Subgroups — C₂²×#3676⋊C₃ #13153765

Order 1728.

2396 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂³×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×#3676⋊C₃	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×#217	—	3	1	no	maximal
5	C₂²×#3676	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×3₋¹⁺²	—	2	8	no	—
7	C₂⁴×3₋¹⁺²	C₄	1	8	yes	complemented
8	C₂⁴×3₋¹⁺²	C₂²	1	6	yes	—
9	C₂⁴×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂²×3₋¹⁺²⋊C₄	—	2	8	no	—
11	C₂²×3₋¹⁺²⋊C₄	C₂²	1	12	yes	complemented
12	#3676⋊C₃	C₂²	1	16	yes	direct product factor
13	C₂⁵×C₃²	D₃	1	1	yes	characteristic
14	C₂³×C₃×Dic₃	—	3	2	no	—
15	C₂⁵×C₉	—	2	1	no	—
16	C₂⁵×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#217	—	3	12	no	—
18	C₂³×Dic₉	C₂×C₃	1	2	yes	complemented
19	C₂×#3676	C₂×C₃	1	12	yes	complemented
20	C₂³×3₋¹⁺²	—	2	32	no	—
21	C₂³×3₋¹⁺²	—	2	24	no	—
22	C₂³×3₋¹⁺²	C₂×C₄	1	4	yes	—
23	C₂³×3₋¹⁺²	C₂×C₄	1	24	yes	complemented
24	C₂³×3₋¹⁺²	D₄	1	8	yes	—
25	C₂³×3₋¹⁺²	C₂³	1	4	yes	—
26	C₂³×3₋¹⁺²	C₂³	1	3	yes	—
27	C₂×3₋¹⁺²⋊C₄	—	2	24	no	—
28	C₂×3₋¹⁺²⋊C₄	C₂³	1	8	yes	complemented
29	C₂²×C₃×C₂²⋊C₄	—	9	1	no	—
30	C₂²×#217	—	3	1	no	—
31	C₂⁴×C₃²	—	2	8	no	—
32	C₂⁴×C₃²	Dic₃	1	8	yes	—
33	C₂⁴×C₃²	C₂×D₃	1	6	yes	—
34	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃×Dic₃	—	3	12	no	—
36	C₂²×C₃×Dic₃	—	6	8	no	—
37	C₂⁴×C₉	—	2	16	no	—
38	C₂⁴×C₉	—	2	8	no	—
39	C₂⁴×C₉	—	2	6	no	—
40	C₂⁴×C₉	—	2	1	no	—
41	C₂⁴×C₉	—	2	8	no	—
42	C₂⁴×C₉	C₃×C₄	1	8	yes	complemented
43	C₂⁴×C₉	C₂²×C₃	1	6	yes	—
44	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₃×#217	—	3	16	no	—
46	C₂²×Dic₉	—	2	8	no	—
47	C₂²×Dic₉	C₂²×C₃	1	12	yes	complemented
48	#3676	C₂²×C₃	1	16	yes	complemented
49	C₂²×3₋¹⁺²	—	2	48	no	—
50	C₂²×3₋¹⁺²	—	2	8	no	—
51	C₂²×3₋¹⁺²	C₂²⋊C₄	1	16	yes	complemented
52	C₂²×3₋¹⁺²	C₂²×C₄	1	6	yes	—
53	C₂²×3₋¹⁺²	C₂²×C₄	1	8	yes	complemented
54	C₂²×3₋¹⁺²	C₂×D₄	1	12	yes	—
55	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
56	3₋¹⁺²⋊C₄	—	2	8	no	—
57	C₂⁵×C₃	—	3	1	no	—
58	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
59	C₂³×C₃×C₄	—	9	2	no	—
60	C₂³×Dic₃	—	3	2	no	—
61	C₂×C₃×C₂²⋊C₄	—	9	12	no	—
62	C₂×#217	—	3	12	no	—
63	C₂³×C₃²	—	2	32	no	—
64	C₂³×C₃²	—	2	24	no	—
65	C₂³×C₃²	C₂×Dic₃	1	24	yes	—
66	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
67	C₂³×C₃²	#67	1	8	yes	—
68	C₂³×C₃²	C₂²×D₃	1	4	yes	—
69	C₂³×C₃²	C₂²×D₃	1	3	yes	—
70	C₂×C₃×Dic₃	—	3	8	no	—
71	C₂×C₃×Dic₃	—	6	24	no	—
72	C₂³×C₉	—	2	64	no	—
73	C₂³×C₉	—	2	48	no	—
74	C₂³×C₉	—	2	32	no	—
75	C₂³×C₉	—	2	24	no	—
76	C₂³×C₉	—	2	4	no	—
77	C₂³×C₉	—	2	24	no	—
78	C₂³×C₉	—	2	8	no	—
79	C₂³×C₉	—	2	4	no	—
80	C₂³×C₉	—	2	3	no	—
81	C₂³×C₉	C₂×C₃×C₄	1	4	yes	—
82	C₂³×C₉	C₂×C₃×C₄	1	24	yes	complemented
83	C₂³×C₉	C₃×D₄	1	8	yes	—
84	C₂³×C₉	C₂³×C₃	1	4	yes	—
85	C₂³×C₉	C₂³×C₃	1	3	yes	—
86	C₂×Dic₉	—	2	24	no	—
87	C₂×Dic₉	C₂³×C₃	1	8	yes	complemented
88	C₂²×C₂²⋊C₄	—	9	1	no	Sylow
89	C₂×3₋¹⁺²	—	2	8	no	—
90	C₂×3₋¹⁺²	C₂×C₂²⋊C₄	1	12	yes	complemented
91	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
92	C₂×3₋¹⁺²	C₂²×D₄	1	2	yes	—
93	C₂⁴×C₃	—	2	8	no	—
94	C₂⁴×C₃	—	3	8	no	—
95	C₂⁴×C₃	—	3	6	no	—
96	C₂⁴×C₃	—	3	1	no	—
97	C₂⁴×C₃	—	6	8	no	—
98	C₂⁴×C₃	C₃×Dic₃	1	8	yes	—
99	C₂⁴×C₃	C₂×C₃×D₃	1	6	yes	—
100	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	—	9	12	no	—
102	C₂²×C₃×C₄	—	18	8	no	—
103	C₂²×Dic₃	—	3	12	no	—
104	C₂²×Dic₃	—	6	8	no	—
105	C₃×C₂²⋊C₄	—	9	16	no	—
106	#217	—	3	16	no	—
107	C₂²×C₃²	—	2	48	no	—
108	C₂²×C₃²	—	2	8	no	—
109	C₂²×C₃²	#217	1	16	yes	—
110	C₂²×C₃²	C₂²×Dic₃	1	8	yes	—
111	C₂²×C₃²	C₂²×Dic₃	1	6	yes	—
112	C₂²×C₃²	C₂×#67	1	12	yes	—
113	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
114	C₃×Dic₃	—	6	8	no	—
115	C₂²×C₉	—	2	96	no	—
116	C₂²×C₉	—	2	16	no	—
117	C₂²×C₉	—	2	48	no	—
118	C₂²×C₉	—	2	8	no	—
119	C₂²×C₉	—	2	1	no	—
120	C₂²×C₉	—	2	16	no	—
121	C₂²×C₉	—	2	6	no	—
122	C₂²×C₉	—	2	8	no	—
123	C₂²×C₉	—	2	12	no	—
124	C₂²×C₉	C₃×C₂²⋊C₄	1	16	yes	complemented
125	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
126	C₂²×C₉	C₂²×C₃×C₄	1	8	yes	complemented
127	C₂²×C₉	C₂×C₃×D₄	1	12	yes	—
128	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
129	Dic₉	—	2	8	no	—
130	C₂⁵	AGL(1,ℤ/9)	1	1	yes	characteristic
131	C₂³×C₄	—	9	2	no	—
132	C₂×C₂²⋊C₄	—	9	12	no	—
133	3₋¹⁺²	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₃	—	2	32	no	—
135	C₂³×C₃	—	2	24	no	—
136	C₂³×C₃	—	3	4	no	—
137	C₂³×C₃	—	3	24	no	—
138	C₂³×C₃	—	3	8	no	—
139	C₂³×C₃	—	3	3	no	—
140	C₂³×C₃	—	3	4	no	—
141	C₂³×C₃	—	6	24	no	—
142	C₂³×C₃	—	6	32	no	—
143	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
144	C₂³×C₃	C₂×C₃×Dic₃	1	24	yes	—
145	C₂³×C₃	C₃×#67	1	8	yes	—
146	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
147	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
148	C₂×C₃×C₄	—	9	8	no	—
149	C₂×C₃×C₄	—	18	24	no	—
150	C₂×Dic₃	—	3	8	no	—
151	C₂×Dic₃	—	6	24	no	—
152	C₂×C₃²	—	2	8	no	—
153	C₂×C₃²	C₂×#217	1	12	yes	—
154	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
155	C₂×C₃²	C₂²×#67	1	2	yes	—
156	C₂×C₉	—	2	16	no	—
157	C₂×C₉	—	2	8	no	—
158	C₂×C₉	—	2	12	no	—
159	C₂×C₉	—	2	1	no	—
160	C₂×C₉	—	2	2	no	—
161	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	12	yes	complemented
162	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
163	C₂×C₉	C₂²×C₃×D₄	1	2	yes	—
164	C₂⁴	—	2	8	no	—
165	C₂⁴	3₋¹⁺²⋊C₄	1	8	yes	complemented
166	C₂⁴	C₂×AGL(1,ℤ/9)	1	6	yes	—
167	C₂⁴	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
168	C₂²×C₄	—	9	12	no	—
169	C₂²×C₄	—	18	8	no	—
170	C₂²⋊C₄	—	9	16	no	—
171	C₂²×C₃	—	2	48	no	—
172	C₂²×C₃	—	2	8	no	—
173	C₂²×C₃	—	3	1	no	—
174	C₂²×C₃	—	3	6	no	—
175	C₂²×C₃	—	3	8	no	—
176	C₂²×C₃	—	3	12	no	—
177	C₂²×C₃	—	3	16	no	—
178	C₂²×C₃	—	6	8	no	—
179	C₂²×C₃	—	6	48	no	—
180	C₂²×C₃	C₃×#217	1	16	yes	—
181	C₂²×C₃	C₂²×C₃×Dic₃	1	6	yes	—
182	C₂²×C₃	C₂²×C₃×Dic₃	1	8	yes	—
183	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
184	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
185	C₃×C₄	—	18	8	no	—
186	Dic₃	—	6	8	no	—
187	C₃²	C₂²×#217	1	1	yes	characteristic
188	C₉	—	2	1	no	—
189	C₉	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
190	C₂³	—	2	24	no	—
191	C₂³	—	2	32	no	—
192	C₂³	C₂×3₋¹⁺²⋊C₄	1	4	yes	—
193	C₂³	C₂×3₋¹⁺²⋊C₄	1	24	yes	complemented
194	C₂³	#611⋊C₃	1	8	yes	—
195	C₂³	C₂²×AGL(1,ℤ/9)	1	4	yes	—
196	C₂³	C₂²×AGL(1,ℤ/9)	1	3	yes	—
197	C₂×C₄	—	9	8	no	—
198	C₂×C₄	—	18	24	no	—
199	C₂×C₃	—	2	8	no	—
200	C₂×C₃	—	3	1	no	—
201	C₂×C₃	—	3	2	no	—
202	C₂×C₃	—	3	12	no	—
203	C₂×C₃	—	6	8	no	—
204	C₂×C₃	C₂×C₃×#217	1	12	yes	—
205	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
206	C₂×C₃	C₂²×C₃×#67	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	48	no	—
209	C₂²	#3676⋊C₃	1	16	yes	direct product factor
210	C₂²	C₂²×3₋¹⁺²⋊C₄	1	6	yes	—
211	C₂²	C₂²×3₋¹⁺²⋊C₄	1	8	yes	complemented
212	C₂²	C₂×#611⋊C₃	1	12	yes	—
213	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
214	C₄	—	18	8	no	—
215	C₃	—	3	1	no	—
216	C₃	C₂²×C₃×#217	1	1	yes	characteristic
217	C₂	—	2	8	no	—
218	C₂	C₂×#3676⋊C₃	1	12	yes	direct product factor
219	C₂	C₂³×3₋¹⁺²⋊C₄	1	1	yes	characteristic
220	C₂	C₂²×#611⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂⁵×3₋¹⁺²	C₂
socle	C₂⁴×C₃	C₂×C₃×D₃
prime radical	C₂⁵×C₃²	D₃
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₂⁵	AGL(1,ℤ/9)
hypocenter	C₉	C₂²×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂³×3₋¹⁺²⋊C₄	C₂	semidirect
3. C₂×#3676⋊C₃	C₂	direct (N × H)
5. C₂²×#3676	C₃	semidirect
7. C₂⁴×3₋¹⁺²	C₄	semidirect
11. C₂²×3₋¹⁺²⋊C₄	C₂²	semidirect
12. #3676⋊C₃	C₂²	direct (N × H)
18. C₂³×Dic₉	C₂×C₃	semidirect
19. C₂×#3676	C₂×C₃	semidirect
23. C₂³×3₋¹⁺²	C₂×C₄	semidirect
28. C₂×3₋¹⁺²⋊C₄	C₂³	semidirect
42. C₂⁴×C₉	C₃×C₄	semidirect
47. C₂²×Dic₉	C₂²×C₃	semidirect
48. #3676	C₂²×C₃	semidirect
51. C₂²×3₋¹⁺²	C₂²⋊C₄	semidirect
53. C₂²×3₋¹⁺²	C₂²×C₄	semidirect
82. C₂³×C₉	C₂×C₃×C₄	semidirect
87. C₂×Dic₉	C₂³×C₃	semidirect
90. C₂×3₋¹⁺²	C₂×C₂²⋊C₄	semidirect
124. C₂²×C₉	C₃×C₂²⋊C₄	semidirect
126. C₂²×C₉	C₂²×C₃×C₄	semidirect
133. 3₋¹⁺²	C₂²×C₂²⋊C₄	semidirect
161. C₂×C₉	C₂×C₃×C₂²⋊C₄	semidirect
165. C₂⁴	3₋¹⁺²⋊C₄	semidirect
189. C₉	C₂²×C₃×C₂²⋊C₄	semidirect
193. C₂³	C₂×3₋¹⁺²⋊C₄	semidirect
209. C₂²	#3676⋊C₃	direct (N × H)
211. C₂²	C₂²×3₋¹⁺²⋊C₄	semidirect
218. C₂	C₂×#3676⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×3₋¹⁺²	2
C₂³×3₋¹⁺²⋊C₄	2
C₂×#3676⋊C₃	2
C₂²×C₃×#217	3
C₂²×#3676	3

Subgroups — C22×#3676⋊C3 #13153765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#3676⋊C₃ #13153765