Subgroups — #13144849

Order 1728.

750 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	2	yes	maximal, complemented
3	#11720581	C₂	1	2	yes	maximal, complemented
4	#11720575	C₂	1	2	yes	maximal, complemented
5	#10592936	—	3	1	no	maximal
6	#10588915	—	3	3	no	maximal
7	C₃×C₉×#41	—	2	2	no	—
8	C₃×C₉×#41	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₄×C₉	C₄	1	2	yes	—
11	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	#88532	—	2	2	no	—
15	#88532	C₂²	1	4	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
17	C₂×C₃²×#41	D₃	1	1	yes	characteristic
18	C₂×#3748	—	3	2	no	—
19	C₂×C₉×#41	D₃	1	3	yes	—
20	#63663	—	3	2	no	—
21	#63660	—	3	2	no	—
22	C₂×#3667	—	3	6	no	—
23	#63394	—	3	6	no	—
24	#63391	—	3	6	no	—
25	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
26	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
27	C₃×Q₈×C₉	—	2	2	no	—
28	C₃×D₄×C₉	—	2	4	no	—
29	C₃×D₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₃⋊(C₉⋊C₈)	—	2	4	no	—
38	#5237	—	9	3	no	—
39	#5237	—	9	1	no	—
40	C₃²×#41	—	2	2	no	—
41	C₃²×#41	—	2	2	no	—
42	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃²×D₄	Dic₃	1	2	yes	—
44	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
46	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
47	#3748	—	3	4	no	—
48	#3748	—	6	2	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
50	C₉×#41	—	2	6	no	—
51	C₉×#41	—	2	6	no	—
52	C₂×Q₈×C₉	C₂×D₃	1	3	yes	—
53	C₂×D₄×C₉	Dic₃	1	6	yes	—
54	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
55	C₂²×C₄×C₉	Dic₃	1	6	yes	—
56	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
57	#3667	—	3	12	no	—
58	#3667	—	6	6	no	—
59	C₂×C₉⋊C₈	—	6	6	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	—	4	1	no	—
62	C₂²×C₃×C₉	—	4	2	no	—
63	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	—	2	2	no	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
69	C₂×C₃×#41	D₉	1	3	yes	—
70	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
71	C₂×#208	—	9	6	no	—
72	C₂×#208	—	9	2	no	—
73	#836	—	9	6	no	—
74	#836	—	9	2	no	—
75	#833	—	9	6	no	—
76	#833	—	9	2	no	—
77	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
78	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
79	C₃²×Q₈	—	2	2	no	—
80	C₃²×D₄	—	2	4	no	—
81	C₃²×D₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	#67	1	2	yes	—
87	C₂×C₃²×C₄	#67	1	2	yes	—
88	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂³×C₉	C₂×Dic₃	1	6	yes	—
90	C₂³×C₉	C₂×Dic₃	1	3	yes	—
91	C₃⋊(C₃⋊C₈)	—	6	4	no	—
92	Q₈×C₉	—	2	6	no	—
93	D₄×C₉	—	2	12	no	—
94	D₄×C₉	—	2	6	no	—
95	C₂×C₄×C₉	—	2	6	no	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
98	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
99	C₂×C₄×C₉	#67	1	6	yes	—
100	C₂×C₄×C₉	#67	1	6	yes	—
101	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
102	C₉⋊C₈	—	6	12	no	—
103	#413	—	27	1	no	Sylow
104	C₂×C₃×C₉	—	2	1	no	—
105	C₂×C₃×C₉	—	2	2	no	—
106	C₂×C₃×C₉	—	4	2	no	—
107	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₃×#41	—	2	6	no	—
109	C₃×#41	—	2	6	no	—
110	C₃×#41	—	2	2	no	—
111	C₃×#41	—	2	2	no	—
112	C₂×C₃×Q₈	C₂×D₉	1	3	yes	—
113	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
114	C₂×C₃×D₄	Dic₉	1	6	yes	—
115	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
116	C₂×C₃×D₄	C₃⋊Dic₃	1	2	yes	—
117	C₂×C₃×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	Dic₉	1	6	yes	—
119	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
120	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
121	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
122	#208	—	9	12	no	—
123	#208	—	9	4	no	—
124	#208	—	18	6	no	—
125	#208	—	18	2	no	—
126	C₂×C₃⋊C₈	—	18	6	no	—
127	C₂×C₃⋊C₈	—	18	2	no	—
128	C₂²×C₃²	—	2	2	no	—
129	C₂²×C₃²	—	4	1	no	—
130	C₂²×C₃²	—	4	2	no	—
131	C₂²×C₃²	#217	1	2	yes	—
132	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
133	C₃²×C₄	—	2	2	no	—
134	C₃²×C₄	#217	1	1	yes	characteristic
135	C₃²×C₄	#217	1	1	yes	characteristic
136	C₃²×C₄	C₂×#67	1	2	yes	—
137	C₂²×C₉	—	2	6	no	—
138	C₂²×C₉	—	4	3	no	—
139	C₂²×C₉	—	4	6	no	—
140	C₂²×C₉	#217	1	6	yes	—
141	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
142	C₄×C₉	—	2	6	no	—
143	C₄×C₉	#217	1	3	yes	—
144	C₄×C₉	#217	1	3	yes	—
145	C₄×C₉	C₂×#67	1	6	yes	—
146	C₂×#41	C₃⋊D₉	1	1	yes	characteristic
147	C₂×M₁₆	—	27	2	no	—
148	#101	—	27	2	no	—
149	#100	—	27	2	no	—
150	C₃×C₉	#413	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	C₂×Dic₉	1	6	yes	—
152	C₂³×C₃	C₂×Dic₉	1	3	yes	—
153	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
154	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
155	C₃×Q₈	—	2	6	no	—
156	C₃×Q₈	—	2	2	no	—
157	C₃×D₄	—	2	12	no	—
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	2	6	no	—
160	C₃×D₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	6	no	—
162	C₂×C₃×C₄	—	2	6	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
166	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
167	C₂×C₃×C₄	#611	1	6	yes	—
168	C₂×C₃×C₄	#611	1	6	yes	—
169	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
170	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
172	C₂×C₃×C₄	#638	1	2	yes	—
173	C₂×C₃×C₄	#638	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
175	C₃⋊C₈	—	18	12	no	—
176	C₃⋊C₈	—	18	4	no	—
177	C₂×C₃²	—	2	1	no	—
178	C₂×C₃²	—	2	2	no	—
179	C₂×C₃²	—	4	2	no	—
180	C₂×C₃²	C₂×#217	1	1	yes	characteristic
181	C₂×C₉	—	2	3	no	—
182	C₂×C₉	—	2	6	no	—
183	C₂×C₉	—	4	6	no	—
184	C₂×C₉	C₂×#217	1	3	yes	—
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	C₂×Q₈	C₂×C₃⋊D₉	1	1	yes	characteristic
188	C₂×D₄	C₃⋊Dic₉	1	2	yes	—
189	C₂×D₄	C₂×C₃⋊D₉	1	1	yes	characteristic
190	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
191	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
192	M₁₆	—	27	4	no	—
193	M₁₆	—	54	2	no	—
194	C₂×C₈	—	54	2	no	—
195	C₂²×C₃	—	2	6	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	4	3	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	6	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	#3676	1	6	yes	—
202	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
203	C₂²×C₃	#3757	1	2	yes	—
204	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
205	C₃×C₄	—	2	6	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	#3676	1	3	yes	—
208	C₃×C₄	#3676	1	3	yes	—
209	C₃×C₄	C₂×#611	1	6	yes	—
210	C₃×C₄	#3757	1	1	yes	characteristic
211	C₃×C₄	#3757	1	1	yes	characteristic
212	C₃×C₄	C₂×#638	1	2	yes	—
213	C₃²	#5237	1	1	yes	characteristic
214	C₉	#5237	1	3	yes	complemented
215	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
216	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
223	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
224	C₂×C₄	#6233	1	2	yes	—
225	C₂×C₄	#6233	1	2	yes	—
226	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
227	C₈	—	54	4	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	3	no	—
231	C₂×C₃	—	2	6	no	—
232	C₂×C₃	—	4	2	no	—
233	C₂×C₃	—	4	6	no	—
234	C₂×C₃	C₂×#3676	1	3	yes	—
235	C₂×C₃	C₂×#3757	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#88551	1	2	yes	—
240	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	#88551	1	1	yes	characteristic
243	C₄	#88551	1	1	yes	characteristic
244	C₄	C₂×#6233	1	2	yes	—
245	C₃	#10588915	1	3	yes	complemented
246	C₃	#10592936	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	2	no	—
250	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88551
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂×C₃²	C₂×#217
prime radical	C₂×C₃²×#41	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×#41	C₃⋊D₉
hypocenter	C₃×C₉	#413
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#88532	C₂	semidirect
3. #11720581	C₂	semidirect
4. #11720575	C₂	semidirect
150. C₃×C₉	#413	semidirect
214. C₉	#5237	semidirect
245. C₃	#10588915	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×#41	2
C₂×#88532	2
#11720581	2
#11720575	2
#10592936	3
#10588915	3