Subgroups — #10589648

Order 576.

441 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3699	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#611	C₂	1	2	yes	maximal, complemented
5	C₂×Dic₉⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×D₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#63449	C₂	1	1	yes	maximal, characteristic, complemented
8	#611⋊C₄	C₂	1	2	yes	maximal, complemented
9	#63439	C₂	1	2	yes	maximal, complemented
10	#63434	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#5546	—	3	1	no	maximal
13	C₂×D₄×C₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₉	C₂²	1	2	yes	—
15	C₂×#611	C₂²	1	2	yes	—
16	C₂²×Dic₉	C₂²	1	2	yes	—
17	C₂²×Dic₉	C₂²	1	2	yes	—
18	#3699	C₄	1	8	yes	complemented
19	C₂×C₄×D₉	C₂²	1	2	yes	—
20	C₂×C₄×D₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
22	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₉×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₄²×C₉	C₂²	1	1	yes	characteristic
25	#3676	C₂²	1	2	yes	—
26	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
27	(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
29	Dic₉⋊C₄	C₂²	1	2	yes	—
30	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₉⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₉⋊C₄	C₂²	1	2	yes	—
33	Dic₉⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₉	C₂²	1	2	yes	—
35	C₄×Dic₉	C₂²	1	1	yes	characteristic
36	C₄×Dic₉	C₂²	1	2	yes	—
37	C₂×#237	—	3	1	no	—
38	C₃×C₄×D₄	D₃	1	1	yes	characteristic
39	D₄×Dic₃	—	3	1	no	—
40	C₄×#67	—	3	2	no	—
41	C₂×Dic₃⋊C₄	—	3	2	no	—
42	D₃×C₄⋊C₄	—	3	1	no	—
43	#887	—	3	1	no	—
44	#67⋊C₄	—	3	2	no	—
45	#877	—	3	2	no	—
46	#872	—	3	1	no	—
47	C₄×Dic₆	—	3	1	no	—
48	C₂³×C₉	C₂³	1	2	yes	—
49	C₂²×D₉	C₂³	1	1	yes	characteristic
50	D₄×C₉	C₂×C₄	1	4	yes	complemented
51	C₂×C₄×C₉	—	4	2	no	—
52	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₉	C₂³	1	2	yes	—
54	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
56	#611	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₉	—	4	2	no	—
58	C₂×Dic₉	C₂×C₄	1	8	yes	complemented
59	C₂×Dic₉	C₂³	1	1	yes	characteristic
60	C₂×Dic₉	C₂³	1	1	yes	characteristic
61	C₂×Dic₉	C₂³	1	2	yes	—
62	C₂×Dic₉	C₂³	1	2	yes	—
63	C₂×Dic₉	C₂³	1	1	yes	characteristic
64	C₄×D₉	—	4	2	no	—
65	C₄×D₉	C₂×C₄	1	4	yes	complemented
66	Dic₁₈	C₂×C₄	1	4	yes	complemented
67	#520	—	9	1	no	Sylow
68	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
70	C₂×#67	—	3	2	no	—
71	C₂²×Dic₃	—	3	2	no	—
72	C₂²×Dic₃	—	3	2	no	—
73	#237	—	3	8	no	—
74	C₂×C₄×D₃	—	3	1	no	—
75	C₂×C₄×D₃	—	3	2	no	—
76	C₂×Dic₆	—	3	1	no	—
77	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
79	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
80	#217	—	3	2	no	—
81	(C₂×D₃)⋊C₄	—	3	1	no	—
82	(C₂×D₃)⋊C₄	—	3	1	no	—
83	C₄⋊Dic₃	—	3	1	no	—
84	Dic₃⋊C₄	—	3	2	no	—
85	Dic₃⋊C₄	—	3	1	no	—
86	Dic₃⋊C₄	—	3	1	no	—
87	Dic₃⋊C₄	—	3	2	no	—
88	Dic₃⋊C₄	—	3	2	no	—
89	C₄×Dic₃	—	3	2	no	—
90	C₄×Dic₃	—	3	1	no	—
91	C₄×Dic₃	—	3	2	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	2	2	no	—
94	C₂²×C₉	C₂²×C₄	1	4	yes	—
95	C₂²×C₉	C₂⁴	1	1	yes	characteristic
96	C₂×D₉	—	2	1	no	—
97	C₂×D₉	—	2	1	no	—
98	C₂×D₉	C₂²×C₄	1	2	yes	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	1	no	—
101	C₄×C₉	—	2	1	no	—
102	C₄×C₉	C₂²×C₄	1	2	yes	—
103	Dic₉	—	2	2	no	—
104	Dic₉	—	2	1	no	—
105	Dic₉	—	2	1	no	—
106	Dic₉	C₂²×C₄	1	4	yes	—
107	Dic₉	C₂²×C₄	1	2	yes	—
108	C₂×#41	—	9	1	no	—
109	C₄×Q₈	—	9	1	no	—
110	C₄×Q₈	—	9	1	no	—
111	C₄×D₄	—	9	1	no	—
112	C₄×D₄	—	9	2	no	—
113	C₄×D₄	—	9	2	no	—
114	C₄×D₄	D₉	1	1	yes	characteristic, complemented
115	#117	—	9	2	no	—
116	#117	—	9	1	no	—
117	C₂×C₄⋊C₄	—	9	2	no	—
118	C₂×C₄⋊C₄	—	9	1	no	—
119	C₂³×C₃	C₂²×D₃	1	2	yes	—
120	C₂²×D₃	—	3	1	no	—
121	C₃×D₄	C₄×D₃	1	4	yes	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
125	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
127	#67	—	3	8	no	—
128	C₂×Dic₃	—	3	8	no	—
129	C₂×Dic₃	—	3	2	no	—
130	C₂×Dic₃	—	3	1	no	—
131	C₂×Dic₃	—	3	1	no	—
132	C₂×Dic₃	—	3	1	no	—
133	C₂×Dic₃	—	3	2	no	—
134	C₂×Dic₃	—	12	2	no	—
135	C₄×D₃	—	3	4	no	—
136	C₄×D₃	—	12	2	no	—
137	Dic₆	—	3	4	no	—
138	C₂×C₉	—	2	4	no	—
139	C₂×C₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₉	2₋¹⁺⁴	1	1	yes	characteristic
142	D₉	—	2	2	no	—
143	#41	—	9	8	no	—
144	C₂×Q₈	—	9	1	no	—
145	C₂×D₄	—	9	2	no	—
146	C₂×D₄	C₂×D₉	1	1	yes	characteristic
147	C₂²×C₄	—	9	2	no	—
148	C₂²×C₄	—	9	1	no	—
149	C₂²×C₄	—	9	2	no	—
150	C₂²×C₄	—	9	2	no	—
151	C₂²×C₄	C₂×D₉	1	2	yes	—
152	C₄⋊C₄	—	9	2	no	—
153	C₄⋊C₄	—	9	1	no	—
154	C₄⋊C₄	—	9	2	no	—
155	C₄⋊C₄	—	9	2	no	—
156	C₄⋊C₄	—	9	1	no	—
157	C₄⋊C₄	—	9	1	no	—
158	C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
159	C₂²⋊C₄	—	9	2	no	—
160	C₂²⋊C₄	—	9	1	no	—
161	C₂²⋊C₄	—	9	1	no	—
162	C₂²⋊C₄	C₂×D₉	1	2	yes	—
163	C₄²	—	9	2	no	—
164	C₄²	—	9	2	no	—
165	C₄²	—	9	1	no	—
166	C₄²	C₂×D₉	1	1	yes	characteristic
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	C₂×C₄×D₃	1	4	yes	—
170	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
171	C₂×D₃	—	3	2	no	—
172	C₂×D₃	—	6	1	no	—
173	C₂×D₃	—	6	1	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	C₂×C₄×D₃	1	2	yes	—
178	Dic₃	—	3	4	no	—
179	Dic₃	—	3	2	no	—
180	Dic₃	—	6	2	no	—
181	Dic₃	—	6	1	no	—
182	Dic₃	—	6	1	no	—
183	C₉	#520	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	9	1	no	—
185	C₂³	C₂²×D₉	1	2	yes	—
186	Q₈	—	9	4	no	—
187	D₄	—	9	8	no	—
188	D₄	C₄×D₉	1	4	yes	complemented
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	9	8	no	—
191	C₂×C₄	—	9	2	no	—
192	C₂×C₄	—	9	1	no	—
193	C₂×C₄	—	9	4	no	—
194	C₂×C₄	—	9	1	no	—
195	C₂×C₄	—	9	2	no	—
196	C₂×C₄	—	9	1	no	—
197	C₂×C₄	—	36	2	no	—
198	C₂×C₄	—	36	2	no	—
199	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
200	C₂×C₄	C₂²×D₉	1	2	yes	—
201	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
203	C₂×C₃	—	2	4	no	—
204	C₂×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃	#1004	1	1	yes	characteristic
206	C₂×C₃	#1010	1	1	yes	characteristic
207	D₃	—	6	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	9	2	no	—
211	C₂²	—	18	1	no	—
212	C₂²	—	18	1	no	—
213	C₂²	C₂×C₄×D₉	1	4	yes	—
214	C₂²	C₂³×D₉	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	9	2	no	—
219	C₄	—	9	4	no	—
220	C₄	—	18	2	no	—
221	C₄	—	18	1	no	—
222	C₄	—	18	1	no	—
223	C₄	C₂×C₄×D₉	1	2	yes	—
224	C₃	#5546	1	1	yes	characteristic
225	C₂	—	2	4	no	—
226	C₂	—	18	2	no	—
227	C₂	C₂²×C₄×D₉	1	1	yes	characteristic
228	C₂	#63710	1	1	yes	characteristic
229	C₂	#63716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator	C₂×C₉	C₂³×C₄
Frattini, socle	C₂²×C₃	C₂³×D₃
Fitting	C₄×D₄×C₉	C₂
prime radical	C₂×#3699	C₂
cosocle	C₂²×C₉	C₂⁴
hypercenter	C₄×D₄	D₉
hypocenter	C₉	#520
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×C₉	C₂	semidirect
3. D₄×Dic₉	C₂	semidirect
4. C₄×#611	C₂	semidirect
5. C₂×Dic₉⋊C₄	C₂	semidirect
6. C₄⋊C₄×D₉	C₂	semidirect
7. #63449	C₂	semidirect
8. #611⋊C₄	C₂	semidirect
9. #63439	C₂	semidirect
10. #63434	C₂	semidirect
11. C₄×Dic₁₈	C₂	semidirect
18. #3699	C₄	semidirect
50. D₄×C₉	C₂×C₄	semidirect
56. #611	C₂×C₄	semidirect
58. C₂×Dic₉	C₂×C₄	semidirect
65. C₄×D₉	C₂×C₄	semidirect
66. Dic₁₈	C₂×C₄	semidirect
114. C₄×D₄	D₉	semidirect
183. C₉	#520	semidirect
188. D₄	C₄×D₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3699	2
C₄×D₄×C₉	2
D₄×Dic₉	2
C₄×#611	2
C₂×Dic₉⋊C₄	2
C₄⋊C₄×D₉	2
#63449	2
#611⋊C₄	2
#63439	2
#63434	2
C₄×Dic₁₈	2
#5546	3