Subgroups — D₃×C₉×#101 #13141690

Order 1728.

476 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×M₁₆	C₂	1	2	yes	maximal
3	C₃×C₉×#101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#836	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#824	C₂	1	2	yes	maximal, complemented
6	C₃×D₃×#101	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#101	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	D₃×Q₈×C₉	—	2	2	no	—
10	D₃×Q₈×C₉	—	2	2	no	—
11	C₂×C₄×D₃×C₉	C₄	1	2	yes	—
12	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×Dic₆	C₄	1	2	yes	—
14	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×C₉×M₁₆	C₂²	1	2	yes	—
16	C₉×#208	C₂²	1	2	yes	—
17	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
18	D₃×C₈×C₉	—	2	2	no	—
19	C₂×C₃×D₃×Q₈	C₂×C₃	1	1	yes	characteristic
20	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
21	C₂²×Q₈×C₉	—	3	1	no	—
22	C₃²×#101	C₂×C₃	1	1	yes	characteristic
23	C₃×#836	C₂×C₃	1	1	yes	characteristic
24	C₃×#824	C₂×C₃	1	2	yes	—
25	C₂×C₉×M₁₆	—	3	2	no	—
26	C₉×#101	—	2	1	no	—
27	C₉×#101	—	3	1	no	—
28	C₉×#101	—	3	2	no	—
29	C₉×#101	D₃	1	1	yes	characteristic, direct product factor
30	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
31	C₃×Q₈×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₄×D₃×C₉	—	2	2	no	—
37	C₄×D₃×C₉	—	2	2	no	—
38	C₄×D₃×C₉	D₄	1	2	yes	—
39	C₄×D₃×C₉	D₄	1	2	yes	—
40	C₉×Dic₆	—	2	2	no	—
41	C₉×Dic₆	—	2	4	no	—
42	C₃×C₈×C₉	—	2	2	no	—
43	C₉×C₃⋊C₈	—	2	2	no	—
44	C₂×C₃×#101	—	3	1	no	—
45	D₃×#101	C₉	1	1	yes	characteristic, direct product factor
46	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
47	C₃×D₃×Q₈	—	2	2	no	—
48	C₃×D₃×Q₈	—	2	2	no	—
49	C₂×C₃×C₄×D₃	C₃×C₄	1	2	yes	—
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×Dic₆	C₃×C₄	1	2	yes	—
52	C₂×C₃×Dic₆	C₂²×C₃	1	1	yes	characteristic
53	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
54	C₃×#208	C₂²×C₃	1	2	yes	—
55	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
56	C₃×D₃×C₈	—	2	2	no	—
57	C₂×Q₈×C₉	—	2	1	no	—
58	C₂×Q₈×C₉	—	3	2	no	—
59	C₂×Q₈×C₉	—	3	1	no	—
60	C₂×Q₈×C₉	—	6	2	no	—
61	C₂×Q₈×C₉	—	6	2	no	—
62	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₂²×C₄×C₉	—	3	2	no	—
65	C₉×M₁₆	—	2	2	no	—
66	C₉×M₁₆	—	3	2	no	—
67	C₉×M₁₆	—	6	2	no	—
68	C₉×M₁₆	C₂×D₃	1	2	yes	—
69	C₂×C₈×C₉	—	6	2	no	—
70	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×D₃×C₉	—	2	2	no	—
72	C₂×D₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₃×C₄×C₉	—	2	2	no	—
75	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₉×Dic₃	—	2	2	no	—
77	C₉×Dic₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₃×Q₈	—	3	1	no	—
79	C₂×D₃×Q₈	C₂×C₉	1	1	yes	characteristic
80	C₂×C₃×M₁₆	—	3	2	no	—
81	D₃×M₁₆	C₂×C₉	1	2	yes	—
82	C₃×#101	—	2	1	no	—
83	C₃×#101	—	3	1	no	—
84	C₃×#101	—	3	2	no	—
85	C₃×#101	C₂×C₉	1	1	yes	characteristic, complemented
86	C₃×#101	C₃×D₃	1	1	yes	characteristic
87	#836	C₂×C₉	1	1	yes	characteristic, complemented
88	#824	C₂×C₉	1	2	yes	complemented
89	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
90	C₃²×Q₈	—	2	2	no	—
91	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
95	C₃×C₄×D₃	—	2	2	no	—
96	C₃×C₄×D₃	—	2	2	no	—
97	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
98	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
99	C₃×Dic₆	—	2	2	no	—
100	C₃×Dic₆	—	2	4	no	—
101	C₂³×C₉	—	3	1	no	—
102	C₃²×C₈	—	2	2	no	—
103	C₃×C₃⋊C₈	—	2	2	no	—
104	Q₈×C₉	—	2	2	no	—
105	Q₈×C₉	—	4	2	no	—
106	Q₈×C₉	—	6	4	no	—
107	Q₈×C₉	—	6	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	3	1	no	—
111	C₂×C₄×C₉	—	3	2	no	—
112	C₂×C₄×C₉	—	3	2	no	—
113	C₂×C₄×C₉	—	3	2	no	—
114	C₂×C₄×C₉	—	6	2	no	—
115	C₂×C₄×C₉	—	6	2	no	—
116	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
117	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	—	4	2	no	—
120	C₈×C₉	—	6	2	no	—
121	C₂×#101	—	3	1	no	Sylow
122	C₂×C₃×C₉	—	2	1	no	—
123	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	D₃×C₉	—	2	1	no	—
125	D₃×C₉	#101	1	2	yes	direct product factor
126	C₂×C₃×Q₈	—	2	1	no	—
127	C₂×C₃×Q₈	—	3	2	no	—
128	C₂×C₃×Q₈	—	3	1	no	—
129	C₂×C₃×Q₈	—	6	2	no	—
130	C₂×C₃×Q₈	—	6	2	no	—
131	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
132	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
133	C₂²×C₃×C₄	—	3	1	no	—
134	C₂²×C₃×C₄	—	3	2	no	—
135	D₃×Q₈	—	2	2	no	—
136	D₃×Q₈	—	2	2	no	—
137	C₂×C₄×D₃	C₄×C₉	1	2	yes	—
138	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
139	C₂×Dic₆	C₄×C₉	1	2	yes	—
140	C₂×Dic₆	C₂²×C₉	1	1	yes	characteristic
141	C₃×M₁₆	—	2	2	no	—
142	C₃×M₁₆	—	3	2	no	—
143	C₃×M₁₆	—	6	2	no	—
144	C₃×M₁₆	C₂²×C₉	1	2	yes	—
145	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
146	C₂×C₃×C₈	—	6	2	no	—
147	#208	C₂²×C₉	1	2	yes	—
148	ΓL(1,𝔽₂₅)	—	2	2	no	—
149	D₃×C₈	—	2	2	no	—
150	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
151	C₂×C₃×D₃	—	2	2	no	—
152	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₃²×C₄	—	2	2	no	—
155	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
156	C₃×Dic₃	—	2	2	no	—
157	C₃×Dic₃	C₃×C₂²⋊C₄	1	2	yes	—
158	C₂²×C₉	—	2	1	no	—
159	C₂²×C₉	—	3	1	no	—
160	C₂²×C₉	—	3	1	no	—
161	C₂²×C₉	—	6	2	no	—
162	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
163	C₄×C₉	—	2	2	no	—
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	3	2	no	—
166	C₄×C₉	—	4	2	no	—
167	C₄×C₉	—	6	2	no	—
168	C₄×C₉	D₃×D₄	1	2	yes	—
169	C₂²×Q₈	—	3	1	no	—
170	C₂×M₁₆	—	3	2	no	—
171	#101	—	3	1	no	—
172	#101	—	3	2	no	—
173	#101	D₃×C₉	1	1	yes	characteristic, direct product factor
174	C₃×C₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	3	1	no	—
176	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
177	C₃×Q₈	—	2	2	no	—
178	C₃×Q₈	—	2	2	no	—
179	C₃×Q₈	—	4	2	no	—
180	C₃×Q₈	—	6	4	no	—
181	C₃×Q₈	—	6	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	3	1	no	—
185	C₂×C₃×C₄	—	3	2	no	—
186	C₂×C₃×C₄	—	3	2	no	—
187	C₂×C₃×C₄	—	3	2	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	6	2	no	—
190	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
191	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
192	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
193	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
194	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
195	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
196	C₄×D₃	—	2	2	no	—
197	C₄×D₃	—	2	2	no	—
198	C₄×D₃	D₄×C₉	1	2	yes	—
199	C₄×D₃	D₄×C₉	1	2	yes	—
200	Dic₆	—	2	2	no	—
201	Dic₆	—	2	4	no	—
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	2	no	—
205	C₃×C₈	—	6	2	no	—
206	C₃⋊C₈	—	2	2	no	—
207	C₂×C₃²	—	2	1	no	—
208	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₃×D₃	—	2	1	no	—
210	C₃×D₃	C₃×#101	1	2	yes	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	3	2	no	—
214	C₂×C₉	—	4	1	no	—
215	C₂×C₉	—	6	1	no	—
216	C₂×C₉	D₃×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×Q₈	—	3	2	no	—
218	C₂×Q₈	—	3	1	no	—
219	C₂×Q₈	—	6	2	no	—
220	C₂×Q₈	—	6	2	no	—
221	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
222	C₂²×C₄	—	3	1	no	—
223	C₂²×C₄	—	3	2	no	—
224	M₁₆	—	3	2	no	—
225	M₁₆	—	6	2	no	—
226	M₁₆	C₂×D₃×C₉	1	2	yes	—
227	C₂×C₈	—	6	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	3	1	no	—
230	C₂²×C₃	—	3	1	no	—
231	C₂²×C₃	—	6	2	no	—
232	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
233	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
234	C₂×D₃	—	2	2	no	—
235	C₂×D₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×D₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	3	2	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	—	6	2	no	—
243	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
244	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
245	Dic₃	—	2	2	no	—
246	Dic₃	C₉×C₂²⋊C₄	1	2	yes	—
247	C₃²	C₂×C₃×#101	1	1	yes	characteristic
248	C₉	—	2	1	no	—
249	C₉	D₃×#101	1	1	yes	characteristic, direct product factor
250	C₂³	—	3	1	no	—
251	Q₈	—	2	2	no	—
252	Q₈	—	6	4	no	—
253	Q₈	—	6	2	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
261	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	6	2	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	3	2	no	—
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	—	6	1	no	—
270	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
272	D₃	—	2	1	no	—
273	D₃	C₉×#101	1	2	yes	direct product factor
274	C₂²	—	3	1	no	—
275	C₂²	—	3	1	no	—
276	C₂²	—	6	2	no	—
277	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	3	2	no	—
280	C₄	—	6	2	no	—
281	C₄	D₃×D₄×C₉	1	2	yes	—
282	C₃	—	2	1	no	—
283	C₃	C₂×C₉×#101	1	1	yes	characteristic, complemented
284	C₃	C₃×D₃×#101	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	3	2	no	—
287	C₂	—	6	1	no	—
288	C₂	D₃×C₉×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#101	C₂
socle	C₂×C₃²	C₂×C₃×C₂²⋊C₄
prime radical	C₂²×C₃×D₃	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#101	D₃
hypocenter	C₃	C₂×C₉×#101
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#101

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#101	C₂	semidirect
4. C₉×#836	C₂	semidirect
5. C₉×#824	C₂	semidirect
29. C₉×#101	D₃	direct (N × H)
45. D₃×#101	C₉	direct (N × H)
85. C₃×#101	C₂×C₉	semidirect
87. #836	C₂×C₉	semidirect
88. #824	C₂×C₉	semidirect
125. D₃×C₉	#101	direct (N × H)
173. #101	D₃×C₉	direct (N × H)
174. C₃×C₉	C₂×#101	semidirect
249. C₉	D₃×#101	direct (N × H)
273. D₃	C₉×#101	direct (N × H)
283. C₃	C₂×C₉×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Q₈×C₉	2
D₃×C₉×M₁₆	2
C₃×C₉×#101	2
C₉×#836	2
C₉×#824	2
C₃×D₃×#101	3
C₂×C₉×#101	3

Subgroups — D3×C9×#101 #13141690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×#101 #13141690