Subgroups — #13144427

Order 1728.

480 conjugacy classes of subgroups.

The subgroup lattice has 222 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	#11721706	C₂	1	1	yes	maximal, characteristic
3	#11720582	C₂	1	2	yes	maximal
4	#11720525	C₂	1	2	yes	maximal
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
6	#10592720	—	3	1	no	maximal
7	#10588703	—	3	3	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×#6226	—	2	1	no	—
10	C₃×C₉×C₄⋊C₄	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
17	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
18	#64078	—	3	1	no	—
19	#63662	—	3	2	no	—
20	#63634	—	3	2	no	—
21	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
22	C₄×Q₈×C₉	D₃	1	3	yes	—
23	#63431	—	3	3	no	—
24	#63395	—	3	6	no	—
25	#63366	—	3	6	no	—
26	C₄⋊(C₉⋊C₈)	—	3	3	no	—
27	C₃×Q₈×C₉	—	2	1	no	—
28	C₃×Q₈×C₉	Q₈	1	2	yes	complemented
29	C₂×C₃×C₄×C₉	—	2	1	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊Dic₉	—	2	2	no	—
35	#6226	—	4	2	no	—
36	C₃⋊(C₉⋊C₈)	—	2	2	no	—
37	#5021	—	9	3	no	—
38	#5021	—	9	1	no	—
39	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
40	C₂×#634	—	6	1	no	—
41	C₃²×C₄⋊C₄	—	2	1	no	—
42	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
43	C₃²×C₄²	—	2	1	no	—
44	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
46	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
48	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
49	C₂×Dic₁₈	—	6	3	no	—
50	C₉×C₄⋊C₄	—	2	3	no	—
51	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
52	C₄²×C₉	—	2	3	no	—
53	C₄²×C₉	C₂×D₃	1	3	yes	—
54	C₄⋊Dic₉	—	3	6	no	—
55	C₄⋊Dic₉	—	6	3	no	—
56	C₂×C₉⋊C₈	—	3	6	no	—
57	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	2	2	no	—
59	C₃×C₄×C₉	—	2	1	no	—
60	C₃×C₄×C₉	—	4	1	no	—
61	C₃×C₄×C₉	Q₁₆	1	2	yes	—
62	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₄×C₉	#41	1	1	yes	characteristic
64	C₃⋊Dic₉	—	4	2	no	—
65	C₃×C₄×Q₈	D₉	1	3	yes	—
66	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
67	#869	—	9	3	no	—
68	#869	—	9	1	no	—
69	#835	—	9	6	no	—
70	#835	—	9	2	no	—
71	#807	—	9	6	no	—
72	#807	—	9	2	no	—
73	C₄⋊(C₃⋊C₈)	—	9	3	no	—
74	C₄⋊(C₃⋊C₈)	—	9	1	no	—
75	C₃²×Q₈	—	2	1	no	—
76	C₃²×Q₈	Dic₆	1	2	yes	—
77	C₂×C₃²×C₄	—	2	1	no	—
78	C₂×C₃²×C₄	—	2	1	no	—
79	C₂×C₃²×C₄	#67	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	—	6	2	no	—
83	#634	—	12	2	no	—
84	C₃⋊(C₃⋊C₈)	—	6	2	no	—
85	Q₈×C₉	—	2	3	no	—
86	Q₈×C₉	Dic₆	1	6	yes	complemented
87	C₂×C₄×C₉	—	2	3	no	—
88	C₂×C₄×C₉	—	2	3	no	—
89	C₂×C₄×C₉	#67	1	3	yes	—
90	C₂×C₄×C₉	#67	1	3	yes	—
91	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
92	C₂×Dic₉	—	6	6	no	—
93	Dic₁₈	—	12	6	no	—
94	C₉⋊C₈	—	6	6	no	—
95	#477	—	27	1	no	Sylow
96	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
100	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₂×Dic₆	—	18	3	no	—
102	C₂×Dic₆	—	18	1	no	—
103	C₃×C₄⋊C₄	—	2	3	no	—
104	C₃×C₄⋊C₄	—	2	1	no	—
105	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
106	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₃×C₄²	—	2	3	no	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	C₂×D₉	1	3	yes	—
110	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
111	C₄⋊Dic₃	—	9	6	no	—
112	C₄⋊Dic₃	—	9	2	no	—
113	C₄⋊Dic₃	—	18	3	no	—
114	C₄⋊Dic₃	—	18	1	no	—
115	C₂×C₃⋊C₈	—	9	6	no	—
116	C₂×C₃⋊C₈	—	9	2	no	—
117	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	—	2	1	no	—
120	C₃²×C₄	—	4	1	no	—
121	C₃²×C₄	#216	1	2	yes	—
122	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
123	C₃²×C₄	#235	1	1	yes	characteristic
124	C₃⋊Dic₃	—	12	2	no	—
125	C₂²×C₉	C₂×#67	1	3	yes	—
126	C₄×C₉	—	2	6	no	—
127	C₄×C₉	—	2	3	no	—
128	C₄×C₉	—	4	3	no	—
129	C₄×C₉	#216	1	6	yes	—
130	C₄×C₉	C₂×Dic₆	1	3	yes	—
131	C₄×C₉	#235	1	3	yes	—
132	Dic₉	—	12	6	no	—
133	C₄⋊Q₈	—	27	1	no	—
134	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
135	#107	—	27	2	no	—
136	C₄⋊C₈	—	27	1	no	—
137	#103	—	27	2	no	—
138	C₃×C₉	#477	1	1	yes	characteristic, complemented, Sylow
139	C₃×Q₈	—	2	3	no	—
140	C₃×Q₈	—	2	1	no	—
141	C₃×Q₈	Dic₁₈	1	6	yes	complemented
142	C₃×Q₈	#634	1	2	yes	—
143	C₂×C₃×C₄	—	2	3	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	3	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	#611	1	3	yes	—
148	C₂×C₃×C₄	#611	1	3	yes	—
149	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
150	C₂×C₃×C₄	#638	1	1	yes	characteristic
151	C₂×C₃×C₄	#638	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
153	C₂×Dic₃	—	18	6	no	—
154	C₂×Dic₃	—	18	2	no	—
155	Dic₆	—	36	2	no	—
156	Dic₆	—	36	6	no	—
157	C₃⋊C₈	—	18	6	no	—
158	C₃⋊C₈	—	18	2	no	—
159	C₂×C₃²	#924	1	1	yes	characteristic
160	C₂×C₃²	C₂×#216	1	1	yes	characteristic
161	C₂×C₃²	#949	1	1	yes	characteristic
162	C₂×C₉	#924	1	3	yes	—
163	C₂×C₉	C₂×#216	1	3	yes	—
164	C₂×C₉	#949	1	3	yes	—
165	C₂×Q₈	—	54	1	no	—
166	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
167	C₂×C₈	—	27	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	27	2	no	—
170	C₄⋊C₄	—	54	1	no	—
171	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
174	C₂²×C₃	C₂×#611	1	3	yes	—
175	C₂²×C₃	C₂×#638	1	1	yes	characteristic
176	C₃×C₄	—	2	3	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	6	no	—
180	C₃×C₄	—	4	3	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	#3674	1	6	yes	—
183	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
184	C₃×C₄	#3697	1	3	yes	—
185	C₃×C₄	#3756	1	2	yes	—
186	C₃×C₄	C₂×#634	1	1	yes	characteristic
187	C₃×C₄	#3828	1	1	yes	characteristic
188	Dic₃	—	36	6	no	—
189	Dic₃	—	36	2	no	—
190	C₃²	#5021	1	1	yes	characteristic
191	C₉	#5021	1	3	yes	complemented
192	Q₈	—	2	1	no	—
193	Q₈	—	108	2	no	—
194	Q₈	#6226	1	2	yes	complemented
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	54	2	no	—
198	C₂×C₄	#6233	1	1	yes	characteristic
199	C₂×C₄	#6233	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
201	C₈	—	54	2	no	—
202	C₂×C₃	#63486	1	3	yes	—
203	C₂×C₃	C₂×#3674	1	3	yes	—
204	C₂×C₃	#63512	1	3	yes	—
205	C₂×C₃	#64133	1	1	yes	characteristic
206	C₂×C₃	C₂×#3756	1	1	yes	characteristic
207	C₂×C₃	#64158	1	1	yes	characteristic
208	C₂²	C₂×#6233	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	1	no	—
212	C₄	—	108	2	no	—
213	C₄	#88547	1	2	yes	—
214	C₄	C₂×#6226	1	1	yes	characteristic
215	C₄	#88736	1	1	yes	characteristic
216	C₃	#10588703	1	3	yes	complemented
217	C₃	#10592720	1	1	yes	characteristic
218	C₂	#11721813	1	1	yes	characteristic
219	C₂	C₂×#88547	1	1	yes	characteristic
220	C₂	#11721867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6233
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#477
LCS	G ▷ C₂×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
28. C₃×Q₈×C₉	Q₈	semidirect
86. Q₈×C₉	Dic₆	semidirect
138. C₃×C₉	#477	semidirect
141. C₃×Q₈	Dic₁₈	semidirect
191. C₉	#5021	semidirect
194. Q₈	#6226	semidirect
216. C₃	#10588703	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
#11721706	2
#11720582	2
#11720525	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592720	3
#10588703	3