Subgroups — #13153749

Order 1728.

887 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#11722641	C₂	1	1	yes	maximal, characteristic, complemented
3	#63513⋊C₃	C₂	1	3	yes	maximal, complemented
4	#63512⋊C₃	C₂	1	3	yes	maximal, complemented
5	#11721693	C₂	1	1	yes	maximal, characteristic, complemented
6	#63495⋊C₃	C₂	1	3	yes	maximal, complemented
7	C₂×#3673⋊C₃	C₂	1	3	yes	maximal, complemented
8	C₃×#5835	—	3	1	no	maximal
9	#10589937	C₃	1	1	yes	maximal, characteristic, complemented
10	#41×3₋¹⁺²	—	2	1	no	—
11	#41×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	#41×3₋¹⁺²	C₂²	1	3	yes	complemented
13	C₂×D₄×3₋¹⁺²	—	2	3	no	—
14	C₂×D₄×3₋¹⁺²	C₂²	1	3	yes	complemented
15	#88724	—	2	1	no	—
16	D₄×AGL(1,ℤ/9)	—	2	3	no	—
17	#3697⋊C₃	C₂²	1	3	yes	complemented
18	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	3	yes	complemented
19	#3675⋊C₃	C₂²	1	3	yes	complemented
20	#3674⋊C₃	C₂²	1	1	yes	characteristic, complemented
21	#3673⋊C₃	C₂²	1	3	yes	complemented
22	#3673⋊C₃	C₂²	1	6	yes	complemented
23	#3672⋊C₃	C₂²	1	3	yes	complemented
24	#3667⋊C₃	C₂²	1	3	yes	complemented
25	C₂×3₋¹⁺²⋊C₈	C₂²	1	3	yes	complemented
26	C₃²×H(5,𝔽₂)	D₃	1	1	yes	characteristic
27	C₃×#1009	—	3	1	no	—
28	C₃×#950	—	3	3	no	—
29	C₃×#949	—	3	3	no	—
30	C₃×#948	—	3	1	no	—
31	C₃×#932	—	3	3	no	—
32	C₂×C₃×#213	—	3	3	no	—
33	C₉×H(5,𝔽₂)	—	2	1	no	—
34	C₉×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
35	#63715	C₂×C₃	1	1	yes	characteristic, complemented
36	#63513	C₂×C₃	1	3	yes	complemented
37	#63512	C₂×C₃	1	3	yes	complemented
38	#63510	C₂×C₃	1	1	yes	characteristic, complemented
39	#63495	C₂×C₃	1	3	yes	complemented
40	C₂×#3673	C₂×C₃	1	3	yes	complemented
41	C₂³×3₋¹⁺²	—	2	3	no	—
42	C₂²×AGL(1,ℤ/9)	—	2	3	no	—
43	Q₈×3₋¹⁺²	D₄	1	1	yes	characteristic
44	Q₈×3₋¹⁺²	C₂³	1	1	yes	characteristic
45	D₄×3₋¹⁺²	—	2	3	no	—
46	D₄×3₋¹⁺²	—	2	3	no	—
47	D₄×3₋¹⁺²	D₄	1	3	yes	—
48	D₄×3₋¹⁺²	C₂³	1	3	yes	—
49	C₂×C₄×3₋¹⁺²	—	2	3	no	—
50	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
51	#611⋊C₃	—	2	3	no	—
52	C₄⋊AGL(1,ℤ/9)	—	2	3	no	—
53	C₄⋊AGL(1,ℤ/9)	C₂³	1	3	yes	—
54	C₄×AGL(1,ℤ/9)	—	2	3	no	—
55	3₋¹⁺²⋊Q₈	C₂³	1	1	yes	characteristic
56	3₋¹⁺²⋊C₈	C₂³	1	1	yes	characteristic
57	3₋¹⁺²⋊C₈	C₂³	1	3	yes	—
58	C₃×#576	—	9	1	no	—
59	#5835	—	3	1	no	—
60	C₃²×#41	—	2	1	no	—
61	C₃²×#41	C₂×D₃	1	3	yes	—
62	C₃²×#41	C₂×D₃	1	1	yes	characteristic
63	C₂×C₃²×D₄	—	2	3	no	—
64	C₂×C₃²×D₄	C₂×D₃	1	3	yes	—
65	C₃×#239	—	6	1	no	—
66	C₃×D₃×D₄	—	6	3	no	—
67	C₃×#235	—	3	3	no	—
68	C₂×C₃×D₁₂	—	3	3	no	—
69	C₃×#216	—	3	1	no	—
70	C₃×#215	—	3	3	no	—
71	C₃×#214	—	3	3	no	—
72	C₃×#213	—	3	3	no	—
73	C₃×#213	—	3	6	no	—
74	C₃×#208	—	3	3	no	—
75	C₂×C₃×C₃⋊C₈	—	3	3	no	—
76	C₉×#41	—	2	2	no	—
77	C₉×#41	—	2	1	no	—
78	C₉×#41	—	2	3	no	—
79	C₉×#41	—	2	1	no	—
80	C₉×#41	C₂²×C₃	1	1	yes	characteristic
81	C₉×#41	C₂²×C₃	1	3	yes	complemented
82	C₂×D₄×C₉	—	2	6	no	—
83	C₂×D₄×C₉	—	2	3	no	—
84	C₂×D₄×C₉	—	2	3	no	—
85	C₂×D₄×C₉	C₂²×C₃	1	3	yes	complemented
86	#3701	—	2	1	no	—
87	D₄×D₉	—	2	3	no	—
88	#3697	C₂²×C₃	1	3	yes	complemented
89	C₂×D₃₆	C₂²×C₃	1	3	yes	complemented
90	#3675	C₂²×C₃	1	3	yes	complemented
91	#3674	C₂²×C₃	1	1	yes	characteristic, complemented
92	#3673	C₂²×C₃	1	3	yes	complemented
93	#3673	C₂²×C₃	1	6	yes	complemented
94	#3672	C₂²×C₃	1	3	yes	complemented
95	#3667	C₂²×C₃	1	3	yes	complemented
96	C₂×C₉⋊C₈	C₂²×C₃	1	3	yes	complemented
97	C₂²×3₋¹⁺²	—	2	3	no	—
98	C₂²×3₋¹⁺²	—	8	3	no	—
99	C₂²×3₋¹⁺²	C₂×D₄	1	3	yes	—
100	C₂×AGL(1,ℤ/9)	—	2	3	no	—
101	C₂×AGL(1,ℤ/9)	—	8	3	no	—
102	C₄×3₋¹⁺²	—	2	1	no	—
103	C₄×3₋¹⁺²	C₂×D₄	1	3	yes	—
104	C₄×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
105	3₋¹⁺²⋊C₄	—	2	1	no	—
106	C₃×H(5,𝔽₂)	—	3	1	no	—
107	C₃×H(5,𝔽₂)	—	9	1	no	—
108	C₃×H(5,𝔽₂)	C₃×D₃	1	1	yes	characteristic
109	#1009	—	3	1	no	—
110	C₃×AGL(1,ℤ/8)	—	9	3	no	—
111	C₃×AGL(1,ℤ/8)	—	9	3	no	—
112	C₃×#135	—	9	3	no	—
113	C₂×C₃×D₈	—	9	3	no	—
114	C₃×#131	—	9	1	no	—
115	#950	—	3	3	no	—
116	#949	—	3	3	no	—
117	#948	—	3	1	no	—
118	#932	—	3	3	no	—
119	C₂×#213	—	3	3	no	—
120	C₂³×C₃²	—	2	3	no	—
121	C₂²×C₃×D₃	—	6	3	no	—
122	C₃²×Q₈	#67	1	1	yes	characteristic
123	C₃²×Q₈	C₂²×D₃	1	1	yes	characteristic
124	C₃²×D₄	—	2	3	no	—
125	C₃²×D₄	—	2	3	no	—
126	C₃²×D₄	#67	1	3	yes	—
127	C₃²×D₄	C₂²×D₃	1	3	yes	—
128	C₂×C₃²×C₄	—	2	3	no	—
129	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
130	C₃×#67	—	6	3	no	—
131	C₃×D₁₂	—	3	3	no	—
132	C₃×D₁₂	—	6	3	no	—
133	C₃×C₄×D₃	—	6	3	no	—
134	C₃×Dic₆	—	3	1	no	—
135	C₂³×C₉	—	2	6	no	—
136	C₂³×C₉	—	2	3	no	—
137	C₂²×D₉	—	2	3	no	—
138	C₃×C₃⋊C₈	—	3	1	no	—
139	C₃×C₃⋊C₈	—	3	3	no	—
140	Q₈×C₉	—	2	1	no	—
141	Q₈×C₉	—	2	1	no	—
142	Q₈×C₉	C₃×D₄	1	1	yes	characteristic
143	Q₈×C₉	C₂³×C₃	1	1	yes	characteristic
144	D₄×C₉	—	2	6	no	—
145	D₄×C₉	—	2	6	no	—
146	D₄×C₉	—	2	3	no	—
147	D₄×C₉	—	2	3	no	—
148	D₄×C₉	—	2	3	no	—
149	D₄×C₉	—	2	3	no	—
150	D₄×C₉	C₃×D₄	1	3	yes	—
151	D₄×C₉	C₂³×C₃	1	3	yes	—
152	C₂×C₄×C₉	—	2	6	no	—
153	C₂×C₄×C₉	—	2	3	no	—
154	C₂×C₄×C₉	—	2	3	no	—
155	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
156	#611	—	2	3	no	—
157	D₃₆	—	2	3	no	—
158	D₃₆	C₂³×C₃	1	3	yes	—
159	C₄×D₉	—	2	3	no	—
160	Dic₁₈	C₂³×C₃	1	1	yes	characteristic
161	C₉⋊C₈	C₂³×C₃	1	1	yes	characteristic
162	C₉⋊C₈	C₂³×C₃	1	3	yes	—
163	#576	—	9	1	no	Sylow
164	C₂×3₋¹⁺²	—	2	3	no	—
165	C₂×3₋¹⁺²	—	4	3	no	—
166	C₂×3₋¹⁺²	C₂²×D₄	1	1	yes	characteristic
167	AGL(1,ℤ/9)	—	4	3	no	—
168	C₃×#41	—	2	1	no	—
169	C₃×#41	—	3	1	no	—
170	C₃×#41	—	3	3	no	—
171	C₃×#41	—	6	1	no	—
172	C₃×#41	—	9	3	no	—
173	C₃×#41	—	18	1	no	—
174	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
175	C₃×#41	C₂×C₃×D₃	1	3	yes	—
176	C₂×C₃×D₄	—	2	3	no	—
177	C₂×C₃×D₄	—	3	3	no	—
178	C₂×C₃×D₄	—	6	3	no	—
179	C₂×C₃×D₄	—	9	3	no	—
180	C₂×C₃×D₄	—	18	3	no	—
181	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
182	#239	—	6	1	no	—
183	D₃×D₄	—	6	3	no	—
184	#235	—	3	3	no	—
185	C₂×D₁₂	—	3	3	no	—
186	C₃×Q₁₆	—	9	1	no	—
187	C₃×SD₁₆	—	9	3	no	—
188	C₃×SD₁₆	—	9	3	no	—
189	C₃×D₈	—	9	3	no	—
190	C₃×D₈	—	9	6	no	—
191	C₃×M₁₆	—	9	3	no	—
192	C₂×C₃×C₈	—	9	3	no	—
193	#216	—	3	1	no	—
194	#215	—	3	3	no	—
195	#214	—	3	3	no	—
196	#213	—	3	3	no	—
197	#213	—	3	6	no	—
198	#208	—	3	3	no	—
199	C₂×C₃⋊C₈	—	3	3	no	—
200	C₂²×C₃²	—	2	3	no	—
201	C₂²×C₃²	—	8	3	no	—
202	C₂²×C₃²	C₂×#67	1	3	yes	—
203	C₂×C₃×D₃	—	6	3	no	—
204	C₂×C₃×D₃	—	24	3	no	—
205	C₃²×C₄	—	2	1	no	—
206	C₃²×C₄	C₂×#67	1	3	yes	—
207	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
208	C₃×Dic₃	—	6	1	no	—
209	C₂²×C₉	—	2	6	no	—
210	C₂²×C₉	—	2	3	no	—
211	C₂²×C₉	—	2	3	no	—
212	C₂²×C₉	—	8	3	no	—
213	C₂²×C₉	—	8	6	no	—
214	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
215	C₂×D₉	—	2	3	no	—
216	C₂×D₉	—	8	3	no	—
217	C₄×C₉	—	2	2	no	—
218	C₄×C₉	—	2	1	no	—
219	C₄×C₉	—	2	1	no	—
220	C₄×C₉	—	2	3	no	—
221	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
222	C₄×C₉	C₂⁴×C₃	1	1	yes	characteristic
223	Dic₉	—	2	1	no	—
224	H(5,𝔽₂)	—	9	1	no	—
225	H(5,𝔽₂)	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
226	AGL(1,ℤ/8)	—	9	3	no	—
227	AGL(1,ℤ/8)	—	9	3	no	—
228	#135	—	9	3	no	—
229	C₂×D₈	—	9	3	no	—
230	#131	—	9	1	no	—
231	3₋¹⁺²	#576	1	1	yes	characteristic, complemented, Sylow
232	C₂³×C₃	—	2	3	no	—
233	C₂³×C₃	—	6	3	no	—
234	C₂³×C₃	—	18	3	no	—
235	C₂²×D₃	—	6	3	no	—
236	C₃×Q₈	—	3	1	no	—
237	C₃×Q₈	—	3	1	no	—
238	C₃×Q₈	—	9	1	no	—
239	C₃×Q₈	C₃×#67	1	1	yes	characteristic
240	C₃×Q₈	C₂²×C₃×D₃	1	1	yes	characteristic
241	C₃×D₄	—	2	3	no	—
242	C₃×D₄	—	2	3	no	—
243	C₃×D₄	—	3	3	no	—
244	C₃×D₄	—	3	3	no	—
245	C₃×D₄	—	6	3	no	—
246	C₃×D₄	—	6	3	no	—
247	C₃×D₄	—	9	3	no	—
248	C₃×D₄	—	18	3	no	—
249	C₃×D₄	—	18	3	no	—
250	C₃×D₄	C₃×#67	1	3	yes	—
251	C₃×D₄	C₂²×C₃×D₃	1	3	yes	—
252	C₂×C₃×C₄	—	2	3	no	—
253	C₂×C₃×C₄	—	3	3	no	—
254	C₂×C₃×C₄	—	6	3	no	—
255	C₂×C₃×C₄	—	18	3	no	—
256	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
257	#67	—	6	3	no	—
258	D₁₂	—	3	3	no	—
259	D₁₂	—	6	3	no	—
260	C₄×D₃	—	6	3	no	—
261	Dic₆	—	3	1	no	—
262	C₃×C₈	—	9	1	no	—
263	C₃×C₈	—	9	3	no	—
264	C₃⋊C₈	—	3	1	no	—
265	C₃⋊C₈	—	3	3	no	—
266	C₂×C₃²	—	2	3	no	—
267	C₂×C₃²	—	4	3	no	—
268	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
269	C₃×D₃	—	12	3	no	—
270	C₂×C₉	—	2	3	no	—
271	C₂×C₉	—	2	1	no	—
272	C₂×C₉	—	4	3	no	—
273	C₂×C₉	—	4	6	no	—
274	C₂×C₉	—	4	3	no	—
275	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
276	D₉	—	4	3	no	—
277	#41	—	2	1	no	—
278	#41	—	9	3	no	—
279	#41	—	18	1	no	—
280	#41	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
281	#41	C₂×AGL(1,ℤ/9)	1	3	yes	complemented
282	C₂×D₄	—	2	3	no	—
283	C₂×D₄	—	9	3	no	—
284	C₂×D₄	—	18	3	no	—
285	C₂×D₄	C₂×AGL(1,ℤ/9)	1	3	yes	complemented
286	Q₁₆	—	9	1	no	—
287	SD₁₆	—	9	3	no	—
288	SD₁₆	—	9	3	no	—
289	D₈	—	9	3	no	—
290	D₈	—	9	6	no	—
291	M₁₆	—	9	3	no	—
292	C₂×C₈	—	9	3	no	—
293	C₂²×C₃	—	2	3	no	—
294	C₂²×C₃	—	3	3	no	—
295	C₂²×C₃	—	6	3	no	—
296	C₂²×C₃	—	8	3	no	—
297	C₂²×C₃	—	18	3	no	—
298	C₂²×C₃	—	24	3	no	—
299	C₂²×C₃	—	72	3	no	—
300	C₂²×C₃	C₂×C₃×#67	1	3	yes	—
301	C₂×D₃	—	6	3	no	—
302	C₂×D₃	—	24	3	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	3	1	no	—
305	C₃×C₄	—	3	3	no	—
306	C₃×C₄	—	6	1	no	—
307	C₃×C₄	—	18	1	no	—
308	C₃×C₄	C₂×C₃×#67	1	3	yes	—
309	C₃×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
310	Dic₃	—	6	1	no	—
311	C₃²	#5835	1	1	yes	characteristic
312	C₉	—	2	1	no	—
313	C₉	C₃×#576	1	1	yes	characteristic, complemented
314	C₂³	—	2	3	no	—
315	C₂³	—	18	3	no	—
316	Q₈	—	9	1	no	—
317	Q₈	#611⋊C₃	1	1	yes	characteristic
318	Q₈	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
319	D₄	—	2	3	no	—
320	D₄	—	2	3	no	—
321	D₄	—	9	3	no	—
322	D₄	—	18	3	no	—
323	D₄	—	18	3	no	—
324	D₄	#611⋊C₃	1	3	yes	—
325	D₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
326	C₂×C₄	—	2	3	no	—
327	C₂×C₄	—	18	3	no	—
328	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
329	C₈	—	9	1	no	—
330	C₈	—	9	3	no	—
331	C₂×C₃	—	2	3	no	—
332	C₂×C₃	—	3	1	no	—
333	C₂×C₃	—	4	3	no	—
334	C₂×C₃	—	6	3	no	—
335	C₂×C₃	—	12	3	no	—
336	C₂×C₃	—	36	3	no	—
337	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
338	D₃	—	12	3	no	—
339	C₂²	—	2	3	no	—
340	C₂²	—	8	3	no	—
341	C₂²	—	18	3	no	—
342	C₂²	—	72	3	no	—
343	C₂²	C₂×#611⋊C₃	1	3	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	18	1	no	—
346	C₄	C₂×#611⋊C₃	1	3	yes	—
347	C₄	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
348	C₃	—	3	1	no	—
349	C₃	C₃×#5835	1	1	yes	characteristic
350	C₂	—	2	3	no	—
351	C₂	—	4	3	no	—
352	C₂	—	36	3	no	—
353	C₂	C₂²×#611⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#611⋊C₃
commutator, cosocle	C₄×C₉	C₂⁴×C₃
Frattini	C₃×C₄	C₂³×C₃×D₃
Fitting	3₋¹⁺²×H(5,𝔽₂)	C₂
socle	C₂×C₃	C₂²×C₃×#67
hypercenter	H(5,𝔽₂)	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#576
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×H(5,𝔽₂)	C₂	semidirect
2. #11722641	C₂	semidirect
3. #63513⋊C₃	C₂	semidirect
4. #63512⋊C₃	C₂	semidirect
5. #11721693	C₂	semidirect
6. #63495⋊C₃	C₂	semidirect
7. C₂×#3673⋊C₃	C₂	semidirect
9. #10589937	C₃	semidirect
12. #41×3₋¹⁺²	C₂²	semidirect
14. C₂×D₄×3₋¹⁺²	C₂²	semidirect
17. #3697⋊C₃	C₂²	semidirect
18. C₂×C₄⋊AGL(1,ℤ/9)	C₂²	semidirect
19. #3675⋊C₃	C₂²	semidirect
20. #3674⋊C₃	C₂²	semidirect
21. #3673⋊C₃	C₂²	semidirect
22. #3673⋊C₃	C₂²	semidirect
23. #3672⋊C₃	C₂²	semidirect
24. #3667⋊C₃	C₂²	semidirect
25. C₂×3₋¹⁺²⋊C₈	C₂²	semidirect
34. C₉×H(5,𝔽₂)	C₂×C₃	semidirect
35. #63715	C₂×C₃	semidirect
36. #63513	C₂×C₃	semidirect
37. #63512	C₂×C₃	semidirect
38. #63510	C₂×C₃	semidirect
39. #63495	C₂×C₃	semidirect
40. C₂×#3673	C₂×C₃	semidirect
81. C₉×#41	C₂²×C₃	semidirect
85. C₂×D₄×C₉	C₂²×C₃	semidirect
88. #3697	C₂²×C₃	semidirect
89. C₂×D₃₆	C₂²×C₃	semidirect
90. #3675	C₂²×C₃	semidirect
91. #3674	C₂²×C₃	semidirect
92. #3673	C₂²×C₃	semidirect
93. #3673	C₂²×C₃	semidirect
94. #3672	C₂²×C₃	semidirect
95. #3667	C₂²×C₃	semidirect
96. C₂×C₉⋊C₈	C₂²×C₃	semidirect
225. H(5,𝔽₂)	AGL(1,ℤ/9)	semidirect
231. 3₋¹⁺²	#576	semidirect
281. #41	C₂×AGL(1,ℤ/9)	semidirect
285. C₂×D₄	C₂×AGL(1,ℤ/9)	semidirect
313. C₉	C₃×#576	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×H(5,𝔽₂)	2
#11722641	2
#63513⋊C₃	2
#63512⋊C₃	2
#11721693	2
#63495⋊C₃	2
C₂×#3673⋊C₃	2
C₃×#5835	3
#10589937	3