Subgroups — C₉×#4842 #13142038

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#821	C₂	1	2	yes	maximal
5	C₉×#817	C₂	1	2	yes	maximal, complemented
6	C₃×#4842	C₃	1	1	yes	maximal, characteristic
7	C₉×#476	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃×C₉	—	2	1	no	—
10	C₂×C₉×Dic₆	—	2	1	no	—
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
14	C₉×C₄⋊Dic₃	—	2	1	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
18	C₃×#869	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×#821	C₂×C₃	1	2	yes	—
21	C₃×#817	C₂×C₃	1	2	yes	—
22	C₉×C₄⋊Q₈	—	3	1	no	—
23	C₄×D₄×C₉	—	3	1	no	—
24	C₉×#106	—	3	2	no	—
25	C₉×C₄⋊C₈	—	2	1	no	—
26	C₉×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
27	C₉×#102	—	3	2	no	—
28	C₂²×D₃×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	—	2	2	no	—
33	C₂×C₉×Dic₃	—	2	1	no	—
34	C₉×D₁₂	—	2	1	no	—
35	C₉×D₁₂	Q₈	1	2	yes	complemented
36	C₄×D₃×C₉	—	4	2	no	—
37	C₉×Dic₆	—	4	2	no	—
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃×#476	—	3	1	no	—
40	#4842	C₉	1	1	yes	characteristic, direct product factor
41	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃×C₄×D₃	—	2	1	no	—
43	C₂×C₃×Dic₆	—	2	1	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
45	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₃×C₄⋊Dic₃	—	2	1	no	—
48	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
50	C₂×Q₈×C₉	—	6	1	no	—
51	C₂×D₄×C₉	—	3	1	no	—
52	C₂²×C₄×C₉	—	6	1	no	—
53	C₂×C₈×C₉	—	2	2	no	—
54	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
55	C₉×C₄⋊C₄	—	3	1	no	—
56	C₉×C₄⋊C₄	—	3	2	no	—
57	C₉×C₄⋊C₄	—	6	1	no	—
58	C₉×C₂²⋊C₄	—	6	1	no	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₂×D₃×C₉	—	2	2	no	—
63	C₂×D₃×C₉	—	4	1	no	—
64	C₂×D₃×C₉	—	4	1	no	—
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	SD₁₆	1	2	yes	—
67	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
68	C₃×C₄×C₉	#41	1	1	yes	characteristic
69	C₉×Dic₃	—	4	1	no	—
70	C₉×Dic₃	—	4	2	no	—
71	C₃×C₄⋊Q₈	—	3	1	no	—
72	C₃×C₄×D₄	—	3	1	no	—
73	C₄×D₁₂	C₂×C₉	1	1	yes	characteristic
74	#869	C₂×C₉	1	1	yes	characteristic, complemented
75	C₃×#106	—	3	2	no	—
76	C₃×C₄⋊C₈	—	2	1	no	—
77	C₃×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
78	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
79	C₃×#102	—	3	2	no	—
80	#821	C₂×C₉	1	2	yes	—
81	#817	C₂×C₉	1	2	yes	complemented
82	C₂²×C₃×D₃	—	2	1	no	—
83	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃×Dic₃	—	2	2	no	—
87	C₂×C₃×Dic₃	—	2	1	no	—
88	C₃×D₁₂	—	2	1	no	—
89	C₃×D₁₂	C₃×Q₈	1	2	yes	—
90	C₃×C₄×D₃	—	4	2	no	—
91	C₃×Dic₆	—	4	2	no	—
92	C₂³×C₉	—	6	1	no	—
93	C₃²×C₈	—	2	2	no	—
94	Q₈×C₉	—	12	2	no	—
95	D₄×C₉	—	3	2	no	—
96	D₄×C₉	—	6	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	1	no	—
100	C₂×C₄×C₉	—	6	1	no	—
101	C₂×C₄×C₉	—	6	2	no	—
102	C₂×C₄×C₉	—	12	2	no	—
103	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
104	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
105	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
106	C₈×C₉	—	2	2	no	—
107	C₈×C₉	—	4	2	no	—
108	#476	—	3	1	no	Sylow
109	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	D₃×C₉	—	4	2	no	—
113	C₂×C₃×Q₈	—	6	1	no	—
114	C₂×C₃×D₄	—	3	1	no	—
115	C₂²×C₃×C₄	—	6	1	no	—
116	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
117	C₂×C₄×D₃	—	2	1	no	—
118	C₂×Dic₆	—	2	1	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
121	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
122	C₃×C₄⋊C₄	—	3	2	no	—
123	C₃×C₄⋊C₄	—	3	1	no	—
124	C₃×C₄⋊C₄	—	6	1	no	—
125	C₃×C₂²⋊C₄	—	6	1	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
128	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
129	(C₂×D₃)⋊C₄	—	2	1	no	—
130	C₄⋊Dic₃	—	2	1	no	—
131	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
132	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
133	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₃×D₃	—	2	2	no	—
135	C₂×C₃×D₃	—	4	1	no	—
136	C₂×C₃×D₃	—	4	1	no	—
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
139	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
140	C₃²×C₄	C₃×#41	1	1	yes	characteristic
141	C₃×Dic₃	—	4	1	no	—
142	C₃×Dic₃	—	4	2	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	—	6	2	no	—
145	C₂²×C₉	—	12	1	no	—
146	C₂²×C₉	—	12	1	no	—
147	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	4	1	no	—
153	C₄×C₉	—	12	1	no	—
154	C₄×C₉	—	12	2	no	—
155	C₄×C₉	#204	1	2	yes	—
156	C₄×C₉	#237	1	1	yes	characteristic
157	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
158	C₄⋊Q₈	—	3	1	no	—
159	C₄×D₄	—	3	1	no	—
160	#106	—	3	2	no	—
161	C₄⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
162	#102	—	3	2	no	—
163	C₃×C₉	#476	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₃	—	6	1	no	—
165	C₂²×D₃	—	2	1	no	—
166	C₃×Q₈	—	12	2	no	—
167	C₃×D₄	—	3	2	no	—
168	C₃×D₄	—	6	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	6	1	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	—	12	2	no	—
175	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
176	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
178	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₂×Dic₃	—	2	2	no	—
182	C₂×Dic₃	—	2	1	no	—
183	D₁₂	—	2	1	no	—
184	D₁₂	Q₈×C₉	1	2	yes	complemented
185	C₄×D₃	—	4	2	no	—
186	Dic₆	—	4	2	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	4	2	no	—
190	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
191	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
192	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₃×D₃	—	4	2	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	12	2	no	—
198	C₂×C₉	#897	1	1	yes	characteristic
199	C₂×C₉	C₂×#204	1	1	yes	characteristic
200	C₂×C₉	#908	1	1	yes	characteristic
201	C₂×Q₈	—	6	1	no	—
202	C₂×D₄	—	3	1	no	—
203	C₂²×C₄	—	6	1	no	—
204	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	3	1	no	—
207	C₄⋊C₄	—	6	1	no	—
208	C₂²⋊C₄	—	6	1	no	—
209	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	6	2	no	—
212	C₂²×C₃	—	12	1	no	—
213	C₂²×C₃	—	12	1	no	—
214	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
215	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
216	C₂×D₃	—	2	2	no	—
217	C₂×D₃	—	4	1	no	—
218	C₂×D₃	—	4	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	12	1	no	—
226	C₃×C₄	—	12	2	no	—
227	C₃×C₄	C₉×SD₁₆	1	2	yes	—
228	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
229	C₃×C₄	C₉×#41	1	1	yes	characteristic
230	C₃×C₄	C₃×#204	1	2	yes	—
231	C₃×C₄	C₃×#237	1	1	yes	characteristic
232	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
233	Dic₃	—	4	1	no	—
234	Dic₃	—	4	2	no	—
235	C₃²	C₃×#476	1	1	yes	characteristic
236	C₉	—	2	1	no	—
237	C₉	#4842	1	1	yes	characteristic, direct product factor
238	C₂³	—	6	1	no	—
239	Q₈	—	12	2	no	—
240	D₄	—	3	2	no	—
241	D₄	—	6	1	no	—
242	C₂×C₄	—	6	1	no	—
243	C₂×C₄	—	6	2	no	—
244	C₂×C₄	—	12	2	no	—
245	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
246	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	12	2	no	—
253	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
255	C₂×C₃	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂×C₃	C₃×#897	1	1	yes	characteristic
257	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
258	C₂×C₃	C₃×#908	1	1	yes	characteristic
259	D₃	—	4	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	—	12	1	no	—
262	C₂²	—	12	1	no	—
263	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	12	1	no	—
266	C₄	—	12	2	no	—
267	C₄	C₉×#204	1	2	yes	—
268	C₄	C₉×#237	1	1	yes	characteristic
269	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
270	C₃	—	2	1	no	—
271	C₃	C₉×#476	1	1	yes	characteristic, complemented
272	C₃	C₃×#4842	1	1	yes	characteristic
273	C₂	—	12	2	no	—
274	C₂	C₉×#897	1	1	yes	characteristic
275	C₂	C₂×C₉×#204	1	1	yes	characteristic
276	C₂	C₉×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃×D₁₂	C₂²×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₄⋊C₈	D₃
hypocenter	C₃	C₉×#476
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₉×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₉×#869	C₂	semidirect
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₉×#817	C₂	semidirect
26. C₉×C₄⋊C₈	D₃	semidirect
35. C₉×D₁₂	Q₈	semidirect
40. #4842	C₉	direct (N × H)
74. #869	C₂×C₉	semidirect
77. C₃×C₄⋊C₈	C₂×C₉	semidirect
81. #817	C₂×C₉	semidirect
161. C₄⋊C₈	D₃×C₉	semidirect
163. C₃×C₉	#476	semidirect
184. D₁₂	Q₈×C₉	semidirect
237. C₉	#4842	direct (N × H)
271. C₃	C₉×#476	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×D₁₂	2
C₉×#869	2
C₃×C₉×C₄⋊C₈	2
C₉×#821	2
C₉×#817	2
C₃×#4842	3
C₉×#476	3

Subgroups — C9×#4842 #13142038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4842 #13142038