Subgroups — #11858332

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 243 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#63510	C₂	1	1	yes	maximal, characteristic
4	#10588803	C₂	1	3	yes	maximal, complemented
5	#10588802	C₂	1	3	yes	maximal, complemented
6	#10588783	C₂	1	3	yes	maximal, complemented
7	C₂×Dic₉⋊C₈	C₂	1	3	yes	maximal, complemented
8	#83783	—	3	1	no	maximal
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₉×#131	—	2	4	no	—
11	C₂×C₉×M₁₆	C₂²	1	3	yes	—
12	C₂²×C₈×C₉	C₂²	1	3	yes	—
13	#63510	—	2	4	no	—
14	Q₈×Dic₉	C₄	1	1	yes	characteristic
15	Q₈×Dic₉	C₄	1	1	yes	characteristic
16	D₄×Dic₉	C₄	1	3	yes	—
17	D₄×Dic₉	C₄	1	3	yes	—
18	#63488	C₂²	1	3	yes	—
19	C₂×C₄×Dic₉	C₂²	1	3	yes	—
20	C₂×#3667	C₂²	1	3	yes	—
21	C₂²×C₉⋊C₈	C₂²	1	3	yes	—
22	Dic₉⋊C₈	C₂²	1	1	yes	characteristic
23	Dic₉⋊C₈	C₂²	1	3	yes	—
24	Dic₉⋊C₈	C₂²	1	3	yes	—
25	Dic₉⋊C₈	C₂²	1	3	yes	—
26	Dic₉⋊C₈	C₂²	1	6	yes	—
27	C₂×C₃×#131	D₃	1	1	yes	characteristic
28	Dic₃×#41	—	3	1	no	—
29	C₂×#948	—	3	1	no	—
30	#5119	—	3	3	no	—
31	#5118	—	3	3	no	—
32	#5102	—	3	3	no	—
33	C₂×Dic₃⋊C₈	—	3	3	no	—
34	C₉×#41	D₄	1	4	yes	—
35	C₉×#41	Q₈	1	4	yes	—
36	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₉	C₂×C₄	1	3	yes	—
38	C₂²×C₄×C₉	C₂³	1	3	yes	—
39	C₂²×Dic₉	C₂×C₄	1	3	yes	—
40	C₂²×Dic₉	C₂×C₄	1	3	yes	—
41	C₉×M₁₆	—	2	6	no	—
42	C₂×C₈×C₉	—	2	6	no	—
43	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₉	C₂³	1	3	yes	—
45	#3676	C₂×C₄	1	6	yes	—
46	C₄⋊Dic₉	C₂×C₄	1	3	yes	—
47	C₄⋊Dic₉	C₂×C₄	1	3	yes	—
48	C₄×Dic₉	C₂×C₄	1	3	yes	—
49	C₄×Dic₉	C₂×C₄	1	3	yes	—
50	C₄×Dic₉	C₂³	1	3	yes	—
51	C₄×Dic₉	C₂³	1	1	yes	characteristic
52	#3667	—	2	6	no	—
53	C₂×C₉⋊C₈	—	2	6	no	—
54	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₉⋊C₈	C₂³	1	3	yes	—
56	#2906	—	9	1	no	Sylow
57	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
58	C₃×#131	—	2	4	no	—
59	C₂×C₃×M₁₆	C₂×D₃	1	3	yes	—
60	C₂²×C₃×C₈	C₂×D₃	1	3	yes	—
61	#948	—	6	4	no	—
62	Q₈×Dic₃	—	3	1	no	—
63	Q₈×Dic₃	—	3	1	no	—
64	D₄×Dic₃	—	3	3	no	—
65	D₄×Dic₃	—	3	3	no	—
66	#926	—	3	3	no	—
67	C₂×C₄×Dic₃	—	3	3	no	—
68	C₂×#208	—	3	3	no	—
69	C₂²×C₃⋊C₈	—	3	3	no	—
70	Dic₃⋊C₈	—	3	1	no	—
71	Dic₃⋊C₈	—	3	3	no	—
72	Dic₃⋊C₈	—	3	3	no	—
73	Dic₃⋊C₈	—	3	3	no	—
74	Dic₃⋊C₈	—	3	6	no	—
75	C₂³×C₉	C₂²×C₄	1	3	yes	—
76	Q₈×C₉	C₄⋊C₄	1	4	yes	—
77	D₄×C₉	C₄⋊C₄	1	12	yes	—
78	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
79	C₂×C₄×C₉	C₂×D₄	1	6	yes	—
80	C₂×C₄×C₉	C₂×Q₈	1	6	yes	—
81	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₉	—	2	6	no	—
83	C₂×Dic₉	—	4	3	no	—
84	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₉	C₂²×C₄	1	3	yes	—
86	C₂×Dic₉	C₂²×C₄	1	3	yes	—
87	C₂×Dic₉	C₂²×C₄	1	1	yes	characteristic
88	C₈×C₉	—	2	1	no	—
89	C₈×C₉	—	2	3	no	—
90	C₉⋊C₈	—	2	1	no	—
91	C₉⋊C₈	—	2	3	no	—
92	C₂×#131	—	9	1	no	—
93	C₂×#131	D₉	1	1	yes	characteristic
94	C₄×#41	—	9	1	no	—
95	#424	—	9	3	no	—
96	#424	—	9	3	no	—
97	#423	—	9	3	no	—
98	C₂×C₄⋊C₈	—	9	3	no	—
99	C₃×#41	Dic₆	1	4	yes	—
100	C₃×#41	#67	1	4	yes	—
101	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
102	C₂×C₃×D₄	C₄×D₃	1	3	yes	—
103	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
104	C₂²×Dic₃	—	3	3	no	—
105	C₂²×Dic₃	—	3	3	no	—
106	C₃×M₁₆	—	2	6	no	—
107	C₂×C₃×C₈	—	2	6	no	—
108	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂²×D₃	1	3	yes	—
110	#217	—	3	6	no	—
111	C₄⋊Dic₃	—	3	3	no	—
112	C₄⋊Dic₃	—	3	3	no	—
113	C₄×Dic₃	—	3	3	no	—
114	C₄×Dic₃	—	3	3	no	—
115	C₄×Dic₃	—	3	1	no	—
116	C₄×Dic₃	—	3	3	no	—
117	#208	—	6	6	no	—
118	C₂×C₃⋊C₈	—	3	1	no	—
119	C₂×C₃⋊C₈	—	3	3	no	—
120	C₂×C₃⋊C₈	—	6	6	no	—
121	C₂²×C₉	—	2	3	no	—
122	C₂²×C₉	—	2	3	no	—
123	C₂²×C₉	C₂×C₄⋊C₄	1	6	yes	—
124	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₉	C₂×C₄⋊C₄	1	6	yes	—
126	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
128	Dic₉	—	2	3	no	—
129	Dic₉	—	2	1	no	—
130	Dic₉	—	2	3	no	—
131	Dic₉	#131	1	2	yes	complemented
132	C₂×#41	C₂×D₉	1	1	yes	characteristic
133	#131	—	2	4	no	—
134	#131	—	18	4	no	—
135	C₂×M₁₆	—	9	3	no	—
136	C₂×M₁₆	C₂×D₉	1	3	yes	—
137	C₂²×C₈	—	9	3	no	—
138	C₂²×C₈	C₂×D₉	1	3	yes	—
139	C₄×Q₈	—	9	1	no	—
140	C₄×Q₈	—	9	1	no	—
141	C₄×D₄	—	9	3	no	—
142	C₄×D₄	—	9	3	no	—
143	#117	—	9	3	no	—
144	C₂×C₄²	—	9	3	no	—
145	C₄⋊C₈	—	9	1	no	—
146	C₄⋊C₈	—	9	3	no	—
147	C₄⋊C₈	—	9	3	no	—
148	C₄⋊C₈	—	9	3	no	—
149	C₄⋊C₈	—	9	6	no	—
150	C₂³×C₃	C₂×C₄×D₃	1	3	yes	—
151	C₃×Q₈	Dic₃⋊C₄	1	4	yes	—
152	C₃×D₄	Dic₃⋊C₄	1	12	yes	—
153	C₂×C₃×C₄	C₂×Dic₆	1	6	yes	—
154	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
155	C₂×C₃×C₄	C₂×#67	1	6	yes	—
156	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
157	C₂×Dic₃	—	3	1	no	—
158	C₂×Dic₃	—	3	1	no	—
159	C₂×Dic₃	—	3	3	no	—
160	C₂×Dic₃	—	3	3	no	—
161	C₂×Dic₃	—	6	6	no	—
162	C₂×Dic₃	—	12	3	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	3	no	—
165	C₃⋊C₈	—	6	1	no	—
166	C₃⋊C₈	—	6	3	no	—
167	C₂×C₉	—	2	6	no	—
168	C₂×C₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₉	C₂×#131	1	1	yes	characteristic
170	C₂×C₉	#568	1	1	yes	characteristic
171	#41	Dic₁₈	1	4	yes	—
172	#41	#611	1	4	yes	—
173	C₂×Q₈	C₄×D₉	1	1	yes	characteristic
174	C₂×D₄	C₄×D₉	1	3	yes	—
175	C₂²×C₄	—	9	3	no	—
176	C₂²×C₄	—	9	3	no	—
177	C₂²×C₄	C₂²×D₉	1	3	yes	—
178	M₁₆	—	2	6	no	—
179	M₁₆	—	18	6	no	—
180	C₂×C₈	—	2	6	no	—
181	C₂×C₈	—	9	1	no	—
182	C₂×C₈	—	9	3	no	—
183	C₂×C₈	—	18	6	no	—
184	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₉	1	3	yes	—
186	C₄⋊C₄	—	9	3	no	—
187	C₄⋊C₄	—	9	3	no	—
188	C₂²⋊C₄	—	9	6	no	—
189	C₄²	—	9	3	no	—
190	C₄²	—	9	3	no	—
191	C₄²	—	9	1	no	—
192	C₄²	—	9	3	no	—
193	C₂²×C₃	—	2	3	no	—
194	C₂²×C₃	—	2	3	no	—
195	C₂²×C₃	C₂×Dic₃⋊C₄	1	6	yes	—
196	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
197	C₃×C₄	C₂×Dic₃⋊C₄	1	6	yes	—
198	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
199	C₃×C₄	C₂²×#67	1	1	yes	characteristic
200	Dic₃	—	3	2	no	—
201	Dic₃	—	6	3	no	—
202	Dic₃	—	6	1	no	—
203	Dic₃	—	6	3	no	—
204	C₉	#2906	1	1	yes	characteristic, complemented, Sylow
205	C₂³	C₂×C₄×D₉	1	3	yes	—
206	Q₈	Dic₉⋊C₄	1	4	yes	—
207	D₄	Dic₉⋊C₄	1	12	yes	—
208	C₂×C₄	—	9	1	no	—
209	C₂×C₄	—	9	1	no	—
210	C₂×C₄	—	9	3	no	—
211	C₂×C₄	—	9	3	no	—
212	C₂×C₄	—	18	6	no	—
213	C₂×C₄	—	36	3	no	—
214	C₂×C₄	C₂×Dic₁₈	1	6	yes	—
215	C₂×C₄	C₂×C₄×D₉	1	3	yes	—
216	C₂×C₄	C₂×#611	1	6	yes	—
217	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	3	no	—
220	C₈	—	18	1	no	—
221	C₈	—	18	3	no	—
222	C₂×C₃	—	2	6	no	—
223	C₂×C₃	D₃×#131	1	1	yes	characteristic
224	C₂×C₃	#5750	1	1	yes	characteristic
225	C₂×C₃	C₂²×Dic₃⋊C₄	1	1	yes	characteristic
226	C₂²	—	2	3	no	—
227	C₂²	—	2	3	no	—
228	C₂²	C₂×Dic₉⋊C₄	1	6	yes	—
229	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
230	C₄	—	9	2	no	—
231	C₄	—	18	3	no	—
232	C₄	—	18	1	no	—
233	C₄	—	18	3	no	—
234	C₄	C₂×Dic₉⋊C₄	1	6	yes	—
235	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
236	C₄	C₂²×#611	1	1	yes	characteristic
237	C₃	#83783	1	1	yes	characteristic
238	C₂	—	2	6	no	—
239	C₂	D₉×#131	1	1	yes	characteristic
240	C₂	#10589852	1	1	yes	characteristic
241	C₂	C₂²×Dic₉⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×#131	D₉
hypocenter	C₉	#2906
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
4. #10588803	C₂	semidirect
5. #10588802	C₂	semidirect
6. #10588783	C₂	semidirect
7. C₂×Dic₉⋊C₈	C₂	semidirect
131. Dic₉	#131	semidirect
204. C₉	#2906	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
#41×Dic₉	2
C₂×#63510	2
#10588803	2
#10588802	2
#10588783	2
C₂×Dic₉⋊C₈	2
#83783	3