Subgroups — #13138170

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88523	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₃⋊D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#88414	C₂	1	1	yes	maximal, characteristic
4	C₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63375	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720274	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#10590664	—	3	1	no	maximal
9	C₄×#3663	—	3	1	no	maximal
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×(C₃⋊D₉)⋊C₄	C₄	1	2	yes	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#88523	C₄	1	4	yes	complemented
15	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
16	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
19	Dic₃×Dic₉	C₄	1	2	yes	—
20	#88414	C₄	1	4	yes	complemented
21	C₂×#3743	—	3	1	no	—
22	C₄×#624	—	3	1	no	—
23	C₂×#3712	—	3	1	no	—
24	C₃×#815	—	3	1	no	—
25	C₃×C₈×Dic₃	D₃	1	1	yes	characteristic
26	#63554	—	3	1	no	—
27	Dic₃×C₃⋊C₈	—	3	1	no	—
28	C₂×#3663	—	3	1	no	—
29	C₂×#3663	—	3	1	no	—
30	C₂×#3663	—	6	1	no	—
31	C₄²×D₉	—	3	1	no	—
32	C₄×C₈×C₉	—	3	1	no	—
33	#63375	—	3	1	no	—
34	#63375	D₃	1	1	yes	characteristic, complemented
35	C₄×C₉⋊C₈	—	3	1	no	—
36	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₃⋊D₉	C₂×C₄	1	2	yes	—
40	C₄×C₃⋊D₉	C₂×C₄	1	2	yes	—
41	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
42	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
43	(C₃⋊D₉)⋊C₄	—	2	4	no	—
44	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
45	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	complemented
46	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	complemented
47	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
48	#4707	—	9	1	no	—
49	C₄×ΓL(1,𝔽₂₅)	—	9	1	no	—
50	C₂×C₄×C₃⋊D₃	—	3	1	no	—
51	C₂×#624	—	3	2	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
54	#3743	—	3	4	no	—
55	C₃×C₄×Dic₃	—	3	1	no	—
56	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₃⋊C₈	—	3	1	no	—
58	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
59	Dic₃²	—	3	2	no	—
60	#3712	—	3	4	no	—
61	C₂×C₄×D₉	—	3	1	no	—
62	C₂×C₄×D₉	—	3	2	no	—
63	C₂×C₄×D₉	—	6	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	3	1	no	—
66	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
67	C₄²×C₉	—	3	1	no	—
68	C₄×Dic₉	—	3	2	no	—
69	C₄×Dic₉	C₂×D₃	1	1	yes	characteristic
70	C₂×C₉⋊C₈	—	3	1	no	—
71	C₂×C₉⋊C₈	—	6	1	no	—
72	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
73	#3663	—	3	4	no	—
74	#3663	—	3	4	no	—
75	#3663	—	6	4	no	—
76	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃⋊D₉	—	2	2	no	—
78	C₂×C₃⋊D₉	C₄²	1	2	yes	—
79	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
81	C₃⋊Dic₉	C₄²	1	2	yes	—
82	C₉×Dic₃	M₁₆	1	4	yes	complemented
83	C₃×Dic₉	—	2	2	no	—
84	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
85	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
87	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
88	C₂×ΓL(1,𝔽₂₅)	—	18	1	no	—
89	C₄²×D₃	—	9	1	no	—
90	C₄²×D₃	—	9	1	no	—
91	C₃×#97	—	9	1	no	—
92	C₃×C₄×C₈	—	3	1	no	—
93	#815	—	3	1	no	—
94	#815	—	9	1	no	—
95	C₈×Dic₃	—	9	1	no	—
96	C₈×Dic₃	D₉	1	1	yes	characteristic, complemented
97	#803	—	9	1	no	—
98	C₄×C₃⋊C₈	—	9	1	no	—
99	C₂²×C₃⋊D₃	—	3	1	no	—
100	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂×C₃⋊Dic₃	—	3	1	no	—
102	C₄×C₃⋊D₃	—	3	2	no	—
103	C₄×C₃⋊D₃	—	3	2	no	—
104	C₂×C₃×Dic₃	—	3	2	no	—
105	C₂×C₃×Dic₃	C₄×D₃	1	2	yes	—
106	#624	—	6	4	no	—
107	C₂²×D₉	—	3	1	no	—
108	C₂²×D₉	—	6	1	no	—
109	C₃²×C₈	C₄×D₃	1	2	yes	—
110	C₃⋊(C₃⋊C₈)	—	3	2	no	—
111	C₃×C₃⋊C₈	—	3	2	no	—
112	C₃×C₃⋊C₈	C₄×D₃	1	2	yes	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	3	2	no	—
115	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
116	C₂×Dic₉	—	3	1	no	—
117	C₂×Dic₉	—	6	1	no	—
118	C₂×Dic₉	C₄×D₃	1	2	yes	—
119	C₄×D₉	—	3	2	no	—
120	C₄×D₉	—	3	2	no	—
121	C₄×D₉	—	6	2	no	—
122	C₄×D₉	—	6	2	no	—
123	C₄×D₉	—	6	4	no	—
124	C₈×C₉	—	2	2	no	—
125	C₈×C₉	—	3	2	no	—
126	C₈×C₉	C₄×D₃	1	2	yes	complemented
127	C₉⋊C₈	—	3	2	no	—
128	C₉⋊C₈	—	6	2	no	—
129	C₉⋊C₈	C₄×D₃	1	2	yes	complemented
130	C₄×M₁₆	—	27	1	no	Sylow
131	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
133	C₃⋊D₉	—	2	2	no	—
134	C₂×C₄×D₃	—	9	1	no	—
135	C₂×C₄×D₃	—	9	1	no	—
136	C₂×C₄×D₃	—	9	2	no	—
137	C₂×C₄×D₃	—	9	2	no	—
138	C₂×C₄×D₃	—	18	1	no	—
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	3	1	no	—
141	C₂×C₃×C₈	—	9	1	no	—
142	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
143	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
144	C₃×C₄²	—	3	1	no	—
145	C₃×C₄²	—	9	1	no	—
146	C₄×Dic₃	—	3	1	no	—
147	C₄×Dic₃	—	9	2	no	—
148	C₄×Dic₃	—	9	2	no	—
149	C₄×Dic₃	C₂×D₉	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	3	1	no	—
151	C₂×C₃⋊C₈	—	9	1	no	—
152	C₂×C₃⋊C₈	—	9	1	no	—
153	C₂×C₃⋊C₈	—	18	1	no	—
154	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
155	ΓL(1,𝔽₂₅)	—	9	4	no	—
156	ΓL(1,𝔽₂₅)	—	9	4	no	—
157	ΓL(1,𝔽₂₅)	—	9	4	no	—
158	ΓL(1,𝔽₂₅)	—	9	4	no	—
159	ΓL(1,𝔽₂₅)	—	18	4	no	—
160	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₃⋊D₃	—	3	2	no	—
162	C₂×C₃⋊D₃	—	6	2	no	—
163	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
165	C₃⋊Dic₃	—	3	2	no	—
166	C₃×Dic₃	—	6	2	no	—
167	C₃×Dic₃	ΓL(1,𝔽₂₅)	1	4	yes	—
168	C₂²×C₉	—	2	1	no	—
169	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×D₉	—	3	2	no	—
171	C₂×D₉	—	6	2	no	—
172	C₂×D₉	—	6	2	no	—
173	C₂×D₉	—	12	2	no	—
174	C₄×C₉	—	2	1	no	—
175	C₄×C₉	—	2	1	no	—
176	C₄×C₉	—	3	4	no	—
177	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
178	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
179	Dic₉	—	2	2	no	—
180	Dic₉	—	3	2	no	—
181	Dic₉	—	6	2	no	—
182	C₂×M₁₆	—	27	1	no	—
183	C₂×M₁₆	—	27	1	no	—
184	C₂×C₄²	—	27	1	no	—
185	#97	—	9	1	no	—
186	#97	—	27	1	no	—
187	C₄×C₈	—	3	1	no	—
188	C₄×C₈	—	27	1	no	—
189	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₃	—	9	1	no	—
191	C₂²×D₃	—	9	1	no	—
192	C₂²×D₃	—	18	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	3	2	no	—
195	C₂×C₃×C₄	—	9	2	no	—
196	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
198	C₂×Dic₃	—	3	2	no	—
199	C₂×Dic₃	—	9	1	no	—
200	C₂×Dic₃	—	9	1	no	—
201	C₂×Dic₃	—	18	1	no	—
202	C₂×Dic₃	C₄×D₉	1	2	yes	—
203	C₄×D₃	—	9	2	no	—
204	C₄×D₃	—	9	2	no	—
205	C₄×D₃	—	9	2	no	—
206	C₄×D₃	—	9	2	no	—
207	C₄×D₃	—	18	2	no	—
208	C₄×D₃	—	18	2	no	—
209	C₄×D₃	—	18	4	no	—
210	C₄×D₃	—	18	4	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	—	3	2	no	—
213	C₃×C₈	—	9	2	no	—
214	C₃×C₈	C₄×D₉	1	2	yes	complemented
215	C₃×C₈	#624	1	2	yes	—
216	C₃⋊C₈	—	3	2	no	—
217	C₃⋊C₈	—	9	2	no	—
218	C₃⋊C₈	—	9	2	no	—
219	C₃⋊C₈	—	18	2	no	—
220	C₃⋊C₈	C₄×D₉	1	2	yes	complemented
221	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
222	C₂×C₃²	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
223	C₃⋊D₃	—	6	2	no	—
224	C₂×C₉	—	2	2	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	C₄²×D₃	1	1	yes	characteristic
227	C₂×C₉	D₃×M₁₆	1	2	yes	—
228	D₉	—	6	2	no	—
229	D₉	—	12	2	no	—
230	C₂²×C₄	—	27	2	no	—
231	C₂²×C₄	—	27	1	no	—
232	M₁₆	—	27	4	no	—
233	M₁₆	—	27	4	no	—
234	C₂×C₈	—	3	1	no	—
235	C₂×C₈	—	9	1	no	—
236	C₂×C₈	—	27	1	no	—
237	C₂×C₈	D₃×D₉	1	1	yes	characteristic
238	C₄²	—	3	1	no	—
239	C₄²	—	9	1	no	—
240	C₄²	—	27	2	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
243	C₂²×C₃	C₄×D₃²	1	1	yes	characteristic
244	C₂×D₃	—	9	2	no	—
245	C₂×D₃	—	9	2	no	—
246	C₂×D₃	—	18	2	no	—
247	C₂×D₃	—	18	2	no	—
248	C₂×D₃	—	18	2	no	—
249	C₂×D₃	—	36	2	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	3	4	no	—
253	C₃×C₄	—	18	2	no	—
254	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
255	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
256	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
257	C₃×C₄	C₂×#624	1	1	yes	characteristic
258	Dic₃	—	6	2	no	—
259	Dic₃	—	9	2	no	—
260	Dic₃	—	9	2	no	—
261	Dic₃	—	18	2	no	—
262	Dic₃	#3663	1	4	yes	complemented
263	C₃²	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
264	C₉	—	2	1	no	—
265	C₉	#4707	1	1	yes	characteristic, complemented
266	C₂³	—	27	1	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	9	2	no	—
269	C₂×C₄	—	27	1	no	—
270	C₂×C₄	—	27	2	no	—
271	C₂×C₄	—	27	2	no	—
272	C₂×C₄	—	54	4	no	—
273	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
274	C₈	—	3	2	no	—
275	C₈	—	9	2	no	—
276	C₈	—	27	2	no	—
277	C₈	(C₃⋊D₉)⋊C₄	1	2	yes	complemented
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	C₄²×D₉	1	1	yes	characteristic
281	C₂×C₃	C₂×#3663	1	2	yes	—
282	C₂×C₃	D₃×ΓL(1,𝔽₂₅)	1	2	yes	—
283	C₂×C₃	C₄×#624	1	1	yes	characteristic
284	D₃	—	18	2	no	—
285	D₃	—	18	2	no	—
286	D₃	—	36	2	no	—
287	C₂²	—	27	2	no	—
288	C₂²	—	54	2	no	—
289	C₂²	C₄×D₃×D₉	1	1	yes	characteristic
290	C₄	—	3	4	no	—
291	C₄	—	18	2	no	—
292	C₄	—	27	2	no	—
293	C₄	C₄×D₃×D₉	1	1	yes	characteristic
294	C₄	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
295	C₃	—	2	1	no	—
296	C₃	C₄×#3663	1	1	yes	characteristic, complemented
297	C₃	#10590664	1	1	yes	characteristic
298	C₂	—	54	2	no	—
299	C₂	D₃×#3663	1	2	yes	—
300	C₂	C₄×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃⋊D₉	C₂×C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₈	D₃×D₉
hypocenter	C₃×C₉	C₄×M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₉×Dic₃	C₂	semidirect
5. C₃×#63375	C₂	semidirect
6. #11720274	C₂	semidirect
7. Dic₃×C₉⋊C₈	C₂	semidirect
14. #88523	C₄	semidirect
20. #88414	C₄	semidirect
34. #63375	D₃	semidirect
44. C₃×C₈×C₉	C₂×C₄	semidirect
45. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
46. C₉×C₃⋊C₈	C₂×C₄	semidirect
47. C₃×C₉⋊C₈	C₂×C₄	semidirect
82. C₉×Dic₃	M₁₆	semidirect
96. C₈×Dic₃	D₉	semidirect
126. C₈×C₉	C₄×D₃	semidirect
129. C₉⋊C₈	C₄×D₃	semidirect
189. C₃×C₉	C₄×M₁₆	semidirect
214. C₃×C₈	C₄×D₉	semidirect
220. C₃⋊C₈	C₄×D₉	semidirect
262. Dic₃	#3663	semidirect
265. C₉	#4707	semidirect
277. C₈	(C₃⋊D₉)⋊C₄	semidirect
296. C₃	C₄×#3663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88523	2
C₄×(C₃⋊D₉)⋊C₄	2
C₂×#88414	2
C₈×C₉×Dic₃	2
C₃×#63375	2
#11720274	2
Dic₃×C₉⋊C₈	2
#10590664	3
C₄×#3663	3