Subgroups — Q₈×C₉×C₃⋊C₈ #13142772

Order 1728.

376 conjugacy classes of subgroups.

The subgroup lattice has 178 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	C₉×C₄⋊(C₃⋊C₈)	C₂	1	3	yes	maximal
3	C₄×C₉×C₃⋊C₈	C₂	1	3	yes	maximal
4	C₃×Q₈×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₉	—	3	1	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×C₉×C₃⋊C₈	C₂²	1	3	yes	—
10	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
12	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	3	yes	—
13	C₃×C₄×C₃⋊C₈	C₂×C₃	1	3	yes	—
14	C₄×Q₈×C₉	—	2	1	no	—
15	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
16	C₉×C₄⋊C₈	—	3	3	no	—
17	C₄×C₈×C₉	—	3	3	no	—
18	C₃×Q₈×C₉	C₈	1	4	yes	complemented
19	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
20	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
21	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
22	C₉×C₃⋊C₈	—	2	3	no	—
23	C₉×C₃⋊C₈	Q₈	1	2	yes	direct product factor
24	C₃×C₈×Q₈	—	3	1	no	—
25	Q₈×C₃⋊C₈	C₉	1	1	yes	characteristic, direct product factor
26	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
27	C₃²×C₄⋊C₄	C₃×C₄	1	3	yes	—
28	C₃²×C₄²	C₂²×C₃	1	3	yes	—
29	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	3	yes	—
30	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
31	C₂×Q₈×C₉	—	2	1	no	—
32	C₂×Q₈×C₉	Dic₃	1	1	yes	characteristic
33	C₂×C₈×C₉	—	3	1	no	—
34	C₂×C₈×C₉	—	3	3	no	—
35	C₉×C₄⋊C₄	—	2	3	no	—
36	C₉×C₄⋊C₄	Dic₃	1	3	yes	—
37	C₄²×C₉	—	2	3	no	—
38	C₄²×C₉	C₂×D₃	1	3	yes	—
39	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
40	C₃×C₄×C₉	—	2	3	no	—
41	C₃×C₄×C₉	C₂×C₈	1	6	yes	—
42	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
43	C₃×C₄×C₉	#41	1	1	yes	characteristic
44	C₃×C₄×Q₈	—	2	1	no	—
45	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
46	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
47	C₃×C₄⋊C₈	—	3	3	no	—
48	C₃×C₄×C₈	—	3	3	no	—
49	C₄⋊(C₃⋊C₈)	C₂×C₉	1	3	yes	—
50	C₄×C₃⋊C₈	C₂×C₉	1	3	yes	—
51	C₃²×Q₈	C₃×C₈	1	4	yes	—
52	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
53	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
54	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
55	C₃×C₃⋊C₈	—	2	3	no	—
56	C₃×C₃⋊C₈	C₃×Q₈	1	2	yes	—
57	Q₈×C₉	—	2	4	no	—
58	Q₈×C₉	C₃⋊C₈	1	4	yes	direct product factor
59	C₂×C₄×C₉	—	2	3	no	—
60	C₂×C₄×C₉	—	2	3	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
63	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
64	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
65	C₈×C₉	—	3	2	no	—
66	C₈×C₉	—	6	3	no	—
67	C₈×Q₈	—	3	1	no	Sylow
68	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₃×C₉	#131	1	1	yes	characteristic
71	C₂×C₃×Q₈	—	2	1	no	—
72	C₂×C₃×Q₈	C₄×C₉	1	1	yes	characteristic
73	C₂×C₃×Q₈	C₃×Dic₃	1	1	yes	characteristic
74	C₂×C₃×C₈	—	3	1	no	—
75	C₂×C₃×C₈	—	3	3	no	—
76	C₃×C₄⋊C₄	—	2	3	no	—
77	C₃×C₄⋊C₄	C₄×C₉	1	3	yes	—
78	C₃×C₄⋊C₄	C₃×Dic₃	1	3	yes	—
79	C₃×C₄²	—	2	3	no	—
80	C₃×C₄²	C₂²×C₉	1	3	yes	—
81	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
82	C₂×C₃⋊C₈	C₂²×C₉	1	3	yes	—
83	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
84	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
85	C₃²×C₄	—	2	3	no	—
86	C₃²×C₄	C₂×C₃×C₈	1	6	yes	—
87	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
88	C₃²×C₄	C₃×#41	1	1	yes	characteristic
89	C₂²×C₉	—	2	1	no	—
90	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
91	C₄×C₉	—	2	3	no	—
92	C₄×C₉	—	2	6	no	—
93	C₄×C₉	—	2	1	no	—
94	C₄×C₉	—	2	1	no	—
95	C₄×C₉	—	4	3	no	—
96	C₄×C₉	C₂×C₃⋊C₈	1	6	yes	—
97	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
98	C₄×C₉	#239	1	1	yes	characteristic
99	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
100	C₄⋊C₈	—	3	3	no	—
101	C₄×C₈	—	3	3	no	—
102	C₃×C₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
103	C₃×Q₈	—	2	4	no	—
104	C₃×Q₈	C₈×C₉	1	4	yes	complemented
105	C₃×Q₈	C₃×C₃⋊C₈	1	4	yes	—
106	C₂×C₃×C₄	—	2	3	no	—
107	C₂×C₃×C₄	—	2	3	no	—
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
110	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
111	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
113	C₂×C₃×C₄	C₂×C₃×Dic₃	1	3	yes	—
114	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
115	C₃×C₈	—	3	2	no	—
116	C₃×C₈	—	6	3	no	—
117	C₃⋊C₈	—	2	3	no	—
118	C₃⋊C₈	Q₈×C₉	1	2	yes	direct product factor
119	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
120	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
121	C₂×C₃²	C₃×#131	1	1	yes	characteristic
122	C₂×C₉	—	2	1	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	—	2	1	no	—
125	C₂×C₉	C₂²×C₃⋊C₈	1	1	yes	characteristic
126	C₂×C₉	Q₈×Dic₃	1	1	yes	characteristic
127	C₂×C₉	#948	1	1	yes	characteristic
128	C₂×Q₈	C₉×Dic₃	1	1	yes	characteristic
129	C₂×C₈	—	3	1	no	—
130	C₂×C₈	—	3	3	no	—
131	C₄⋊C₄	C₉×Dic₃	1	3	yes	—
132	C₄²	C₂×D₃×C₉	1	3	yes	—
133	C₂²×C₃	—	2	1	no	—
134	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
135	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
136	C₃×C₄	—	2	3	no	—
137	C₃×C₄	—	2	3	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	2	6	no	—
141	C₃×C₄	—	4	3	no	—
142	C₃×C₄	C₂×C₈×C₉	1	6	yes	—
143	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
144	C₃×C₄	C₉×#41	1	1	yes	characteristic
145	C₃×C₄	C₂×C₃×C₃⋊C₈	1	6	yes	—
146	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
147	C₃×C₄	C₃×#239	1	1	yes	characteristic
148	C₃²	C₃×C₈×Q₈	1	1	yes	characteristic
149	C₉	—	2	1	no	—
150	C₉	Q₈×C₃⋊C₈	1	1	yes	characteristic, direct product factor
151	Q₈	C₉×C₃⋊C₈	1	4	yes	direct product factor
152	C₂×C₄	C₂×C₉×Dic₃	1	3	yes	—
153	C₂×C₄	C₂×C₉×Dic₃	1	3	yes	—
154	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
155	C₈	—	3	2	no	—
156	C₈	—	6	3	no	—
157	C₂×C₃	—	2	1	no	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	C₄×Q₈×C₉	1	1	yes	characteristic
161	C₂×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
162	C₂×C₃	C₉×#131	1	1	yes	characteristic
163	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
164	C₂×C₃	C₃×Q₈×Dic₃	1	1	yes	characteristic
165	C₂×C₃	C₃×#948	1	1	yes	characteristic
166	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
167	C₄	—	2	3	no	—
168	C₄	C₂×C₉×C₃⋊C₈	1	6	yes	—
169	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
170	C₄	C₉×#239	1	1	yes	characteristic
171	C₃	—	2	1	no	—
172	C₃	C₈×Q₈×C₉	1	1	yes	characteristic, complemented
173	C₃	C₃×Q₈×C₃⋊C₈	1	1	yes	characteristic
174	C₂	C₂²×C₉×C₃⋊C₈	1	1	yes	characteristic
175	C₂	Q₈×C₉×Dic₃	1	1	yes	characteristic
176	C₂	C₉×#948	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
18. C₃×Q₈×C₉	C₈	semidirect
23. C₉×C₃⋊C₈	Q₈	direct (N × H)
25. Q₈×C₃⋊C₈	C₉	direct (N × H)
58. Q₈×C₉	C₃⋊C₈	direct (N × H)
102. C₃×C₉	C₈×Q₈	semidirect
104. C₃×Q₈	C₈×C₉	semidirect
118. C₃⋊C₈	Q₈×C₉	direct (N × H)
150. C₉	Q₈×C₃⋊C₈	direct (N × H)
151. Q₈	C₉×C₃⋊C₈	direct (N × H)
172. C₃	C₈×Q₈×C₉	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
C₉×C₄⋊(C₃⋊C₈)	2
C₄×C₉×C₃⋊C₈	2
C₃×Q₈×C₃⋊C₈	3
C₈×Q₈×C₉	3

Subgroups — Q8×C9×C3⋊C8 #13142772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×C₃⋊C₈ #13142772