Subgroups — C₂×#10588782 #11857175

Order 1152.

888 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	#10588782	C₂	1	8	yes	maximal, direct product factor
5	C₂×#63375	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
7	C₂×#5101	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₄	1	8	yes	complemented
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉	C₂²	1	4	yes	—
13	C₂×#3676	C₄	1	2	yes	—
14	#63488	C₄	1	4	yes	—
15	#63488	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₉	C₄	1	2	yes	—
17	C₂×C₄×Dic₉	C₂²	1	2	yes	—
18	C₂×#3667	C₄	1	8	yes	complemented
19	C₂×#3667	C₂²	1	4	yes	—
20	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
21	#63375	C₂²	1	8	yes	complemented
22	C₈×Dic₉	C₂²	1	8	yes	complemented
23	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
24	C₂×#926	—	3	1	no	—
25	C₂²×#208	—	3	1	no	—
26	#5101	—	3	8	no	—
27	C₂×#815	—	3	2	no	—
28	C₂×C₈×Dic₃	—	3	2	no	—
29	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
39	C₂×C₈×C₉	C₂×C₄	1	16	yes	complemented
40	C₂×C₈×C₉	C₂³	1	2	yes	—
41	C₂×C₈×C₉	C₂³	1	2	yes	—
42	#3676	C₂×C₄	1	8	yes	—
43	C₄⋊Dic₉	C₂×C₄	1	8	yes	—
44	C₄×Dic₉	C₂×C₄	1	4	yes	—
45	C₄×Dic₉	C₂³	1	4	yes	—
46	#3667	C₂×C₄	1	16	yes	complemented
47	C₂×C₉⋊C₈	C₂×C₄	1	8	yes	—
48	C₂×C₉⋊C₈	C₂³	1	4	yes	—
49	C₂×#405	—	9	1	no	Sylow
50	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	Dic₃	1	8	yes	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×#217	—	3	2	no	—
56	#926	—	3	4	no	—
57	#926	—	3	4	no	—
58	C₂×C₄⋊Dic₃	—	3	2	no	—
59	C₂×C₄×Dic₃	—	3	2	no	—
60	C₂×#208	—	3	8	no	—
61	C₂×#208	—	3	4	no	—
62	C₂²×C₃⋊C₈	—	3	2	no	—
63	#815	—	3	8	no	—
64	C₈×Dic₃	—	3	8	no	—
65	C₂³×C₉	—	2	4	no	—
66	C₂³×C₉	C₄²	1	4	yes	—
67	C₂³×C₉	C₂²×C₄	1	2	yes	—
68	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₄²	1	8	yes	—
70	C₂×C₄×C₉	C₄²	1	4	yes	—
71	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₉	—	2	8	no	—
79	C₂×Dic₉	C₂²×C₄	1	8	yes	—
80	C₈×C₉	C₂²×C₄	1	4	yes	—
81	C₈×C₉	C₂²×C₄	1	4	yes	—
82	C₉⋊C₈	C₂²×C₄	1	8	yes	—
83	C₂²×M₁₆	—	9	1	no	—
84	C₂³×C₈	D₉	1	1	yes	characteristic
85	C₂×#117	—	9	1	no	—
86	#405	—	9	8	no	—
87	C₂×#97	—	9	2	no	—
88	C₂×C₄×C₈	—	9	2	no	—
89	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
92	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
93	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
94	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
96	C₂²×Dic₃	—	3	4	no	—
97	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
98	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
99	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
100	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
101	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
102	#217	—	3	8	no	—
103	C₄⋊Dic₃	—	3	8	no	—
104	C₄×Dic₃	—	3	4	no	—
105	C₄×Dic₃	—	3	4	no	—
106	#208	—	3	16	no	—
107	C₂×C₃⋊C₈	—	3	8	no	—
108	C₂×C₃⋊C₈	—	3	4	no	—
109	C₂²×C₉	—	2	4	no	—
110	C₂²×C₉	—	2	8	no	—
111	C₂²×C₉	C₂×C₄²	1	4	yes	—
112	C₂²×C₉	C₂×C₄²	1	2	yes	—
113	C₂²×C₉	#131	1	4	yes	—
114	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
115	C₄×C₉	C₂×C₄²	1	2	yes	—
116	C₄×C₉	C₂×C₄²	1	4	yes	—
117	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
119	Dic₉	—	2	8	no	—
120	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
121	C₂×M₁₆	—	9	8	no	—
122	C₂×M₁₆	—	9	4	no	—
123	C₂²×C₈	—	9	2	no	—
124	C₂²×C₈	Dic₉	1	8	yes	complemented
125	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
126	C₂²×C₈	C₂×D₉	1	4	yes	—
127	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
128	#117	—	9	4	no	—
129	#117	—	9	4	no	—
130	C₂×C₄⋊C₄	—	9	2	no	—
131	C₂×C₂²⋊C₄	—	9	2	no	—
132	C₂×C₄²	—	9	2	no	—
133	#97	—	9	8	no	—
134	C₄×C₈	—	9	8	no	—
135	C₂³×C₃	—	2	4	no	—
136	C₂³×C₃	C₄×Dic₃	1	4	yes	—
137	C₂³×C₃	C₂×C₄×D₃	1	2	yes	—
138	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₄×Dic₃	1	8	yes	—
140	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
141	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
143	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄×D₃	1	4	yes	—
145	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
146	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
147	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
148	C₂×Dic₃	—	3	8	no	—
149	C₂×Dic₃	—	6	8	no	—
150	C₃×C₈	C₂²×Dic₃	1	4	yes	—
151	C₃×C₈	C₂²×Dic₃	1	4	yes	—
152	C₃⋊C₈	—	3	8	no	—
153	C₂×C₉	—	2	4	no	—
154	C₂×C₉	#405	1	4	yes	complemented
155	C₂×C₉	C₂²×C₄²	1	1	yes	characteristic
156	C₂×C₉	C₂×#131	1	2	yes	—
157	C₂⁴	C₄×D₉	1	1	yes	characteristic
158	C₂²×C₄	—	9	4	no	—
159	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
160	C₂²×C₄	C₄×D₉	1	2	yes	—
161	C₂²×C₄	C₄×D₉	1	4	yes	—
162	C₂²×C₄	C₂×Dic₉	1	4	yes	—
163	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₉	1	2	yes	—
165	M₁₆	—	9	16	no	—
166	C₂×C₈	—	9	8	no	—
167	C₂×C₈	—	9	4	no	—
168	C₂×C₈	C₂×Dic₉	1	4	yes	—
169	C₂×C₈	C₂×Dic₉	1	4	yes	—
170	C₂×C₈	C₂×Dic₉	1	16	yes	complemented
171	C₂×C₈	C₂²×D₉	1	2	yes	—
172	C₂×C₈	C₂²×D₉	1	2	yes	—
173	C₄⋊C₄	—	9	8	no	—
174	C₂²⋊C₄	—	9	8	no	—
175	C₄²	—	9	4	no	—
176	C₄²	—	9	4	no	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	8	no	—
179	C₂²×C₃	#901	1	4	yes	—
180	C₂²×C₃	C₂×C₄×Dic₃	1	4	yes	—
181	C₂²×C₃	C₂×C₄×Dic₃	1	2	yes	—
182	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
183	C₃×C₄	C₂×C₄×Dic₃	1	2	yes	—
184	C₃×C₄	C₂×C₄×Dic₃	1	4	yes	—
185	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
186	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
187	Dic₃	—	6	8	no	—
188	C₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	2	4	no	—
190	C₂³	C₄×Dic₉	1	4	yes	—
191	C₂³	C₂×C₄×D₉	1	2	yes	—
192	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
193	C₂×C₄	—	9	8	no	—
194	C₂×C₄	—	18	8	no	—
195	C₂×C₄	C₄×Dic₉	1	8	yes	—
196	C₂×C₄	C₄×Dic₉	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₉	1	4	yes	—
199	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
201	C₂×C₄	C₂²×Dic₉	1	2	yes	—
202	C₂×C₄	C₂²×Dic₉	1	4	yes	—
203	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
204	C₈	—	9	8	no	—
205	C₈	C₂²×Dic₉	1	4	yes	—
206	C₈	C₂²×Dic₉	1	4	yes	—
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	#5101	1	4	yes	—
209	C₂×C₃	C₂×#901	1	2	yes	—
210	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
211	C₂²	—	2	8	no	—
212	C₂²	—	2	4	no	—
213	C₂²	#63464	1	4	yes	—
214	C₂²	C₂×C₄×Dic₉	1	4	yes	—
215	C₂²	C₂×C₄×Dic₉	1	2	yes	—
216	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
217	C₄	—	18	8	no	—
218	C₄	C₂×C₄×Dic₉	1	2	yes	—
219	C₄	C₂×C₄×Dic₉	1	4	yes	—
220	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
221	C₄	C₂³×Dic₉	1	1	yes	characteristic
222	C₃	C₂×#5101	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#10588782	1	4	yes	direct product factor
225	C₂	C₂×#63464	1	2	yes	—
226	C₂	C₂²×C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₉
commutator	C₂×C₉	C₂²×C₄²
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂³×C₈×C₉	C₂
socle	C₂³×C₃	C₂×C₄×D₃
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂³×C₈	D₉
hypocenter	C₉	C₂×#405
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
4. #10588782	C₂	direct (N × H)
5. C₂×#63375	C₂	semidirect
6. C₂×C₈×Dic₉	C₂	semidirect
9. C₂²×C₈×C₉	C₄	semidirect
18. C₂×#3667	C₄	semidirect
21. #63375	C₂²	semidirect
22. C₈×Dic₉	C₂²	semidirect
39. C₂×C₈×C₉	C₂×C₄	semidirect
46. #3667	C₂×C₄	semidirect
124. C₂²×C₈	Dic₉	semidirect
154. C₂×C₉	#405	semidirect
170. C₂×C₈	C₂×Dic₉	semidirect
188. C₉	C₂×#405	semidirect
224. C₂	#10588782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×#63488	2
C₂²×#3667	2
#10588782	2
C₂×#63375	2
C₂×C₈×Dic₉	2
C₂×#5101	3

Subgroups — C2×#10588782 #11857175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588782 #11857175