Subgroups — C₂×#10589585 #11901862

Order 1152.

1944 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3676	C₂	1	2	yes	maximal
3	C₂×#63486	C₂	1	4	yes	maximal, complemented
4	#10589585	C₂	1	16	yes	maximal, direct product factor
5	C₂×#63440	C₂	1	8	yes	maximal, complemented
6	C₂×#5483	—	3	1	no	maximal
7	C₂⁵×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	C₂²	1	2	yes	—
9	C₂³×Dic₉	C₂²	1	4	yes	—
10	C₂²×Dic₁₈	C₂²	1	4	yes	complemented
11	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
12	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
13	C₂×#3676	C₂²	1	4	yes	—
14	C₂×#3676	C₂²	1	16	yes	—
15	C₂×#3676	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₉	C₂²	1	4	yes	complemented
17	#63486	C₂²	1	32	yes	complemented
18	C₂×Dic₉⋊C₄	C₂²	1	8	yes	complemented
19	#63440	C₂²	1	64	yes	complemented
20	C₂²×C₃×C₂²⋊C₄	D₃	1	1	yes	characteristic
21	C₂²×#217	—	3	2	no	—
22	C₂×#924	—	3	4	no	—
23	#5483	—	3	16	no	—
24	C₂×#878	—	3	8	no	—
25	C₂⁴×C₉	—	4	4	no	—
26	C₂⁴×C₉	Q₈	1	8	yes	complemented
27	C₂⁴×C₉	C₂³	1	1	yes	characteristic
28	C₂⁴×C₉	C₂³	1	2	yes	—
29	C₂⁴×C₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₉	—	4	4	no	—
31	C₂²×C₄×C₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₉	C₂³	1	8	yes	—
33	C₂²×Dic₉	—	4	8	no	—
34	C₂²×Dic₉	C₂³	1	8	yes	—
35	C₂²×Dic₉	C₂³	1	16	yes	—
36	C₂×Dic₁₈	—	4	8	no	—
37	C₂×Dic₁₈	C₂³	1	16	yes	complemented
38	C₉×C₂²⋊C₄	C₂³	1	16	yes	—
39	#3676	C₂³	1	16	yes	—
40	#3676	C₂³	1	16	yes	—
41	C₄⋊Dic₉	C₂³	1	16	yes	complemented
42	Dic₉⋊C₄	C₂³	1	32	yes	complemented
43	C₂×#543	—	9	1	no	Sylow
44	C₂⁵×C₃	C₂×D₃	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂×D₃	1	2	yes	—
46	C₂³×Dic₃	—	3	4	no	—
47	C₂²×Dic₆	—	3	4	no	—
48	C₂×C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
49	C₂×C₃×C₂²⋊C₄	C₂×D₃	1	8	yes	—
50	C₂×#217	—	3	4	no	—
51	C₂×#217	—	3	16	no	—
52	C₂×#217	—	3	4	no	—
53	C₂×C₄⋊Dic₃	—	3	4	no	—
54	#924	—	3	32	no	—
55	C₂×Dic₃⋊C₄	—	3	8	no	—
56	#878	—	3	64	no	—
57	C₂³×C₉	—	2	8	no	—
58	C₂³×C₉	—	2	4	no	—
59	C₂³×C₉	—	4	16	no	—
60	C₂³×C₉	—	4	8	no	—
61	C₂³×C₉	C₂×Q₈	1	4	yes	—
62	C₂³×C₉	C₂×Q₈	1	8	yes	—
63	C₂³×C₉	C₂×Q₈	1	16	yes	complemented
64	C₂³×C₉	C₂⁴	1	2	yes	—
65	C₂³×C₉	C₂⁴	1	4	yes	—
66	C₂³×C₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₉	—	2	8	no	—
68	C₂×C₄×C₉	—	4	8	no	—
69	C₂×C₄×C₉	C₂⁴	1	8	yes	—
70	C₂×Dic₉	—	2	16	no	—
71	C₂×Dic₉	—	4	16	no	—
72	C₂×Dic₉	C₂⁴	1	16	yes	—
73	Dic₁₈	—	4	16	no	—
74	#543	—	9	16	no	—
75	C₂×C₂²⋊Q₈	—	9	4	no	—
76	C₂×C₂²⋊Q₈	—	9	8	no	—
77	C₂²×C₂²⋊C₄	—	9	2	no	—
78	C₂²×C₂²⋊C₄	D₉	1	1	yes	characteristic
79	C₂⁴×C₃	—	4	4	no	—
80	C₂⁴×C₃	Dic₆	1	8	yes	—
81	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
82	C₂⁴×C₃	C₂²×D₃	1	2	yes	—
83	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
84	C₂²×C₃×C₄	—	4	4	no	—
85	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×D₃	1	8	yes	—
87	C₂²×Dic₃	—	3	8	no	—
88	C₂²×Dic₃	—	3	16	no	—
89	C₂²×Dic₃	—	12	8	no	—
90	C₂×Dic₆	—	3	16	no	—
91	C₂×Dic₆	—	12	8	no	—
92	C₃×C₂²⋊C₄	C₂²×D₃	1	16	yes	—
93	#217	—	3	16	no	—
94	#217	—	3	16	no	—
95	C₄⋊Dic₃	—	3	16	no	—
96	Dic₃⋊C₄	—	3	32	no	—
97	C₂²×C₉	—	2	16	no	—
98	C₂²×C₉	—	2	8	no	—
99	C₂²×C₉	—	2	8	no	—
100	C₂²×C₉	—	2	4	no	—
101	C₂²×C₉	—	4	16	no	—
102	C₂²×C₉	C₂²×Q₈	1	4	yes	—
103	C₂²×C₉	C₂²×Q₈	1	8	yes	—
104	C₂²×C₉	C₂²×Q₈	1	2	yes	—
105	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
106	C₂²×C₉	C₂⁵	1	1	yes	characteristic
107	C₄×C₉	—	2	8	no	—
108	Dic₉	—	2	16	no	—
109	C₂⁵	C₂×D₉	1	1	yes	characteristic
110	C₂²×Q₈	—	9	4	no	—
111	C₂³×C₄	—	9	4	no	—
112	C₂³×C₄	C₂×D₉	1	2	yes	—
113	C₂²⋊Q₈	—	9	32	no	—
114	C₂²⋊Q₈	—	9	64	no	—
115	C₂×C₄⋊C₄	—	9	8	no	—
116	C₂×C₄⋊C₄	—	9	4	no	—
117	C₂×C₂²⋊C₄	—	9	4	no	—
118	C₂×C₂²⋊C₄	—	9	4	no	—
119	C₂×C₂²⋊C₄	—	9	16	no	—
120	C₂×C₂²⋊C₄	C₂×D₉	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂×D₉	1	8	yes	—
122	C₂³×C₃	—	2	8	no	—
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	4	16	no	—
125	C₂³×C₃	—	4	8	no	—
126	C₂³×C₃	C₂×Dic₆	1	4	yes	—
127	C₂³×C₃	C₂×Dic₆	1	8	yes	—
128	C₂³×C₃	C₂×Dic₆	1	16	yes	—
129	C₂³×C₃	C₂³×D₃	1	2	yes	—
130	C₂³×C₃	C₂³×D₃	1	4	yes	—
131	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	8	no	—
133	C₂×C₃×C₄	—	4	8	no	—
134	C₂×C₃×C₄	C₂³×D₃	1	8	yes	—
135	C₂×Dic₃	—	3	16	no	—
136	C₂×Dic₃	—	6	16	no	—
137	C₂×Dic₃	—	12	16	no	—
138	Dic₆	—	12	16	no	—
139	C₂×C₉	—	2	8	no	—
140	C₂×C₉	—	2	4	no	—
141	C₂×C₉	#543	1	4	yes	complemented
142	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
143	C₂×C₉	C₂×H(5,𝔽₂)	1	2	yes	—
144	C₂⁴	—	4	4	no	—
145	C₂⁴	Dic₁₈	1	8	yes	complemented
146	C₂⁴	C₂²×D₉	1	1	yes	characteristic
147	C₂⁴	C₂²×D₉	1	2	yes	—
148	C₂⁴	C₂²×D₉	1	4	yes	—
149	C₂×Q₈	—	9	16	no	—
150	C₂×Q₈	—	36	8	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	—	9	8	no	—
153	C₂²×C₄	—	9	16	no	—
154	C₂²×C₄	—	36	8	no	—
155	C₂²×C₄	C₂²×D₉	1	4	yes	—
156	C₂²×C₄	C₂²×D₉	1	8	yes	—
157	C₄⋊C₄	—	9	32	no	—
158	C₄⋊C₄	—	9	16	no	—
159	C₂²⋊C₄	—	9	16	no	—
160	C₂²⋊C₄	—	9	16	no	—
161	C₂²⋊C₄	C₂²×D₉	1	16	yes	—
162	C₂²×C₃	—	2	8	no	—
163	C₂²×C₃	—	2	4	no	—
164	C₂²×C₃	—	2	16	no	—
165	C₂²×C₃	—	2	8	no	—
166	C₂²×C₃	—	4	16	no	—
167	C₂²×C₃	C₂²×Dic₆	1	4	yes	—
168	C₂²×C₃	C₂²×Dic₆	1	8	yes	—
169	C₂²×C₃	C₂²×Dic₆	1	2	yes	—
170	C₂²×C₃	#1004	1	4	yes	—
171	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
172	C₃×C₄	—	2	8	no	—
173	Dic₃	—	6	16	no	—
174	C₉	C₂×#543	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	2	4	no	—
176	C₂³	—	2	8	no	—
177	C₂³	—	4	16	no	—
178	C₂³	—	4	8	no	—
179	C₂³	C₂×Dic₁₈	1	4	yes	—
180	C₂³	C₂×Dic₁₈	1	8	yes	—
181	C₂³	C₂×Dic₁₈	1	16	yes	complemented
182	C₂³	C₂³×D₉	1	2	yes	—
183	C₂³	C₂³×D₉	1	4	yes	—
184	C₂³	C₂³×D₉	1	1	yes	characteristic
185	Q₈	—	36	16	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	—	9	16	no	—
189	C₂×C₄	—	18	16	no	—
190	C₂×C₄	—	36	16	no	—
191	C₂×C₄	C₂³×D₉	1	8	yes	—
192	C₂×C₃	—	2	8	no	—
193	C₂×C₃	—	2	4	no	—
194	C₂×C₃	#5483	1	4	yes	—
195	C₂×C₃	C₂³×Dic₆	1	1	yes	characteristic
196	C₂×C₃	C₂×#1004	1	2	yes	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	16	no	—
201	C₂²	—	4	16	no	—
202	C₂²	C₂²×Dic₁₈	1	4	yes	—
203	C₂²	C₂²×Dic₁₈	1	8	yes	—
204	C₂²	C₂²×Dic₁₈	1	2	yes	—
205	C₂²	#63710	1	4	yes	—
206	C₂²	C₂⁴×D₉	1	1	yes	characteristic
207	C₄	—	2	8	no	—
208	C₄	—	18	16	no	—
209	C₃	C₂×#5483	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	2	8	no	—
212	C₂	#10589585	1	4	yes	direct product factor
213	C₂	C₂³×Dic₁₈	1	1	yes	characteristic
214	C₂	C₂×#63710	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁵
Frattini	C₂²×C₃	C₂⁴×D₃
Fitting	C₂²×C₉×C₂²⋊C₄	C₂
socle	C₂³×C₃	C₂³×D₃
prime radical	C₂⁵×C₃	C₂×D₃
hypercenter	C₂²×C₂²⋊C₄	D₉
hypocenter	C₉	C₂×#543
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#63486	C₂	semidirect
4. #10589585	C₂	direct (N × H)
5. C₂×#63440	C₂	semidirect
10. C₂²×Dic₁₈	C₂²	semidirect
16. C₂×C₄⋊Dic₉	C₂²	semidirect
17. #63486	C₂²	semidirect
18. C₂×Dic₉⋊C₄	C₂²	semidirect
19. #63440	C₂²	semidirect
26. C₂⁴×C₉	Q₈	semidirect
37. C₂×Dic₁₈	C₂³	semidirect
41. C₄⋊Dic₉	C₂³	semidirect
42. Dic₉⋊C₄	C₂³	semidirect
63. C₂³×C₉	C₂×Q₈	semidirect
141. C₂×C₉	#543	semidirect
145. C₂⁴	Dic₁₈	semidirect
174. C₉	C₂×#543	semidirect
181. C₂³	C₂×Dic₁₈	semidirect
212. C₂	#10589585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₂²⋊C₄	2
C₂²×#3676	2
C₂×#63486	2
#10589585	2
C₂×#63440	2
C₂×#5483	3

Subgroups — C2×#10589585 #11901862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589585 #11901862