no	—
102	C₄²×D₃	—	9	1	no	—
103	C₃×C₄⋊C₈	—	3	1	no	—
104	C₃×C₄⋊C₈	—	9	1	no	—
105	Dic₃⋊C₈	—	3	1	no	—
106	Dic₃⋊C₈	—	9	1	no	—
107	Dic₃⋊C₈	—	9	1	no	—
108	Dic₃⋊C₈	D₉	1	1	yes	characteristic, complemented
109	C₄⋊(C₃⋊C₈)	—	9	1	no	—
110	C₄⋊(C₃⋊C₈)	—	9	1	no	—
111	C₂²×C₃⋊D₃	—	3	1	no	—
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	—	3	1	no	—
114	C₄×C₃⋊D₃	—	3	2	no	—
115	C₄×C₃⋊D₃	—	3	2	no	—
116	C₂×C₃×Dic₃	—	3	1	no	—
117	C₂×C₃×Dic₃	—	3	1	no	—
118	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
119	C₂×C₃×Dic₃	C₄×D₃	1	1	yes	characteristic
120	#624	—	6	2	no	—
121	#624	—	6	2	no	—
122	C₂²×D₉	—	3	1	no	—
123	C₂²×D₉	—	6	1	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃⋊(C₃⋊C₈)	—	6	1	no	—
126	C₃×C₃⋊C₈	—	2	1	no	—
127	C₃×C₃⋊C₈	—	6	1	no	—
128	C₂×C₄×C₉	—	2	1	no	—
129	C₂×C₄×C₉	—	3	1	no	—
130	C₂×C₄×C₉	—	3	1	no	—
131	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
132	C₂×Dic₉	—	3	1	no	—
133	C₂×Dic₉	—	6	1	no	—
134	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
135	C₂×Dic₉	C₄×D₃	1	1	yes	characteristic
136	C₄×D₉	—	3	2	no	—
137	C₄×D₉	—	3	2	no	—
138	C₄×D₉	—	6	2	no	—
139	C₄×D₉	—	6	2	no	—
140	C₄×D₉	—	6	2	no	—
141	C₄×D₉	—	6	2	no	—
142	C₈×C₉	—	2	2	no	—
143	C₈×C₉	—	2	1	no	—
144	C₈×C₉	—	6	1	no	—
145	C₉⋊C₈	—	2	1	no	—
146	C₉⋊C₈	—	6	2	no	—
147	C₉⋊C₈	—	6	1	no	—
148	#423	—	27	1	no	Sylow
149	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
152	C₃⋊D₉	—	2	2	no	—
153	C₂×C₄×D₃	—	9	1	no	—
154	C₂×C₄×D₃	—	9	1	no	—
155	C₂×C₄×D₃	—	9	1	no	—
156	C₂×C₄×D₃	—	9	1	no	—
157	C₂×C₄×D₃	—	9	1	no	—
158	C₂×C₄×D₃	—	9	1	no	—
159	C₂×C₄×D₃	—	18	1	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	3	1	no	—
162	C₂×C₃×C₈	—	9	1	no	—
163	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
164	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
165	C₃×C₄²	—	3	1	no	—
166	C₃×C₄²	—	9	1	no	—
167	C₄×Dic₃	—	3	1	no	—
168	C₄×Dic₃	—	9	1	no	—
169	C₄×Dic₃	—	9	1	no	—
170	C₄×Dic₃	—	9	1	no	—
171	C₄×Dic₃	—	9	1	no	—
172	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
173	C₂×C₃⋊C₈	—	3	1	no	—
174	C₂×C₃⋊C₈	—	9	1	no	—
175	C₂×C₃⋊C₈	—	9	1	no	—
176	C₂×C₃⋊C₈	—	18	1	no	—
177	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
178	ΓL(1,𝔽₂₅)	—	18	2	no	—
179	ΓL(1,𝔽₂₅)	—	18	2	no	—
180	ΓL(1,𝔽₂₅)	—	18	2	no	—
181	ΓL(1,𝔽₂₅)	—	18	2	no	—
182	ΓL(1,𝔽₂₅)	—	18	4	no	—
183	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₃⋊D₃	—	3	2	no	—
185	C₂×C₃⋊D₃	—	6	1	no	—
186	C₂×C₃⋊D₃	—	6	1	no	—
187	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
188	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
189	C₃⋊Dic₃	—	3	2	no	—
190	C₃×Dic₃	—	2	1	no	—
191	C₃×Dic₃	—	3	2	no	—
192	C₃×Dic₃	—	6	1	no	—
193	C₃×Dic₃	ΓL(1,𝔽₂₅)	1	2	yes	—
194	C₂²×C₉	—	2	1	no	—
195	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×D₉	—	3	2	no	—
197	C₂×D₉	—	6	2	no	—
198	C₂×D₉	—	6	1	no	—
199	C₂×D₉	—	6	1	no	—
200	C₂×D₉	—	12	1	no	—
201	C₂×D₉	—	12	1	no	—
202	C₄×C₉	—	2	1	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	3	2	no	—
205	C₄×C₉	—	6	1	no	—
206	C₄×C₉	D₃×D₄	1	1	yes	characteristic
207	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
208	Dic₉	—	2	1	no	—
209	Dic₉	—	3	2	no	—
210	Dic₉	—	6	2	no	—
211	Dic₉	ΓL(1,𝔽₂₅)	1	2	yes	complemented
212	C₂×M₁₆	—	27	1	no	—
213	C₂×M₁₆	—	27	1	no	—
214	C₂×C₄²	—	27	1	no	—
215	C₄⋊C₈	—	3	1	no	—
216	C₄⋊C₈	—	9	1	no	—
217	C₄⋊C₈	—	27	1	no	—
218	C₄⋊C₈	—	27	1	no	—
219	C₃×C₉	#423	1	1	yes	characteristic, complemented, Sylow
220	C₂²×D₃	—	9	1	no	—
221	C₂²×D₃	—	9	1	no	—
222	C₂²×D₃	—	18	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	3	1	no	—
225	C₂×C₃×C₄	—	3	1	no	—
226	C₂×C₃×C₄	—	9	1	no	—
227	C₂×C₃×C₄	—	9	1	no	—
228	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
230	C₂×Dic₃	—	3	1	no	—
231	C₂×Dic₃	—	3	1	no	—
232	C₂×Dic₃	—	9	1	no	—
233	C₂×Dic₃	—	9	1	no	—
234	C₂×Dic₃	—	18	1	no	—
235	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
236	C₂×Dic₃	C₄×D₉	1	1	yes	characteristic
237	C₄×D₃	—	9	2	no	—
238	C₄×D₃	—	9	2	no	—
239	C₄×D₃	—	9	2	no	—
240	C₄×D₃	—	9	2	no	—
241	C₄×D₃	—	18	2	no	—
242	C₄×D₃	—	18	2	no	—
243	C₄×D₃	—	18	2	no	—
244	C₄×D₃	—	18	2	no	—
245	C₄×D₃	—	18	2	no	—
246	C₄×D₃	—	18	2	no	—
247	C₃×C₈	—	2	2	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	2	1	no	—
250	C₃×C₈	—	6	1	no	—
251	C₃×C₈	—	18	1	no	—
252	C₃⋊C₈	—	2	1	no	—
253	C₃⋊C₈	—	6	1	no	—
254	C₃⋊C₈	—	18	2	no	—
255	C₃⋊C₈	—	18	1	no	—
256	C₃⋊C₈	—	18	1	no	—
257	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
259	C₂×C₃²	D₃×M₁₆	1	1	yes	characteristic
260	C₃⋊D₃	—	6	2	no	—
261	C₂×C₉	—	2	1	no	—
262	C₂×C₉	—	2	1	no	—
263	C₂×C₉	—	2	1	no	—
264	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
265	C₂×C₉	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
266	C₂×C₉	D₃×M₁₆	1	1	yes	characteristic
267	D₉	—	6	2	no	—
268	D₉	—	12	2	no	—
269	C₂²×C₄	—	27	1	no	—
270	C₂²×C₄	—	27	1	no	—
271	C₂²×C₄	—	27	1	no	—
272	M₁₆	—	54	2	no	—
273	M₁₆	—	54	2	no	—
274	C₂×C₈	—	3	1	no	—
275	C₂×C₈	—	9	1	no	—
276	C₂×C₈	—	27	1	no	—
277	C₂×C₈	D₃×D₉	1	1	yes	characteristic
278	C₄²	—	3	1	no	—
279	C₄²	—	9	1	no	—
280	C₄²	—	27	1	no	—
281	C₄²	—	27	1	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
284	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
285	C₂×D₃	—	9	2	no	—
286	C₂×D₃	—	9	2	no	—
287	C₂×D₃	—	18	2	no	—
288	C₂×D₃	—	18	1	no	—
289	C₂×D₃	—	18	1	no	—
290	C₂×D₃	—	18	1	no	—
291	C₂×D₃	—	18	1	no	—
292	C₂×D₃	—	36	1	no	—
293	C₂×D₃	—	36	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	3	2	no	—
297	C₃×C₄	—	6	1	no	—
298	C₃×C₄	—	9	2	no	—
299	C₃×C₄	—	18	1	no	—
300	C₃×C₄	D₄×D₉	1	1	yes	characteristic
301	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
302	C₃×C₄	#3797	1	1	yes	characteristic
303	C₃×C₄	#3811	1	1	yes	characteristic
304	Dic₃	—	2	1	no	—
305	Dic₃	—	3	2	no	—
306	Dic₃	—	6	1	no	—
307	Dic₃	—	9	2	no	—
308	Dic₃	—	9	2	no	—
309	Dic₃	—	18	2	no	—
310	Dic₃	#3663	1	2	yes	complemented
311	C₃²	#4837	1	1	yes	characteristic
312	C₉	—	2	1	no	—
313	C₉	#4837	1	1	yes	characteristic, complemented
314	C₂³	—	27	1	no	—
315	C₂×C₄	—	3	1	no	—
316	C₂×C₄	—	3	1	no	—
317	C₂×C₄	—	9	1	no	—
318	C₂×C₄	—	9	1	no	—
319	C₂×C₄	—	27	1	no	—
320	C₂×C₄	—	27	2	no	—
321	C₂×C₄	—	27	2	no	—
322	C₂×C₄	—	54	2	no	—
323	C₂×C₄	—	54	2	no	—
324	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	6	1	no	—
327	C₈	—	18	1	no	—
328	C₈	—	54	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
333	C₂×C₃	C₂×#3663	1	1	yes	characteristic
334	C₂×C₃	M₁₆×D₉	1	1	yes	characteristic
335	C₂×C₃	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
336	C₂×C₃	D₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
337	C₂×C₃	#63883	1	1	yes	characteristic
338	D₃	—	18	2	no	—
339	D₃	—	18	2	no	—
340	D₃	—	36	2	no	—
341	C₂²	—	27	2	no	—
342	C₂²	—	54	1	no	—
343	C₂²	—	54	1	no	—
344	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
345	C₄	—	3	2	no	—
346	C₄	—	6	1	no	—
347	C₄	—	9	2	no	—
348	C₄	—	18	1	no	—
349	C₄	—	27	2	no	—
350	C₄	(C₃⋊D₉)⋊Q₈	1	1	yes	characteristic
351	C₄	#88668	1	1	yes	characteristic
352	C₃	—	2	1	no	—
353	C₃	#10588521	1	1	yes	characteristic, complemented
354	C₃	#10590666	1	1	yes	characteristic
355	C₂	—	54	2	no	—
356	C₂	D₉×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
357	C₂	D₃×#3663	1	1	yes	characteristic
358	C₂	#11720980	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	C₂²×C₃⋊D₉	C₂×C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	#423
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
4. C₉×Dic₃⋊C₈	C₂	semidirect
5. C₃×Dic₉⋊C₈	C₂	semidirect
6. Dic₉⋊(C₃⋊C₈)	C₂	semidirect
7. Dic₃⋊(C₉⋊C₈)	C₂	semidirect
36. Dic₉⋊C₈	D₃	semidirect
93. C₉×Dic₃	M₁₆	semidirect
95. C₃×Dic₉	M₁₆	semidirect
108. Dic₃⋊C₈	D₉	semidirect
211. Dic₉	ΓL(1,𝔽₂₅)	semidirect
219. C₃×C₉	#423	semidirect
310. Dic₃	#3663	semidirect
313. C₉	#4837	semidirect
353. C₃	#10588521	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
C₄×(C₃⋊D₉)⋊C₄	2
C₂×#88414	2
C₉×Dic₃⋊C₈	2
C₃×Dic₉⋊C₈	2
Dic₉⋊(C₃⋊C₈)	2
Dic₃⋊(C₉⋊C₈)	2
#10590666	3
#10588521	3