Subgroups — #13135935

Order 1728.

450 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	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10589424	—	3	1	no	maximal
5	#10588231	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×M₁₆	—	2	1	no	—
8	C₃×C₉×M₁₆	C₄	1	2	yes	—
9	C₂×C₃×C₈×C₉	—	2	1	no	—
10	#88532	—	2	1	no	—
11	#88532	C₄	1	2	yes	—
12	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
13	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
14	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	—
16	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
17	C₂×#3748	—	3	1	no	—
18	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
19	C₂×C₉×M₁₆	D₃	1	3	yes	—
20	C₂×#3667	—	3	3	no	—
21	C₂²×C₉⋊C₈	—	3	3	no	—
22	C₂³×C₃×C₉	Q₈	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₃×C₈×C₉	—	2	2	no	—
30	C₃⋊(C₉⋊C₈)	—	2	2	no	—
31	C₃⋊(C₉⋊C₈)	—	2	2	no	—
32	#4558	—	9	3	no	—
33	#4558	—	9	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×M₁₆	—	2	1	no	—
36	C₃²×M₁₆	Dic₃	1	2	yes	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	#3748	—	3	2	no	—
39	#3748	—	6	1	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
45	C₉×M₁₆	—	2	3	no	—
46	C₉×M₁₆	Dic₃	1	6	yes	—
47	C₂×C₈×C₉	—	2	3	no	—
48	#3667	—	3	6	no	—
49	#3667	—	6	3	no	—
50	C₂×C₉⋊C₈	—	3	6	no	—
51	C₂×C₉⋊C₈	—	6	3	no	—
52	C₂×C₉⋊C₈	—	6	3	no	—
53	C₂×C₉⋊C₈	—	6	3	no	—
54	C₂²×C₃×C₉	—	2	2	no	—
55	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
59	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₃×M₁₆	D₉	1	3	yes	—
63	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
64	C₂×#208	—	9	3	no	—
65	C₂×#208	—	9	1	no	—
66	C₂²×C₃⋊C₈	—	9	3	no	—
67	C₂²×C₃⋊C₈	—	9	1	no	—
68	C₂³×C₃²	Dic₆	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	#67	1	1	yes	characteristic
74	C₂×C₃²×C₄	#67	1	1	yes	characteristic
75	C₂³×C₉	Dic₆	1	3	yes	—
76	C₃²×C₈	—	2	2	no	—
77	C₃⋊(C₃⋊C₈)	—	6	2	no	—
78	C₃⋊(C₃⋊C₈)	—	6	2	no	—
79	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
80	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
81	C₂×C₄×C₉	D₁₂	1	3	yes	—
82	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
83	C₂×C₄×C₉	#67	1	3	yes	—
84	C₂×C₄×C₉	#67	1	3	yes	—
85	C₈×C₉	—	2	6	no	—
86	C₉⋊C₈	—	6	6	no	—
87	C₉⋊C₈	—	6	6	no	—
88	#341	—	27	1	no	Sylow
89	C₂×C₃×C₉	—	2	1	no	—
90	C₂×C₃×C₉	—	2	1	no	—
91	C₂×C₃×C₉	#95	1	1	yes	characteristic
92	C₂×C₃×C₉	#108	1	2	yes	—
93	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
94	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₃×M₁₆	—	2	3	no	—
96	C₃×M₁₆	—	2	1	no	—
97	C₃×M₁₆	Dic₉	1	6	yes	—
98	C₃×M₁₆	C₃⋊Dic₃	1	2	yes	—
99	C₂×C₃×C₈	—	2	3	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	#208	—	9	6	no	—
102	#208	—	9	2	no	—
103	#208	—	18	3	no	—
104	#208	—	18	1	no	—
105	C₂×C₃⋊C₈	—	9	6	no	—
106	C₂×C₃⋊C₈	—	9	2	no	—
107	C₂×C₃⋊C₈	—	18	3	no	—
108	C₂×C₃⋊C₈	—	18	3	no	—
109	C₂×C₃⋊C₈	—	18	3	no	—
110	C₂×C₃⋊C₈	—	18	1	no	—
111	C₂×C₃⋊C₈	—	18	1	no	—
112	C₂×C₃⋊C₈	—	18	1	no	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
115	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
116	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
117	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
118	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
119	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
120	C₃²×C₄	#217	1	1	yes	characteristic
121	C₂²×C₉	—	2	6	no	—
122	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
123	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
124	C₂²×C₉	C₄⋊Dic₃	1	3	yes	—
125	C₄×C₉	C₄×Dic₃	1	3	yes	—
126	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
127	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
128	C₄×C₉	#217	1	3	yes	—
129	C₂×M₁₆	—	27	1	no	—
130	C₂×M₁₆	C₃⋊D₉	1	1	yes	characteristic
131	C₂²×C₈	—	27	1	no	—
132	C₃×C₉	#341	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₃	Dic₁₈	1	3	yes	—
134	C₂³×C₃	#634	1	1	yes	characteristic
135	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
136	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
137	C₂×C₃×C₄	D₃₆	1	3	yes	—
138	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
139	C₂×C₃×C₄	#611	1	3	yes	—
140	C₂×C₃×C₄	#611	1	3	yes	—
141	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	#638	1	1	yes	characteristic
146	C₂×C₃×C₄	#638	1	1	yes	characteristic
147	C₃×C₈	—	2	6	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃⋊C₈	—	18	6	no	—
150	C₃⋊C₈	—	18	6	no	—
151	C₃⋊C₈	—	18	2	no	—
152	C₃⋊C₈	—	18	2	no	—
153	C₂×C₃²	—	2	1	no	—
154	C₂×C₃²	—	2	1	no	—
155	C₂×C₃²	#822	1	2	yes	—
156	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
157	C₂×C₉	—	2	3	no	—
158	C₂×C₉	—	2	3	no	—
159	C₂×C₉	#822	1	6	yes	—
160	C₂×C₉	C₃⋊#95	1	3	yes	—
161	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
162	M₁₆	—	2	1	no	—
163	M₁₆	—	27	2	no	—
164	M₁₆	—	54	1	no	—
165	M₁₆	C₃⋊Dic₉	1	2	yes	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	27	2	no	—
168	C₂×C₈	—	54	1	no	—
169	C₂×C₈	—	54	1	no	—
170	C₂×C₈	—	54	1	no	—
171	C₂²×C₃	—	2	6	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
174	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
175	C₂²×C₃	C₄⋊Dic₉	1	3	yes	—
176	C₂²×C₃	#3750	1	1	yes	characteristic
177	C₂²×C₃	#3750	1	1	yes	characteristic
178	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
179	C₃×C₄	C₄×Dic₉	1	3	yes	—
180	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
181	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
182	C₃×C₄	#3676	1	3	yes	—
183	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
184	C₃×C₄	#3752	1	1	yes	characteristic
185	C₃×C₄	#3752	1	1	yes	characteristic
186	C₃×C₄	#3757	1	1	yes	characteristic
187	C₃²	#4558	1	1	yes	characteristic
188	C₉	#4558	1	3	yes	complemented
189	C₂³	#6226	1	1	yes	characteristic
190	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
191	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
192	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
194	C₂×C₄	#6233	1	1	yes	characteristic
195	C₂×C₄	#6233	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	54	2	no	—
198	C₈	—	54	2	no	—
199	C₂×C₃	—	2	3	no	—
200	C₂×C₃	—	2	3	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	#63381	1	6	yes	—
204	C₂×C₃	C₉⋊#95	1	3	yes	—
205	C₂×C₃	#63649	1	2	yes	—
206	C₂×C₃	#63658	1	1	yes	characteristic
207	C₂²	—	2	2	no	—
208	C₂²	#88534	1	1	yes	characteristic
209	C₂²	#88534	1	1	yes	characteristic
210	C₂²	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
211	C₄	C₄×C₃⋊Dic₉	1	1	yes	characteristic
212	C₄	#88536	1	1	yes	characteristic
213	C₄	#88536	1	1	yes	characteristic
214	C₄	#88551	1	1	yes	characteristic
215	C₃	#10588231	1	3	yes	complemented
216	C₃	#10589424	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	#11720555	1	2	yes	—
220	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	Dic₃⋊C₄
prime radical	C₂³×C₃²	Dic₆
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#341
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
132. C₃×C₉	#341	semidirect
188. C₉	#4558	semidirect
215. C₃	#10588231	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×M₁₆	2
C₂×#88532	2
C₂²×C₃⋊(C₉⋊C₈)	2
#10589424	3
#10588231	3