Subgroups — #10589620⋊C₃ #13152532

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 281 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#117	C₂	1	1	yes	maximal, characteristic
2	#63486⋊C₃	C₂	1	2	yes	maximal
3	#63451⋊C₃	C₂	1	4	yes	maximal, complemented
4	#63441⋊C₃	C₂	1	4	yes	maximal
5	#63432⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#63431⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×3₋¹⁺²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₃×#5518	—	3	1	no	maximal
9	#10589620	C₃	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
12	C₄⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
13	C₂²⋊C₄×3₋¹⁺²	C₂²	1	2	yes	—
14	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
15	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	#3676⋊C₃	C₂²	1	4	yes	—
17	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	4	yes	—
18	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
19	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	4	yes	—
20	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	4	yes	—
21	(3₋¹⁺²⋊C₄)⋊C₄	C₂²	1	4	yes	—
22	C₄×3₋¹⁺²⋊C₄	C₂²	1	4	yes	—
23	C₃²×#117	D₃	1	1	yes	characteristic
24	C₃×#924	—	3	2	no	—
25	C₃×#889	—	3	4	no	—
26	C₃×#879	—	3	4	no	—
27	C₃×#870	—	3	1	no	—
28	C₃×#869	—	3	1	no	—
29	C₃×C₄×Dic₆	—	3	2	no	—
30	C₉×#117	—	2	1	no	—
31	C₉×#117	C₂×C₃	1	1	yes	characteristic
32	#63486	C₂×C₃	1	2	yes	—
33	#63451	C₂×C₃	1	4	yes	complemented
34	#63441	C₂×C₃	1	4	yes	—
35	#63432	C₂×C₃	1	1	yes	characteristic, complemented
36	#63431	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄×Dic₁₈	C₂×C₃	1	2	yes	complemented
38	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
39	C₂×C₄×3₋¹⁺²	—	2	2	no	—
40	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
41	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
42	C₂×C₄×3₋¹⁺²	C₂³	1	2	yes	—
43	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
44	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
45	C₂×3₋¹⁺²⋊C₄	C₂³	1	4	yes	—
46	3₋¹⁺²⋊Q₈	—	2	4	no	—
47	C₃×#541	—	9	1	no	—
48	#5518	—	3	1	no	—
49	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×Dic₆	—	3	2	no	—
51	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₃²×C₂²⋊C₄	C₂×D₃	1	2	yes	—
53	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₃×#217	—	3	4	no	—
56	C₃×C₄⋊Dic₃	—	3	4	no	—
57	C₃×C₄⋊Dic₃	—	3	2	no	—
58	C₃×Dic₃⋊C₄	—	3	4	no	—
59	C₃×Dic₃⋊C₄	—	3	4	no	—
60	C₃×Dic₃⋊C₄	—	3	4	no	—
61	C₃×C₄×Dic₃	—	3	4	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
64	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
65	C₉×C₄⋊C₄	—	2	2	no	—
66	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₉×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄²×C₉	—	2	1	no	—
70	C₄²×C₉	—	2	1	no	—
71	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
72	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
73	#3676	C₂²×C₃	1	4	yes	—
74	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
75	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
76	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
77	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
78	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
79	C₄×Dic₉	C₂²×C₃	1	4	yes	—
80	C₂²×3₋¹⁺²	—	2	1	no	—
81	C₂²×3₋¹⁺²	—	2	2	no	—
82	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
83	C₄×3₋¹⁺²	—	2	1	no	—
84	C₄×3₋¹⁺²	—	2	2	no	—
85	C₄×3₋¹⁺²	—	2	2	no	—
86	C₄×3₋¹⁺²	#41	1	2	yes	—
87	3₋¹⁺²⋊C₄	—	2	4	no	—
88	3₋¹⁺²⋊C₄	—	2	4	no	—
89	C₃×C₄⋊Q₈	—	9	1	no	—
90	C₃×#126	—	9	4	no	—
91	C₃×#125	—	9	1	no	—
92	C₃×#125	—	9	4	no	—
93	C₃×C₂²⋊Q₈	—	9	2	no	—
94	C₃×C₄×Q₈	—	9	2	no	—
95	C₃×#117	—	3	1	no	—
96	C₃×#117	C₃×D₃	1	1	yes	characteristic
97	#924	—	3	2	no	—
98	#889	—	3	4	no	—
99	#879	—	3	4	no	—
100	#870	—	3	1	no	—
101	#869	—	3	1	no	—
102	C₄×Dic₆	—	3	2	no	—
103	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₃²×C₄	—	2	2	no	—
105	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
106	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×Dic₃	—	3	4	no	—
110	C₂×C₃×Dic₃	—	3	4	no	—
111	C₃×Dic₆	—	6	4	no	—
112	C₂³×C₉	—	2	1	no	—
113	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₄×C₉	—	2	4	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
121	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
123	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₉	C₂³×C₃	1	4	yes	—
125	C₂×Dic₉	C₂³×C₃	1	4	yes	—
126	Dic₁₈	—	2	4	no	—
127	#541	—	9	1	no	Sylow
128	C₂×3₋¹⁺²	—	4	1	no	—
129	C₂×3₋¹⁺²	C₂×#41	1	1	yes	characteristic
130	C₂×3₋¹⁺²	2₋¹⁺⁴	1	2	yes	—
131	C₂×C₃×Q₈	—	9	2	no	—
132	C₂²×C₃×C₄	—	3	1	no	—
133	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×Dic₆	—	3	2	no	—
135	C₃×C₄⋊C₄	—	3	2	no	—
136	C₃×C₄⋊C₄	—	9	4	no	—
137	C₃×C₄⋊C₄	—	9	4	no	—
138	C₃×C₄⋊C₄	—	9	4	no	—
139	C₃×C₄⋊C₄	—	9	2	no	—
140	C₃×C₄⋊C₄	—	9	4	no	—
141	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
142	C₃×C₂²⋊C₄	—	3	2	no	—
143	C₃×C₂²⋊C₄	—	9	4	no	—
144	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
145	C₃×C₄²	—	3	1	no	—
146	C₃×C₄²	—	3	1	no	—
147	C₃×C₄²	—	9	4	no	—
148	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
150	#217	—	3	4	no	—
151	C₄⋊Dic₃	—	3	2	no	—
152	C₄⋊Dic₃	—	3	4	no	—
153	Dic₃⋊C₄	—	3	4	no	—
154	Dic₃⋊C₄	—	3	4	no	—
155	Dic₃⋊C₄	—	3	4	no	—
156	C₄×Dic₃	—	3	4	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	2	no	—
159	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
160	C₃²×C₄	—	2	1	no	—
161	C₃²×C₄	—	2	2	no	—
162	C₃²×C₄	—	2	2	no	—
163	C₃²×C₄	#235	1	2	yes	—
164	C₃×Dic₃	—	6	4	no	—
165	C₃×Dic₃	—	6	4	no	—
166	C₂²×C₉	—	2	2	no	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	4	2	no	—
171	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	2	2	no	—
175	C₄×C₉	—	2	2	no	—
176	C₄×C₉	—	2	2	no	—
177	C₄×C₉	—	4	2	no	—
178	C₄×C₉	—	4	2	no	—
179	C₄×C₉	C₃×#41	1	2	yes	—
180	Dic₉	—	2	4	no	—
181	Dic₉	—	2	4	no	—
182	C₄⋊Q₈	—	9	1	no	—
183	#126	—	9	4	no	—
184	#125	—	9	4	no	—
185	#125	—	9	1	no	—
186	C₂²⋊Q₈	—	9	2	no	—
187	C₄×Q₈	—	9	2	no	—
188	#117	AGL(1,ℤ/9)	1	1	yes	characteristic
189	3₋¹⁺²	#541	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₃	—	3	1	no	—
191	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
192	C₃×Q₈	—	18	4	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	3	2	no	—
196	C₂×C₃×C₄	—	3	2	no	—
197	C₂×C₃×C₄	—	3	1	no	—
198	C₂×C₃×C₄	—	6	2	no	—
199	C₂×C₃×C₄	—	9	4	no	—
200	C₂×C₃×C₄	—	9	4	no	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	C₂×Dic₃	—	3	4	no	—
206	C₂×Dic₃	—	3	4	no	—
207	Dic₆	—	6	4	no	—
208	C₂×C₃²	—	4	1	no	—
209	C₂×C₃²	C₂×#235	1	1	yes	characteristic
210	C₂×C₃²	#1010	1	2	yes	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	2	2	no	—
213	C₂×C₉	—	4	2	no	—
214	C₂×C₉	—	4	1	no	—
215	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
216	C₂×C₉	C₃×2₋¹⁺⁴	1	2	yes	—
217	C₂×Q₈	—	9	2	no	—
218	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
219	C₄⋊C₄	—	9	2	no	—
220	C₄⋊C₄	—	9	4	no	—
221	C₄⋊C₄	—	9	4	no	—
222	C₄⋊C₄	—	9	4	no	—
223	C₄⋊C₄	—	9	4	no	—
224	C₄⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
225	C₂²⋊C₄	—	9	4	no	—
226	C₂²⋊C₄	C₂×AGL(1,ℤ/9)	1	2	yes	—
227	C₄²	—	9	4	no	—
228	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
229	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	3	1	no	—
233	C₂²×C₃	—	6	1	no	—
234	C₂²×C₃	—	6	2	no	—
235	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	3	2	no	—
240	C₃×C₄	—	6	1	no	—
241	C₃×C₄	—	6	2	no	—
242	C₃×C₄	—	6	2	no	—
243	C₃×C₄	—	18	4	no	—
244	C₃×C₄	—	18	4	no	—
245	C₃×C₄	C₃×#235	1	2	yes	—
246	Dic₃	—	6	4	no	—
247	Dic₃	—	6	4	no	—
248	C₃²	#5518	1	1	yes	characteristic
249	C₉	—	2	1	no	—
250	C₉	C₃×#541	1	1	yes	characteristic, complemented
251	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
252	Q₈	—	18	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	9	4	no	—
255	C₂×C₄	—	9	4	no	—
256	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
257	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	2	yes	—
258	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
259	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
260	C₂×C₃	—	3	1	no	—
261	C₂×C₃	—	3	2	no	—
262	C₂×C₃	—	4	1	no	—
263	C₂×C₃	—	12	1	no	—
264	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
265	C₂×C₃	C₃×#1010	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	18	4	no	—
273	C₄	—	18	4	no	—
274	C₄	#3697⋊C₃	1	2	yes	—
275	C₃	—	3	1	no	—
276	C₃	C₃×#5518	1	1	yes	characteristic
277	C₂	—	4	1	no	—
278	C₂	C₂×#3697⋊C₃	1	1	yes	characteristic
279	C₂	#11722642	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	3₋¹⁺²×#117	C₂
prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	#117	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#541
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
3. #63451⋊C₃	C₂	semidirect
5. #63432⋊C₃	C₂	semidirect
6. #63431⋊C₃	C₂	semidirect
7. C₄×3₋¹⁺²⋊Q₈	C₂	semidirect
9. #10589620	C₃	semidirect
33. #63451	C₂×C₃	semidirect
35. #63432	C₂×C₃	semidirect
36. #63431	C₂×C₃	semidirect
37. C₄×Dic₁₈	C₂×C₃	semidirect
189. 3₋¹⁺²	#541	semidirect
250. C₉	C₃×#541	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×#117	2
#63486⋊C₃	2
#63451⋊C₃	2
#63441⋊C₃	2
#63432⋊C₃	2
#63431⋊C₃	2
C₄×3₋¹⁺²⋊Q₈	2
C₃×#5518	3
#10589620	3

Subgroups — #10589620⋊C3 #13152532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10589620⋊C₃ #13152532