Subgroups — #13144430

Order 1728.

540 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88547	C₂	1	1	yes	maximal, characteristic
3	C₄×#6226	C₂	1	1	yes	maximal, characteristic
4	#11720582	C₂	1	1	yes	maximal, characteristic
5	#11720527	C₂	1	1	yes	maximal, characteristic
6	#11720526	C₂	1	1	yes	maximal, characteristic
7	#11720516	C₂	1	1	yes	maximal, characteristic
8	#10592729	—	3	1	no	maximal
9	#10588706	—	3	3	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×#6226	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₄⋊C₄	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄²×C₉	—	2	1	no	—
15	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
16	#88547	C₄	1	4	yes	complemented
17	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
18	#88534	—	2	1	no	—
19	C₄×C₃⋊Dic₉	—	2	1	no	—
20	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
22	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
23	C₂×#3756	—	3	1	no	—
24	C₄×#634	—	3	1	no	—
25	#63662	—	3	1	no	—
26	#63637	—	3	1	no	—
27	#63635	—	3	1	no	—
28	#63630	—	3	1	no	—
29	C₄×Q₈×C₉	D₃	1	3	yes	—
30	C₂×#3674	—	3	3	no	—
31	C₄×Dic₁₈	—	3	3	no	—
32	#63395	—	3	3	no	—
33	#63368	—	3	3	no	—
34	#63367	—	3	3	no	—
35	#63362	—	3	3	no	—
36	C₃×Q₈×C₉	—	2	1	no	—
37	C₃×Q₈×C₉	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊Dic₉	—	2	1	no	—
44	C₂×C₃⋊Dic₉	—	2	1	no	—
45	#6226	—	2	1	no	—
46	#6226	C₂×C₄	1	2	yes	—
47	C₃⋊(C₉⋊C₈)	—	2	1	no	—
48	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
49	#5030	—	9	3	no	—
50	#5030	—	9	1	no	—
51	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
52	C₂×#634	—	3	1	no	—
53	C₃²×C₄⋊C₄	—	2	1	no	—
54	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
55	C₃²×C₄²	—	2	1	no	—
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	#3756	—	3	4	no	—
58	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
59	#3750	—	6	1	no	—
60	C₄×C₃⋊Dic₃	—	6	1	no	—
61	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
62	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
63	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
64	C₂×Dic₁₈	—	3	3	no	—
65	C₉×C₄⋊C₄	—	2	3	no	—
66	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
67	C₄²×C₉	—	2	3	no	—
68	C₄²×C₉	C₂×D₃	1	3	yes	—
69	#3674	—	3	12	no	—
70	C₄⋊Dic₉	—	3	3	no	—
71	Dic₉⋊C₄	—	6	3	no	—
72	C₄×Dic₉	—	6	3	no	—
73	C₂×C₉⋊C₈	—	3	3	no	—
74	C₂×C₉⋊C₈	—	3	3	no	—
75	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×C₉	—	2	2	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	—	2	1	no	—
79	C₃×C₄×C₉	—	4	1	no	—
80	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₄×C₉	#41	1	1	yes	characteristic
82	C₃⋊Dic₉	—	2	2	no	—
83	C₃⋊Dic₉	—	4	1	no	—
84	C₃×C₄×Q₈	D₉	1	3	yes	—
85	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
86	C₂×#216	—	9	3	no	—
87	C₂×#216	—	9	1	no	—
88	C₄×Dic₆	—	9	3	no	—
89	C₄×Dic₆	—	9	1	no	—
90	#835	—	9	3	no	—
91	#835	—	9	1	no	—
92	#810	—	9	3	no	—
93	#810	—	9	1	no	—
94	#808	—	9	3	no	—
95	#808	—	9	1	no	—
96	#803	—	9	3	no	—
97	#803	—	9	1	no	—
98	C₃²×Q₈	—	2	1	no	—
99	C₃²×Q₈	C₄×D₃	1	2	yes	—
100	C₂×C₃²×C₄	—	2	1	no	—
101	C₂×C₃²×C₄	—	2	1	no	—
102	C₂×C₃²×C₄	#67	1	1	yes	characteristic
103	C₂×C₃²×C₄	#67	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃⋊Dic₃	—	6	1	no	—
106	C₂×C₃⋊Dic₃	—	6	1	no	—
107	#634	—	3	2	no	—
108	#634	—	6	1	no	—
109	C₃⋊(C₃⋊C₈)	—	3	2	no	—
110	C₃⋊(C₃⋊C₈)	—	6	1	no	—
111	Q₈×C₉	—	2	3	no	—
112	Q₈×C₉	C₄×D₃	1	6	yes	—
113	C₂×C₄×C₉	—	2	3	no	—
114	C₂×C₄×C₉	—	2	3	no	—
115	C₂×C₄×C₉	#67	1	3	yes	—
116	C₂×C₄×C₉	#67	1	3	yes	—
117	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
118	C₂×Dic₉	—	6	3	no	—
119	C₂×Dic₉	—	6	3	no	—
120	Dic₁₈	—	3	6	no	—
121	Dic₁₈	—	6	3	no	—
122	C₉⋊C₈	—	3	6	no	—
123	C₉⋊C₈	—	6	3	no	—
124	#441	—	27	1	no	Sylow
125	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₃×C₉	#137	1	1	yes	characteristic
127	C₂×C₃×C₉	#137	1	1	yes	characteristic
128	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
129	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
130	C₂×Dic₆	—	9	3	no	—
131	C₂×Dic₆	—	9	1	no	—
132	C₃×C₄⋊C₄	—	2	3	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
135	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
136	C₃×C₄²	—	2	3	no	—
137	C₃×C₄²	—	2	1	no	—
138	C₃×C₄²	C₂×D₉	1	3	yes	—
139	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
140	#216	—	9	12	no	—
141	#216	—	9	4	no	—
142	C₄⋊Dic₃	—	9	3	no	—
143	C₄⋊Dic₃	—	9	1	no	—
144	Dic₃⋊C₄	—	18	3	no	—
145	Dic₃⋊C₄	—	18	1	no	—
146	C₄×Dic₃	—	18	3	no	—
147	C₄×Dic₃	—	18	1	no	—
148	C₂×C₃⋊C₈	—	9	3	no	—
149	C₂×C₃⋊C₈	—	9	1	no	—
150	C₂×C₃⋊C₈	—	9	3	no	—
151	C₂×C₃⋊C₈	—	9	1	no	—
152	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
153	C₃²×C₄	—	2	2	no	—
154	C₃²×C₄	—	2	1	no	—
155	C₃²×C₄	—	2	1	no	—
156	C₃²×C₄	—	4	1	no	—
157	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₃²×C₄	#235	1	1	yes	characteristic
159	C₃⋊Dic₃	—	6	2	no	—
160	C₃⋊Dic₃	—	12	1	no	—
161	C₂²×C₉	C₂×#67	1	3	yes	—
162	C₄×C₉	—	2	6	no	—
163	C₄×C₉	—	2	3	no	—
164	C₄×C₉	—	2	3	no	—
165	C₄×C₉	—	4	3	no	—
166	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
167	C₄×C₉	#235	1	3	yes	—
168	Dic₉	—	6	6	no	—
169	Dic₉	—	12	3	no	—
170	C₂×Q₁₆	—	27	1	no	—
171	C₄×Q₈	—	27	1	no	—
172	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
173	#106	—	27	1	no	—
174	#103	—	27	1	no	—
175	#103	—	27	1	no	—
176	#97	—	27	1	no	—
177	C₃×C₉	#441	1	1	yes	characteristic, complemented, Sylow
178	C₃×Q₈	—	2	3	no	—
179	C₃×Q₈	—	2	1	no	—
180	C₃×Q₈	C₄×D₉	1	6	yes	—
181	C₃×Q₈	C₄×C₃⋊D₃	1	2	yes	—
182	C₂×C₃×C₄	—	2	3	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	3	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	#611	1	3	yes	—
187	C₂×C₃×C₄	#611	1	3	yes	—
188	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
189	C₂×C₃×C₄	#638	1	1	yes	characteristic
190	C₂×C₃×C₄	#638	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
192	C₂×Dic₃	—	18	3	no	—
193	C₂×Dic₃	—	18	1	no	—
194	C₂×Dic₃	—	18	3	no	—
195	C₂×Dic₃	—	18	1	no	—
196	Dic₆	—	9	6	no	—
197	Dic₆	—	9	2	no	—
198	Dic₆	—	18	3	no	—
199	Dic₆	—	18	1	no	—
200	C₃⋊C₈	—	9	6	no	—
201	C₃⋊C₈	—	9	2	no	—
202	C₃⋊C₈	—	18	3	no	—
203	C₃⋊C₈	—	18	1	no	—
204	C₂×C₃²	C₄×#67	1	1	yes	characteristic
205	C₂×C₃²	#942	1	1	yes	characteristic
206	C₂×C₃²	#951	1	1	yes	characteristic
207	C₂×C₉	C₄×#67	1	3	yes	—
208	C₂×C₉	#942	1	3	yes	—
209	C₂×C₉	#951	1	3	yes	—
210	C₂×Q₈	—	27	1	no	—
211	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
212	Q₁₆	—	27	4	no	—
213	C₂×C₈	—	27	1	no	—
214	C₂×C₈	—	27	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	27	1	no	—
217	C₄⋊C₄	—	54	1	no	—
218	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	—	54	1	no	—
221	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
222	C₂²×C₃	C₂×#611	1	3	yes	—
223	C₂²×C₃	C₂×#638	1	1	yes	characteristic
224	C₃×C₄	—	2	6	no	—
225	C₃×C₄	—	2	3	no	—
226	C₃×C₄	—	2	3	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	4	3	no	—
231	C₃×C₄	—	4	1	no	—
232	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
233	C₃×C₄	#3697	1	3	yes	—
234	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
235	C₃×C₄	#3828	1	1	yes	characteristic
236	Dic₃	—	18	6	no	—
237	Dic₃	—	18	2	no	—
238	Dic₃	—	36	3	no	—
239	Dic₃	—	36	1	no	—
240	C₃²	#5030	1	1	yes	characteristic
241	C₉	#5030	1	3	yes	complemented
242	Q₈	—	2	1	no	—
243	Q₈	—	27	2	no	—
244	Q₈	—	54	1	no	—
245	Q₈	C₄×C₃⋊D₉	1	2	yes	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	54	1	no	—
249	C₂×C₄	—	54	1	no	—
250	C₂×C₄	#6233	1	1	yes	characteristic
251	C₂×C₄	#6233	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
253	C₈	—	27	2	no	—
254	C₈	—	54	1	no	—
255	C₂×C₃	C₄×#611	1	3	yes	—
256	C₂×C₃	#63505	1	3	yes	—
257	C₂×C₃	#63511	1	3	yes	—
258	C₂×C₃	C₄×#638	1	1	yes	characteristic
259	C₂×C₃	#64151	1	1	yes	characteristic
260	C₂×C₃	#64160	1	1	yes	characteristic
261	C₂²	C₂×#6233	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	54	2	no	—
267	C₄	—	108	1	no	—
268	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
269	C₄	#88736	1	1	yes	characteristic
270	C₃	#10588706	1	3	yes	complemented
271	C₃	#10592729	1	1	yes	characteristic
272	C₂	C₄×#6233	1	1	yes	characteristic
273	C₂	#11721853	1	1	yes	characteristic
274	C₂	#11721866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#441
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. #88547	C₄	semidirect
177. C₃×C₉	#441	semidirect
241. C₉	#5030	semidirect
270. C₃	#10588706	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₂×#88547	2
C₄×#6226	2
#11720582	2
#11720527	2
#11720526	2
#11720516	2
#10592729	3
#10588706	3