Subgroups — #13144834

Order 1728.

750 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#11721815	C₂	1	1	yes	maximal, characteristic
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic
4	#11720582	C₂	1	4	yes	maximal, complemented
5	#10592924	—	3	1	no	maximal
6	#10588906	—	3	3	no	maximal
7	C₂×C₃×Q₈×C₉	—	2	2	no	—
8	C₂×C₃×Q₈×C₉	—	2	1	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	—	2	1	no	—
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	#88551	—	2	1	no	—
14	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
15	C₄×C₃⋊Dic₉	—	2	1	no	—
16	#88532	—	2	2	no	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
19	#64135	—	3	1	no	—
20	C₂×#3748	—	3	1	no	—
21	C₂²×Q₈×C₉	D₃	1	3	yes	—
22	#63662	—	3	4	no	—
23	#63488	—	3	3	no	—
24	C₂×#3667	—	3	3	no	—
25	#63395	—	3	12	no	—
26	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
27	C₃×Q₈×C₉	—	2	4	no	—
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×Q₈×C₉	C₂×C₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	—	2	4	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×C₃⋊Dic₉	—	2	2	no	—
37	C₃⋊(C₉⋊C₈)	—	2	2	no	—
38	#5225	—	9	3	no	—
39	#5225	—	9	1	no	—
40	C₂×C₃²×Q₈	—	2	2	no	—
41	C₂×C₃²×Q₈	—	2	1	no	—
42	C₂×C₃²×Q₈	Dic₃	1	4	yes	—
43	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
46	#3757	—	6	1	no	—
47	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
48	C₄×C₃⋊Dic₃	—	6	1	no	—
49	#3748	—	6	2	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×Q₈×C₉	—	2	6	no	—
52	C₂×Q₈×C₉	—	2	3	no	—
53	C₂×Q₈×C₉	Dic₃	1	12	yes	complemented
54	C₂×Q₈×C₉	C₂×D₃	1	6	yes	—
55	C₂²×C₄×C₉	—	2	3	no	—
56	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
57	#3676	—	6	3	no	—
58	C₄⋊Dic₉	—	3	6	no	—
59	C₄×Dic₉	—	6	3	no	—
60	#3667	—	6	6	no	—
61	C₂×C₉⋊C₈	—	3	6	no	—
62	C₂²×C₃×C₉	—	2	2	no	—
63	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	—	2	4	no	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
67	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
69	C₃⋊Dic₉	—	4	2	no	—
70	C₂²×C₃×Q₈	D₉	1	3	yes	—
71	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
72	#926	—	9	3	no	—
73	#926	—	9	1	no	—
74	C₂×#208	—	9	3	no	—
75	C₂×#208	—	9	1	no	—
76	#835	—	9	12	no	—
77	#835	—	9	4	no	—
78	C₂³×C₃²	#67	1	1	yes	characteristic
79	C₃²×Q₈	—	2	4	no	—
80	C₃²×Q₈	—	2	2	no	—
81	C₃²×Q₈	C₂×Dic₃	1	4	yes	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
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	—	6	2	no	—
89	C₂³×C₉	#67	1	3	yes	—
90	C₃⋊(C₃⋊C₈)	—	6	2	no	—
91	Q₈×C₉	—	2	12	no	—
92	Q₈×C₉	—	2	6	no	—
93	Q₈×C₉	C₂×Dic₃	1	12	yes	—
94	C₂×C₄×C₉	—	2	12	no	—
95	C₂×C₄×C₉	—	2	6	no	—
96	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
97	C₂×C₄×C₉	#67	1	3	yes	—
98	C₂×C₄×C₉	#67	1	6	yes	—
99	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
100	C₂×Dic₉	—	6	6	no	—
101	C₉⋊C₈	—	6	6	no	—
102	#419	—	27	1	no	Sylow
103	C₂×C₃×C₉	—	2	2	no	—
104	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₉	#137	1	2	yes	—
106	C₂×C₃×Q₈	—	2	6	no	—
107	C₂×C₃×Q₈	—	2	3	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	1	no	—
110	C₂×C₃×Q₈	Dic₉	1	12	yes	complemented
111	C₂×C₃×Q₈	C₂×D₉	1	6	yes	—
112	C₂×C₃×Q₈	C₃⋊Dic₃	1	4	yes	—
113	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
114	C₂²×C₃×C₄	—	2	3	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
117	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
118	#217	—	18	3	no	—
119	#217	—	18	1	no	—
120	C₄⋊Dic₃	—	9	6	no	—
121	C₄⋊Dic₃	—	9	2	no	—
122	C₄×Dic₃	—	18	3	no	—
123	C₄×Dic₃	—	18	1	no	—
124	#208	—	18	6	no	—
125	#208	—	18	2	no	—
126	C₂×C₃⋊C₈	—	9	6	no	—
127	C₂×C₃⋊C₈	—	9	2	no	—
128	C₂²×C₃²	—	2	2	no	—
129	C₂²×C₃²	#217	1	2	yes	—
130	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
131	C₃²×C₄	—	2	4	no	—
132	C₃²×C₄	#217	1	2	yes	—
133	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₃²×C₄	C₂×#67	1	1	yes	characteristic
135	C₃⋊Dic₃	—	12	2	no	—
136	C₂²×C₉	—	2	6	no	—
137	C₂²×C₉	#217	1	6	yes	—
138	C₂²×C₉	C₂×#67	1	3	yes	—
139	C₄×C₉	—	2	12	no	—
140	C₄×C₉	#217	1	6	yes	—
141	C₄×C₉	C₂²×Dic₃	1	3	yes	—
142	C₄×C₉	C₂×#67	1	3	yes	—
143	Dic₉	—	12	6	no	—
144	C₂²×Q₈	C₃⋊D₉	1	1	yes	characteristic
145	C₂×M₁₆	—	27	1	no	—
146	#117	—	27	1	no	—
147	#103	—	27	4	no	—
148	C₃×C₉	#419	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	#611	1	3	yes	—
150	C₂³×C₃	#638	1	1	yes	characteristic
151	C₃×Q₈	—	2	12	no	—
152	C₃×Q₈	—	2	6	no	—
153	C₃×Q₈	—	2	4	no	—
154	C₃×Q₈	—	2	2	no	—
155	C₃×Q₈	C₂×Dic₉	1	12	yes	—
156	C₃×Q₈	C₂×C₃⋊Dic₃	1	4	yes	—
157	C₂×C₃×C₄	—	2	12	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
162	C₂×C₃×C₄	#611	1	3	yes	—
163	C₂×C₃×C₄	#611	1	6	yes	—
164	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
165	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
166	C₂×C₃×C₄	#638	1	1	yes	characteristic
167	C₂×C₃×C₄	#638	1	2	yes	—
168	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
169	C₂×Dic₃	—	18	6	no	—
170	C₂×Dic₃	—	18	2	no	—
171	C₃⋊C₈	—	18	6	no	—
172	C₃⋊C₈	—	18	2	no	—
173	C₂×C₃²	—	2	2	no	—
174	C₂×C₃²	#942	1	2	yes	—
175	C₂×C₃²	C₂×#217	1	1	yes	characteristic
176	C₂×C₉	—	2	6	no	—
177	C₂×C₉	#942	1	6	yes	—
178	C₂×C₉	C₂×#217	1	3	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₃⋊Dic₉	1	4	yes	complemented
182	C₂×Q₈	C₂×C₃⋊D₉	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
185	M₁₆	—	54	2	no	—
186	C₂×C₈	—	27	2	no	—
187	C₄⋊C₄	—	27	2	no	—
188	C₂²⋊C₄	—	54	1	no	—
189	C₄²	—	54	1	no	—
190	C₂²×C₃	—	2	6	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	#3676	1	6	yes	—
193	C₂²×C₃	C₂×#611	1	3	yes	—
194	C₂²×C₃	#3757	1	2	yes	—
195	C₂²×C₃	C₂×#638	1	1	yes	characteristic
196	C₃×C₄	—	2	12	no	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	#3676	1	6	yes	—
199	C₃×C₄	C₂²×Dic₉	1	3	yes	—
200	C₃×C₄	C₂×#611	1	3	yes	—
201	C₃×C₄	#3757	1	2	yes	—
202	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
203	C₃×C₄	C₂×#638	1	1	yes	characteristic
204	Dic₃	—	36	6	no	—
205	Dic₃	—	36	2	no	—
206	C₃²	#5225	1	1	yes	characteristic
207	C₉	#5225	1	3	yes	complemented
208	C₂³	#6233	1	1	yes	characteristic
209	Q₈	—	2	2	no	—
210	Q₈	—	2	4	no	—
211	Q₈	C₂×C₃⋊Dic₉	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	54	2	no	—
215	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
216	C₂×C₄	#6233	1	2	yes	—
217	C₂×C₄	#6233	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
219	C₈	—	54	2	no	—
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	2	6	no	—
222	C₂×C₃	#63505	1	6	yes	—
223	C₂×C₃	C₂×#3676	1	3	yes	—
224	C₂×C₃	#64151	1	2	yes	—
225	C₂×C₃	C₂×#3757	1	1	yes	characteristic
226	C₂²	—	2	2	no	—
227	C₂²	#88551	1	2	yes	—
228	C₂²	C₂×#6233	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	108	2	no	—
231	C₄	#88551	1	2	yes	—
232	C₄	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
233	C₄	C₂×#6233	1	1	yes	characteristic
234	C₃	#10588906	1	3	yes	complemented
235	C₃	#10592924	1	1	yes	characteristic
236	C₂	—	2	2	no	—
237	C₂	#11721853	1	2	yes	—
238	C₂	C₂×#88551	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	C₂²×C₃×Q₈×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#419
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
4. #11720582	C₂	semidirect
9. C₂×C₃×Q₈×C₉	C₄	semidirect
53. C₂×Q₈×C₉	Dic₃	semidirect
110. C₂×C₃×Q₈	Dic₉	semidirect
148. C₃×C₉	#419	semidirect
181. C₂×Q₈	C₃⋊Dic₉	semidirect
207. C₉	#5225	semidirect
234. 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×C₃×Q₈×C₉	2
#11721815	2
C₂×#88532	2
#11720582	2
#10592924	3
#10588906	3