Subgroups — #13152733

Order 1728.

833 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×AGL(1,ℤ/9)	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	#63454⋊C₃	C₂	1	3	yes	maximal
5	#63449⋊C₃	C₂	1	3	yes	maximal, complemented
6	#63434⋊C₃	C₂	1	3	yes	maximal
7	C₄×3₋¹⁺²⋊Q₈	C₂	1	3	yes	maximal, complemented
8	C₃×#5572	—	3	1	no	maximal
9	#10589674	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	Q₈×AGL(1,ℤ/9)	C₄	1	8	yes	complemented
12	C₂×C₄×AGL(1,ℤ/9)	C₂²	1	3	yes	—
13	C₂×3₋¹⁺²⋊Q₈	C₂²	1	3	yes	—
14	C₄⋊C₄×3₋¹⁺²	C₂²	1	3	yes	—
15	C₄²×3₋¹⁺²	C₂²	1	3	yes	—
16	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	3	yes	—
17	(C₂×AGL(1,ℤ/9))⋊C₄	C₂²	1	1	yes	characteristic
18	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	3	yes	—
19	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	3	yes	—
20	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	3	yes	—
21	C₄×3₋¹⁺²⋊C₄	C₂²	1	6	yes	—
22	C₄×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
23	C₂×C₃×D₃×Q₈	—	3	1	no	—
24	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
25	C₃×Q₈×Dic₃	—	3	1	no	—
26	C₃×#892	—	3	3	no	—
27	C₃×#887	—	3	3	no	—
28	C₃×#872	—	3	3	no	—
29	C₃×C₄×Dic₆	—	3	3	no	—
30	C₂×Q₈×D₉	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	#63454	C₂×C₃	1	3	yes	—
35	#63449	C₂×C₃	1	3	yes	complemented
36	#63434	C₂×C₃	1	3	yes	—
37	C₄×Dic₁₈	C₂×C₃	1	3	yes	complemented
38	C₂²×AGL(1,ℤ/9)	C₂³	1	1	yes	characteristic
39	Q₈×3₋¹⁺²	C₂×C₄	1	4	yes	—
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	1	yes	characteristic
44	C₂×3₋¹⁺²⋊C₄	C₂³	1	3	yes	—
45	C₂×3₋¹⁺²⋊C₄	C₂³	1	3	yes	—
46	C₄×AGL(1,ℤ/9)	C₂×C₄	1	12	yes	—
47	3₋¹⁺²⋊Q₈	C₂×C₄	1	12	yes	—
48	C₃×#519	—	9	1	no	—
49	#5572	—	3	1	no	—
50	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
51	C₃×D₃×Q₈	—	3	8	no	—
52	C₂×C₃×C₄×D₃	—	3	3	no	—
53	C₂×C₃×Dic₆	—	3	3	no	—
54	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
55	C₃²×C₄²	C₂×D₃	1	3	yes	—
56	C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
57	C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
58	C₃×C₄⋊Dic₃	—	3	3	no	—
59	C₃×Dic₃⋊C₄	—	3	3	no	—
60	C₃×Dic₃⋊C₄	—	3	3	no	—
61	C₃×C₄×Dic₃	—	3	6	no	—
62	C₃×C₄×Dic₃	—	3	3	no	—
63	C₂×Q₈×C₉	—	2	1	no	—
64	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
65	Q₈×D₉	C₃×C₄	1	8	yes	complemented
66	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
67	C₂×Dic₁₈	C₂²×C₃	1	3	yes	—
68	C₉×C₄⋊C₄	—	2	3	no	—
69	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
70	C₄²×C₉	—	2	3	no	—
71	C₄²×C₉	C₂²×C₃	1	3	yes	—
72	(C₂×D₉)⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	(C₂×D₉)⋊C₄	C₂²×C₃	1	3	yes	—
74	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
75	Dic₉⋊C₄	C₂²×C₃	1	3	yes	—
76	Dic₉⋊C₄	C₂²×C₃	1	3	yes	—
77	C₄×Dic₉	C₂²×C₃	1	3	yes	—
78	C₄×Dic₉	C₂²×C₃	1	6	yes	—
79	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
80	C₂×AGL(1,ℤ/9)	—	2	1	no	—
81	C₂×AGL(1,ℤ/9)	—	2	1	no	—
82	C₂×AGL(1,ℤ/9)	C₂²×C₄	1	2	yes	—
83	C₄×3₋¹⁺²	—	2	1	no	—
84	C₄×3₋¹⁺²	—	2	3	no	—
85	C₄×3₋¹⁺²	C₂²×C₄	1	6	yes	—
86	3₋¹⁺²⋊C₄	—	2	3	no	—
87	3₋¹⁺²⋊C₄	—	2	1	no	—
88	3₋¹⁺²⋊C₄	C₂²×C₄	1	6	yes	—
89	C₂²×C₃×Q₈	—	9	1	no	—
90	C₂×D₃×Q₈	—	3	1	no	—
91	C₃×C₄×Q₈	—	3	1	no	—
92	C₃×C₄×Q₈	—	9	3	no	—
93	C₃×C₄×Q₈	—	9	3	no	—
94	C₃×C₄×Q₈	—	9	1	no	—
95	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
96	C₃×#117	—	9	3	no	—
97	C₃×#117	—	9	3	no	—
98	Q₈×Dic₃	—	3	1	no	—
99	#892	—	3	3	no	—
100	#887	—	3	3	no	—
101	#872	—	3	3	no	—
102	C₄×Dic₆	—	3	3	no	—
103	C₂²×C₃×D₃	—	3	1	no	—
104	C₃²×Q₈	C₄×D₃	1	4	yes	—
105	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
108	C₂×C₃×Dic₃	—	3	3	no	—
109	C₂×C₃×Dic₃	—	3	1	no	—
110	C₂×C₃×Dic₃	—	3	3	no	—
111	C₃×C₄×D₃	—	3	12	no	—
112	C₃×Dic₆	—	3	12	no	—
113	C₂²×D₉	C₂³×C₃	1	1	yes	characteristic
114	Q₈×C₉	—	2	4	no	—
115	Q₈×C₉	C₂×C₃×C₄	1	4	yes	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	3	no	—
118	C₂×C₄×C₉	—	2	3	no	—
119	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
120	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
122	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
123	C₂×Dic₉	C₂³×C₃	1	3	yes	—
124	C₂×Dic₉	C₂³×C₃	1	3	yes	—
125	C₄×D₉	C₂×C₃×C₄	1	12	yes	—
126	Dic₁₈	C₂×C₃×C₄	1	12	yes	—
127	#519	—	9	1	no	Sylow
128	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
129	C₂×3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic
131	AGL(1,ℤ/9)	—	2	2	no	—
132	C₂×C₃×Q₈	—	3	1	no	—
133	C₂×C₃×Q₈	—	9	3	no	—
134	C₂×C₃×Q₈	—	9	8	no	—
135	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
136	C₂²×C₃×C₄	—	9	3	no	—
137	D₃×Q₈	—	3	8	no	—
138	C₂×C₄×D₃	—	3	3	no	—
139	C₂×Dic₆	—	3	3	no	—
140	C₃×C₄⋊C₄	—	3	3	no	—
141	C₃×C₄⋊C₄	—	9	3	no	—
142	C₃×C₄⋊C₄	—	9	3	no	—
143	C₃×C₄⋊C₄	—	9	3	no	—
144	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
145	C₃×C₂²⋊C₄	—	9	1	no	—
146	C₃×C₂²⋊C₄	—	9	3	no	—
147	C₃×C₄²	—	3	3	no	—
148	C₃×C₄²	—	9	3	no	—
149	C₃×C₄²	—	9	6	no	—
150	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
151	(C₂×D₃)⋊C₄	—	3	1	no	—
152	(C₂×D₃)⋊C₄	—	3	3	no	—
153	C₄⋊Dic₃	—	3	3	no	—
154	Dic₃⋊C₄	—	3	3	no	—
155	Dic₃⋊C₄	—	3	3	no	—
156	C₄×Dic₃	—	3	6	no	—
157	C₄×Dic₃	—	3	3	no	—
158	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
159	C₂×C₃×D₃	—	3	2	no	—
160	C₂×C₃×D₃	—	6	1	no	—
161	C₂×C₃×D₃	—	6	1	no	—
162	C₃²×C₄	—	2	1	no	—
163	C₃²×C₄	—	2	3	no	—
164	C₃²×C₄	C₂×C₄×D₃	1	6	yes	—
165	C₃×Dic₃	—	3	6	no	—
166	C₃×Dic₃	—	6	3	no	—
167	C₃×Dic₃	—	6	1	no	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×D₉	—	2	1	no	—
171	C₂×D₉	—	2	1	no	—
172	C₂×D₉	C₂²×C₃×C₄	1	2	yes	—
173	C₄×C₉	—	2	2	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	—	2	3	no	—
176	C₄×C₉	—	2	6	no	—
177	C₄×C₉	—	4	3	no	—
178	C₄×C₉	C₂²×C₃×C₄	1	6	yes	—
179	Dic₉	—	2	3	no	—
180	Dic₉	—	2	1	no	—
181	Dic₉	C₂²×C₃×C₄	1	6	yes	—
182	C₂²×Q₈	—	9	1	no	—
183	C₄×Q₈	—	9	1	no	—
184	C₄×Q₈	—	9	3	no	—
185	C₄×Q₈	—	9	3	no	—
186	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
187	#117	—	9	3	no	—
188	#117	—	9	3	no	—
189	3₋¹⁺²	#519	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₃	—	9	1	no	—
191	C₂²×D₃	—	3	1	no	—
192	C₃×Q₈	—	3	4	no	—
193	C₃×Q₈	—	9	12	no	—
194	C₃×Q₈	C₃×C₄×D₃	1	4	yes	—
195	C₂×C₃×C₄	—	3	3	no	—
196	C₂×C₃×C₄	—	3	1	no	—
197	C₂×C₃×C₄	—	3	3	no	—
198	C₂×C₃×C₄	—	9	12	no	—
199	C₂×C₃×C₄	—	9	3	no	—
200	C₂×C₃×C₄	—	9	1	no	—
201	C₂×C₃×C₄	—	9	3	no	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
205	C₂×Dic₃	—	3	3	no	—
206	C₂×Dic₃	—	3	1	no	—
207	C₂×Dic₃	—	3	3	no	—
208	C₄×D₃	—	3	12	no	—
209	Dic₆	—	3	12	no	—
210	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
211	C₂×C₃²	#1007	1	1	yes	characteristic
212	C₂×C₃²	#1010	1	1	yes	characteristic
213	C₃×D₃	—	6	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
218	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
219	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
220	D₉	—	2	2	no	—
221	C₂×Q₈	—	9	8	no	—
222	C₂×Q₈	—	9	3	no	—
223	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
224	C₂²×C₄	—	9	3	no	—
225	C₄⋊C₄	—	9	3	no	—
226	C₄⋊C₄	—	9	3	no	—
227	C₄⋊C₄	—	9	3	no	—
228	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
229	C₂²⋊C₄	—	9	3	no	—
230	C₂²⋊C₄	—	9	1	no	—
231	C₄²	—	9	6	no	—
232	C₄²	—	9	3	no	—
233	C₄²	C₂×AGL(1,ℤ/9)	1	3	yes	—
234	C₂²×C₃	—	3	1	no	—
235	C₂²×C₃	—	9	2	no	—
236	C₂²×C₃	—	18	1	no	—
237	C₂²×C₃	—	18	1	no	—
238	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
239	C₂×D₃	—	3	2	no	—
240	C₂×D₃	—	6	1	no	—
241	C₂×D₃	—	6	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	3	no	—
244	C₃×C₄	—	3	6	no	—
245	C₃×C₄	—	6	1	no	—
246	C₃×C₄	—	6	3	no	—
247	C₃×C₄	—	9	6	no	—
248	C₃×C₄	—	18	3	no	—
249	C₃×C₄	—	18	1	no	—
250	C₃×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
251	Dic₃	—	3	6	no	—
252	Dic₃	—	6	3	no	—
253	Dic₃	—	6	1	no	—
254	C₃²	#5572	1	1	yes	characteristic
255	C₉	—	2	1	no	—
256	C₉	C₃×#519	1	1	yes	characteristic, complemented
257	C₂³	—	9	1	no	—
258	Q₈	—	9	12	no	—
259	Q₈	C₄×AGL(1,ℤ/9)	1	4	yes	—
260	C₂×C₄	—	9	3	no	—
261	C₂×C₄	—	9	12	no	—
262	C₂×C₄	—	9	3	no	—
263	C₂×C₄	—	9	1	no	—
264	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
265	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
266	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
267	C₂×C₃	—	3	1	no	—
268	C₂×C₃	—	3	1	no	—
269	C₂×C₃	—	3	1	no	—
270	C₂×C₃	—	18	2	no	—
271	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
272	C₂×C₃	C₃×#1007	1	1	yes	characteristic
273	C₂×C₃	C₃×#1010	1	1	yes	characteristic
274	D₃	—	6	2	no	—
275	C₂²	—	9	2	no	—
276	C₂²	—	18	1	no	—
277	C₂²	—	18	1	no	—
278	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	3	no	—
281	C₄	—	9	6	no	—
282	C₄	—	18	3	no	—
283	C₄	—	18	1	no	—
284	C₄	C₂×C₄×AGL(1,ℤ/9)	1	6	yes	—
285	C₃	—	3	1	no	—
286	C₃	C₃×#5572	1	1	yes	characteristic
287	C₂	—	18	2	no	—
288	C₂	C₂²×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
289	C₂	#11722630	1	1	yes	characteristic
290	C₂	#11722642	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	C₂²×AGL(1,ℤ/9)	C₂³
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₄×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#519
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
5. #63449⋊C₃	C₂	semidirect
7. C₄×3₋¹⁺²⋊Q₈	C₂	semidirect
9. #10589674	C₃	semidirect
11. Q₈×AGL(1,ℤ/9)	C₄	semidirect
32. C₄×Q₈×C₉	C₂×C₃	semidirect
33. Q₈×Dic₉	C₂×C₃	semidirect
35. #63449	C₂×C₃	semidirect
37. C₄×Dic₁₈	C₂×C₃	semidirect
65. Q₈×D₉	C₃×C₄	semidirect
186. C₄×Q₈	AGL(1,ℤ/9)	semidirect
189. 3₋¹⁺²	#519	semidirect
256. C₉	C₃×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×AGL(1,ℤ/9)	2
C₄×Q₈×3₋¹⁺²	2
Q₈×3₋¹⁺²⋊C₄	2
#63454⋊C₃	2
#63449⋊C₃	2
#63434⋊C₃	2
C₄×3₋¹⁺²⋊Q₈	2
C₃×#5572	3
#10589674	3