Subgroups — Q₈×C₃⋊(C₉⋊C₈) #13144424

Order 1728.

510 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	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	3	yes	maximal
3	C₄×C₃⋊(C₉⋊C₈)	C₂	1	3	yes	maximal
4	Q₈×C₃⋊(C₃⋊C₈)	—	3	1	no	maximal
5	Q₈×C₉⋊C₈	—	3	3	no	maximal
6	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₃×C₉×C₄⋊C₄	C₄	1	3	yes	—
8	C₃×C₄²×C₉	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	3	yes	—
10	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	3	no	—
13	C₄×C₃⋊(C₃⋊C₈)	—	3	3	no	—
14	C₄×Q₈×C₉	D₃	1	3	yes	—
15	C₄⋊(C₉⋊C₈)	—	3	9	no	—
16	C₄×C₉⋊C₈	—	3	9	no	—
17	C₃×Q₈×C₉	C₈	1	4	yes	complemented
18	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
19	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
20	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
21	C₃⋊(C₉⋊C₈)	—	2	3	no	—
22	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	direct product factor
23	Q₈×C₃⋊C₈	—	9	3	no	—
24	Q₈×C₃⋊C₈	—	9	1	no	—
25	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
26	C₃²×C₄⋊C₄	Dic₃	1	3	yes	—
27	C₃²×C₄²	C₂×D₃	1	3	yes	—
28	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
29	C₂×C₃⋊(C₃⋊C₈)	—	3	3	no	—
30	C₂×Q₈×C₉	Dic₃	1	3	yes	—
31	C₉×C₄⋊C₄	Dic₃	1	9	yes	—
32	C₄²×C₉	C₂×D₃	1	9	yes	—
33	C₂×C₉⋊C₈	—	3	3	no	—
34	C₂×C₉⋊C₈	—	3	9	no	—
35	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
36	C₃×C₄×C₉	—	2	3	no	—
37	C₃×C₄×C₉	C₂×C₈	1	6	yes	—
38	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
39	C₃×C₄×C₉	#41	1	1	yes	characteristic
40	C₃×C₄×Q₈	D₉	1	3	yes	—
41	C₃×C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
42	C₄⋊(C₃⋊C₈)	—	9	9	no	—
43	C₄⋊(C₃⋊C₈)	—	9	3	no	—
44	C₄×C₃⋊C₈	—	9	9	no	—
45	C₄×C₃⋊C₈	—	9	3	no	—
46	C₃²×Q₈	C₃⋊C₈	1	4	yes	—
47	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
48	C₂×C₃²×C₄	C₂×Dic₃	1	3	yes	—
49	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
50	C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₃⋊(C₃⋊C₈)	—	6	3	no	—
52	Q₈×C₉	C₃⋊C₈	1	12	yes	complemented
53	C₂×C₄×C₉	C₂×Dic₃	1	9	yes	—
54	C₂×C₄×C₉	C₂×Dic₃	1	9	yes	—
55	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
56	C₉⋊C₈	—	3	6	no	—
57	C₉⋊C₈	—	6	9	no	—
58	C₈×Q₈	—	27	1	no	Sylow
59	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₃×C₉	#131	1	1	yes	characteristic
62	C₂×C₃×Q₈	Dic₉	1	3	yes	—
63	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
64	C₃×C₄⋊C₄	Dic₉	1	9	yes	—
65	C₃×C₄⋊C₄	C₃⋊Dic₃	1	3	yes	—
66	C₃×C₄²	C₂×D₉	1	9	yes	—
67	C₃×C₄²	C₂×C₃⋊D₃	1	3	yes	—
68	C₂×C₃⋊C₈	—	9	3	no	—
69	C₂×C₃⋊C₈	—	9	1	no	—
70	C₂×C₃⋊C₈	—	9	9	no	—
71	C₂×C₃⋊C₈	—	9	3	no	—
72	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
73	C₃²×C₄	—	2	3	no	—
74	C₃²×C₄	C₂×C₃⋊C₈	1	6	yes	—
75	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
76	C₃²×C₄	#239	1	1	yes	characteristic
77	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
78	C₄×C₉	—	2	9	no	—
79	C₄×C₉	C₂×C₃⋊C₈	1	18	yes	—
80	C₄×C₉	D₃×Q₈	1	3	yes	—
81	C₄×C₉	#239	1	3	yes	—
82	C₄×Q₈	C₃⋊D₉	1	1	yes	characteristic
83	C₄⋊C₈	—	27	3	no	—
84	C₄×C₈	—	27	3	no	—
85	C₃×C₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
86	C₃×Q₈	C₉⋊C₈	1	12	yes	complemented
87	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	4	yes	—
88	C₂×C₃×C₄	C₂×Dic₉	1	9	yes	—
89	C₂×C₃×C₄	C₂×Dic₉	1	9	yes	—
90	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
91	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
92	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
93	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
94	C₃⋊C₈	—	9	6	no	—
95	C₃⋊C₈	—	9	2	no	—
96	C₃⋊C₈	—	18	9	no	—
97	C₃⋊C₈	—	18	3	no	—
98	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
99	C₂×C₃²	Q₈×Dic₃	1	1	yes	characteristic
100	C₂×C₃²	#948	1	1	yes	characteristic
101	C₂×C₉	C₂²×C₃⋊C₈	1	3	yes	—
102	C₂×C₉	Q₈×Dic₃	1	3	yes	—
103	C₂×C₉	#948	1	3	yes	—
104	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic
105	C₂×C₈	—	27	1	no	—
106	C₂×C₈	—	27	3	no	—
107	C₄⋊C₄	C₃⋊Dic₉	1	3	yes	—
108	C₄²	C₂×C₃⋊D₉	1	3	yes	—
109	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
110	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
111	C₃×C₄	—	2	9	no	—
112	C₃×C₄	—	2	3	no	—
113	C₃×C₄	C₂×C₉⋊C₈	1	18	yes	—
114	C₃×C₄	Q₈×D₉	1	3	yes	—
115	C₃×C₄	#3701	1	3	yes	—
116	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
117	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
118	C₃×C₄	#3832	1	1	yes	characteristic
119	C₃²	Q₈×C₃⋊C₈	1	1	yes	characteristic
120	C₉	Q₈×C₃⋊C₈	1	3	yes	complemented
121	Q₈	C₃⋊(C₉⋊C₈)	1	4	yes	direct product factor
122	C₂×C₄	C₂×C₃⋊Dic₉	1	3	yes	—
123	C₂×C₄	C₂×C₃⋊Dic₉	1	3	yes	—
124	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
125	C₈	—	27	2	no	—
126	C₈	—	54	3	no	—
127	C₂×C₃	C₂²×C₉⋊C₈	1	3	yes	—
128	C₂×C₃	Q₈×Dic₉	1	3	yes	—
129	C₂×C₃	#63510	1	3	yes	—
130	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
131	C₂×C₃	Q₈×C₃⋊Dic₃	1	1	yes	characteristic
132	C₂×C₃	#64157	1	1	yes	characteristic
133	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
134	C₄	—	2	3	no	—
135	C₄	C₂×C₃⋊(C₉⋊C₈)	1	6	yes	—
136	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
137	C₄	#88746	1	1	yes	characteristic
138	C₃	Q₈×C₉⋊C₈	1	3	yes	complemented
139	C₃	Q₈×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
140	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
141	C₂	Q₈×C₃⋊Dic₉	1	1	yes	characteristic
142	C₂	#11721865	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂²×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄×Q₈	C₃⋊D₉
hypocenter	C₃×C₉	C₈×Q₈
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
17. C₃×Q₈×C₉	C₈	semidirect
22. C₃⋊(C₉⋊C₈)	Q₈	direct (N × H)
52. Q₈×C₉	C₃⋊C₈	semidirect
85. C₃×C₉	C₈×Q₈	semidirect
86. C₃×Q₈	C₉⋊C₈	semidirect
120. C₉	Q₈×C₃⋊C₈	semidirect
121. Q₈	C₃⋊(C₉⋊C₈)	direct (N × H)
138. C₃	Q₈×C₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₄×C₃⋊(C₉⋊C₈)	2
Q₈×C₃⋊(C₃⋊C₈)	3
Q₈×C₉⋊C₈	3

Subgroups — Q8×C3⋊(C9⋊C8) #13144424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊(C₉⋊C₈) #13144424