Subgroups — C₂×C₃×C₄×Q₈×C₉ #13154530

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₈×C₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₉×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂×C₃×C₄²×C₉	C₂	1	3	yes	maximal
5	C₂×C₃²×C₄×Q₈	C₃	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₉	C₃	1	3	yes	maximal, direct product factor
7	C₂×C₃×Q₈×C₉	C₄	1	8	yes	direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	4	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₄⋊C₄	C₂²	1	12	yes	—
13	C₃×C₄²×C₉	C₂²	1	12	yes	—
14	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
15	C₃²×C₄×Q₈	C₂×C₃	1	8	yes	—
16	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	3	yes	—
17	C₂×C₃²×C₄²	C₂×C₃	1	3	yes	—
18	C₂²×Q₈×C₉	C₂×C₃	1	3	yes	—
19	C₄×Q₈×C₉	C₂×C₃	1	24	yes	direct product factor
20	C₂×C₉×C₄⋊C₄	C₂×C₃	1	9	yes	—
21	C₂×C₄²×C₉	C₂×C₃	1	9	yes	—
22	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
23	C₃×Q₈×C₉	C₂×C₄	1	16	yes	direct product factor
24	C₂×C₃×C₄×C₉	—	2	6	no	—
25	C₂×C₃×C₄×C₉	C₂×C₄	1	12	yes	—
26	C₂×C₃×C₄×C₉	Q₈	1	4	yes	direct product factor
27	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
30	C₂×C₃×C₄×Q₈	C₉	1	3	yes	direct product factor
31	C₂×C₃×C₄×Q₈	C₃²	1	1	yes	characteristic
32	C₂×C₃²×Q₈	C₃×C₄	1	8	yes	—
33	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
34	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
35	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
36	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₃²×C₄⋊C₄	C₂²×C₃	1	12	yes	—
38	C₃²×C₄²	C₂²×C₃	1	12	yes	—
39	C₂×Q₈×C₉	C₃×C₄	1	24	yes	direct product factor
40	C₂×Q₈×C₉	C₂²×C₃	1	12	yes	—
41	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	—
42	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	—
43	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
44	C₉×C₄⋊C₄	C₂²×C₃	1	36	yes	—
45	C₄²×C₉	C₂²×C₃	1	36	yes	—
46	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₃×C₉	C₂×Q₈	1	2	yes	—
48	C₂²×C₃×C₉	#41	1	2	yes	—
49	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
50	C₃×C₄×C₉	—	2	6	no	—
51	C₃×C₄×C₉	C₂²×C₄	1	12	yes	—
52	C₃×C₄×C₉	C₂×Q₈	1	4	yes	direct product factor
53	C₂²×C₃×Q₈	C₂×C₉	1	3	yes	—
54	C₂²×C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
55	C₃×C₄×Q₈	C₂×C₉	1	24	yes	direct product factor
56	C₃×C₄×Q₈	C₂×C₃²	1	8	yes	—
57	C₂×C₃×C₄⋊C₄	C₂×C₉	1	9	yes	—
58	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	3	yes	—
59	C₂×C₃×C₄²	C₂×C₉	1	9	yes	—
60	C₂×C₃×C₄²	C₂×C₃²	1	3	yes	—
61	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
62	C₃²×Q₈	C₂×C₃×C₄	1	16	yes	—
63	C₂×C₃²×C₄	—	2	6	no	—
64	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
65	C₂×C₃²×C₄	C₃×Q₈	1	4	yes	—
66	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
67	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
68	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
69	C₂³×C₉	C₂³×C₃	1	3	yes	—
70	Q₈×C₉	C₂×C₃×C₄	1	48	yes	direct product factor
71	C₂×C₄×C₉	—	2	18	no	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	36	yes	—
73	C₂×C₄×C₉	C₃×Q₈	1	12	yes	direct product factor
74	C₂×C₄×C₉	C₂³×C₃	1	18	yes	—
75	C₂×C₄×C₉	C₂³×C₃	1	18	yes	—
76	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
77	C₂×C₄×Q₈	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×C₉	C₄×Q₈	1	4	yes	direct product factor
79	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
82	C₂×C₃×Q₈	C₄×C₉	1	24	yes	direct product factor
83	C₂×C₃×Q₈	C₂²×C₉	1	12	yes	—
84	C₂×C₃×Q₈	C₃²×C₄	1	8	yes	—
85	C₂×C₃×Q₈	C₂²×C₃²	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×C₉	1	9	yes	—
87	C₂²×C₃×C₄	C₂²×C₉	1	9	yes	—
88	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
89	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
90	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
91	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
92	C₃×C₄⋊C₄	C₂²×C₉	1	36	yes	—
93	C₃×C₄⋊C₄	C₂²×C₃²	1	12	yes	—
94	C₃×C₄²	C₂²×C₉	1	36	yes	—
95	C₃×C₄²	C₂²×C₃²	1	12	yes	—
96	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
97	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
98	C₂²×C₃²	C₃×#41	1	2	yes	—
99	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
100	C₃²×C₄	—	2	6	no	—
101	C₃²×C₄	C₂²×C₃×C₄	1	12	yes	—
102	C₃²×C₄	C₂×C₃×Q₈	1	4	yes	—
103	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
104	C₂²×C₉	C₂×C₃×Q₈	1	6	yes	—
105	C₂²×C₉	C₃×#41	1	6	yes	—
106	C₂²×C₉	C₂⁴×C₃	1	3	yes	—
107	C₄×C₉	—	2	18	no	—
108	C₄×C₉	C₂²×C₃×C₄	1	36	yes	—
109	C₄×C₉	C₂×C₃×Q₈	1	12	yes	direct product factor
110	C₂²×Q₈	C₂×C₃×C₉	1	1	yes	characteristic
111	C₄×Q₈	C₂×C₃×C₉	1	8	yes	direct product factor
112	C₂×C₄⋊C₄	C₂×C₃×C₉	1	3	yes	—
113	C₂×C₄²	C₂×C₃×C₉	1	3	yes	—
114	C₃×C₉	C₂×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
115	C₂³×C₃	C₂³×C₉	1	3	yes	—
116	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
117	C₃×Q₈	C₂×C₄×C₉	1	48	yes	direct product factor
118	C₃×Q₈	C₂×C₃²×C₄	1	16	yes	—
119	C₂×C₃×C₄	—	2	18	no	—
120	C₂×C₃×C₄	—	2	6	no	—
121	C₂×C₃×C₄	C₂×C₄×C₉	1	36	yes	—
122	C₂×C₃×C₄	Q₈×C₉	1	12	yes	direct product factor
123	C₂×C₃×C₄	C₂³×C₉	1	18	yes	—
124	C₂×C₃×C₄	C₂³×C₉	1	18	yes	—
125	C₂×C₃×C₄	C₂³×C₉	1	6	yes	—
126	C₂×C₃×C₄	C₂×C₃²×C₄	1	12	yes	—
127	C₂×C₃×C₄	C₃²×Q₈	1	4	yes	—
128	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
129	C₂×C₃×C₄	C₂³×C₃²	1	6	yes	—
130	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
131	C₂×C₃²	C₃×C₄×Q₈	1	4	yes	—
132	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
133	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
134	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
135	C₂×C₉	C₃×C₄×Q₈	1	12	yes	direct product factor
136	C₂×C₉	C₂³×C₃×C₄	1	3	yes	—
137	C₂×C₉	C₂²×C₃×Q₈	1	3	yes	—
138	C₂×C₉	C₂×C₃×#41	1	3	yes	—
139	C₂×Q₈	C₃×C₄×C₉	1	8	yes	direct product factor
140	C₂×Q₈	C₂²×C₃×C₉	1	4	yes	—
141	C₂²×C₄	C₂²×C₃×C₉	1	3	yes	—
142	C₂²×C₄	C₂²×C₃×C₉	1	3	yes	—
143	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
144	C₄⋊C₄	C₂²×C₃×C₉	1	12	yes	—
145	C₄²	C₂²×C₃×C₉	1	12	yes	—
146	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
147	C₂²×C₃	C₂×Q₈×C₉	1	6	yes	—
148	C₂²×C₃	C₉×#41	1	6	yes	—
149	C₂²×C₃	C₂⁴×C₉	1	3	yes	—
150	C₂²×C₃	C₂²×C₃²×C₄	1	2	yes	—
151	C₂²×C₃	C₂×C₃²×Q₈	1	2	yes	—
152	C₂²×C₃	C₃²×#41	1	2	yes	—
153	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
154	C₃×C₄	—	2	6	no	—
155	C₃×C₄	—	2	18	no	—
156	C₃×C₄	C₂²×C₄×C₉	1	36	yes	—
157	C₃×C₄	C₂×Q₈×C₉	1	12	yes	direct product factor
158	C₃×C₄	C₂²×C₃²×C₄	1	12	yes	—
159	C₃×C₄	C₂×C₃²×Q₈	1	4	yes	—
160	C₃²	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
161	C₉	C₂×C₃×C₄×Q₈	1	3	yes	direct product factor
162	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
163	Q₈	C₂×C₃×C₄×C₉	1	16	yes	direct product factor
164	C₂×C₄	—	2	6	no	—
165	C₂×C₄	C₂×C₃×C₄×C₉	1	12	yes	—
166	C₂×C₄	C₃×Q₈×C₉	1	4	yes	direct product factor
167	C₂×C₄	C₂³×C₃×C₉	1	6	yes	—
168	C₂×C₄	C₂³×C₃×C₉	1	6	yes	—
169	C₂×C₄	C₂³×C₃×C₉	1	2	yes	—
170	C₂×C₃	C₄×Q₈×C₉	1	12	yes	direct product factor
171	C₂×C₃	C₂³×C₄×C₉	1	3	yes	—
172	C₂×C₃	C₂²×Q₈×C₉	1	3	yes	—
173	C₂×C₃	C₂×C₉×#41	1	3	yes	—
174	C₂×C₃	C₃²×C₄×Q₈	1	4	yes	—
175	C₂×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
176	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
177	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
178	C₂²	C₂²×C₃×C₄×C₉	1	2	yes	—
179	C₂²	C₂×C₃×Q₈×C₉	1	2	yes	—
180	C₂²	C₃×C₉×#41	1	2	yes	—
181	C₂²	C₂⁴×C₃×C₉	1	1	yes	characteristic
182	C₄	—	2	6	no	—
183	C₄	C₂²×C₃×C₄×C₉	1	12	yes	—
184	C₄	C₂×C₃×Q₈×C₉	1	4	yes	direct product factor
185	C₃	C₂×C₄×Q₈×C₉	1	3	yes	direct product factor
186	C₃	C₂×C₃²×C₄×Q₈	1	1	yes	characteristic
187	C₂	C₃×C₄×Q₈×C₉	1	4	yes	direct product factor
188	C₂	C₂³×C₃×C₄×C₉	1	1	yes	characteristic
189	C₂	C₂²×C₃×Q₈×C₉	1	1	yes	characteristic
190	C₂	C₂×C₃×C₉×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄×C₉	C₂²
commutator	C₂	C₂³×C₃×C₄×C₉
Frattini, cosocle	C₂²×C₃	C₂⁴×C₃²
socle, prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₈×C₉	C₂	direct (N × H)
6. C₂×C₄×Q₈×C₉	C₃	direct (N × H)
7. C₂×C₃×Q₈×C₉	C₄	direct (N × H)
19. C₄×Q₈×C₉	C₂×C₃	direct (N × H)
23. C₃×Q₈×C₉	C₂×C₄	direct (N × H)
26. C₂×C₃×C₄×C₉	Q₈	direct (N × H)
30. C₂×C₃×C₄×Q₈	C₉	direct (N × H)
39. C₂×Q₈×C₉	C₃×C₄	direct (N × H)
52. C₃×C₄×C₉	C₂×Q₈	direct (N × H)
55. C₃×C₄×Q₈	C₂×C₉	direct (N × H)
70. Q₈×C₉	C₂×C₃×C₄	direct (N × H)
73. C₂×C₄×C₉	C₃×Q₈	direct (N × H)
77. C₂×C₄×Q₈	C₃×C₉	direct (N × H)
78. C₂×C₃×C₉	C₄×Q₈	direct (N × H)
82. C₂×C₃×Q₈	C₄×C₉	direct (N × H)
109. C₄×C₉	C₂×C₃×Q₈	direct (N × H)
111. C₄×Q₈	C₂×C₃×C₉	direct (N × H)
114. C₃×C₉	C₂×C₄×Q₈	direct (N × H)
117. C₃×Q₈	C₂×C₄×C₉	direct (N × H)
122. C₂×C₃×C₄	Q₈×C₉	direct (N × H)
135. C₂×C₉	C₃×C₄×Q₈	direct (N × H)
139. C₂×Q₈	C₃×C₄×C₉	direct (N × H)
157. C₃×C₄	C₂×Q₈×C₉	direct (N × H)
161. C₉	C₂×C₃×C₄×Q₈	direct (N × H)
163. Q₈	C₂×C₃×C₄×C₉	direct (N × H)
166. C₂×C₄	C₃×Q₈×C₉	direct (N × H)
170. C₂×C₃	C₄×Q₈×C₉	direct (N × H)
184. C₄	C₂×C₃×Q₈×C₉	direct (N × H)
185. C₃	C₂×C₄×Q₈×C₉	direct (N × H)
187. C₂	C₃×C₄×Q₈×C₉	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×C₃×Q₈×C₉	2
C₃×C₄×Q₈×C₉	2
C₂×C₃×C₉×C₄⋊C₄	2
C₂×C₃×C₄²×C₉	2
C₂×C₃²×C₄×Q₈	3
C₂×C₄×Q₈×C₉	3

Subgroups — C2×C3×C4×Q8×C9 #13154530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×Q₈×C₉ #13154530