Subgroups — C₃²×#1580 #11768629

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×D₄	C₂	1	2	yes	maximal, complemented
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1580	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×D₈	C₄	1	1	yes	characteristic
8	C₂²×C₃²×C₈	—	2	2	no	—
9	C₃²×C₄×D₄	C₂²	1	2	yes	—
10	C₃²×#107	C₄	1	1	yes	characteristic
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃²×#102	C₄	1	2	yes	—
13	C₃²×C₂²⋊C₈	—	2	2	no	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₈	C₂×C₃	1	4	yes	—
18	C₃×C₈×D₄	C₂×C₃	1	8	yes	complemented
19	C₃×#334	C₂×C₃	1	4	yes	complemented
20	C₃×D₄⋊C₈	C₂×C₃	1	8	yes	complemented
21	C₃×#322	C₂×C₃	1	4	yes	complemented
22	C₂×C₃²×D₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃²×C₄	—	2	2	no	—
24	C₃²×D₈	C₈	1	4	yes	complemented
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₂×C₃²×C₈	—	2	1	no	—
27	C₂×C₃²×C₈	—	4	4	no	—
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	D₄	1	2	yes	—
31	C₃²×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₃²×C₂²⋊C₄	—	2	2	no	—
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	#1580	C₃²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×D₈	C₃×C₄	1	4	yes	—
36	C₂²×C₃×C₈	—	2	8	no	—
37	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
38	C₃×#107	C₃×C₄	1	4	yes	—
39	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
40	C₃×#102	C₃×C₄	1	8	yes	—
41	C₃×C₂²⋊C₈	—	2	8	no	—
42	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
45	C₂³×C₃²	—	2	2	no	—
46	C₃²×D₄	—	2	2	no	—
47	C₃²×D₄	C₂×C₈	1	4	yes	complemented
48	C₂×C₃²×C₄	—	2	2	no	—
49	C₂×C₃²×C₄	—	4	2	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	#41	1	1	yes	characteristic
54	C₃²×C₈	—	2	1	no	—
55	C₃²×C₈	—	2	2	no	—
56	C₃²×C₈	—	4	1	no	—
57	C₃²×C₈	—	4	2	no	—
58	C₃²×C₈	C₂×C₈	1	2	yes	complemented
59	C₄×D₈	C₂×C₃²	1	1	yes	characteristic
60	C₈×D₄	C₂×C₃²	1	2	yes	complemented
61	#334	C₂×C₃²	1	1	yes	characteristic, complemented
62	D₄⋊C₈	C₂×C₃²	1	2	yes	complemented
63	#322	C₂×C₃²	1	1	yes	characteristic, complemented
64	C₂×C₃×D₄	C₂×C₃×C₄	1	8	yes	—
65	C₂²×C₃×C₄	—	2	8	no	—
66	C₃×D₈	C₃×C₈	1	16	yes	complemented
67	C₂×C₃×C₈	—	2	8	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	4	16	no	—
70	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
71	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
73	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
74	C₃×C₂²⋊C₄	—	2	8	no	—
75	C₃×C₄²	C₂³×C₃	1	4	yes	—
76	C₂²×C₃²	—	2	4	no	—
77	C₂²×C₃²	—	4	2	no	—
78	C₂²×C₃²	—	4	2	no	—
79	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
80	C₃²×C₄	—	2	1	no	—
81	C₃²×C₄	—	4	2	no	—
82	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
83	C₃²×C₄	#131	1	1	yes	characteristic
84	C₃²×C₄	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×D₈	C₃²×C₄	1	1	yes	characteristic
86	C₂²×C₈	—	2	2	no	—
87	C₄×D₄	C₂²×C₃²	1	2	yes	—
88	#107	C₃²×C₄	1	1	yes	characteristic
89	C₄⋊C₈	C₂²×C₃²	1	2	yes	—
90	#102	C₃²×C₄	1	2	yes	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
93	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
94	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
95	C₂³×C₃	—	2	8	no	—
96	C₃×D₄	—	2	8	no	—
97	C₃×D₄	C₂×C₃×C₈	1	16	yes	complemented
98	C₂×C₃×C₄	—	2	8	no	—
99	C₂×C₃×C₄	—	4	8	no	—
100	C₂×C₃×C₄	—	4	8	no	—
101	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
102	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
103	C₂×C₃×C₄	C₃×#41	1	4	yes	—
104	C₃×C₈	—	2	4	no	—
105	C₃×C₈	—	2	8	no	—
106	C₃×C₈	—	4	4	no	—
107	C₃×C₈	—	4	8	no	—
108	C₃×C₈	C₂×C₃×C₈	1	8	yes	complemented
109	C₂×C₃²	—	4	4	no	—
110	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
111	C₂×C₃²	#442	1	1	yes	characteristic
112	C₂×C₃²	#444	1	1	yes	characteristic
113	C₂×D₄	C₂×C₃²×C₄	1	2	yes	—
114	C₂²×C₄	—	2	2	no	—
115	D₈	C₃²×C₈	1	4	yes	complemented
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
121	C₂×C₈	C₃²×D₄	1	2	yes	—
122	C₄⋊C₄	C₂×C₃²×C₄	1	2	yes	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₄²	C₂³×C₃²	1	1	yes	characteristic
125	C₂²×C₃	—	2	16	no	—
126	C₂²×C₃	—	4	8	no	—
127	C₂²×C₃	—	4	8	no	—
128	C₂²×C₃	C₃×C₄×D₄	1	4	yes	—
129	C₃×C₄	—	2	4	no	—
130	C₃×C₄	—	4	8	no	—
131	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
132	C₃×C₄	C₃×#131	1	4	yes	—
133	C₃×C₄	C₃×AGL(1,ℤ/8)	1	8	yes	—
134	C₃²	#1580	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³	—	2	2	no	—
136	D₄	—	2	2	no	—
137	D₄	C₂×C₃²×C₈	1	4	yes	complemented
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
143	C₂×C₄	C₃²×#41	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	2	no	—
146	C₈	—	4	1	no	—
147	C₈	—	4	2	no	—
148	C₈	C₂×C₃²×C₈	1	2	yes	complemented
149	C₂×C₃	—	4	16	no	—
150	C₂×C₃	C₃×C₈×D₄	1	4	yes	—
151	C₂×C₃	C₃×#442	1	4	yes	—
152	C₂×C₃	C₃×#444	1	4	yes	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	2	no	—
156	C₂²	C₃²×C₄×D₄	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
160	C₄	C₃²×#131	1	1	yes	characteristic
161	C₄	C₃²×AGL(1,ℤ/8)	1	2	yes	—
162	C₃	C₃×#1580	1	4	yes	direct product factor
163	C₂	—	4	4	no	—
164	C₂	C₃²×C₈×D₄	1	1	yes	characteristic
165	C₂	C₃²×#442	1	1	yes	characteristic
166	C₂	C₃²×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₈	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₈×D₄	C₂	semidirect
3. C₃²×#334	C₂	semidirect
4. C₃²×D₄⋊C₈	C₂	semidirect
5. C₃²×#322	C₂	semidirect
6. C₃×#1580	C₃	direct (N × H)
18. C₃×C₈×D₄	C₂×C₃	semidirect
19. C₃×#334	C₂×C₃	semidirect
20. C₃×D₄⋊C₈	C₂×C₃	semidirect
21. C₃×#322	C₂×C₃	semidirect
24. C₃²×D₈	C₈	semidirect
34. #1580	C₃²	direct (N × H)
47. C₃²×D₄	C₂×C₈	semidirect
58. C₃²×C₈	C₂×C₈	semidirect
60. C₈×D₄	C₂×C₃²	semidirect
61. #334	C₂×C₃²	semidirect
62. D₄⋊C₈	C₂×C₃²	semidirect
63. #322	C₂×C₃²	semidirect
66. C₃×D₈	C₃×C₈	semidirect
97. C₃×D₄	C₂×C₃×C₈	semidirect
108. C₃×C₈	C₂×C₃×C₈	semidirect
115. D₈	C₃²×C₈	semidirect
134. C₃²	#1580	direct (N × H)
137. D₄	C₂×C₃²×C₈	semidirect
148. C₈	C₂×C₃²×C₈	semidirect
162. C₃	C₃×#1580	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₈	2
C₃²×C₈×D₄	2
C₃²×#334	2
C₃²×D₄⋊C₈	2
C₃²×#322	2
C₃×#1580	3

Subgroups — C32×#1580 #11768629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1580 #11768629