Subgroups — C₂×C₄×3₋¹⁺²⋊C₈ #13142427

Order 1728.

540 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄²	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂×C₃×C₄×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₄²×3₋¹⁺²	C₂
socle	C₂³×C₃	C₂×C₃×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×3₋¹⁺²⋊C₈	C₂	direct (N × H)
5. C₂×C₄×C₉⋊C₈	C₃	semidirect
10. C₂×3₋¹⁺²⋊C₈	C₄	direct (N × H)
18. C₄×C₉⋊C₈	C₂×C₃	semidirect
20. C₂×C₄×3₋¹⁺²	C₈	semidirect
25. 3₋¹⁺²⋊C₈	C₂×C₄	direct (N × H)
42. C₂×C₉⋊C₈	C₃×C₄	semidirect
48. C₄×3₋¹⁺²	C₂×C₈	semidirect
70. C₂×C₄×C₉	C₃×C₈	semidirect
75. C₉⋊C₈	C₂×C₃×C₄	semidirect
77. C₂×3₋¹⁺²	C₄×C₈	semidirect
108. C₄×C₉	C₂×C₃×C₈	semidirect
113. 3₋¹⁺²	C₂×C₄×C₈	semidirect
134. C₂×C₉	C₃×C₄×C₈	semidirect
157. C₉	C₂×C₃×C₄×C₈	semidirect
159. C₂×C₄	3₋¹⁺²⋊C₈	direct (N × H)
174. C₄	C₂×3₋¹⁺²⋊C₈	direct (N × H)
179. C₂	C₄×3₋¹⁺²⋊C₈	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×C₄²×3₋¹⁺²	2
C₂²×3₋¹⁺²⋊C₈	2
C₄×3₋¹⁺²⋊C₈	2
C₂×C₃×C₄×C₃⋊C₈	3
C₂×C₄×C₉⋊C₈	3

Subgroups — C2×C4×3−1+2⋊C8 #13142427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×3₋¹⁺²⋊C₈ #13142427