Subgroups — #13144425

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 204 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	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	#11720516	C₂	1	2	yes	maximal
5	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10592723	—	3	1	no	maximal
7	#10588701	—	3	3	no	maximal
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	2	yes	—
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	2	no	—
18	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
19	#63630	—	3	2	no	—
20	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	C₄×Q₈×C₉	D₃	1	3	yes	—
22	C₄⋊(C₉⋊C₈)	—	3	6	no	—
23	C₄⋊(C₉⋊C₈)	—	3	3	no	—
24	#63362	—	3	6	no	—
25	C₄×C₉⋊C₈	—	3	3	no	—
26	C₃×Q₈×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₃⋊(C₉⋊C₈)	—	2	1	no	—
33	C₃⋊(C₉⋊C₈)	—	2	2	no	—
34	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	complemented
35	#5024	—	9	3	no	—
36	#5024	—	9	1	no	—
37	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
38	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
39	C₃²×C₄⋊C₄	Dic₃	1	1	yes	characteristic
40	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
41	C₃²×C₄²	C₂×D₃	1	2	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
45	C₂×Q₈×C₉	Dic₃	1	3	yes	—
46	C₉×C₄⋊C₄	Dic₃	1	6	yes	—
47	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
48	C₄²×C₉	C₂×D₃	1	3	yes	—
49	C₄²×C₉	C₂×D₃	1	6	yes	—
50	C₂×C₉⋊C₈	—	3	3	no	—
51	C₂×C₉⋊C₈	—	3	3	no	—
52	C₂×C₉⋊C₈	—	3	6	no	—
53	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₄×C₉	—	2	2	no	—
55	C₃×C₄×C₉	—	2	1	no	—
56	C₃×C₄×C₉	—	2	2	no	—
57	C₃×C₄×C₉	M₁₆	1	2	yes	—
58	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₃×C₄×C₉	#41	1	1	yes	characteristic
60	C₃×C₄×Q₈	D₉	1	3	yes	—
61	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
62	C₄⋊(C₃⋊C₈)	—	9	6	no	—
63	C₄⋊(C₃⋊C₈)	—	9	3	no	—
64	C₄⋊(C₃⋊C₈)	—	9	2	no	—
65	C₄⋊(C₃⋊C₈)	—	9	1	no	—
66	#803	—	9	6	no	—
67	#803	—	9	2	no	—
68	C₄×C₃⋊C₈	—	9	3	no	—
69	C₄×C₃⋊C₈	—	9	1	no	—
70	C₃²×Q₈	—	2	2	no	—
71	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₃⋊(C₃⋊C₈)	—	3	2	no	—
77	C₃⋊(C₃⋊C₈)	—	6	1	no	—
78	C₃⋊(C₃⋊C₈)	—	6	2	no	—
79	Q₈×C₉	—	2	6	no	—
80	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
81	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
82	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
83	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
84	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
85	C₉⋊C₈	—	3	6	no	—
86	C₉⋊C₈	—	6	3	no	—
87	C₉⋊C₈	—	6	6	no	—
88	#446	—	27	1	no	Sylow
89	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₃×C₉	#131	1	1	yes	characteristic
92	C₂×C₃×Q₈	Dic₉	1	3	yes	—
93	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
94	C₃×C₄⋊C₄	Dic₉	1	6	yes	—
95	C₃×C₄⋊C₄	Dic₉	1	3	yes	—
96	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
97	C₃×C₄⋊C₄	C₃⋊Dic₃	1	1	yes	characteristic
98	C₃×C₄²	C₂×D₉	1	3	yes	—
99	C₃×C₄²	C₂×D₉	1	6	yes	—
100	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
102	C₂×C₃⋊C₈	—	9	3	no	—
103	C₂×C₃⋊C₈	—	9	1	no	—
104	C₂×C₃⋊C₈	—	9	3	no	—
105	C₂×C₃⋊C₈	—	9	6	no	—
106	C₂×C₃⋊C₈	—	9	1	no	—
107	C₂×C₃⋊C₈	—	9	2	no	—
108	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
109	C₃²×C₄	—	2	2	no	—
110	C₃²×C₄	—	2	1	no	—
111	C₃²×C₄	—	2	2	no	—
112	C₃²×C₄	#208	1	2	yes	—
113	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
114	C₃²×C₄	#239	1	1	yes	characteristic
115	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
116	C₄×C₉	—	2	6	no	—
117	C₄×C₉	—	2	3	no	—
118	C₄×C₉	—	2	6	no	—
119	C₄×C₉	#208	1	6	yes	—
120	C₄×C₉	D₃×Q₈	1	3	yes	—
121	C₄×C₉	#239	1	3	yes	—
122	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
123	C₄⋊C₈	—	27	1	no	—
124	C₄⋊C₈	—	27	2	no	—
125	#97	—	27	2	no	—
126	C₄×C₈	—	27	1	no	—
127	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
128	C₃×Q₈	—	2	6	no	—
129	C₃×Q₈	—	2	2	no	—
130	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
131	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
132	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
133	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
134	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
135	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
138	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
140	C₃⋊C₈	—	9	6	no	—
141	C₃⋊C₈	—	9	2	no	—
142	C₃⋊C₈	—	18	3	no	—
143	C₃⋊C₈	—	18	6	no	—
144	C₃⋊C₈	—	18	1	no	—
145	C₃⋊C₈	—	18	2	no	—
146	C₂×C₃²	C₂×#208	1	1	yes	characteristic
147	C₂×C₃²	Q₈×Dic₃	1	1	yes	characteristic
148	C₂×C₃²	#948	1	1	yes	characteristic
149	C₂×C₉	C₂×#208	1	3	yes	—
150	C₂×C₉	Q₈×Dic₃	1	3	yes	—
151	C₂×C₉	#948	1	3	yes	—
152	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic
153	C₂×C₈	—	27	1	no	—
154	C₂×C₈	—	27	1	no	—
155	C₂×C₈	—	27	2	no	—
156	C₄⋊C₄	C₃⋊Dic₉	1	2	yes	—
157	C₄⋊C₄	C₃⋊Dic₉	1	1	yes	characteristic
158	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
159	C₄²	C₂×C₃⋊D₉	1	2	yes	—
160	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
161	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
162	C₃×C₄	—	2	3	no	—
163	C₃×C₄	—	2	6	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	2	6	no	—
168	C₃×C₄	#3667	1	6	yes	—
169	C₃×C₄	Q₈×D₉	1	3	yes	—
170	C₃×C₄	#3701	1	3	yes	—
171	C₃×C₄	#3748	1	2	yes	—
172	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
173	C₃×C₄	#3832	1	1	yes	characteristic
174	C₃²	#5024	1	1	yes	characteristic
175	C₉	#5024	1	3	yes	complemented
176	Q₈	—	2	2	no	—
177	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
179	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
180	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
182	C₈	—	27	2	no	—
183	C₈	—	54	1	no	—
184	C₈	—	54	2	no	—
185	C₂×C₃	C₂×#3667	1	3	yes	—
186	C₂×C₃	Q₈×Dic₉	1	3	yes	—
187	C₂×C₃	#63510	1	3	yes	—
188	C₂×C₃	C₂×#3748	1	1	yes	characteristic
189	C₂×C₃	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
190	C₂×C₃	#64157	1	1	yes	characteristic
191	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	2	no	—
195	C₄	#88532	1	2	yes	—
196	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
197	C₄	#88746	1	1	yes	characteristic
198	C₃	#10588701	1	3	yes	complemented
199	C₃	#10592723	1	1	yes	characteristic
200	C₂	C₂×#88532	1	1	yes	characteristic
201	C₂	Q₈×C₃⋊Dic₉	1	1	yes	characteristic
202	C₂	#11721865	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂²×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	#446
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
34. C₃⋊(C₉⋊C₈)	Q₈	semidirect
127. C₃×C₉	#446	semidirect
175. C₉	#5024	semidirect
198. C₃	#10588701	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
C₄⋊(C₃⋊(C₉⋊C₈))	2
#11720516	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592723	3
#10588701	3