Subgroups — C₂×C₉×#494 #11767866

Order 1152.

828 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Q₁₆	C₂	1	4	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₉×#494	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#494	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	C₂²	1	4	yes	—
7	C₂×C₉×Q₁₆	—	2	16	no	—
8	C₂×C₉×Q₁₆	C₂²	1	16	yes	—
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	—	2	4	no	—
11	C₉×C₄⋊Q₈	C₂²	1	8	yes	—
12	C₂×C₉×C₄⋊C₄	—	2	4	no	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉	C₂²	1	4	yes	—
15	C₂²×C₃×Q₁₆	C₂×C₃	1	4	yes	—
16	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
17	C₃×#494	C₂×C₃	1	8	yes	—
18	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×Q₈×C₉	—	2	4	no	—
20	C₂×Q₈×C₉	—	2	16	no	—
21	C₂×Q₈×C₉	C₂³	1	8	yes	—
22	C₂²×C₄×C₉	—	2	4	no	—
23	C₂²×C₄×C₉	D₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₉×Q₁₆	—	2	32	no	—
26	C₂×C₈×C₉	D₄	1	8	yes	—
27	C₂×C₈×C₉	C₂³	1	4	yes	—
28	C₉×C₄⋊C₄	—	2	16	no	—
29	C₄²×C₉	D₄	1	2	yes	—
30	C₄²×C₉	C₂³	1	2	yes	—
31	C₂×#494	C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
33	C₂×C₃×Q₁₆	—	2	16	no	—
34	C₂×C₃×Q₁₆	C₂²×C₃	1	16	yes	—
35	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
36	C₃×C₄⋊Q₈	—	2	4	no	—
37	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
38	C₂×C₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
40	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
41	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₉	—	2	16	no	—
43	Q₈×C₉	—	4	8	no	—
44	C₂×C₄×C₉	—	2	8	no	—
45	C₂×C₄×C₉	—	4	8	no	—
46	C₂×C₄×C₉	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₉	C₂×D₄	1	8	yes	—
52	C₂²×Q₁₆	C₂×C₉	1	4	yes	—
53	C₂×C₄⋊Q₈	C₂×C₉	1	2	yes	—
54	#494	C₂×C₉	1	8	yes	direct product factor
55	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	2	16	no	—
58	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
59	C₂²×C₃×C₄	—	2	4	no	—
60	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
61	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
62	C₃×Q₁₆	—	2	32	no	—
63	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
64	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
65	C₃×C₄⋊C₄	—	2	16	no	—
66	C₃×C₄²	C₃×D₄	1	2	yes	—
67	C₃×C₄²	C₂³×C₃	1	2	yes	—
68	C₂²×C₉	#127	1	2	yes	—
69	C₂²×C₉	C₂×Q₁₆	1	4	yes	—
70	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₉	—	4	8	no	—
72	C₄×C₉	#127	1	2	yes	—
73	C₄×C₉	C₂×Q₁₆	1	8	yes	complemented
74	C₄×C₉	C₂²×D₄	1	2	yes	—
75	C₂²×Q₈	C₂²×C₉	1	4	yes	—
76	C₂×Q₁₆	—	2	16	no	—
77	C₂×Q₁₆	C₂²×C₉	1	16	yes	—
78	C₂²×C₈	C₂²×C₉	1	2	yes	—
79	C₄⋊Q₈	—	2	4	no	—
80	C₄⋊Q₈	C₂²×C₉	1	8	yes	—
81	C₂×C₄⋊C₄	—	2	4	no	—
82	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
83	C₄×C₈	C₂²×C₉	1	4	yes	—
84	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
85	C₃×Q₈	—	2	16	no	—
86	C₃×Q₈	—	4	8	no	—
87	C₂×C₃×C₄	—	2	8	no	—
88	C₂×C₃×C₄	—	4	8	no	—
89	C₂×C₃×C₄	C₃×Q₁₆	1	8	yes	—
90	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
92	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
93	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
94	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
95	C₂×C₉	#494	1	4	yes	direct product factor
96	C₂×C₉	C₂×#127	1	1	yes	characteristic
97	C₂×C₉	C₂²×Q₁₆	1	2	yes	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	2	16	no	—
100	C₂×Q₈	C₂³×C₉	1	8	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	D₄×C₉	1	2	yes	—
103	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
104	Q₁₆	—	2	32	no	—
105	C₂×C₈	D₄×C₉	1	8	yes	—
106	C₂×C₈	C₂³×C₉	1	4	yes	—
107	C₄⋊C₄	—	2	16	no	—
108	C₄²	D₄×C₉	1	2	yes	—
109	C₄²	C₂³×C₉	1	2	yes	—
110	C₂²×C₃	C₃×#127	1	2	yes	—
111	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
112	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
113	C₃×C₄	—	4	8	no	—
114	C₃×C₄	C₃×#127	1	2	yes	—
115	C₃×C₄	C₂×C₃×Q₁₆	1	8	yes	—
116	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
117	C₉	C₂×#494	1	1	yes	characteristic, direct product factor, Sylow
118	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
119	Q₈	—	2	16	no	—
120	Q₈	—	4	8	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	4	8	no	—
123	C₂×C₄	C₉×Q₁₆	1	8	yes	complemented
124	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
126	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
127	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
128	C₈	C₂×D₄×C₉	1	8	yes	—
129	C₂×C₃	C₃×#494	1	4	yes	—
130	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
131	C₂×C₃	C₂²×C₃×Q₁₆	1	2	yes	—
132	C₂²	C₉×#127	1	2	yes	—
133	C₂²	C₂×C₉×Q₁₆	1	4	yes	—
134	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
135	C₄	—	4	8	no	—
136	C₄	C₉×#127	1	2	yes	—
137	C₄	C₂×C₉×Q₁₆	1	8	yes	complemented
138	C₄	C₂²×D₄×C₉	1	2	yes	—
139	C₃	C₂×C₃×#494	1	1	yes	characteristic
140	C₂	C₉×#494	1	4	yes	direct product factor
141	C₂	C₂×C₉×#127	1	1	yes	characteristic
142	C₂	C₂²×C₉×Q₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉×#494	C₂	direct (N × H)
31. C₂×#494	C₉	direct (N × H)
46. C₂×C₄×C₉	Q₁₆	semidirect
54. #494	C₂×C₉	direct (N × H)
73. C₄×C₉	C₂×Q₁₆	semidirect
95. C₂×C₉	#494	direct (N × H)
117. C₉	C₂×#494	direct (N × H)
123. C₂×C₄	C₉×Q₁₆	semidirect
137. C₄	C₂×C₉×Q₁₆	semidirect
140. C₂	C₉×#494	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×C₉×Q₁₆	2
C₂×C₉×C₄⋊Q₈	2
C₉×#494	2
C₂×C₄×C₈×C₉	2
C₂×C₃×#494	3

Subgroups — C2×C9×#494 #11767866

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#494 #11767866