Subgroups — C₉×#1848 #11766569

Order 1152.

330 conjugacy classes of subgroups.

The subgroup lattice has 129 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#108	C₂	1	4	yes	maximal
2	C₉×#423	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
4	C₃×#1848	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₈×C₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₉×#108	—	2	8	no	—
9	C₉×C₄⋊C₈	—	2	4	no	—
10	C₄×C₈×C₉	C₄	1	2	yes	—
11	C₄×C₈×C₉	C₄	1	2	yes	—
12	C₂×C₃×#108	C₂×C₃	1	4	yes	—
13	C₃×#423	C₂×C₃	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₄×C₉	Q₈	1	2	yes	—
16	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
17	C₉×M₁₆	—	2	8	no	—
18	C₂×C₈×C₉	—	2	4	no	—
19	C₂×C₈×C₉	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₉	D₄	1	4	yes	—
21	C₂×C₈×C₉	Q₈	1	4	yes	—
22	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉	D₄	1	2	yes	—
25	#1848	C₉	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
27	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
28	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
29	C₃×#108	—	2	8	no	—
30	C₃×C₄⋊C₈	—	2	4	no	—
31	C₃×C₄×C₈	C₃×C₄	1	2	yes	—
32	C₃×C₄×C₈	C₃×C₄	1	2	yes	—
33	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₉	—	2	4	no	—
35	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
41	C₈×C₉	—	4	4	no	—
42	C₈×C₉	C₄⋊C₄	1	8	yes	—
43	C₂×#108	C₂×C₉	1	4	yes	—
44	#423	C₂×C₉	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
47	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
48	C₃×M₁₆	—	2	8	no	—
49	C₂×C₃×C₈	—	2	4	no	—
50	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
51	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
52	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
53	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
54	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
55	C₃×C₄²	C₃×D₄	1	2	yes	—
56	C₂²×C₉	—	2	2	no	—
57	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₉	C₄⋊Q₈	1	2	yes	—
59	C₄×C₉	—	2	2	no	—
60	C₄×C₉	#108	1	4	yes	complemented
61	C₄×C₉	C₂×C₄⋊C₄	1	2	yes	—
62	C₄×C₉	#127	1	1	yes	characteristic
63	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×M₁₆	C₂²×C₉	1	4	yes	—
65	C₂²×C₈	C₂²×C₉	1	2	yes	—
66	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
67	#108	—	2	8	no	—
68	C₄⋊C₈	—	2	4	no	—
69	C₄×C₈	C₄×C₉	1	2	yes	—
70	C₄×C₈	C₄×C₉	1	2	yes	—
71	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₄	—	2	4	no	—
73	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
74	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
75	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
77	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
79	C₃×C₈	—	4	4	no	—
80	C₃×C₈	C₃×C₄⋊C₄	1	8	yes	—
81	C₂×C₉	—	2	2	no	—
82	C₂×C₉	#384	1	1	yes	characteristic
83	C₂×C₉	C₂×#108	1	2	yes	—
84	C₂²×C₄	Q₈×C₉	1	2	yes	—
85	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
86	M₁₆	—	2	8	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	C₂×C₄×C₉	1	4	yes	—
89	C₂×C₈	D₄×C₉	1	4	yes	—
90	C₂×C₈	Q₈×C₉	1	4	yes	—
91	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
92	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
93	C₄²	D₄×C₉	1	2	yes	—
94	C₂²×C₃	—	2	2	no	—
95	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃	C₃×C₄⋊Q₈	1	2	yes	—
97	C₃×C₄	—	2	2	no	—
98	C₃×C₄	C₃×#108	1	4	yes	—
99	C₃×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
100	C₃×C₄	C₃×#127	1	1	yes	characteristic
101	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
102	C₉	#1848	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³	C₂×Q₈×C₉	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
106	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
107	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
108	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
109	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
110	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
111	C₈	—	4	4	no	—
112	C₈	C₉×C₄⋊C₄	1	8	yes	—
113	C₂×C₃	—	2	2	no	—
114	C₂×C₃	C₃×#384	1	1	yes	characteristic
115	C₂×C₃	C₂×C₃×#108	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
118	C₂²	C₉×C₄⋊Q₈	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₉×#108	1	4	yes	complemented
121	C₄	C₂×C₉×C₄⋊C₄	1	2	yes	—
122	C₄	C₉×#127	1	1	yes	characteristic
123	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
124	C₃	C₃×#1848	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	C₉×#384	1	1	yes	characteristic
127	C₂	C₂×C₉×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₄⋊C₄
prime radical	C₂³×C₃	C₂×C₃×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
25. #1848	C₉	direct (N × H)
60. C₄×C₉	#108	semidirect
102. C₉	#1848	direct (N × H)
120. C₄	C₉×#108	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#108	2
C₉×#423	2
C₂×C₄×C₈×C₉	2
C₃×#1848	3

Subgroups — C9×#1848 #11766569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1848 #11766569