Subgroups — C₈×C₉×Dic₆ #13141402

Order 1728.

350 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×Dic₃⋊C₈	C₂	1	2	yes	maximal
4	C₈×C₉×Dic₃	C₂	1	2	yes	maximal
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₃×C₈×Dic₆	C₃	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₉	—	3	1	no	maximal
8	C₂×C₉×Dic₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₉×Dic₃⋊C₄	C₄	1	2	yes	—
14	C₄×C₉×Dic₃	C₂²	1	2	yes	—
15	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
17	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	C₂×C₃	1	2	yes	—
19	C₃×C₈×Dic₃	C₂×C₃	1	2	yes	—
20	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
21	C₄×Q₈×C₉	—	3	1	no	—
22	C₉×C₄⋊C₈	—	3	2	no	—
23	C₉×C₄⋊C₈	—	3	1	no	—
24	C₄×C₈×C₉	—	2	1	no	—
25	C₄×C₈×C₉	—	3	2	no	—
26	C₄×C₈×C₉	D₃	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
31	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
32	C₉×Dic₆	C₈	1	4	yes	direct product factor
33	C₃×C₈×C₉	—	2	1	no	—
34	C₃×C₈×C₉	Q₈	1	2	yes	complemented
35	C₉×C₃⋊C₈	—	2	2	no	—
36	C₃×C₈×Q₈	—	3	1	no	—
37	C₈×Dic₆	C₉	1	1	yes	characteristic, direct product factor
38	C₂×C₃×Dic₆	C₃×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₃×C₄⋊Dic₃	C₃×C₄	1	1	yes	characteristic
43	C₃×Dic₃⋊C₄	C₃×C₄	1	2	yes	—
44	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
46	C₂×Q₈×C₉	—	3	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	3	2	no	—
50	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
51	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
52	C₉×C₄⋊C₄	—	3	2	no	—
53	C₉×C₄⋊C₄	—	3	1	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	—	3	2	no	—
56	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	2	1	no	—
59	C₃×C₄×C₉	C₂×C₈	1	2	yes	—
60	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₄×C₉	#41	1	1	yes	characteristic
62	C₉×Dic₃	—	2	2	no	—
63	C₉×Dic₃	C₂×C₈	1	4	yes	—
64	C₃×C₄×Q₈	—	3	1	no	—
65	C₄×Dic₆	C₂×C₉	1	1	yes	characteristic
66	C₃×C₄⋊C₈	—	3	2	no	—
67	C₃×C₄⋊C₈	—	3	1	no	—
68	C₃×C₄×C₈	—	2	1	no	—
69	C₃×C₄×C₈	—	3	2	no	—
70	C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic
71	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
72	Dic₃⋊C₈	C₂×C₉	1	2	yes	—
73	C₈×Dic₃	C₂×C₉	1	2	yes	—
74	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
78	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
79	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
80	C₃×Dic₆	C₃×C₈	1	4	yes	—
81	C₃²×C₈	—	2	1	no	—
82	C₃²×C₈	C₃×Q₈	1	2	yes	—
83	C₃×C₃⋊C₈	—	2	2	no	—
84	Q₈×C₉	—	3	4	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	3	2	no	—
89	C₂×C₄×C₉	—	3	2	no	—
90	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
91	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₈×C₉	—	2	2	no	—
94	C₈×C₉	—	2	1	no	—
95	C₈×C₉	—	2	2	no	—
96	C₈×C₉	—	6	2	no	—
97	C₈×C₉	Dic₆	1	2	yes	direct product factor
98	C₈×Q₈	—	3	1	no	Sylow
99	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₃×C₉	#131	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	3	1	no	—
103	C₂×Dic₆	C₄×C₉	1	1	yes	characteristic
104	C₂×C₃×C₈	—	2	1	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	3	2	no	—
112	C₃×C₄⋊C₄	—	3	1	no	—
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	—	3	2	no	—
115	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
116	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
117	C₄⋊Dic₃	C₄×C₉	1	1	yes	characteristic
118	Dic₃⋊C₄	C₄×C₉	1	2	yes	—
119	C₄×Dic₃	C₂²×C₉	1	2	yes	—
120	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
121	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	C₂×C₃×C₈	1	2	yes	—
124	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
125	C₃²×C₄	C₃×#41	1	1	yes	characteristic
126	C₃×Dic₃	—	2	2	no	—
127	C₃×Dic₃	C₂×C₃×C₈	1	4	yes	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	2	2	no	—
133	C₄×C₉	—	2	1	no	—
134	C₄×C₉	—	2	1	no	—
135	C₄×C₉	—	3	4	no	—
136	C₄×C₉	—	6	2	no	—
137	C₄×C₉	D₃×C₈	1	2	yes	—
138	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
139	C₄×C₉	#235	1	1	yes	characteristic
140	C₄×Q₈	—	3	1	no	—
141	C₄⋊C₈	—	3	2	no	—
142	C₄⋊C₈	—	3	1	no	—
143	C₄×C₈	—	3	2	no	—
144	C₄×C₈	D₃×C₉	1	1	yes	characteristic
145	C₃×C₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
146	C₃×Q₈	—	3	4	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	3	2	no	—
151	C₂×C₃×C₄	—	3	2	no	—
152	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
158	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
159	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
160	Dic₆	C₈×C₉	1	4	yes	direct product factor
161	C₃×C₈	—	2	2	no	—
162	C₃×C₈	—	2	1	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	—	6	2	no	—
166	C₃×C₈	Q₈×C₉	1	2	yes	complemented
167	C₃×C₈	C₃×Dic₆	1	2	yes	—
168	C₃⋊C₈	—	2	2	no	—
169	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
170	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
171	C₂×C₃²	C₃×#131	1	1	yes	characteristic
172	C₂×C₉	—	2	1	no	—
173	C₂×C₉	—	2	1	no	—
174	C₂×C₉	—	2	1	no	—
175	C₂×C₉	C₄×Dic₆	1	1	yes	characteristic
176	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
177	C₂×C₉	#901	1	1	yes	characteristic
178	C₂×Q₈	—	3	1	no	—
179	C₂×C₈	—	3	2	no	—
180	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
181	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
182	C₄⋊C₄	—	3	2	no	—
183	C₄⋊C₄	—	3	1	no	—
184	C₄²	—	3	2	no	—
185	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
188	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	3	4	no	—
196	C₃×C₄	—	6	2	no	—
197	C₃×C₄	C₂×C₈×C₉	1	2	yes	—
198	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
199	C₃×C₄	C₉×#41	1	1	yes	characteristic
200	C₃×C₄	C₃×D₃×C₈	1	2	yes	—
201	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
202	C₃×C₄	C₃×#235	1	1	yes	characteristic
203	Dic₃	—	2	2	no	—
204	Dic₃	C₂×C₈×C₉	1	4	yes	—
205	C₃²	C₃×C₈×Q₈	1	1	yes	characteristic
206	C₉	—	2	1	no	—
207	C₉	C₈×Dic₆	1	1	yes	characteristic, direct product factor
208	Q₈	—	3	4	no	—
209	C₂×C₄	—	3	2	no	—
210	C₂×C₄	—	3	2	no	—
211	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
212	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	6	2	no	—
216	C₈	C₉×Dic₆	1	2	yes	direct product factor
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	C₄×Q₈×C₉	1	1	yes	characteristic
221	C₂×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
222	C₂×C₃	C₉×#131	1	1	yes	characteristic
223	C₂×C₃	C₃×C₄×Dic₆	1	1	yes	characteristic
224	C₂×C₃	C₂×C₃×D₃×C₈	1	1	yes	characteristic
225	C₂×C₃	C₃×#901	1	1	yes	characteristic
226	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	3	4	no	—
229	C₄	—	6	2	no	—
230	C₄	D₃×C₈×C₉	1	2	yes	—
231	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
232	C₄	C₉×#235	1	1	yes	characteristic
233	C₃	—	2	1	no	—
234	C₃	C₈×Q₈×C₉	1	1	yes	characteristic, complemented
235	C₃	C₃×C₈×Dic₆	1	1	yes	characteristic
236	C₂	C₄×C₉×Dic₆	1	1	yes	characteristic
237	C₂	C₂×D₃×C₈×C₉	1	1	yes	characteristic
238	C₂	C₉×#901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×C₈×C₉	D₃
hypocenter	C₃	C₈×Q₈×C₉
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉

Complements

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

N	H ≅ G/N	type
32. C₉×Dic₆	C₈	direct (N × H)
34. C₃×C₈×C₉	Q₈	semidirect
37. C₈×Dic₆	C₉	direct (N × H)
97. C₈×C₉	Dic₆	direct (N × H)
145. C₃×C₉	C₈×Q₈	semidirect
160. Dic₆	C₈×C₉	direct (N × H)
166. C₃×C₈	Q₈×C₉	semidirect
207. C₉	C₈×Dic₆	direct (N × H)
216. C₈	C₉×Dic₆	direct (N × H)
234. C₃	C₈×Q₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×Dic₆	2
C₃×C₄×C₈×C₉	2
C₉×Dic₃⋊C₈	2
C₈×C₉×Dic₃	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×C₈×Dic₆	3
C₈×Q₈×C₉	3

Subgroups — C8×C9×Dic6 #13141402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₉×Dic₆ #13141402