Subgroups — D₃×C₉×#100 #13141666

Order 1728.

604 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	D₃×C₉×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#833	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#823	C₂	1	2	yes	maximal, complemented
6	C₃×D₃×#100	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#100	—	3	1	no	maximal
8	C₂³×D₃×C₉	C₄	1	2	yes	—
9	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₉×#67	C₄	1	2	yes	—
11	D₃×D₄×C₉	—	2	2	no	—
12	D₃×D₄×C₉	—	2	2	no	—
13	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₉×M₁₆	C₂²	1	2	yes	complemented
16	C₉×#208	C₂²	1	2	yes	complemented
17	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
18	D₃×C₈×C₉	—	2	2	no	—
19	C₂×C₃×D₃×D₄	C₂×C₃	1	1	yes	characteristic
20	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
21	C₂²×D₄×C₉	—	3	1	no	—
22	C₃²×#100	C₂×C₃	1	1	yes	characteristic
23	C₃×#833	C₂×C₃	1	1	yes	characteristic
24	C₃×#823	C₂×C₃	1	2	yes	—
25	C₂×C₉×M₁₆	—	3	2	no	—
26	C₉×#100	—	2	1	no	—
27	C₉×#100	—	3	1	no	—
28	C₉×#100	—	3	2	no	—
29	C₉×#100	D₃	1	1	yes	characteristic, direct product factor
30	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
31	C₂²×D₃×C₉	—	2	2	no	—
32	C₂²×D₃×C₉	—	2	2	no	—
33	C₂²×D₃×C₉	—	4	4	no	—
34	C₂²×D₃×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₃×C₉	C₂×C₄	1	2	yes	—
36	C₃×D₄×C₉	—	2	2	no	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₉×#67	—	2	4	no	—
39	C₂×C₉×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₉×D₁₂	—	2	2	no	—
41	C₄×D₃×C₉	D₄	1	2	yes	—
42	C₄×D₃×C₉	D₄	1	2	yes	—
43	C₃×C₈×C₉	—	2	2	no	—
44	C₉×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×#100	—	3	1	no	—
46	D₃×#100	C₉	1	1	yes	characteristic, direct product factor
47	C₂³×C₃×D₃	C₃×C₄	1	2	yes	—
48	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×#67	C₃×C₄	1	2	yes	—
50	C₃×D₃×D₄	—	2	2	no	—
51	C₃×D₃×D₄	—	2	2	no	—
52	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
54	C₂⁴×C₉	—	3	2	no	—
55	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
56	C₃×#208	C₂²×C₃	1	2	yes	—
57	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
58	C₃×D₃×C₈	—	2	2	no	—
59	C₂×D₄×C₉	—	2	1	no	—
60	C₂×D₄×C₉	—	3	2	no	—
61	C₂×D₄×C₉	—	3	1	no	—
62	C₂×D₄×C₉	—	6	2	no	—
63	C₂×D₄×C₉	—	6	2	no	—
64	C₂×D₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₂²×C₄×C₉	—	3	1	no	—
66	C₉×M₁₆	—	2	2	no	—
67	C₉×M₁₆	—	3	2	no	—
68	C₉×M₁₆	—	6	2	no	—
69	C₉×M₁₆	C₂×D₃	1	2	yes	complemented
70	C₂×C₈×C₉	—	6	2	no	—
71	C₂²×C₃×C₉	—	2	2	no	—
72	C₂²×C₃×C₉	—	4	2	no	—
73	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×D₃×C₉	—	2	2	no	—
75	C₂×D₃×C₉	—	2	2	no	—
76	C₂×D₃×C₉	—	4	4	no	—
77	C₂×D₃×C₉	—	4	4	no	—
78	C₂×D₃×C₉	—	4	2	no	—
79	C₂×D₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
82	C₉×Dic₃	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₃×D₄	—	3	1	no	—
84	C₂×D₃×D₄	C₂×C₉	1	1	yes	characteristic
85	C₂×C₃×M₁₆	—	3	2	no	—
86	D₃×M₁₆	C₂×C₉	1	2	yes	complemented
87	C₃×#100	—	2	1	no	—
88	C₃×#100	—	3	1	no	—
89	C₃×#100	—	3	2	no	—
90	C₃×#100	C₂×C₉	1	1	yes	characteristic, complemented
91	C₃×#100	C₃×D₃	1	1	yes	characteristic
92	#833	C₂×C₉	1	1	yes	characteristic, complemented
93	#823	C₂×C₉	1	2	yes	complemented
94	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
95	C₂²×C₃×D₃	—	2	2	no	—
96	C₂²×C₃×D₃	—	2	2	no	—
97	C₂²×C₃×D₃	—	4	4	no	—
98	C₂²×C₃×D₃	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₃×D₃	C₂×C₃×C₄	1	2	yes	—
100	C₃²×D₄	—	2	2	no	—
101	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×#67	—	2	4	no	—
103	C₂×C₃×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
104	C₃×D₁₂	—	2	2	no	—
105	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
106	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
107	C₂³×C₉	—	2	2	no	—
108	C₂³×C₉	—	3	1	no	—
109	C₂³×C₉	—	3	2	no	—
110	C₂³×C₉	—	6	2	no	—
111	C₂³×C₉	—	6	2	no	—
112	C₂³×C₉	—	12	4	no	—
113	C₂³×C₉	C₄×D₃	1	2	yes	—
114	C₃²×C₈	—	2	2	no	—
115	C₃×C₃⋊C₈	—	2	2	no	—
116	D₄×C₉	—	2	2	no	—
117	D₄×C₉	—	4	2	no	—
118	D₄×C₉	—	6	4	no	—
119	D₄×C₉	—	6	2	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	3	1	no	—
122	C₂×C₄×C₉	—	3	2	no	—
123	C₂×C₄×C₉	—	3	2	no	—
124	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
125	C₈×C₉	—	2	2	no	—
126	C₈×C₉	—	4	2	no	—
127	C₈×C₉	—	6	2	no	—
128	C₂×#100	—	3	1	no	Sylow
129	C₂×C₃×C₉	—	2	1	no	—
130	C₂×C₃×C₉	—	4	2	no	—
131	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	D₃×C₉	—	2	1	no	—
133	D₃×C₉	—	4	2	no	—
134	D₃×C₉	#100	1	2	yes	direct product factor
135	C₂⁴×C₃	—	3	2	no	—
136	C₂³×D₃	C₄×C₉	1	2	yes	—
137	C₂×C₃×D₄	—	2	1	no	—
138	C₂×C₃×D₄	—	3	2	no	—
139	C₂×C₃×D₄	—	3	1	no	—
140	C₂×C₃×D₄	—	6	2	no	—
141	C₂×C₃×D₄	—	6	2	no	—
142	C₂×C₃×D₄	C₂²×C₉	1	1	yes	characteristic
143	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂²×C₃×C₄	—	3	1	no	—
145	C₂×#67	C₄×C₉	1	2	yes	—
146	D₃×D₄	—	2	2	no	—
147	D₃×D₄	—	2	2	no	—
148	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
149	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
150	C₃×M₁₆	—	2	2	no	—
151	C₃×M₁₆	—	3	2	no	—
152	C₃×M₁₆	—	6	2	no	—
153	C₃×M₁₆	C₂²×C₉	1	2	yes	complemented
154	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
155	C₂×C₃×C₈	—	6	2	no	—
156	#208	C₂²×C₉	1	2	yes	complemented
157	ΓL(1,𝔽₂₅)	—	2	2	no	—
158	D₃×C₈	—	2	2	no	—
159	C₂²×C₃²	—	2	2	no	—
160	C₂²×C₃²	—	4	2	no	—
161	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	2	no	—
163	C₂×C₃×D₃	—	2	2	no	—
164	C₂×C₃×D₃	—	4	4	no	—
165	C₂×C₃×D₃	—	4	4	no	—
166	C₂×C₃×D₃	—	4	2	no	—
167	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
170	C₃×Dic₃	C₃×C₂²⋊C₄	1	2	yes	—
171	C₂²×C₉	—	2	2	no	—
172	C₂²×C₉	—	2	1	no	—
173	C₂²×C₉	—	3	1	no	—
174	C₂²×C₉	—	3	1	no	—
175	C₂²×C₉	—	4	2	no	—
176	C₂²×C₉	—	4	2	no	—
177	C₂²×C₉	—	6	2	no	—
178	C₂²×C₉	—	6	2	no	—
179	C₂²×C₉	—	8	2	no	—
180	C₂²×C₉	—	12	2	no	—
181	C₂²×C₉	—	12	4	no	—
182	C₂²×C₉	—	12	4	no	—
183	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
184	C₄×C₉	—	2	2	no	—
185	C₄×C₉	—	3	2	no	—
186	C₄×C₉	D₃×D₄	1	2	yes	—
187	C₂²×D₄	—	3	1	no	—
188	C₂×M₁₆	—	3	2	no	—
189	#100	—	3	1	no	—
190	#100	—	3	2	no	—
191	#100	D₃×C₉	1	1	yes	characteristic, direct product factor
192	C₃×C₉	C₂×#100	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₃	—	2	2	no	—
194	C₂³×C₃	—	3	1	no	—
195	C₂³×C₃	—	3	2	no	—
196	C₂³×C₃	—	6	2	no	—
197	C₂³×C₃	—	6	2	no	—
198	C₂³×C₃	—	12	4	no	—
199	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
200	C₂³×C₃	C₃×C₄×D₃	1	2	yes	—
201	C₂²×D₃	—	2	2	no	—
202	C₂²×D₃	—	2	2	no	—
203	C₂²×D₃	—	4	4	no	—
204	C₂²×D₃	C₂×C₄×C₉	1	1	yes	characteristic
205	C₂²×D₃	C₂×C₄×C₉	1	2	yes	—
206	C₃×D₄	—	2	2	no	—
207	C₃×D₄	—	2	2	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	6	4	no	—
210	C₃×D₄	—	6	2	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	3	1	no	—
213	C₂×C₃×C₄	—	3	2	no	—
214	C₂×C₃×C₄	—	3	2	no	—
215	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
217	#67	—	2	4	no	—
218	C₂×Dic₃	C₂×C₄×C₉	1	1	yes	characteristic
219	D₁₂	—	2	2	no	—
220	C₄×D₃	D₄×C₉	1	2	yes	—
221	C₄×D₃	D₄×C₉	1	2	yes	—
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	—	2	2	no	—
224	C₃×C₈	—	4	2	no	—
225	C₃×C₈	—	6	2	no	—
226	C₃⋊C₈	—	2	2	no	—
227	C₂×C₃²	—	2	1	no	—
228	C₂×C₃²	—	4	2	no	—
229	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₃×D₃	—	2	1	no	—
231	C₃×D₃	—	4	2	no	—
232	C₃×D₃	C₃×#100	1	2	yes	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	2	1	no	—
235	C₂×C₉	—	3	2	no	—
236	C₂×C₉	—	4	1	no	—
237	C₂×C₉	—	4	2	no	—
238	C₂×C₉	—	6	1	no	—
239	C₂×C₉	—	8	2	no	—
240	C₂×C₉	—	12	2	no	—
241	C₂×C₉	D₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₂⁴	—	3	2	no	—
243	C₂×D₄	—	3	2	no	—
244	C₂×D₄	—	3	1	no	—
245	C₂×D₄	—	6	2	no	—
246	C₂×D₄	—	6	2	no	—
247	C₂×D₄	C₂×D₃×C₉	1	1	yes	characteristic
248	C₂²×C₄	—	3	1	no	—
249	M₁₆	—	3	2	no	—
250	M₁₆	—	6	2	no	—
251	M₁₆	C₂×D₃×C₉	1	2	yes	complemented
252	C₂×C₈	—	6	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	3	1	no	—
257	C₂²×C₃	—	3	1	no	—
258	C₂²×C₃	—	4	2	no	—
259	C₂²×C₃	—	4	2	no	—
260	C₂²×C₃	—	4	2	no	—
261	C₂²×C₃	—	6	2	no	—
262	C₂²×C₃	—	6	2	no	—
263	C₂²×C₃	—	8	2	no	—
264	C₂²×C₃	—	12	2	no	—
265	C₂²×C₃	—	12	4	no	—
266	C₂²×C₃	—	12	4	no	—
267	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
268	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
269	C₂×D₃	—	2	2	no	—
270	C₂×D₃	—	2	2	no	—
271	C₂×D₃	—	4	4	no	—
272	C₂×D₃	—	4	4	no	—
273	C₂×D₃	—	4	2	no	—
274	C₂×D₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×D₃	C₉×C₂²⋊C₄	1	1	yes	characteristic
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	3	2	no	—
278	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
279	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
280	Dic₃	C₉×C₂²⋊C₄	1	2	yes	—
281	C₃²	C₂×C₃×#100	1	1	yes	characteristic
282	C₉	—	2	1	no	—
283	C₉	D₃×#100	1	1	yes	characteristic, direct product factor
284	C₂³	—	3	1	no	—
285	C₂³	—	3	2	no	—
286	C₂³	—	6	2	no	—
287	C₂³	—	6	2	no	—
288	C₂³	—	12	4	no	—
289	C₂³	C₄×D₃×C₉	1	2	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	6	4	no	—
292	D₄	—	6	2	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	3	2	no	—
296	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	6	2	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	3	2	no	—
303	C₂×C₃	—	4	1	no	—
304	C₂×C₃	—	4	2	no	—
305	C₂×C₃	—	4	2	no	—
306	C₂×C₃	—	6	1	no	—
307	C₂×C₃	—	8	2	no	—
308	C₂×C₃	—	12	2	no	—
309	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
310	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	1	yes	characteristic
311	D₃	—	2	1	no	—
312	D₃	—	4	2	no	—
313	D₃	C₉×#100	1	2	yes	direct product factor
314	C₂²	—	2	2	no	—
315	C₂²	—	3	1	no	—
316	C₂²	—	3	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	6	2	no	—
319	C₂²	—	6	2	no	—
320	C₂²	—	12	2	no	—
321	C₂²	—	12	4	no	—
322	C₂²	—	12	4	no	—
323	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
324	C₄	—	3	2	no	—
325	C₄	D₃×D₄×C₉	1	2	yes	—
326	C₃	—	2	1	no	—
327	C₃	C₂×C₉×#100	1	1	yes	characteristic, complemented
328	C₃	C₃×D₃×#100	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	3	2	no	—
331	C₂	—	4	2	no	—
332	C₂	—	6	1	no	—
333	C₂	—	12	2	no	—
334	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×#100	C₂
socle	C₂×C₃²	C₂×C₃×C₂²⋊C₄
prime radical	C₂×C₃×D₃×D₄	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#100	D₃
hypocenter	C₃	C₂×C₉×#100
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#100

Complements

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

N	H ≅ G/N	type
2. D₃×C₉×M₁₆	C₂	semidirect
3. C₃×C₉×#100	C₂	semidirect
4. C₉×#833	C₂	semidirect
5. C₉×#823	C₂	semidirect
15. C₃×C₉×M₁₆	C₂²	semidirect
16. C₉×#208	C₂²	semidirect
29. C₉×#100	D₃	direct (N × H)
46. D₃×#100	C₉	direct (N × H)
69. C₉×M₁₆	C₂×D₃	semidirect
86. D₃×M₁₆	C₂×C₉	semidirect
90. C₃×#100	C₂×C₉	semidirect
92. #833	C₂×C₉	semidirect
93. #823	C₂×C₉	semidirect
134. D₃×C₉	#100	direct (N × H)
153. C₃×M₁₆	C₂²×C₉	semidirect
156. #208	C₂²×C₉	semidirect
191. #100	D₃×C₉	direct (N × H)
192. C₃×C₉	C₂×#100	semidirect
251. M₁₆	C₂×D₃×C₉	semidirect
283. C₉	D₃×#100	direct (N × H)
313. D₃	C₉×#100	direct (N × H)
327. C₃	C₂×C₉×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₄×C₉	2
D₃×C₉×M₁₆	2
C₃×C₉×#100	2
C₉×#833	2
C₉×#823	2
C₃×D₃×#100	3
C₂×C₉×#100	3

Subgroups — D3×C9×#100 #13141666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×#100 #13141666