Subgroups — #13152735

Order 1728.

917 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88724	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#63455⋊C₃	C₂	1	3	yes	maximal, complemented
5	C₄⋊C₄×AGL(1,ℤ/9)	C₂	1	3	yes	maximal, complemented
6	C₄×C₄⋊AGL(1,ℤ/9)	C₂	1	3	yes	maximal, complemented
7	#63434⋊C₃	C₂	1	3	yes	maximal, complemented
8	C₃×#5574	—	3	1	no	maximal
9	#10589676	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	#88724	C₄	1	8	yes	complemented
12	C₂×C₄⋊AGL(1,ℤ/9)	C₂²	1	3	yes	—
13	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	6	yes	—
14	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	3	yes	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	3	yes	—
16	C₄²×3₋¹⁺²	C₂²	1	3	yes	—
17	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	3	yes	—
18	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	3	yes	—
19	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	3	yes	—
20	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	3	yes	—
21	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	1	yes	characteristic
22	C₄×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
23	C₂×C₃×#239	—	3	1	no	—
24	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
25	C₃×Q₈×Dic₃	—	3	1	no	—
26	C₃×#893	—	3	3	no	—
27	C₃×D₃×C₄⋊C₄	—	3	3	no	—
28	C₃×C₄×D₁₂	—	3	3	no	—
29	C₃×#872	—	3	3	no	—
30	C₂×#3701	C₂×C₃	1	1	yes	characteristic
31	C₄×Q₈×C₉	—	2	1	no	—
32	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
33	Q₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
34	#63455	C₂×C₃	1	3	yes	complemented
35	C₄⋊C₄×D₉	C₂×C₃	1	3	yes	complemented
36	C₄×D₃₆	C₂×C₃	1	3	yes	complemented
37	#63434	C₂×C₃	1	3	yes	complemented
38	C₂²×AGL(1,ℤ/9)	C₂³	1	3	yes	—
39	Q₈×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
40	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
41	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
42	C₂×C₄×3₋¹⁺²	C₂³	1	3	yes	—
43	C₂×3₋¹⁺²⋊C₄	C₂³	1	3	yes	—
44	C₂×3₋¹⁺²⋊C₄	C₂³	1	1	yes	characteristic
45	C₂×3₋¹⁺²⋊C₄	C₂³	1	1	yes	characteristic
46	C₄⋊AGL(1,ℤ/9)	C₂×C₄	1	12	yes	complemented
47	C₄×AGL(1,ℤ/9)	—	4	6	no	—
48	C₄×AGL(1,ℤ/9)	C₂×C₄	1	12	yes	complemented
49	C₃×#520	—	9	1	no	—
50	#5574	—	3	1	no	—
51	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
52	C₃×#239	—	3	8	no	—
53	C₂×C₃×D₁₂	—	3	3	no	—
54	C₂×C₃×C₄×D₃	—	3	3	no	—
55	C₂×C₃×C₄×D₃	—	3	6	no	—
56	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
57	C₃²×C₄²	C₂×D₃	1	3	yes	—
58	C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
59	C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
60	C₃×C₄⋊Dic₃	—	3	3	no	—
61	C₃×Dic₃⋊C₄	—	3	3	no	—
62	C₃×Dic₃⋊C₄	—	3	1	no	—
63	C₃×C₄×Dic₃	—	3	3	no	—
64	C₂×Q₈×C₉	—	2	1	no	—
65	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
66	#3701	C₃×C₄	1	8	yes	complemented
67	C₂×D₃₆	C₂²×C₃	1	3	yes	—
68	C₂×C₄×D₉	C₂²×C₃	1	6	yes	—
69	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
70	C₉×C₄⋊C₄	—	2	3	no	—
71	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
72	C₄²×C₉	—	2	3	no	—
73	C₄²×C₉	C₂²×C₃	1	3	yes	—
74	(C₂×D₉)⋊C₄	C₂²×C₃	1	3	yes	—
75	(C₂×D₉)⋊C₄	C₂²×C₃	1	3	yes	—
76	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
77	Dic₉⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	Dic₉⋊C₄	C₂²×C₃	1	3	yes	—
79	C₄×Dic₉	C₂²×C₃	1	3	yes	—
80	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
81	C₂×AGL(1,ℤ/9)	—	2	3	no	—
82	C₂×AGL(1,ℤ/9)	—	2	3	no	—
83	C₂×AGL(1,ℤ/9)	C₂²×C₄	1	6	yes	—
84	C₄×3₋¹⁺²	—	2	3	no	—
85	C₄×3₋¹⁺²	—	2	1	no	—
86	C₄×3₋¹⁺²	C₂²×C₄	1	6	yes	—
87	3₋¹⁺²⋊C₄	—	2	3	no	—
88	3₋¹⁺²⋊C₄	—	2	1	no	—
89	3₋¹⁺²⋊C₄	C₂²×C₄	1	2	yes	—
90	C₂×C₃×#41	—	9	1	no	—
91	C₂×#239	—	3	1	no	—
92	C₃×C₄×Q₈	—	3	1	no	—
93	C₃×C₄×Q₈	—	9	1	no	—
94	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
95	C₃×C₄×D₄	—	9	3	no	—
96	C₃×C₄×D₄	—	9	3	no	—
97	C₃×#117	—	9	3	no	—
98	C₂×C₃×C₄⋊C₄	—	9	3	no	—
99	Q₈×Dic₃	—	3	1	no	—
100	#893	—	3	3	no	—
101	D₃×C₄⋊C₄	—	3	3	no	—
102	C₄×D₁₂	—	3	3	no	—
103	#872	—	3	3	no	—
104	C₂²×C₃×D₃	—	3	3	no	—
105	C₃²×Q₈	C₄×D₃	1	4	yes	—
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
109	C₂×C₃×Dic₃	—	3	1	no	—
110	C₂×C₃×Dic₃	—	3	1	no	—
111	C₂×C₃×Dic₃	—	3	3	no	—
112	C₃×D₁₂	—	3	12	no	—
113	C₃×C₄×D₃	—	3	12	no	—
114	C₃×C₄×D₃	—	12	6	no	—
115	C₂²×D₉	C₂³×C₃	1	3	yes	—
116	Q₈×C₉	—	2	4	no	—
117	Q₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	3	no	—
120	C₂×C₄×C₉	—	2	3	no	—
121	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
123	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
124	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₉	C₂³×C₃	1	3	yes	—
126	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
127	D₃₆	C₂×C₃×C₄	1	12	yes	complemented
128	C₄×D₉	—	4	6	no	—
129	C₄×D₉	C₂×C₃×C₄	1	12	yes	complemented
130	#520	—	9	1	no	Sylow
131	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
132	C₂×3₋¹⁺²	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic
134	AGL(1,ℤ/9)	—	2	6	no	—
135	C₃×#41	—	9	8	no	—
136	C₂×C₃×Q₈	—	3	1	no	—
137	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃×D₄	—	9	3	no	—
139	C₂²×C₃×C₄	—	9	3	no	—
140	C₂²×C₃×C₄	—	9	6	no	—
141	#239	—	3	8	no	—
142	C₂×D₁₂	—	3	3	no	—
143	C₂×C₄×D₃	—	3	3	no	—
144	C₂×C₄×D₃	—	3	6	no	—
145	C₃×C₄⋊C₄	—	3	3	no	—
146	C₃×C₄⋊C₄	—	9	3	no	—
147	C₃×C₄⋊C₄	—	9	1	no	—
148	C₃×C₄⋊C₄	—	9	3	no	—
149	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
150	C₃×C₂²⋊C₄	—	9	3	no	—
151	C₃×C₂²⋊C₄	—	9	3	no	—
152	C₃×C₄²	—	3	3	no	—
153	C₃×C₄²	—	9	3	no	—
154	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
155	(C₂×D₃)⋊C₄	—	3	3	no	—
156	(C₂×D₃)⋊C₄	—	3	3	no	—
157	C₄⋊Dic₃	—	3	3	no	—
158	Dic₃⋊C₄	—	3	1	no	—
159	Dic₃⋊C₄	—	3	3	no	—
160	C₄×Dic₃	—	3	3	no	—
161	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
162	C₂×C₃×D₃	—	3	6	no	—
163	C₂×C₃×D₃	—	6	3	no	—
164	C₂×C₃×D₃	—	6	3	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	—	2	3	no	—
167	C₃²×C₄	C₂×C₄×D₃	1	6	yes	—
168	C₃×Dic₃	—	3	2	no	—
169	C₃×Dic₃	—	6	1	no	—
170	C₃×Dic₃	—	6	3	no	—
171	C₂²×C₉	—	2	1	no	—
172	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
173	C₂×D₉	—	2	3	no	—
174	C₂×D₉	—	2	3	no	—
175	C₂×D₉	C₂²×C₃×C₄	1	6	yes	—
176	C₄×C₉	—	2	2	no	—
177	C₄×C₉	—	2	3	no	—
178	C₄×C₉	—	2	1	no	—
179	C₄×C₉	—	2	6	no	—
180	C₄×C₉	—	4	3	no	—
181	C₄×C₉	C₂²×C₃×C₄	1	6	yes	—
182	Dic₉	—	2	3	no	—
183	Dic₉	—	2	1	no	—
184	Dic₉	C₂²×C₃×C₄	1	2	yes	—
185	C₂×#41	—	9	1	no	—
186	C₄×Q₈	—	9	1	no	—
187	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
188	C₄×D₄	—	9	3	no	—
189	C₄×D₄	—	9	3	no	—
190	#117	—	9	3	no	—
191	C₂×C₄⋊C₄	—	9	3	no	—
192	3₋¹⁺²	#520	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	9	3	no	—
194	C₂²×D₃	—	3	3	no	—
195	C₃×Q₈	—	3	4	no	—
196	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
197	C₃×D₄	—	9	12	no	—
198	C₂×C₃×C₄	—	3	3	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	3	3	no	—
201	C₂×C₃×C₄	—	9	12	no	—
202	C₂×C₃×C₄	—	9	1	no	—
203	C₂×C₃×C₄	—	9	1	no	—
204	C₂×C₃×C₄	—	9	3	no	—
205	C₂×C₃×C₄	—	36	6	no	—
206	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
208	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
209	C₂×Dic₃	—	3	1	no	—
210	C₂×Dic₃	—	3	1	no	—
211	C₂×Dic₃	—	3	3	no	—
212	D₁₂	—	3	12	no	—
213	C₄×D₃	—	3	12	no	—
214	C₄×D₃	—	12	6	no	—
215	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
216	C₂×C₃²	#1007	1	1	yes	characteristic
217	C₂×C₃²	#1009	1	1	yes	characteristic
218	C₃×D₃	—	6	6	no	—
219	C₂×C₉	—	2	1	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	2	1	no	—
222	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
223	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
225	D₉	—	2	6	no	—
226	#41	—	9	8	no	—
227	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
228	C₂×D₄	—	9	3	no	—
229	C₂²×C₄	—	9	3	no	—
230	C₂²×C₄	—	9	6	no	—
231	C₄⋊C₄	—	9	3	no	—
232	C₄⋊C₄	—	9	3	no	—
233	C₄⋊C₄	—	9	1	no	—
234	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
235	C₂²⋊C₄	—	9	3	no	—
236	C₂²⋊C₄	—	9	3	no	—
237	C₄²	—	9	3	no	—
238	C₄²	C₂×AGL(1,ℤ/9)	1	3	yes	—
239	C₂²×C₃	—	3	1	no	—
240	C₂²×C₃	—	9	6	no	—
241	C₂²×C₃	—	18	3	no	—
242	C₂²×C₃	—	18	3	no	—
243	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
244	C₂×D₃	—	3	6	no	—
245	C₂×D₃	—	6	3	no	—
246	C₂×D₃	—	6	3	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	3	no	—
249	C₃×C₄	—	3	6	no	—
250	C₃×C₄	—	6	3	no	—
251	C₃×C₄	—	6	1	no	—
252	C₃×C₄	—	9	2	no	—
253	C₃×C₄	—	18	3	no	—
254	C₃×C₄	—	18	1	no	—
255	C₃×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
256	Dic₃	—	3	2	no	—
257	Dic₃	—	6	1	no	—
258	Dic₃	—	6	3	no	—
259	C₃²	#5574	1	1	yes	characteristic
260	C₉	—	2	1	no	—
261	C₉	C₃×#520	1	1	yes	characteristic, complemented
262	C₂³	—	9	3	no	—
263	Q₈	C₄×AGL(1,ℤ/9)	1	4	yes	complemented
264	D₄	—	9	12	no	—
265	C₂×C₄	—	9	1	no	—
266	C₂×C₄	—	9	12	no	—
267	C₂×C₄	—	9	1	no	—
268	C₂×C₄	—	9	3	no	—
269	C₂×C₄	—	36	6	no	—
270	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
271	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
272	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
273	C₂×C₃	—	3	1	no	—
274	C₂×C₃	—	3	1	no	—
275	C₂×C₃	—	3	1	no	—
276	C₂×C₃	—	18	6	no	—
277	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
278	C₂×C₃	C₃×#1007	1	1	yes	characteristic
279	C₂×C₃	C₃×#1009	1	1	yes	characteristic
280	D₃	—	6	6	no	—
281	C₂²	—	9	6	no	—
282	C₂²	—	18	3	no	—
283	C₂²	—	18	3	no	—
284	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	3	no	—
287	C₄	—	9	2	no	—
288	C₄	—	18	1	no	—
289	C₄	—	18	3	no	—
290	C₄	C₂×C₄×AGL(1,ℤ/9)	1	6	yes	—
291	C₃	—	3	1	no	—
292	C₃	C₃×#5574	1	1	yes	characteristic
293	C₂	—	18	6	no	—
294	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
295	C₂	#11722630	1	1	yes	characteristic
296	C₂	#11722641	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₄×Q₈×3₋¹⁺²	C₂
prime radical	C₂×#88724	C₂
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₄×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#520
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×3₋¹⁺²	C₂	semidirect
3. Q₈×3₋¹⁺²⋊C₄	C₂	semidirect
4. #63455⋊C₃	C₂	semidirect
5. C₄⋊C₄×AGL(1,ℤ/9)	C₂	semidirect
6. C₄×C₄⋊AGL(1,ℤ/9)	C₂	semidirect
7. #63434⋊C₃	C₂	semidirect
9. #10589676	C₃	semidirect
11. #88724	C₄	semidirect
32. C₄×Q₈×C₉	C₂×C₃	semidirect
33. Q₈×Dic₉	C₂×C₃	semidirect
34. #63455	C₂×C₃	semidirect
35. C₄⋊C₄×D₉	C₂×C₃	semidirect
36. C₄×D₃₆	C₂×C₃	semidirect
37. #63434	C₂×C₃	semidirect
39. Q₈×3₋¹⁺²	C₂×C₄	semidirect
46. C₄⋊AGL(1,ℤ/9)	C₂×C₄	semidirect
48. C₄×AGL(1,ℤ/9)	C₂×C₄	semidirect
66. #3701	C₃×C₄	semidirect
117. Q₈×C₉	C₂×C₃×C₄	semidirect
127. D₃₆	C₂×C₃×C₄	semidirect
129. C₄×D₉	C₂×C₃×C₄	semidirect
187. C₄×Q₈	AGL(1,ℤ/9)	semidirect
192. 3₋¹⁺²	#520	semidirect
261. C₉	C₃×#520	semidirect
263. Q₈	C₄×AGL(1,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88724	2
C₄×Q₈×3₋¹⁺²	2
Q₈×3₋¹⁺²⋊C₄	2
#63455⋊C₃	2
C₄⋊C₄×AGL(1,ℤ/9)	2
C₄×C₄⋊AGL(1,ℤ/9)	2
#63434⋊C₃	2
C₃×#5574	3
#10589676	3