Subgroups — #13144402

Order 1728.

540 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	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	#11721706	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720576	C₂	1	2	yes	maximal
4	#11720526	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592707	—	3	1	no	maximal
7	#10588691	—	3	3	no	maximal
8	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	—	2	1	no	—
10	C₂×#6226	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	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×D₄	D₃	1	1	yes	characteristic
18	#64078	—	3	1	no	—
19	#63659	—	3	2	no	—
20	#63635	—	3	2	no	—
21	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	—
22	C₄×D₄×C₉	D₃	1	3	yes	—
23	#63431	—	3	3	no	—
24	#63392	—	3	6	no	—
25	#63367	—	3	6	no	—
26	C₄⋊(C₉⋊C₈)	—	3	3	no	—
27	C₂³×C₃×C₉	—	2	1	no	—
28	C₃×D₄×C₉	—	2	1	no	—
29	C₃×D₄×C₉	Q₈	1	2	yes	complemented
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	4	2	no	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊Dic₉	—	2	2	no	—
36	#6226	—	4	2	no	—
37	C₃⋊(C₉⋊C₈)	—	2	2	no	—
38	#5008	—	9	3	no	—
39	#5008	—	9	1	no	—
40	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂×#634	—	6	1	no	—
43	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
44	C₃²×C₂²⋊C₄	—	2	1	no	—
45	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
46	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
47	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
49	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
50	C₂²×C₄×C₉	—	2	3	no	—
51	C₂×Dic₁₈	—	6	3	no	—
52	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
53	C₉×C₂²⋊C₄	—	2	3	no	—
54	C₄²×C₉	C₂×D₃	1	3	yes	—
55	C₄⋊Dic₉	—	3	6	no	—
56	C₄⋊Dic₉	—	6	3	no	—
57	C₂×C₉⋊C₈	—	3	6	no	—
58	C₂²×C₃×C₉	—	2	2	no	—
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	—	4	1	no	—
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	—	4	1	no	—
64	C₃×C₄×C₉	SD₁₆	1	2	yes	—
65	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃×C₄×C₉	#41	1	1	yes	characteristic
67	C₃⋊Dic₉	—	4	2	no	—
68	C₃×C₄×D₄	D₉	1	3	yes	—
69	C₃×C₄×D₄	C₃⋊D₃	1	1	yes	characteristic
70	#869	—	9	3	no	—
71	#869	—	9	1	no	—
72	#832	—	9	6	no	—
73	#832	—	9	2	no	—
74	#808	—	9	6	no	—
75	#808	—	9	2	no	—
76	C₄⋊(C₃⋊C₈)	—	9	3	no	—
77	C₄⋊(C₃⋊C₈)	—	9	1	no	—
78	C₂³×C₃²	—	2	1	no	—
79	C₃²×D₄	—	2	1	no	—
80	C₃²×D₄	Dic₆	1	2	yes	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	4	2	no	—
83	C₂×C₃²×C₄	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	#67	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃⋊Dic₃	—	6	2	no	—
87	#634	—	12	2	no	—
88	C₂³×C₉	—	2	3	no	—
89	C₃⋊(C₃⋊C₈)	—	6	2	no	—
90	D₄×C₉	—	2	3	no	—
91	D₄×C₉	Dic₆	1	6	yes	complemented
92	C₂×C₄×C₉	—	2	3	no	—
93	C₂×C₄×C₉	—	4	6	no	—
94	C₂×C₄×C₉	#67	1	3	yes	—
95	C₂×C₄×C₉	#67	1	3	yes	—
96	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
97	C₂×Dic₉	—	6	6	no	—
98	Dic₁₈	—	12	6	no	—
99	C₉⋊C₈	—	6	6	no	—
100	#476	—	27	1	no	Sylow
101	C₂×C₃×C₉	—	4	2	no	—
102	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
106	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	2	3	no	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂×Dic₆	—	18	3	no	—
110	C₂×Dic₆	—	18	1	no	—
111	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
112	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₃×C₂²⋊C₄	—	2	3	no	—
114	C₃×C₂²⋊C₄	—	2	1	no	—
115	C₃×C₄²	C₂×D₉	1	3	yes	—
116	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
117	C₄⋊Dic₃	—	9	6	no	—
118	C₄⋊Dic₃	—	9	2	no	—
119	C₄⋊Dic₃	—	18	3	no	—
120	C₄⋊Dic₃	—	18	1	no	—
121	C₂×C₃⋊C₈	—	9	6	no	—
122	C₂×C₃⋊C₈	—	9	2	no	—
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	—	4	1	no	—
125	C₂²×C₃²	—	4	1	no	—
126	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
127	C₃²×C₄	—	2	1	no	—
128	C₃²×C₄	—	4	1	no	—
129	C₃²×C₄	#214	1	2	yes	—
130	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
131	C₃²×C₄	#235	1	1	yes	characteristic
132	C₃⋊Dic₃	—	12	2	no	—
133	C₂²×C₉	—	2	6	no	—
134	C₂²×C₉	—	4	3	no	—
135	C₂²×C₉	—	4	3	no	—
136	C₂²×C₉	C₂×#67	1	3	yes	—
137	C₄×C₉	—	2	3	no	—
138	C₄×C₉	—	4	3	no	—
139	C₄×C₉	#214	1	6	yes	—
140	C₄×C₉	C₂×Dic₆	1	3	yes	—
141	C₄×C₉	#235	1	3	yes	—
142	Dic₉	—	12	6	no	—
143	C₄⋊Q₈	—	27	1	no	—
144	C₄×D₄	C₃⋊D₉	1	1	yes	characteristic
145	#106	—	27	2	no	—
146	C₄⋊C₈	—	27	1	no	—
147	#102	—	27	2	no	—
148	C₃×C₉	#476	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₃	—	2	3	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₃×D₄	—	2	3	no	—
152	C₃×D₄	—	2	1	no	—
153	C₃×D₄	Dic₁₈	1	6	yes	complemented
154	C₃×D₄	#634	1	2	yes	—
155	C₂×C₃×C₄	—	2	3	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	4	6	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	#611	1	3	yes	—
160	C₂×C₃×C₄	#611	1	3	yes	—
161	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
162	C₂×C₃×C₄	#638	1	1	yes	characteristic
163	C₂×C₃×C₄	#638	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
165	C₂×Dic₃	—	18	6	no	—
166	C₂×Dic₃	—	18	2	no	—
167	Dic₆	—	36	2	no	—
168	Dic₆	—	36	6	no	—
169	C₃⋊C₈	—	18	6	no	—
170	C₃⋊C₈	—	18	2	no	—
171	C₂×C₃²	—	4	2	no	—
172	C₂×C₃²	#924	1	1	yes	characteristic
173	C₂×C₃²	C₂×#214	1	1	yes	characteristic
174	C₂×C₃²	#949	1	1	yes	characteristic
175	C₂×C₉	—	4	6	no	—
176	C₂×C₉	#924	1	3	yes	—
177	C₂×C₉	C₂×#214	1	3	yes	—
178	C₂×C₉	#949	1	3	yes	—
179	C₂×Q₈	—	54	1	no	—
180	C₂×D₄	C₂×C₃⋊D₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂×C₈	—	27	2	no	—
183	C₄⋊C₄	—	27	2	no	—
184	C₄⋊C₄	—	54	1	no	—
185	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	1	no	—
187	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	6	no	—
190	C₂²×C₃	—	4	3	no	—
191	C₂²×C₃	—	4	3	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	C₂×#611	1	3	yes	—
195	C₂²×C₃	C₂×#638	1	1	yes	characteristic
196	C₃×C₄	—	2	3	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	4	3	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	#3672	1	6	yes	—
201	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
202	C₃×C₄	#3697	1	3	yes	—
203	C₃×C₄	#3754	1	2	yes	—
204	C₃×C₄	C₂×#634	1	1	yes	characteristic
205	C₃×C₄	#3828	1	1	yes	characteristic
206	Dic₃	—	36	6	no	—
207	Dic₃	—	36	2	no	—
208	C₃²	#5008	1	1	yes	characteristic
209	C₉	#5008	1	3	yes	complemented
210	C₂³	—	2	1	no	—
211	Q₈	—	108	2	no	—
212	D₄	—	2	1	no	—
213	D₄	#6226	1	2	yes	complemented
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	54	2	no	—
217	C₂×C₄	#6233	1	1	yes	characteristic
218	C₂×C₄	#6233	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
220	C₈	—	54	2	no	—
221	C₂×C₃	—	4	6	no	—
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	#63486	1	3	yes	—
224	C₂×C₃	C₂×#3672	1	3	yes	—
225	C₂×C₃	#63512	1	3	yes	—
226	C₂×C₃	#64133	1	1	yes	characteristic
227	C₂×C₃	C₂×#3754	1	1	yes	characteristic
228	C₂×C₃	#64158	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	C₂×#6233	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	108	2	no	—
236	C₄	#88543	1	2	yes	—
237	C₄	C₂×#6226	1	1	yes	characteristic
238	C₄	#88736	1	1	yes	characteristic
239	C₃	#10588691	1	3	yes	complemented
240	C₃	#10592707	1	1	yes	characteristic
241	C₂	—	4	2	no	—
242	C₂	#11721813	1	1	yes	characteristic
243	C₂	C₂×#88543	1	1	yes	characteristic
244	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×D₄×C₉	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₄×D₄	C₃⋊D₉
hypocenter	C₃×C₉	#476
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #11721706	C₂	semidirect
4. #11720526	C₂	semidirect
5. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	semidirect
29. C₃×D₄×C₉	Q₈	semidirect
91. D₄×C₉	Dic₆	semidirect
148. C₃×C₉	#476	semidirect
153. C₃×D₄	Dic₁₈	semidirect
209. C₉	#5008	semidirect
213. D₄	#6226	semidirect
239. C₃	#10588691	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄×C₉	2
#11721706	2
#11720576	2
#11720526	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
#10592707	3
#10588691	3