Subgroups — #13135938

Order 1728.

490 conjugacy classes of subgroups.

The subgroup lattice has 246 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×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic
4	#10589420	—	3	1	no	maximal
5	#10588234	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
8	C₃×C₉×M₁₆	—	2	1	no	—
9	C₃×C₉×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃×C₈×C₉	—	2	1	no	—
11	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
12	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
13	#88532	—	2	1	no	—
14	#88532	C₄	1	2	yes	complemented
15	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
16	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
17	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
18	C₂×#3748	—	3	1	no	—
19	C₂×C₉×M₁₆	D₃	1	3	yes	—
20	C₂×C₄⋊Dic₉	—	3	3	no	—
21	C₂×#3667	—	3	3	no	—
22	C₂³×C₃×C₉	C₂×C₄	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	D₄	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	Q₈	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	—	2	2	no	—
30	C₂×C₃⋊Dic₉	—	4	2	no	—
31	C₃×C₈×C₉	—	2	2	no	—
32	C₃⋊(C₉⋊C₈)	—	2	2	no	—
33	#4554	—	9	3	no	—
34	#4554	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃⋊Dic₃	—	3	2	no	—
37	C₃²×M₁₆	—	2	1	no	—
38	C₃²×M₁₆	Dic₃	1	2	yes	—
39	C₂×C₃²×C₈	—	2	1	no	—
40	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
41	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
42	#3748	—	3	2	no	—
43	#3748	—	6	1	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
45	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
46	C₂²×Dic₉	—	3	6	no	—
47	C₉×M₁₆	—	2	3	no	—
48	C₉×M₁₆	Dic₃	1	6	yes	complemented
49	C₂×C₈×C₉	—	2	3	no	—
50	C₄⋊Dic₉	—	6	3	no	—
51	C₄⋊Dic₉	—	6	3	no	—
52	#3667	—	3	6	no	—
53	#3667	—	6	3	no	—
54	C₂×C₉⋊C₈	—	6	3	no	—
55	C₂²×C₃×C₉	—	2	1	no	—
56	C₂²×C₃×C₉	—	2	1	no	—
57	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×C₉	C₄⋊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	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₃⋊Dic₉	—	4	2	no	—
65	C₂×C₃×M₁₆	D₉	1	3	yes	—
66	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
67	C₂×C₄⋊Dic₃	—	9	3	no	—
68	C₂×C₄⋊Dic₃	—	9	1	no	—
69	C₂×#208	—	9	3	no	—
70	C₂×#208	—	9	1	no	—
71	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	#67	1	1	yes	characteristic
77	C₂×C₃²×C₄	#67	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	6	2	no	—
79	C₂×C₃⋊Dic₃	—	12	2	no	—
80	C₂³×C₉	C₄×D₃	1	3	yes	—
81	C₃²×C₈	—	2	2	no	—
82	C₃⋊(C₃⋊C₈)	—	6	2	no	—
83	C₂×C₄×C₉	Dic₆	1	3	yes	—
84	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
85	C₂×C₄×C₉	D₁₂	1	3	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	#67	1	3	yes	—
88	C₂×C₄×C₉	#67	1	3	yes	—
89	C₂×Dic₉	—	6	6	no	—
90	C₂×Dic₉	—	12	6	no	—
91	C₈×C₉	—	2	6	no	—
92	C₉⋊C₈	—	6	6	no	—
93	#343	—	27	1	no	Sylow
94	C₂×C₃×C₉	—	2	1	no	—
95	C₂×C₃×C₉	—	2	1	no	—
96	C₂×C₃×C₉	#95	1	1	yes	characteristic
97	C₂×C₃×C₉	#100	1	1	yes	characteristic
98	C₂×C₃×C₉	#101	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
100	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	9	6	no	—
102	C₂²×Dic₃	—	9	2	no	—
103	C₃×M₁₆	—	2	3	no	—
104	C₃×M₁₆	—	2	1	no	—
105	C₃×M₁₆	Dic₉	1	6	yes	complemented
106	C₃×M₁₆	C₃⋊Dic₃	1	2	yes	—
107	C₂×C₃×C₈	—	2	3	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₄⋊Dic₃	—	18	3	no	—
110	C₄⋊Dic₃	—	18	3	no	—
111	C₄⋊Dic₃	—	18	1	no	—
112	C₄⋊Dic₃	—	18	1	no	—
113	#208	—	9	6	no	—
114	#208	—	9	2	no	—
115	#208	—	18	3	no	—
116	#208	—	18	1	no	—
117	C₂×C₃⋊C₈	—	18	3	no	—
118	C₂×C₃⋊C₈	—	18	1	no	—
119	C₂²×C₃²	—	2	1	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
122	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
123	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
124	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
125	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
126	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
127	C₃²×C₄	#217	1	1	yes	characteristic
128	C₃⋊Dic₃	—	12	2	no	—
129	C₂²×C₉	—	2	3	no	—
130	C₂²×C₉	—	2	3	no	—
131	C₂²×C₉	C₄×Dic₃	1	3	yes	—
132	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
133	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
134	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
135	C₄×C₉	C₄⋊Dic₃	1	3	yes	—
136	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
137	C₄×C₉	#217	1	3	yes	—
138	Dic₉	—	12	6	no	—
139	C₂×M₁₆	—	27	1	no	—
140	C₂×M₁₆	C₃⋊D₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	27	1	no	—
142	C₃×C₉	#343	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₃	C₄×D₉	1	3	yes	—
144	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
145	C₂×C₃×C₄	Dic₁₈	1	3	yes	—
146	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
147	C₂×C₃×C₄	D₃₆	1	3	yes	—
148	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
149	C₂×C₃×C₄	#611	1	3	yes	—
150	C₂×C₃×C₄	#611	1	3	yes	—
151	C₂×C₃×C₄	#634	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₄	#638	1	1	yes	characteristic
156	C₂×C₃×C₄	#638	1	1	yes	characteristic
157	C₂×Dic₃	—	18	6	no	—
158	C₂×Dic₃	—	18	2	no	—
159	C₂×Dic₃	—	36	2	no	—
160	C₂×Dic₃	—	36	6	no	—
161	C₃×C₈	—	2	6	no	—
162	C₃×C₈	—	2	2	no	—
163	C₃⋊C₈	—	18	6	no	—
164	C₃⋊C₈	—	18	2	no	—
165	C₂×C₃²	—	2	1	no	—
166	C₂×C₃²	—	2	1	no	—
167	C₂×C₃²	#823	1	1	yes	characteristic
168	C₂×C₃²	#824	1	1	yes	characteristic
169	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
170	C₂×C₉	—	2	3	no	—
171	C₂×C₉	—	2	3	no	—
172	C₂×C₉	#823	1	3	yes	—
173	C₂×C₉	#824	1	3	yes	—
174	C₂×C₉	C₃⋊#95	1	3	yes	—
175	C₂²×C₄	—	27	2	no	—
176	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
177	M₁₆	—	2	1	no	—
178	M₁₆	—	27	2	no	—
179	M₁₆	—	54	1	no	—
180	M₁₆	C₃⋊Dic₉	1	2	yes	complemented
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	54	1	no	—
183	C₄⋊C₄	—	54	1	no	—
184	C₄⋊C₄	—	54	1	no	—
185	C₂²×C₃	—	2	3	no	—
186	C₂²×C₃	—	2	3	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	C₄×Dic₉	1	3	yes	—
190	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
191	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
192	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
193	C₂²×C₃	#3750	1	1	yes	characteristic
194	C₂²×C₃	#3752	1	1	yes	characteristic
195	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
196	C₃×C₄	C₄⋊Dic₉	1	3	yes	—
197	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
198	C₃×C₄	#3676	1	3	yes	—
199	C₃×C₄	#3750	1	1	yes	characteristic
200	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
201	C₃×C₄	#3752	1	1	yes	characteristic
202	C₃×C₄	#3757	1	1	yes	characteristic
203	Dic₃	—	36	6	no	—
204	Dic₃	—	36	2	no	—
205	C₃²	#4554	1	1	yes	characteristic
206	C₉	#4554	1	3	yes	complemented
207	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
208	C₂×C₄	—	54	2	no	—
209	C₂×C₄	—	108	2	no	—
210	C₂×C₄	#6226	1	1	yes	characteristic
211	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
212	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
213	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
214	C₂×C₄	#6233	1	1	yes	characteristic
215	C₂×C₄	#6233	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	54	2	no	—
218	C₂×C₃	—	2	3	no	—
219	C₂×C₃	—	2	3	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	#63382	1	3	yes	—
223	C₂×C₃	#63383	1	3	yes	—
224	C₂×C₃	C₉⋊#95	1	3	yes	—
225	C₂×C₃	#63650	1	1	yes	characteristic
226	C₂×C₃	#63651	1	1	yes	characteristic
227	C₂×C₃	#63658	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	C₄×C₃⋊Dic₉	1	1	yes	characteristic
231	C₂²	#88534	1	1	yes	characteristic
232	C₂²	#88536	1	1	yes	characteristic
233	C₄	—	108	2	no	—
234	C₄	#88534	1	1	yes	characteristic
235	C₄	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
236	C₄	#88536	1	1	yes	characteristic
237	C₄	#88551	1	1	yes	characteristic
238	C₃	#10588234	1	3	yes	complemented
239	C₃	#10589420	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	#11720556	1	1	yes	characteristic
243	C₂	#11720557	1	1	yes	characteristic
244	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88536
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#343
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
9. C₃×C₉×M₁₆	C₄	semidirect
14. #88532	C₄	semidirect
48. C₉×M₁₆	Dic₃	semidirect
105. C₃×M₁₆	Dic₉	semidirect
142. C₃×C₉	#343	semidirect
180. M₁₆	C₃⋊Dic₉	semidirect
206. C₉	#4554	semidirect
238. C₃	#10588234	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×C₄⋊(C₃⋊Dic₉)	2
C₂×#88532	2
#10589420	3
#10588234	3