Subgroups — #11783638

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×#3676	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
5	#10588228	C₂	1	2	yes	maximal, complemented
6	#69947	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×#3676	C₄	1	2	yes	—
10	C₂×C₄×Dic₉	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	—	2	4	no	—
12	C₂²×C₉⋊C₈	C₄	1	4	yes	complemented
13	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₉×C₂²⋊C₈	C₄	1	4	yes	complemented
16	C₉⋊#95	C₄	1	2	yes	—
17	C₈×Dic₉	—	2	4	no	—
18	C₂³×C₃⋊C₈	—	3	1	no	—
19	C₂×C₃×C₂²⋊C₈	D₃	1	1	yes	characteristic
20	C₄×#217	—	3	1	no	—
21	C₂×C₈×Dic₃	—	3	2	no	—
22	#4550	—	3	2	no	—
23	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₉	—	2	2	no	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₉	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
34	#3676	C₈	1	8	yes	complemented
35	C₄×Dic₉	—	2	2	no	—
36	C₄×Dic₉	—	2	2	no	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂×C₉⋊C₈	—	2	8	no	—
39	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₉⋊C₈	D₄	1	4	yes	—
42	C₈×C₂²⋊C₄	—	9	1	no	Sylow
43	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
45	C₂×#217	—	3	2	no	—
46	C₂×C₄×Dic₃	—	3	2	no	—
47	C₂²×C₃⋊C₈	—	3	1	no	—
48	C₂²×C₃⋊C₈	—	3	4	no	—
49	C₂²×C₃⋊C₈	—	3	1	no	—
50	C₂²×C₃⋊C₈	—	6	4	no	—
51	C₃×C₂²⋊C₈	Dic₃	1	4	yes	—
52	C₃⋊#95	—	3	2	no	—
53	C₈×Dic₃	—	6	4	no	—
54	C₂³×C₉	—	2	2	no	—
55	C₂³×C₉	—	2	2	no	—
56	C₂³×C₉	C₄²	1	2	yes	—
57	C₂³×C₉	C₂×C₈	1	4	yes	—
58	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	2	4	no	—
63	C₂×C₄×C₉	C₄²	1	2	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₉	#41	1	2	yes	—
68	C₂×Dic₉	—	2	4	no	—
69	C₂×Dic₉	—	2	4	no	—
70	C₂×Dic₉	C₂×C₈	1	8	yes	complemented
71	C₈×C₉	—	2	4	no	—
72	C₉⋊C₈	—	2	2	no	—
73	C₉⋊C₈	C₂²⋊C₄	1	4	yes	complemented
74	C₂³×C₈	—	9	1	no	—
75	C₂×C₂²⋊C₈	D₉	1	1	yes	characteristic
76	C₂×C₄×C₈	—	9	2	no	—
77	C₄×C₂²⋊C₄	—	9	1	no	—
78	#336	—	9	2	no	—
79	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
84	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
85	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₃	—	3	4	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
90	#217	—	3	8	no	—
91	C₄×Dic₃	—	6	2	no	—
92	C₄×Dic₃	—	6	2	no	—
93	C₂×C₃⋊C₈	—	3	2	no	—
94	C₂×C₃⋊C₈	—	3	4	no	—
95	C₂×C₃⋊C₈	—	3	2	no	—
96	C₂×C₃⋊C₈	—	6	8	no	—
97	C₂×C₃⋊C₈	—	6	2	no	—
98	C₂²×C₉	—	2	2	no	—
99	C₂²×C₉	—	2	2	no	—
100	C₂²×C₉	—	2	8	no	—
101	C₂²×C₉	C₄×C₈	1	4	yes	complemented
102	C₂²×C₉	C₂×C₄²	1	1	yes	characteristic
103	C₂²×C₉	C₄×D₄	1	2	yes	—
104	C₂²×C₉	C₂²×C₈	1	2	yes	—
105	C₂²×C₉	#131	1	2	yes	—
106	C₄×C₉	—	2	2	no	—
107	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₉	#117	1	1	yes	characteristic
109	C₄×C₉	C₄×D₄	1	2	yes	—
110	Dic₉	—	2	8	no	—
111	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
112	C₂²×C₈	—	9	1	no	—
113	C₂²×C₈	—	9	4	no	—
114	C₂²×C₈	—	9	1	no	—
115	C₂²×C₈	—	18	4	no	—
116	C₂²×C₈	C₂×D₉	1	2	yes	—
117	C₂×C₂²⋊C₄	—	9	2	no	—
118	C₂×C₄²	—	9	2	no	—
119	C₂²⋊C₈	Dic₉	1	4	yes	complemented
120	C₄×C₈	—	18	4	no	—
121	#95	—	9	2	no	—
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	2	2	no	—
124	C₂³×C₃	D₃×C₈	1	4	yes	—
125	C₂³×C₃	C₄×Dic₃	1	2	yes	—
126	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	4	no	—
131	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
132	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
134	C₂×C₃×C₄	#237	1	2	yes	—
135	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
136	C₂×Dic₃	—	3	8	no	—
137	C₂×Dic₃	—	6	4	no	—
138	C₂×Dic₃	—	6	4	no	—
139	C₃×C₈	—	2	4	no	—
140	C₃⋊C₈	—	3	4	no	—
141	C₃⋊C₈	—	6	2	no	—
142	C₂×C₉	—	2	4	no	—
143	C₂×C₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₉	C₂×C₄×C₈	1	1	yes	characteristic
145	C₂×C₉	#405	1	1	yes	characteristic
146	C₂×C₉	C₈×D₄	1	4	yes	—
147	C₂⁴	C₄×D₉	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	9	4	no	—
151	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
152	C₂²×C₄	C₄×D₉	1	2	yes	—
153	C₂²×C₄	C₂×Dic₉	1	2	yes	—
154	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	9	2	no	—
158	C₂×C₈	—	9	4	no	—
159	C₂×C₈	—	9	2	no	—
160	C₂×C₈	—	18	8	no	—
161	C₂×C₈	—	18	2	no	—
162	C₂×C₈	C₂×Dic₉	1	4	yes	—
163	C₂²⋊C₄	—	9	8	no	—
164	C₄²	—	18	2	no	—
165	C₄²	—	18	2	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	8	no	—
169	C₂²×C₃	C₈×Dic₃	1	4	yes	—
170	C₂²×C₃	#67⋊C₄	1	2	yes	—
171	C₂²×C₃	C₂×D₃×C₈	1	2	yes	—
172	C₂²×C₃	#907	1	2	yes	—
173	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	#877	1	1	yes	characteristic
176	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₃×C₄	D₄×Dic₃	1	2	yes	—
178	Dic₃	—	6	8	no	—
179	C₉	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₈×D₉	1	4	yes	—
183	C₂³	C₄×Dic₉	1	2	yes	—
184	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	9	8	no	—
190	C₂×C₄	—	18	4	no	—
191	C₂×C₄	—	18	4	no	—
192	C₂×C₄	C₄×Dic₉	1	2	yes	—
193	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
194	C₂×C₄	D₄×D₉	1	2	yes	—
195	C₂×C₄	#3699	1	2	yes	—
196	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
197	C₈	—	2	4	no	—
198	C₈	—	9	4	no	—
199	C₈	—	18	2	no	—
200	C₂×C₃	—	2	4	no	—
201	C₂×C₃	#67⋊C₈	1	4	yes	—
202	C₂×C₃	Dic₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
204	C₂×C₃	#5122	1	1	yes	characteristic
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	C₈×Dic₉	1	4	yes	complemented
209	C₂²	#611⋊C₄	1	2	yes	—
210	C₂²	C₂×C₈×D₉	1	2	yes	—
211	C₂²	#63469	1	2	yes	—
212	C₂²	C₂×C₄×Dic₉	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	18	8	no	—
215	C₄	#63439	1	1	yes	characteristic
216	C₄	C₂²⋊C₄×D₉	1	1	yes	characteristic
217	C₄	D₄×Dic₉	1	2	yes	—
218	C₃	#69947	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#611⋊C₈	1	4	yes	—
221	C₂	C₂²⋊C₄×Dic₉	1	1	yes	characteristic
222	C₂	C₂×C₈×Dic₉	1	1	yes	characteristic
223	C₂	#10588797	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂×C₉	C₂×C₄×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×C₂²⋊C₈	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₉
hypocenter	C₉	C₈×C₂²⋊C₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₈×Dic₉	C₂	semidirect
5. #10588228	C₂	semidirect
12. C₂²×C₉⋊C₈	C₄	semidirect
15. C₉×C₂²⋊C₈	C₄	semidirect
34. #3676	C₈	semidirect
70. C₂×Dic₉	C₂×C₈	semidirect
73. C₉⋊C₈	C₂²⋊C₄	semidirect
101. C₂²×C₉	C₄×C₈	semidirect
119. C₂²⋊C₈	Dic₉	semidirect
179. C₉	C₈×C₂²⋊C₄	semidirect
208. C₂²	C₈×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₉⋊C₈	2
C₂×C₉×C₂²⋊C₈	2
C₄×#3676	2
C₂×C₈×Dic₉	2
#10588228	2
#69947	3