Subgroups — C₂×C₈×3₋¹⁺²⋊C₄ #13143134

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×3₋¹⁺²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×3₋¹⁺²⋊C₄	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₈×Dic₃	—	3	1	no	maximal
6	C₂×C₈×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²⋊C₄	C₄	1	2	yes	—
9	C₂×C₈×3₋¹⁺²	C₄	1	4	yes	complemented
10	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
11	C₄×3₋¹⁺²⋊C₄	C₄	1	2	yes	—
12	C₄×3₋¹⁺²⋊C₄	C₂²	1	2	yes	—
13	C₂×3₋¹⁺²⋊C₈	C₄	1	4	yes	complemented
14	C₂×3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄×Dic₃	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	C₃×C₈×Dic₃	—	3	4	no	—
19	C₂²×C₈×C₉	—	2	1	no	—
20	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₈×Dic₉	C₂×C₃	1	4	yes	complemented
24	C₂³×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
27	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
28	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
29	C₂×3₋¹⁺²⋊C₄	C₈	1	8	yes	direct product factor
30	C₂×3₋¹⁺²⋊C₄	C₂×C₄	1	4	yes	—
31	C₈×3₋¹⁺²	C₂×C₄	1	4	yes	complemented
32	3₋¹⁺²⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₃×C₄×C₈	—	9	1	no	—
34	C₂×C₈×Dic₃	—	3	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃×Dic₃	—	3	2	no	—
37	C₂×C₃²×C₈	Dic₃	1	4	yes	—
38	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
39	C₃×C₄×Dic₃	—	3	2	no	—
40	C₃×C₄×Dic₃	—	3	2	no	—
41	C₂×C₃×C₃⋊C₈	—	3	4	no	—
42	C₂×C₃×C₃⋊C₈	—	3	2	no	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₂²×Dic₉	C₃×C₄	1	2	yes	—
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	C₃×C₄	1	4	yes	complemented
49	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
50	C₄×Dic₉	C₃×C₄	1	2	yes	—
51	C₄×Dic₉	C₂²×C₃	1	2	yes	—
52	C₂×C₉⋊C₈	C₃×C₄	1	4	yes	complemented
53	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
54	C₂²×3₋¹⁺²	C₄²	1	2	yes	—
55	C₂²×3₋¹⁺²	C₂×C₈	1	4	yes	—
56	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
57	C₄×3₋¹⁺²	C₄²	1	2	yes	—
58	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
59	C₄×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
60	3₋¹⁺²⋊C₄	C₂×C₈	1	8	yes	direct product factor
61	C₂²×C₃×C₈	—	3	1	no	—
62	C₂²×C₃×C₈	—	9	1	no	—
63	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
64	C₂×C₃×C₄²	—	9	1	no	—
65	C₂×C₄×Dic₃	—	3	1	no	—
66	C₂²×C₃⋊C₈	—	3	1	no	—
67	C₃×C₄×C₈	—	9	4	no	—
68	C₈×Dic₃	—	3	4	no	—
69	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
72	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃×Dic₃	—	3	8	no	—
75	C₂×C₃×Dic₃	—	3	4	no	—
76	C₂³×C₉	—	2	1	no	—
77	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
78	C₃²×C₈	C₂×Dic₃	1	4	yes	—
79	C₃×C₃⋊C₈	—	3	4	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
88	C₂×Dic₉	C₃×C₈	1	8	yes	complemented
89	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
90	C₈×C₉	—	2	4	no	—
91	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
92	C₉⋊C₈	C₂×C₃×C₄	1	4	yes	complemented
93	C₂×C₄×C₈	—	9	1	no	Sylow
94	C₂×3₋¹⁺²	C₄×C₈	1	4	yes	complemented
95	C₂×3₋¹⁺²	C₂×C₄²	1	1	yes	characteristic
96	C₂×3₋¹⁺²	C₂²×C₈	1	2	yes	—
97	C₂²×C₃×C₄	—	3	1	no	—
98	C₂²×C₃×C₄	—	9	2	no	—
99	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
100	C₂²×Dic₃	—	3	2	no	—
101	C₂×C₃×C₈	—	3	4	no	—
102	C₂×C₃×C₈	—	3	2	no	—
103	C₂×C₃×C₈	—	9	2	no	—
104	C₂×C₃×C₈	—	9	4	no	—
105	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
106	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
107	C₃×C₄²	—	9	2	no	—
108	C₃×C₄²	—	9	2	no	—
109	C₄×Dic₃	—	3	2	no	—
110	C₄×Dic₃	—	3	2	no	—
111	C₂×C₃⋊C₈	—	3	4	no	—
112	C₂×C₃⋊C₈	—	3	2	no	—
113	C₂²×C₃²	D₃×C₈	1	4	yes	—
114	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
115	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
116	C₃²×C₄	C₄×Dic₃	1	2	yes	—
117	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
118	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
119	C₃×Dic₃	—	3	8	no	—
120	C₂²×C₉	—	2	4	no	—
121	C₂²×C₉	—	2	2	no	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	C₃×C₄²	1	2	yes	—
124	C₂²×C₉	C₂×C₃×C₈	1	4	yes	—
125	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₄×C₉	—	2	2	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	—	2	1	no	—
129	C₄×C₉	C₃×C₄²	1	2	yes	—
130	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
131	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
132	Dic₉	C₂×C₃×C₈	1	8	yes	complemented
133	C₂²×C₈	—	9	1	no	—
134	C₂²×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
135	C₂×C₄²	—	9	1	no	—
136	C₄×C₈	—	9	4	no	—
137	3₋¹⁺²	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₃	—	3	1	no	—
139	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
140	C₂×C₃×C₄	—	3	1	no	—
141	C₂×C₃×C₄	—	3	2	no	—
142	C₂×C₃×C₄	—	3	2	no	—
143	C₂×C₃×C₄	—	3	1	no	—
144	C₂×C₃×C₄	—	9	4	no	—
145	C₂×C₃×C₄	—	9	8	no	—
146	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
148	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
149	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
150	C₂×Dic₃	—	3	8	no	—
151	C₂×Dic₃	—	3	4	no	—
152	C₃×C₈	—	3	4	no	—
153	C₃×C₈	—	9	4	no	—
154	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
155	C₃⋊C₈	—	3	4	no	—
156	C₂×C₃²	C₈×Dic₃	1	4	yes	—
157	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
158	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	—	2	4	no	—
162	C₂×C₉	C₃×C₄×C₈	1	4	yes	complemented
163	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
164	C₂×C₉	C₂²×C₃×C₈	1	2	yes	—
165	C₂²×C₄	—	9	2	no	—
166	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
167	C₂×C₈	—	9	4	no	—
168	C₂×C₈	—	9	2	no	—
169	C₂×C₈	3₋¹⁺²⋊C₄	1	4	yes	direct product factor
170	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
171	C₄²	—	9	2	no	—
172	C₄²	—	9	2	no	—
173	C₂²×C₃	—	3	1	no	—
174	C₂²×C₃	—	3	4	no	—
175	C₂²×C₃	—	3	2	no	—
176	C₂²×C₃	C₃×D₃×C₈	1	4	yes	—
177	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
178	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
179	C₃×C₄	—	3	1	no	—
180	C₃×C₄	—	3	1	no	—
181	C₃×C₄	—	3	2	no	—
182	C₃×C₄	—	9	8	no	—
183	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
184	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
185	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
186	Dic₃	—	3	8	no	—
187	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic
188	C₉	—	2	1	no	—
189	C₉	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented
190	C₂³	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
191	C₂×C₄	—	9	4	no	—
192	C₂×C₄	—	9	8	no	—
193	C₂×C₄	C₄×AGL(1,ℤ/9)	1	2	yes	—
194	C₂×C₄	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
195	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	2	yes	—
196	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
197	C₈	—	9	4	no	—
198	C₈	C₂×3₋¹⁺²⋊C₄	1	4	yes	direct product factor
199	C₂×C₃	—	3	2	no	—
200	C₂×C₃	—	3	1	no	—
201	C₂×C₃	—	3	4	no	—
202	C₂×C₃	C₃×C₈×Dic₃	1	4	yes	—
203	C₂×C₃	C₂×C₃×D₃×C₈	1	2	yes	—
204	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
205	C₂²	C₈×AGL(1,ℤ/9)	1	4	yes	—
206	C₂²	C₄×3₋¹⁺²⋊C₄	1	2	yes	—
207	C₂²	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
208	C₄	—	9	8	no	—
209	C₄	C₄×3₋¹⁺²⋊C₄	1	2	yes	—
210	C₄	C₂×C₄×AGL(1,ℤ/9)	1	1	yes	characteristic
211	C₄	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
212	C₃	—	3	1	no	—
213	C₃	C₂×C₃×C₈×Dic₃	1	1	yes	characteristic
214	C₂	C₈×3₋¹⁺²⋊C₄	1	4	yes	direct product factor
215	C₂	C₂×C₈×AGL(1,ℤ/9)	1	2	yes	—
216	C₂	C₂×C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂×C₃×C₄×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₈×3₋¹⁺²	C₂
socle	C₂³×C₃	C₃×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×3₋¹⁺²⋊C₄	C₂	direct (N × H)
6. C₂×C₈×Dic₉	C₃	semidirect
9. C₂×C₈×3₋¹⁺²	C₄	semidirect
13. C₂×3₋¹⁺²⋊C₈	C₄	semidirect
23. C₈×Dic₉	C₂×C₃	semidirect
29. C₂×3₋¹⁺²⋊C₄	C₈	direct (N × H)
31. C₈×3₋¹⁺²	C₂×C₄	semidirect
32. 3₋¹⁺²⋊C₈	C₂×C₄	semidirect
48. C₂×C₈×C₉	C₃×C₄	semidirect
52. C₂×C₉⋊C₈	C₃×C₄	semidirect
60. 3₋¹⁺²⋊C₄	C₂×C₈	direct (N × H)
88. C₂×Dic₉	C₃×C₈	semidirect
91. C₈×C₉	C₂×C₃×C₄	semidirect
92. C₉⋊C₈	C₂×C₃×C₄	semidirect
94. C₂×3₋¹⁺²	C₄×C₈	semidirect
132. Dic₉	C₂×C₃×C₈	semidirect
137. 3₋¹⁺²	C₂×C₄×C₈	semidirect
162. C₂×C₉	C₃×C₄×C₈	semidirect
169. C₂×C₈	3₋¹⁺²⋊C₄	direct (N × H)
189. C₉	C₂×C₃×C₄×C₈	semidirect
198. C₈	C₂×3₋¹⁺²⋊C₄	direct (N × H)
214. C₂	C₈×3₋¹⁺²⋊C₄	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×C₈×3₋¹⁺²	2
C₂×C₄×3₋¹⁺²⋊C₄	2
C₂²×3₋¹⁺²⋊C₈	2
C₈×3₋¹⁺²⋊C₄	2
C₂×C₃×C₈×Dic₃	3
C₂×C₈×Dic₉	3

Subgroups — C2×C8×3−1+2⋊C4 #13143134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×3₋¹⁺²⋊C₄ #13143134