Subgroups — C₂×#10588359 #11861284

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#63375	C₂	1	2	yes	maximal
4	C₂×Dic₉⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10588359	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4679	—	3	1	no	maximal
8	C₂²×Dic₁₈	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	C₄	1	1	yes	characteristic
13	C₂×Dic₉⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₉	C₂²	1	2	yes	—
15	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
16	C₄×Dic₁₈	C₄	1	4	yes	—
17	C₄×Dic₁₈	C₂²	1	4	yes	—
18	C₄×C₈×C₉	C₂²	1	4	yes	—
19	#63375	C₂²	1	8	yes	—
20	Dic₉⋊C₈	C₂²	1	8	yes	—
21	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
22	C₂×C₄×Dic₆	—	3	1	no	—
23	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
24	C₂×#815	—	3	2	no	—
25	C₂×Dic₃⋊C₈	—	3	2	no	—
26	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
27	#4679	—	3	8	no	—
28	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₉	C₂×C₄	1	2	yes	—
32	C₂²×Dic₉	C₂×C₄	1	2	yes	—
33	C₂×Dic₁₈	—	2	4	no	—
34	C₂×Dic₁₈	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	Q₈	1	4	yes	complemented
37	C₂×C₈×C₉	C₂³	1	2	yes	—
38	C₂×C₈×C₉	C₂³	1	2	yes	—
39	C₄²×C₉	C₂×C₄	1	2	yes	—
40	C₄²×C₉	C₂³	1	2	yes	—
41	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
42	Dic₉⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₉	C₂×C₄	1	4	yes	—
44	C₄×Dic₉	C₂³	1	4	yes	—
45	C₂×C₉⋊C₈	—	2	4	no	—
46	C₂×C₉⋊C₈	C₂³	1	4	yes	—
47	C₂×#446	—	9	1	no	Sylow
48	C₂²×Dic₆	—	3	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₃	—	3	1	no	—
53	C₂×Dic₃⋊C₄	—	3	2	no	—
54	C₂×C₄×Dic₃	—	3	2	no	—
55	C₂²×C₃⋊C₈	—	3	2	no	—
56	C₄×Dic₆	—	3	4	no	—
57	C₄×Dic₆	—	3	4	no	—
58	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
59	#815	—	3	8	no	—
60	Dic₃⋊C₈	—	3	8	no	—
61	C₄⋊(C₃⋊C₈)	—	3	4	no	—
62	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	M₁₆	1	4	yes	—
65	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₉	#41	1	2	yes	—
70	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₉	—	2	4	no	—
72	C₂×Dic₉	—	2	4	no	—
73	C₂×Dic₉	C₂²×C₄	1	4	yes	—
74	C₂×Dic₉	C₂²×C₄	1	4	yes	—
75	Dic₁₈	—	2	8	no	—
76	C₈×C₉	—	2	2	no	—
77	C₈×C₉	C₂×Q₈	1	4	yes	complemented
78	C₉⋊C₈	—	2	4	no	—
79	C₂×C₄×Q₈	—	9	1	no	—
80	#446	—	9	8	no	—
81	C₂×C₄⋊C₈	—	9	2	no	—
82	C₂×C₄⋊C₈	—	9	1	no	—
83	C₂×#97	—	9	2	no	—
84	C₂×C₄×C₈	D₉	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂²×Dic₃	—	3	2	no	—
89	C₂²×Dic₃	—	3	2	no	—
90	C₂×Dic₆	—	3	4	no	—
91	C₂×Dic₆	—	6	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	Dic₆	1	4	yes	—
94	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
95	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
96	C₃×C₄²	C₄×D₃	1	2	yes	—
97	C₃×C₄²	C₂²×D₃	1	2	yes	—
98	C₄⋊Dic₃	—	3	4	no	—
99	Dic₃⋊C₄	—	3	8	no	—
100	C₄×Dic₃	—	3	4	no	—
101	C₄×Dic₃	—	3	4	no	—
102	C₂×C₃⋊C₈	—	3	4	no	—
103	C₂×C₃⋊C₈	—	6	4	no	—
104	C₂²×C₉	C₄×Q₈	1	2	yes	—
105	C₂²×C₉	C₂×M₁₆	1	2	yes	—
106	C₂²×C₉	#131	1	2	yes	—
107	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₉	—	2	2	no	—
109	C₄×C₉	C₄×Q₈	1	2	yes	—
110	C₄×C₉	C₂×M₁₆	1	4	yes	—
111	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
112	C₄×C₉	C₂×#41	1	1	yes	characteristic
113	Dic₉	—	2	4	no	—
114	Dic₉	—	2	4	no	—
115	C₂²×Q₈	—	9	1	no	—
116	C₂²×C₈	—	9	2	no	—
117	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
119	C₄×Q₈	—	9	4	no	—
120	C₄×Q₈	—	9	4	no	—
121	C₂×C₄⋊C₄	—	9	2	no	—
122	C₂×C₄⋊C₄	—	9	1	no	—
123	C₂×C₄²	—	9	2	no	—
124	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
125	C₄⋊C₈	—	9	8	no	—
126	C₄⋊C₈	—	9	4	no	—
127	#97	—	9	8	no	—
128	C₄×C₈	C₂×D₉	1	4	yes	—
129	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	4	yes	—
132	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
133	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
134	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
136	C₂×C₃×C₄	#235	1	2	yes	—
137	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
138	C₂×Dic₃	—	3	4	no	—
139	C₂×Dic₃	—	3	4	no	—
140	C₂×Dic₃	—	6	4	no	—
141	C₂×Dic₃	—	6	4	no	—
142	Dic₆	—	6	8	no	—
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	C₂×Dic₆	1	4	yes	—
145	C₃⋊C₈	—	6	4	no	—
146	C₂×C₉	#446	1	4	yes	complemented
147	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×C₉	C₂×#131	1	1	yes	characteristic
150	C₂×Q₈	—	9	4	no	—
151	C₂×Q₈	—	18	4	no	—
152	C₂²×C₄	—	9	2	no	—
153	C₂²×C₄	—	9	2	no	—
154	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
155	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	9	4	no	—
159	C₂×C₈	—	18	4	no	—
160	C₂×C₈	Dic₁₈	1	4	yes	complemented
161	C₂×C₈	C₂²×D₉	1	2	yes	—
162	C₂×C₈	C₂²×D₉	1	2	yes	—
163	C₄⋊C₄	—	9	8	no	—
164	C₄⋊C₄	—	9	4	no	—
165	C₄²	—	9	4	no	—
166	C₄²	—	9	4	no	—
167	C₄²	C₄×D₉	1	2	yes	—
168	C₄²	C₂²×D₉	1	2	yes	—
169	C₂²×C₃	C₄×Dic₆	1	2	yes	—
170	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
171	C₂²×C₃	#901	1	2	yes	—
172	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	C₄×Dic₆	1	2	yes	—
175	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
176	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
177	C₃×C₄	C₂×#235	1	1	yes	characteristic
178	Dic₃	—	6	4	no	—
179	Dic₃	—	6	4	no	—
180	C₉	C₂×#446	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
182	Q₈	—	18	8	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	9	4	no	—
185	C₂×C₄	—	9	4	no	—
186	C₂×C₄	—	18	4	no	—
187	C₂×C₄	—	18	4	no	—
188	C₂×C₄	#3663	1	4	yes	—
189	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
190	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
193	C₂×C₄	#3697	1	2	yes	—
194	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	18	4	no	—
197	C₈	C₂×Dic₁₈	1	4	yes	complemented
198	C₂×C₃	#4679	1	4	yes	—
199	C₂×C₃	C₂×C₄×Dic₆	1	1	yes	characteristic
200	C₂×C₃	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
201	C₂×C₃	C₂×#901	1	1	yes	characteristic
202	C₂²	C₄×Dic₁₈	1	2	yes	—
203	C₂²	C₂×#3663	1	2	yes	—
204	C₂²	#63464	1	2	yes	—
205	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	18	4	no	—
208	C₄	—	18	4	no	—
209	C₄	C₄×Dic₁₈	1	2	yes	—
210	C₄	C₂×#3663	1	4	yes	—
211	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
212	C₄	C₂×#3697	1	1	yes	characteristic
213	C₃	C₂×#4679	1	1	yes	characteristic
214	C₂	#10588359	1	4	yes	direct product factor
215	C₂	C₂×C₄×Dic₁₈	1	1	yes	characteristic
216	C₂	C₂²×#3663	1	1	yes	characteristic
217	C₂	C₂×#63464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	C₂×#446
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #10588359	C₂	direct (N × H)
36. C₂×C₈×C₉	Q₈	semidirect
77. C₈×C₉	C₂×Q₈	semidirect
146. C₂×C₉	#446	semidirect
160. C₂×C₈	Dic₁₈	semidirect
180. C₉	C₂×#446	semidirect
197. C₈	C₂×Dic₁₈	semidirect
214. C₂	#10588359	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈	2
C₂×C₄×C₈×C₉	2
C₂×#63375	2
C₂×Dic₉⋊C₈	2
C₂×C₄⋊(C₉⋊C₈)	2
#10588359	2
C₂×#4679	3

Subgroups — C2×#10588359 #11861284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10588359 #11861284