Subgroups — #13139464

Order 1728.

518 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic
3	C₃×#63488	C₂	1	1	yes	maximal, characteristic
4	#11720297	C₂	1	4	yes	maximal, complemented
5	#10591099	—	3	1	no	maximal
6	#10588906	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₉×Dic₃	—	2	1	no	—
9	C₂×C₉×Dic₆	—	2	1	no	—
10	C₂×C₉×Dic₆	—	2	2	no	—
11	C₂×C₉×Dic₆	C₄	1	4	yes	complemented
12	C₂×C₉×Dic₆	C₂²	1	2	yes	—
13	#88532	—	2	2	no	—
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
15	C₃×#3676	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
17	C₃×C₄×Dic₉	—	2	1	no	—
18	C₂²×C₃×Dic₆	D₃	1	1	yes	characteristic
19	C₂×#3748	—	3	1	no	—
20	C₃×#926	—	3	1	no	—
21	C₂²×Q₈×C₉	—	3	1	no	—
22	#63565	—	3	4	no	—
23	#63488	D₃	1	1	yes	characteristic
24	C₂×#3667	—	3	1	no	—
25	C₂×#3667	—	6	1	no	—
26	#63395	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	—	2	4	no	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₉×Dic₆	—	2	4	no	—
36	C₉×Dic₆	—	2	2	no	—
37	C₉×Dic₆	C₂×C₄	1	4	yes	—
38	C₃⋊(C₉⋊C₈)	—	2	2	no	—
39	#5225	—	9	1	no	—
40	#5004	—	9	1	no	—
41	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×Dic₃	—	2	1	no	—
43	C₂×C₃×Dic₆	—	2	1	no	—
44	C₂×C₃×Dic₆	—	2	2	no	—
45	C₂×C₃×Dic₆	Dic₃	1	4	yes	—
46	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
47	#3748	—	6	2	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
49	C₃×#217	—	6	1	no	—
50	C₃×C₄⋊Dic₃	—	3	2	no	—
51	C₃×C₄×Dic₃	—	6	1	no	—
52	C₂×Q₈×C₉	—	3	4	no	—
53	C₂×Q₈×C₉	—	3	2	no	—
54	C₂×Q₈×C₉	—	6	2	no	—
55	C₂×Q₈×C₉	—	6	1	no	—
56	C₂²×C₄×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	—	6	1	no	—
58	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
59	#3676	—	2	1	no	—
60	C₄⋊Dic₉	C₂×D₃	1	2	yes	—
61	C₄×Dic₉	—	2	1	no	—
62	#3667	—	6	4	no	—
63	#3667	—	6	2	no	—
64	C₂×C₉⋊C₈	—	3	2	no	—
65	C₂×C₉⋊C₈	—	6	2	no	—
66	C₂²×C₃×C₉	—	2	2	no	—
67	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
70	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₉×Dic₃	—	2	4	no	—
73	C₃×Dic₉	—	4	2	no	—
74	C₂²×C₃×Q₈	—	3	1	no	—
75	C₂²×Dic₆	D₉	1	1	yes	characteristic
76	C₃×#117	—	9	1	no	—
77	#926	—	3	1	no	—
78	C₂×#208	—	9	1	no	—
79	C₂×#208	—	9	1	no	—
80	C₂×#208	—	18	1	no	—
81	#835	—	9	4	no	—
82	#810	—	9	4	no	—
83	C₂³×C₃²	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	#67	1	1	yes	characteristic
86	C₂×C₃²×C₄	#67	1	2	yes	—
87	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₃×Dic₃	—	2	4	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	6	2	no	—
91	C₃×Dic₆	—	2	4	no	—
92	C₃×Dic₆	—	2	2	no	—
93	C₃×Dic₆	C₂×Dic₃	1	4	yes	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	#67	1	1	yes	characteristic
96	C₃⋊(C₃⋊C₈)	—	6	2	no	—
97	Q₈×C₉	—	3	4	no	—
98	Q₈×C₉	—	6	4	no	—
99	Q₈×C₉	—	6	2	no	—
100	C₂×C₄×C₉	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	6	4	no	—
105	C₂×C₄×C₉	—	6	2	no	—
106	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
107	C₂×C₄×C₉	D₁₂	1	2	yes	—
108	C₂×C₄×C₉	#67	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
110	C₂×Dic₉	—	2	2	no	—
111	C₉⋊C₈	—	6	4	no	—
112	C₉⋊C₈	—	6	2	no	—
113	#419	—	27	1	no	Sylow
114	C₂×C₃×C₉	—	2	2	no	—
115	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₉	#137	1	2	yes	—
117	C₂×C₃×Q₈	—	3	4	no	—
118	C₂×C₃×Q₈	—	3	2	no	—
119	C₂×C₃×Q₈	—	6	1	no	—
120	C₂×C₃×Q₈	—	6	2	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	—	6	1	no	—
123	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
124	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
125	C₂²×Dic₃	—	2	1	no	—
126	C₂×Dic₆	—	2	1	no	—
127	C₂×Dic₆	—	2	2	no	—
128	C₂×Dic₆	Dic₉	1	4	yes	complemented
129	C₂×Dic₆	C₂×D₉	1	2	yes	—
130	C₃×C₄⋊C₄	—	9	2	no	—
131	C₃×C₂²⋊C₄	—	18	1	no	—
132	C₃×C₄²	—	18	1	no	—
133	#217	—	6	1	no	—
134	C₄⋊Dic₃	—	3	2	no	—
135	C₄×Dic₃	—	6	1	no	—
136	#208	—	18	4	no	—
137	#208	—	18	2	no	—
138	#208	—	18	2	no	—
139	C₂×C₃⋊C₈	—	9	2	no	—
140	C₂×C₃⋊C₈	—	9	2	no	—
141	C₂×C₃⋊C₈	—	18	2	no	—
142	C₂²×C₃²	—	2	2	no	—
143	C₂²×C₃²	#217	1	2	yes	—
144	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
145	C₃²×C₄	#217	1	2	yes	—
146	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
147	C₃²×C₄	C₂×#67	1	1	yes	characteristic
148	C₃×Dic₃	—	2	4	no	—
149	C₃×Dic₃	—	12	2	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	4	2	no	—
154	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₂²×C₉	C₂×#67	1	1	yes	characteristic
156	C₄×C₉	—	2	2	no	—
157	C₄×C₉	—	2	1	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	6	4	no	—
160	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
161	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
162	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
163	Dic₉	—	4	2	no	—
164	C₂²×Q₈	—	3	1	no	—
165	C₂×M₁₆	—	27	1	no	—
166	#117	—	9	1	no	—
167	#103	—	27	4	no	—
168	C₃×C₉	#419	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	#611	1	1	yes	characteristic
171	C₂³×C₃	#625	1	1	yes	characteristic
172	C₃×Q₈	—	3	4	no	—
173	C₃×Q₈	—	6	4	no	—
174	C₃×Q₈	—	6	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	6	4	no	—
180	C₂×C₃×C₄	—	6	2	no	—
181	C₂×C₃×C₄	—	18	2	no	—
182	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
183	C₂×C₃×C₄	#611	1	1	yes	characteristic
184	C₂×C₃×C₄	#611	1	2	yes	—
185	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
187	C₂×C₃×C₄	#625	1	1	yes	characteristic
188	C₂×C₃×C₄	#626	1	2	yes	—
189	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
190	C₂×Dic₃	—	2	4	no	—
191	C₂×Dic₃	—	2	2	no	—
192	C₂×Dic₃	—	6	2	no	—
193	Dic₆	—	2	4	no	—
194	Dic₆	—	2	2	no	—
195	Dic₆	C₂×Dic₉	1	4	yes	—
196	C₃⋊C₈	—	18	4	no	—
197	C₃⋊C₈	—	18	2	no	—
198	C₃⋊C₈	—	18	2	no	—
199	C₂×C₃²	—	2	2	no	—
200	C₂×C₃²	#942	1	2	yes	—
201	C₂×C₃²	C₂×#217	1	1	yes	characteristic
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	2	2	no	—
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	4	2	no	—
206	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₉	#951	1	2	yes	—
208	C₂×Q₈	—	3	4	no	—
209	C₂×Q₈	—	3	2	no	—
210	C₂×Q₈	—	6	1	no	—
211	C₂×Q₈	—	6	2	no	—
212	C₂²×C₄	—	6	1	no	—
213	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
214	M₁₆	—	54	2	no	—
215	C₂×C₈	—	27	2	no	—
216	C₄⋊C₄	—	9	2	no	—
217	C₂²⋊C₄	—	18	1	no	—
218	C₄²	—	18	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	4	2	no	—
224	C₂²×C₃	#3676	1	2	yes	—
225	C₂²×C₃	C₂×#611	1	1	yes	characteristic
226	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
227	C₂²×C₃	C₂×#625	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	6	4	no	—
232	C₃×C₄	—	36	2	no	—
233	C₃×C₄	#3676	1	2	yes	—
234	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
235	C₃×C₄	C₂×#611	1	1	yes	characteristic
236	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
237	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
238	C₃×C₄	C₂×#626	1	1	yes	characteristic
239	Dic₃	—	2	4	no	—
240	Dic₃	—	12	2	no	—
241	C₃²	#5225	1	1	yes	characteristic
242	C₉	—	2	1	no	—
243	C₉	#5004	1	1	yes	characteristic, complemented
244	C₂³	#6194	1	1	yes	characteristic
245	Q₈	—	3	4	no	—
246	Q₈	—	6	2	no	—
247	Q₈	—	6	4	no	—
248	C₂×C₄	—	6	2	no	—
249	C₂×C₄	—	6	4	no	—
250	C₂×C₄	—	18	2	no	—
251	C₂×C₄	D₃×Dic₉	1	2	yes	—
252	C₂×C₄	#6194	1	1	yes	characteristic
253	C₂×C₄	#6195	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
255	C₈	—	54	2	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	#63505	1	2	yes	—
262	C₂×C₃	C₂×#3676	1	1	yes	characteristic
263	C₂×C₃	#63827	1	2	yes	—
264	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	(C₂×D₃)⋊Dic₉	1	2	yes	—
267	C₂²	C₂×#6194	1	1	yes	characteristic
268	C₄	—	6	4	no	—
269	C₄	—	36	2	no	—
270	C₄	(C₂×D₃)⋊Dic₉	1	2	yes	—
271	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
272	C₄	C₂×#6195	1	1	yes	characteristic
273	C₃	—	2	1	no	—
274	C₃	#10588906	1	1	yes	characteristic, complemented
275	C₃	#10591099	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	#11720893	1	2	yes	—
278	C₂	C₂×(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6194
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#419
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720297	C₂	semidirect
11. C₂×C₉×Dic₆	C₄	semidirect
128. C₂×Dic₆	Dic₉	semidirect
168. C₃×C₉	#419	semidirect
243. C₉	#5004	semidirect
274. C₃	#10588906	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Dic₆	2
C₂×#88532	2
C₃×#63488	2
#11720297	2
#10591099	3
#10588906	3