Subgroups — #13144322

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#11720572	C₂	1	4	yes	maximal
5	#10592676	—	3	1	no	maximal
6	#10588657	—	3	3	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
10	C₂²×C₃⋊Dic₉	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
13	C₄×C₃⋊Dic₉	C₄	1	4	yes	complemented
14	C₂×C₃²×C₄⋊C₄	D₃	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
16	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
17	#63658	—	3	4	no	—
18	C₂×C₉×C₄⋊C₄	D₃	1	3	yes	—
19	C₂×C₄⋊Dic₉	—	3	3	no	—
20	C₂×C₄×Dic₉	—	3	3	no	—
21	C₉⋊#95	—	3	12	no	—
22	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	—	2	2	no	—
24	C₂×C₃×C₄×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃⋊Dic₉	—	2	2	no	—
29	C₂×C₃⋊Dic₉	—	2	2	no	—
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	—	2	4	no	—
32	C₂×C₃⋊Dic₉	C₂×C₄	1	4	yes	—
33	#4977	—	9	3	no	—
34	#4977	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	2	no	—
38	C₂²×C₃⋊Dic₃	—	3	2	no	—
39	C₃²×C₄⋊C₄	—	2	2	no	—
40	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
41	C₄×C₃⋊Dic₃	—	3	4	no	—
42	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
43	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
44	C₂²×Dic₉	—	3	6	no	—
45	C₂²×Dic₉	—	3	6	no	—
46	C₉×C₄⋊C₄	—	2	6	no	—
47	C₄⋊Dic₉	—	6	6	no	—
48	C₄×Dic₉	—	3	12	no	—
49	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₃×C₉	#41	1	2	yes	—
53	C₂²×C₃×C₉	#41	1	2	yes	—
54	C₃×C₄×C₉	—	4	2	no	—
55	C₃×C₄×C₉	C₄⋊C₄	1	4	yes	—
56	C₃⋊Dic₉	—	2	4	no	—
57	C₃⋊Dic₉	—	4	2	no	—
58	C₂×C₃×C₄⋊C₄	D₉	1	3	yes	—
59	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	9	3	no	—
61	C₂×C₄⋊Dic₃	—	9	1	no	—
62	C₂×C₄×Dic₃	—	9	3	no	—
63	C₂×C₄×Dic₃	—	9	1	no	—
64	C₃⋊#95	—	9	12	no	—
65	C₃⋊#95	—	9	4	no	—
66	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	4	no	—
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	Dic₆	1	2	yes	—
70	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
71	C₂×C₃²×C₄	#67	1	2	yes	—
72	C₂×C₃⋊Dic₃	—	3	4	no	—
73	C₂×C₃⋊Dic₃	—	6	2	no	—
74	C₂×C₃⋊Dic₃	—	6	2	no	—
75	C₂×C₃⋊Dic₃	—	6	4	no	—
76	C₂×C₃⋊Dic₃	—	6	2	no	—
77	C₂³×C₉	C₂²×D₃	1	3	yes	—
78	C₂×C₄×C₉	—	2	12	no	—
79	C₂×C₄×C₉	—	2	6	no	—
80	C₂×C₄×C₉	Dic₆	1	6	yes	—
81	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
82	C₂×C₄×C₉	#67	1	6	yes	—
83	C₂×Dic₉	—	3	12	no	—
84	C₂×Dic₉	—	6	6	no	—
85	C₂×Dic₉	—	6	6	no	—
86	C₂×Dic₉	—	6	12	no	—
87	C₂×Dic₉	—	6	6	no	—
88	#382	—	27	1	no	Sylow
89	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₃×C₉	#117	1	2	yes	—
91	C₂×C₃×C₉	#124	1	1	yes	characteristic
92	C₂×C₃×C₉	#124	1	1	yes	characteristic
93	C₂×C₃×C₉	#125	1	2	yes	—
94	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
95	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
96	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	9	6	no	—
99	C₂²×Dic₃	—	9	2	no	—
100	C₂²×Dic₃	—	9	6	no	—
101	C₂²×Dic₃	—	9	2	no	—
102	C₃×C₄⋊C₄	—	2	6	no	—
103	C₃×C₄⋊C₄	—	2	2	no	—
104	C₄⋊Dic₃	—	18	6	no	—
105	C₄⋊Dic₃	—	18	2	no	—
106	C₄×Dic₃	—	9	12	no	—
107	C₄×Dic₃	—	9	4	no	—
108	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
109	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
110	C₂²×C₃²	#237	1	2	yes	—
111	C₂²×C₃²	#239	1	2	yes	—
112	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
113	C₃²×C₄	—	4	2	no	—
114	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
115	C₃⋊Dic₃	—	6	4	no	—
116	C₃⋊Dic₃	—	12	2	no	—
117	C₂²×C₉	C₂×Dic₆	1	3	yes	—
118	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
119	C₂²×C₉	#237	1	6	yes	—
120	C₂²×C₉	#239	1	6	yes	—
121	C₂²×C₉	C₂×#67	1	3	yes	—
122	C₄×C₉	—	4	6	no	—
123	C₄×C₉	Dic₃⋊C₄	1	12	yes	—
124	Dic₉	—	6	12	no	—
125	Dic₉	—	12	6	no	—
126	C₂×C₄⋊C₄	—	27	1	no	—
127	C₂×C₄⋊C₄	C₃⋊D₉	1	1	yes	characteristic
128	C₂×C₄²	—	27	1	no	—
129	#95	—	27	4	no	—
130	C₃×C₉	#382	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₃	C₂²×D₉	1	3	yes	—
132	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	12	no	—
134	C₂×C₃×C₄	—	2	6	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
138	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
139	C₂×C₃×C₄	#611	1	6	yes	—
140	C₂×C₃×C₄	#634	1	2	yes	—
141	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
142	C₂×C₃×C₄	#638	1	2	yes	—
143	C₂×Dic₃	—	9	12	no	—
144	C₂×Dic₃	—	9	4	no	—
145	C₂×Dic₃	—	18	6	no	—
146	C₂×Dic₃	—	18	6	no	—
147	C₂×Dic₃	—	18	2	no	—
148	C₂×Dic₃	—	18	2	no	—
149	C₂×Dic₃	—	18	12	no	—
150	C₂×Dic₃	—	18	6	no	—
151	C₂×Dic₃	—	18	4	no	—
152	C₂×Dic₃	—	18	2	no	—
153	C₂×C₃²	#890	1	2	yes	—
154	C₂×C₃²	#892	1	2	yes	—
155	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
156	C₂×C₃²	#936	1	1	yes	characteristic
157	C₂×C₃²	#947	1	1	yes	characteristic
158	C₂×C₉	#890	1	6	yes	—
159	C₂×C₉	#892	1	6	yes	—
160	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
161	C₂×C₉	#936	1	3	yes	—
162	C₂×C₉	#947	1	3	yes	—
163	C₂²×C₄	—	27	2	no	—
164	C₂²×C₄	—	27	2	no	—
165	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
166	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	54	2	no	—
169	C₄²	—	27	4	no	—
170	C₂²×C₃	C₂×Dic₁₈	1	3	yes	—
171	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
172	C₂²×C₃	#3699	1	6	yes	—
173	C₂²×C₃	#3701	1	6	yes	—
174	C₂²×C₃	C₂×#611	1	3	yes	—
175	C₂²×C₃	C₂×#634	1	1	yes	characteristic
176	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
177	C₂²×C₃	#3830	1	2	yes	—
178	C₂²×C₃	#3832	1	2	yes	—
179	C₂²×C₃	C₂×#638	1	1	yes	characteristic
180	C₃×C₄	—	4	6	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	Dic₉⋊C₄	1	12	yes	—
183	C₃×C₄	#3750	1	4	yes	—
184	Dic₃	—	18	4	no	—
185	Dic₃	—	18	12	no	—
186	Dic₃	—	36	6	no	—
187	Dic₃	—	36	2	no	—
188	C₃²	#4977	1	1	yes	characteristic
189	C₉	#4977	1	3	yes	complemented
190	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	27	4	no	—
194	C₂×C₄	—	54	2	no	—
195	C₂×C₄	—	54	2	no	—
196	C₂×C₄	—	54	4	no	—
197	C₂×C₄	—	54	2	no	—
198	C₂×C₄	#6226	1	2	yes	—
199	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
200	C₂×C₄	#6233	1	2	yes	—
201	C₂×C₃	#63452	1	6	yes	—
202	C₂×C₃	#63454	1	6	yes	—
203	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
204	C₂×C₃	#63498	1	3	yes	—
205	C₂×C₃	#63509	1	3	yes	—
206	C₂×C₃	#64099	1	2	yes	—
207	C₂×C₃	#64101	1	2	yes	—
208	C₂×C₃	C₂×#3750	1	1	yes	characteristic
209	C₂×C₃	#64145	1	1	yes	characteristic
210	C₂×C₃	#64156	1	1	yes	characteristic
211	C₂²	C₂×#6226	1	1	yes	characteristic
212	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
213	C₂²	#88742	1	2	yes	—
214	C₂²	#88746	1	2	yes	—
215	C₂²	C₂×#6233	1	1	yes	characteristic
216	C₄	—	4	2	no	—
217	C₄	—	54	4	no	—
218	C₄	—	108	2	no	—
219	C₄	#88534	1	4	yes	—
220	C₃	#10588657	1	3	yes	complemented
221	C₃	#10592676	1	1	yes	characteristic
222	C₂	#11721746	1	2	yes	—
223	C₂	#11721748	1	2	yes	—
224	C₂	C₂×#88534	1	1	yes	characteristic
225	C₂	#11721836	1	1	yes	characteristic
226	C₂	#11721857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊D₉
hypocenter	C₃×C₉	#382
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. C₄×C₃⋊Dic₉	C₄	semidirect
130. C₃×C₉	#382	semidirect
189. C₉	#4977	semidirect
220. C₃	#10588657	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×C₄⋊C₄	2
C₂×C₄⋊(C₃⋊Dic₉)	2
C₂×C₄×C₃⋊Dic₉	2
#11720572	2
#10592676	3
#10588657	3