Subgroups — D₃×C₉×#108 #13142259

Order 1728.

348 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₉×M₁₆	C₂	1	2	yes	maximal
2	C₂×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#822	C₂	1	2	yes	maximal, complemented
5	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₃×#108	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#108	—	3	1	no	maximal
8	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#208	C₂²	1	2	yes	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
14	D₃×C₈×C₉	—	2	2	no	—
15	D₃×C₈×C₉	C₄	1	2	yes	—
16	D₃×C₈×C₉	C₄	1	2	yes	—
17	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
18	C₂×C₃×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃²×#108	C₂×C₃	1	1	yes	characteristic
20	C₃×#822	C₂×C₃	1	2	yes	—
21	C₃×#819	C₂×C₃	1	1	yes	characteristic
22	C₂×C₉×M₁₆	—	3	2	no	—
23	C₂²×C₈×C₉	—	3	1	no	—
24	C₉×#108	—	2	1	no	—
25	C₉×#108	—	3	2	no	—
26	C₉×#108	—	3	1	no	—
27	C₉×#108	D₃	1	1	yes	characteristic, direct product factor
28	C₂²×D₃×C₉	Q₈	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₉×Dic₃	Q₈	1	1	yes	characteristic
31	C₄×D₃×C₉	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃×C₉	C₂×C₄	1	1	yes	characteristic
33	C₄×D₃×C₉	D₄	1	1	yes	characteristic
34	C₄×D₃×C₉	D₄	1	1	yes	characteristic
35	C₃×C₈×C₉	—	2	2	no	—
36	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
37	C₉×C₃⋊C₈	—	2	2	no	—
38	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×#108	—	3	1	no	—
40	D₃×#108	C₉	1	1	yes	characteristic, direct product factor
41	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
42	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
43	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₃×#208	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
47	C₃×D₃×C₈	—	2	2	no	—
48	C₃×D₃×C₈	C₃×C₄	1	2	yes	—
49	C₃×D₃×C₈	C₃×C₄	1	2	yes	—
50	C₂²×C₄×C₉	—	3	1	no	—
51	C₉×M₁₆	—	2	2	no	—
52	C₉×M₁₆	—	3	2	no	—
53	C₉×M₁₆	—	6	2	no	—
54	C₉×M₁₆	C₂×D₃	1	2	yes	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₂×C₈×C₉	—	3	1	no	—
57	C₂×C₈×C₉	—	3	2	no	—
58	C₂×C₈×C₉	—	3	2	no	—
59	C₂×C₈×C₉	—	6	2	no	—
60	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×D₃×C₉	—	2	2	no	—
63	C₂×D₃×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×D₃×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₉×Dic₃	C₄⋊C₄	1	1	yes	characteristic
68	C₉×Dic₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃×M₁₆	—	3	2	no	—
70	C₂²×C₃×C₈	—	3	1	no	—
71	D₃×M₁₆	C₂×C₉	1	2	yes	—
72	C₂×D₃×C₈	C₂×C₉	1	1	yes	characteristic
73	C₃×#108	—	2	1	no	—
74	C₃×#108	—	3	2	no	—
75	C₃×#108	—	3	1	no	—
76	C₃×#108	C₂×C₉	1	1	yes	characteristic, complemented
77	C₃×#108	C₃×D₃	1	1	yes	characteristic
78	#822	C₂×C₉	1	2	yes	complemented
79	#819	C₂×C₉	1	1	yes	characteristic, complemented
80	C₂²×C₃×D₃	C₃×Q₈	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	C₃×Q₈	1	1	yes	characteristic
83	C₃×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
84	C₃×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
85	C₃×C₄×D₃	C₃×D₄	1	1	yes	characteristic
86	C₃×C₄×D₃	C₃×D₄	1	1	yes	characteristic
87	C₂³×C₉	—	3	1	no	—
88	C₃²×C₈	—	2	2	no	—
89	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
90	C₃×C₃⋊C₈	—	2	2	no	—
91	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	3	1	no	—
94	C₂×C₄×C₉	—	3	1	no	—
95	C₂×C₄×C₉	—	3	1	no	—
96	C₂×C₄×C₉	—	3	1	no	—
97	C₂×C₄×C₉	—	3	1	no	—
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	—	3	2	no	—
102	C₈×C₉	—	4	2	no	—
103	C₈×C₉	—	6	2	no	—
104	C₈×C₉	C₄×D₃	1	2	yes	—
105	C₂×#108	—	3	1	no	Sylow
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	D₃×C₉	—	2	1	no	—
109	D₃×C₉	#108	1	2	yes	direct product factor
110	C₂²×C₃×C₄	—	3	1	no	—
111	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
112	C₃×M₁₆	—	2	2	no	—
113	C₃×M₁₆	—	3	2	no	—
114	C₃×M₁₆	—	6	2	no	—
115	C₃×M₁₆	C₂²×C₉	1	2	yes	—
116	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	3	1	no	—
119	C₂×C₃×C₈	—	3	2	no	—
120	C₂×C₃×C₈	—	3	2	no	—
121	C₂×C₃×C₈	—	6	2	no	—
122	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	#208	C₂²×C₉	1	2	yes	—
125	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
126	ΓL(1,𝔽₂₅)	—	2	2	no	—
127	D₃×C₈	—	2	2	no	—
128	D₃×C₈	C₄×C₉	1	2	yes	—
129	D₃×C₈	C₄×C₉	1	2	yes	—
130	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
131	C₂×C₃×D₃	—	2	2	no	—
132	C₂×C₃×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₃×D₃	C₃×C₄⋊C₄	1	1	yes	characteristic
134	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
136	C₃×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
137	C₃×Dic₃	C₃×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₉	—	2	1	no	—
139	C₂²×C₉	—	3	1	no	—
140	C₂²×C₉	—	3	1	no	—
141	C₂²×C₉	—	6	2	no	—
142	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
143	C₄×C₉	—	2	1	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	3	1	no	—
146	C₄×C₉	—	3	1	no	—
147	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
148	C₄×C₉	D₃×D₄	1	1	yes	characteristic
149	C₂×M₁₆	—	3	2	no	—
150	C₂²×C₈	—	3	1	no	—
151	#108	—	3	2	no	—
152	#108	—	3	1	no	—
153	#108	D₃×C₉	1	1	yes	characteristic, direct product factor
154	C₃×C₉	C₂×#108	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₃	—	3	1	no	—
156	C₂²×D₃	Q₈×C₉	1	1	yes	characteristic
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	1	no	—
159	C₂×C₃×C₄	—	3	1	no	—
160	C₂×C₃×C₄	—	3	1	no	—
161	C₂×C₃×C₄	—	3	1	no	—
162	C₂×C₃×C₄	—	3	1	no	—
163	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
165	C₂×Dic₃	Q₈×C₉	1	1	yes	characteristic
166	C₄×D₃	C₂×C₄×C₉	1	1	yes	characteristic
167	C₄×D₃	C₂×C₄×C₉	1	1	yes	characteristic
168	C₄×D₃	D₄×C₉	1	1	yes	characteristic
169	C₄×D₃	D₄×C₉	1	1	yes	characteristic
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	3	2	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	—	6	2	no	—
176	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
177	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
178	C₃⋊C₈	—	2	2	no	—
179	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	—
180	C₂×C₃²	—	2	1	no	—
181	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
182	C₃×D₃	—	2	1	no	—
183	C₃×D₃	C₃×#108	1	2	yes	—
184	C₂×C₉	—	2	1	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	3	2	no	—
187	C₂×C₉	—	4	1	no	—
188	C₂×C₉	—	6	1	no	—
189	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	—	3	1	no	—
191	M₁₆	—	3	2	no	—
192	M₁₆	—	6	2	no	—
193	M₁₆	C₂×D₃×C₉	1	2	yes	—
194	C₂×C₈	—	3	1	no	—
195	C₂×C₈	—	3	2	no	—
196	C₂×C₈	—	3	2	no	—
197	C₂×C₈	—	6	2	no	—
198	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	3	1	no	—
201	C₂²×C₃	—	3	1	no	—
202	C₂²×C₃	—	6	2	no	—
203	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
204	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₉×C₄⋊C₄	1	1	yes	characteristic
207	C₂×D₃	C₉×C₄⋊C₄	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	3	1	no	—
211	C₃×C₄	—	3	1	no	—
212	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
213	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
214	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
215	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
216	Dic₃	C₉×C₄⋊C₄	1	1	yes	characteristic
217	Dic₃	C₉×C₄⋊C₄	1	1	yes	characteristic
218	C₃²	C₂×C₃×#108	1	1	yes	characteristic
219	C₉	—	2	1	no	—
220	C₉	D₃×#108	1	1	yes	characteristic, direct product factor
221	C₂³	—	3	1	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	3	1	no	—
225	C₂×C₄	—	3	1	no	—
226	C₂×C₄	—	3	1	no	—
227	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	3	2	no	—
230	C₈	—	6	2	no	—
231	C₈	C₄×D₃×C₉	1	2	yes	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	3	2	no	—
236	C₂×C₃	—	4	1	no	—
237	C₂×C₃	—	6	1	no	—
238	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
240	D₃	—	2	1	no	—
241	D₃	C₉×#108	1	2	yes	direct product factor
242	C₂²	—	3	1	no	—
243	C₂²	—	3	1	no	—
244	C₂²	—	6	2	no	—
245	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
246	C₄	—	3	1	no	—
247	C₄	—	3	1	no	—
248	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
249	C₄	D₃×D₄×C₉	1	1	yes	characteristic
250	C₃	—	2	1	no	—
251	C₃	C₂×C₉×#108	1	1	yes	characteristic, complemented
252	C₃	C₃×D₃×#108	1	1	yes	characteristic
253	C₂	—	2	1	no	—
254	C₂	—	3	2	no	—
255	C₂	—	6	1	no	—
256	C₂	D₃×C₉×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#108	C₂
socle	C₂×C₃²	C₂×C₃×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₃×Q₈
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#108	D₃
hypocenter	C₃	C₂×C₉×#108
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#108

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#108	C₂	semidirect
4. C₉×#822	C₂	semidirect
5. C₉×#819	C₂	semidirect
27. C₉×#108	D₃	direct (N × H)
40. D₃×#108	C₉	direct (N × H)
76. C₃×#108	C₂×C₉	semidirect
78. #822	C₂×C₉	semidirect
79. #819	C₂×C₉	semidirect
109. D₃×C₉	#108	direct (N × H)
153. #108	D₃×C₉	direct (N × H)
154. C₃×C₉	C₂×#108	semidirect
220. C₉	D₃×#108	direct (N × H)
241. D₃	C₉×#108	direct (N × H)
251. C₃	C₂×C₉×#108	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₉×M₁₆	2
C₂×D₃×C₈×C₉	2
C₃×C₉×#108	2
C₉×#822	2
C₉×#819	2
C₃×D₃×#108	3
C₂×C₉×#108	3

Subgroups — D3×C9×#108 #13142259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×#108 #13142259