Subgroups — #13139465

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 284 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	#11721815	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720298	C₂	1	4	yes	maximal, complemented
5	#10591100	—	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	#88551	—	2	1	no	—
14	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
15	C₄×C₃⋊Dic₉	—	2	1	no	—
16	C₃×#3667	—	2	2	no	—
17	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₆	D₃	1	1	yes	characteristic
19	#64135	—	3	1	no	—
20	C₂×C₃×#208	—	3	1	no	—
21	C₂²×Q₈×C₉	—	3	1	no	—
22	#63566	—	3	4	no	—
23	#63488	—	3	1	no	—
24	#63488	—	6	1	no	—
25	C₂×#3667	D₃	1	1	yes	characteristic
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×C₃⋊Dic₉	—	2	2	no	—
33	C₂×C₉×Dic₃	—	2	4	no	—
34	C₂×C₉×Dic₃	—	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	#5120	—	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	#3757	—	6	1	no	—
48	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
49	C₄×C₃⋊Dic₃	—	6	1	no	—
50	C₃×#208	—	6	2	no	—
51	C₂×C₃×C₃⋊C₈	—	3	2	no	—
52	C₂×Q₈×C₉	—	3	4	no	—
53	C₂×Q₈×C₉	—	3	2	no	—
54	C₂×Q₈×C₉	—	6	1	no	—
55	C₂×Q₈×C₉	—	6	2	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	—	6	2	no	—
60	#3676	—	6	1	no	—
61	C₄⋊Dic₉	—	3	2	no	—
62	C₄⋊Dic₉	—	6	2	no	—
63	C₄×Dic₉	—	6	2	no	—
64	C₄×Dic₉	—	6	1	no	—
65	#3667	—	2	2	no	—
66	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
67	C₂²×C₃×C₉	—	2	2	no	—
68	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
71	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃⋊Dic₉	—	4	2	no	—
74	C₉×Dic₃	—	2	4	no	—
75	C₂²×C₃×Q₈	—	3	1	no	—
76	C₂²×Dic₆	D₉	1	1	yes	characteristic
77	C₂×C₃×M₁₆	—	9	1	no	—
78	#926	—	9	1	no	—
79	#926	—	9	1	no	—
80	#926	—	18	1	no	—
81	C₂×#208	—	3	1	no	—
82	#835	—	9	4	no	—
83	#816	—	9	4	no	—
84	C₂³×C₃²	#67	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
86	C₂×C₃²×C₄	#67	1	2	yes	—
87	C₂×C₃²×C₄	#67	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₃⋊Dic₃	—	6	2	no	—
90	C₂×C₃×Dic₃	—	2	4	no	—
91	C₂×C₃×Dic₃	—	2	2	no	—
92	C₃×Dic₆	—	2	4	no	—
93	C₃×Dic₆	—	2	2	no	—
94	C₃×Dic₆	C₂×Dic₃	1	4	yes	—
95	C₂³×C₉	—	2	1	no	—
96	C₂³×C₉	D₁₂	1	1	yes	characteristic
97	C₃×C₃⋊C₈	—	6	2	no	—
98	Q₈×C₉	—	3	4	no	—
99	Q₈×C₉	—	6	4	no	—
100	Q₈×C₉	—	6	2	no	—
101	C₂×C₄×C₉	—	2	2	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	6	4	no	—
106	C₂×C₄×C₉	—	6	2	no	—
107	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
108	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
109	C₂×C₄×C₉	#67	1	2	yes	—
110	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
111	C₂×Dic₉	—	6	4	no	—
112	C₂×Dic₉	—	6	2	no	—
113	C₉⋊C₈	—	2	2	no	—
114	#419	—	27	1	no	Sylow
115	C₂×C₃×C₉	—	2	2	no	—
116	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₉	#137	1	2	yes	—
118	C₂×C₃×Q₈	—	3	4	no	—
119	C₂×C₃×Q₈	—	3	2	no	—
120	C₂×C₃×Q₈	—	6	2	no	—
121	C₂×C₃×Q₈	—	6	1	no	—
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂²×C₃×C₄	—	6	1	no	—
124	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
125	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
126	C₂²×Dic₃	—	2	1	no	—
127	C₂×Dic₆	—	2	2	no	—
128	C₂×Dic₆	—	2	1	no	—
129	C₂×Dic₆	Dic₉	1	4	yes	complemented
130	C₂×Dic₆	C₂×D₉	1	2	yes	—
131	C₃×M₁₆	—	18	2	no	—
132	C₂×C₃×C₈	—	9	2	no	—
133	#217	—	18	2	no	—
134	#217	—	18	1	no	—
135	#217	—	18	1	no	—
136	C₄⋊Dic₃	—	9	2	no	—
137	C₄⋊Dic₃	—	9	2	no	—
138	C₄⋊Dic₃	—	18	2	no	—
139	C₄×Dic₃	—	18	2	no	—
140	C₄×Dic₃	—	18	1	no	—
141	C₄×Dic₃	—	18	1	no	—
142	#208	—	6	2	no	—
143	C₂×C₃⋊C₈	—	3	2	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	#217	1	2	yes	—
146	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
147	C₃²×C₄	#217	1	2	yes	—
148	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
149	C₃²×C₄	C₂×#67	1	1	yes	characteristic
150	C₃⋊Dic₃	—	12	2	no	—
151	C₃×Dic₃	—	2	4	no	—
152	C₂²×C₉	—	2	2	no	—
153	C₂²×C₉	—	2	2	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	4	2	no	—
156	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
157	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
158	C₄×C₉	—	2	2	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	6	4	no	—
162	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
163	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
164	C₄×C₉	C₂×#67	1	1	yes	characteristic
165	Dic₉	—	12	4	no	—
166	Dic₉	—	12	2	no	—
167	C₂²×Q₈	—	3	1	no	—
168	C₂×M₁₆	—	9	1	no	—
169	#117	—	27	1	no	—
170	#103	—	27	4	no	—
171	C₃×C₉	#419	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	#611	1	1	yes	characteristic
174	C₂³×C₃	#626	1	1	yes	characteristic
175	C₃×Q₈	—	3	4	no	—
176	C₃×Q₈	—	6	4	no	—
177	C₃×Q₈	—	6	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	6	4	no	—
183	C₂×C₃×C₄	—	6	2	no	—
184	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
185	C₂×C₃×C₄	#611	1	2	yes	—
186	C₂×C₃×C₄	#611	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
188	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
189	C₂×C₃×C₄	#625	1	2	yes	—
190	C₂×C₃×C₄	#626	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
192	C₂×Dic₃	—	2	4	no	—
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	18	4	no	—
195	C₂×Dic₃	—	18	2	no	—
196	C₂×Dic₃	—	18	2	no	—
197	Dic₆	—	2	4	no	—
198	Dic₆	—	2	2	no	—
199	Dic₆	C₂×Dic₉	1	4	yes	—
200	C₃×C₈	—	18	2	no	—
201	C₃⋊C₈	—	6	2	no	—
202	C₂×C₃²	—	2	2	no	—
203	C₂×C₃²	#942	1	2	yes	—
204	C₂×C₃²	C₂×#217	1	1	yes	characteristic
205	C₂×C₉	—	2	2	no	—
206	C₂×C₉	—	2	2	no	—
207	C₂×C₉	—	2	1	no	—
208	C₂×C₉	—	4	2	no	—
209	C₂×C₉	#909	1	2	yes	—
210	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₂×Q₈	—	3	4	no	—
212	C₂×Q₈	—	3	2	no	—
213	C₂×Q₈	—	6	2	no	—
214	C₂×Q₈	—	6	1	no	—
215	C₂²×C₄	—	6	1	no	—
216	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
217	M₁₆	—	18	2	no	—
218	C₂×C₈	—	9	2	no	—
219	C₄⋊C₄	—	27	2	no	—
220	C₂²⋊C₄	—	54	1	no	—
221	C₄²	—	54	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	#3676	1	2	yes	—
228	C₂²×C₃	C₂×#611	1	1	yes	characteristic
229	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
230	C₂²×C₃	C₂×#626	1	1	yes	characteristic
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	6	4	no	—
235	C₃×C₄	#3676	1	2	yes	—
236	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
237	C₃×C₄	C₂×#611	1	1	yes	characteristic
238	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
239	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
240	C₃×C₄	C₂×#625	1	1	yes	characteristic
241	Dic₃	—	2	4	no	—
242	Dic₃	—	36	4	no	—
243	Dic₃	—	36	2	no	—
244	Dic₃	—	36	2	no	—
245	C₃²	#5225	1	1	yes	characteristic
246	C₉	—	2	1	no	—
247	C₉	#5120	1	1	yes	characteristic, complemented
248	C₂³	#6195	1	1	yes	characteristic
249	Q₈	—	3	4	no	—
250	Q₈	—	6	2	no	—
251	Q₈	—	6	4	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	54	2	no	—
255	C₂×C₄	D₃×Dic₉	1	2	yes	—
256	C₂×C₄	#6194	1	2	yes	—
257	C₂×C₄	#6195	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
259	C₈	—	18	2	no	—
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	4	2	no	—
265	C₂×C₃	#63505	1	2	yes	—
266	C₂×C₃	C₂×#3676	1	1	yes	characteristic
267	C₂×C₃	#63833	1	2	yes	—
268	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	(C₂×D₃)⋊Dic₉	1	2	yes	—
271	C₂²	C₂×#6195	1	1	yes	characteristic
272	C₄	—	6	4	no	—
273	C₄	—	108	2	no	—
274	C₄	(C₂×D₃)⋊Dic₉	1	2	yes	—
275	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
276	C₄	C₂×#6194	1	1	yes	characteristic
277	C₃	—	2	1	no	—
278	C₃	#10588906	1	1	yes	characteristic, complemented
279	C₃	#10591100	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	#11720895	1	2	yes	—
282	C₂	C₂×(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6195
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. #11720298	C₂	semidirect
11. C₂×C₉×Dic₆	C₄	semidirect
129. C₂×Dic₆	Dic₉	semidirect
171. C₃×C₉	#419	semidirect
247. C₉	#5120	semidirect
278. 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
#11721815	2
C₂×C₃×#3667	2
#11720298	2
#10591100	3
#10588906	3