Subgroups — C₉×#2984 #11767705

Order 1152.

546 conjugacy classes of subgroups.

The subgroup lattice has 189 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#446	C₂	1	6	yes	maximal, complemented
3	C₈×Q₈×C₉	C₂	1	2	yes	maximal, complemented
4	C₉×#432	C₂	1	3	yes	maximal
5	C₉×#431	C₂	1	3	yes	maximal
6	C₃×#2984	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₄	1	3	yes	—
9	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₉	C₂²	1	3	yes	—
12	C₂×C₉×C₄⋊C₄	C₄	1	3	yes	—
13	C₂×C₄²×C₉	C₂²	1	3	yes	—
14	C₉×C₄⋊C₈	C₂²	1	6	yes	—
15	C₉×C₄⋊C₈	C₂²	1	6	yes	—
16	C₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₉×C₂²⋊C₈	C₂²	1	3	yes	—
18	C₉×#97	C₂²	1	6	yes	—
19	C₄×C₈×C₉	C₂²	1	6	yes	—
20	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₃×#446	C₂×C₃	1	6	yes	—
22	C₃×C₈×Q₈	C₂×C₃	1	2	yes	—
23	C₃×#432	C₂×C₃	1	3	yes	—
24	C₃×#431	C₂×C₃	1	3	yes	—
25	C₂×Q₈×C₉	—	2	4	no	—
26	C₂×Q₈×C₉	C₂×C₄	1	3	yes	—
27	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉	C₂³	1	2	yes	—
32	C₂×C₈×C₉	C₂³	1	6	yes	—
33	C₉×C₄⋊C₄	C₂×C₄	1	6	yes	—
34	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
35	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
36	C₄²×C₉	C₂×C₄	1	3	yes	—
37	C₄²×C₉	C₂×C₄	1	3	yes	—
38	C₄²×C₉	C₂³	1	3	yes	—
39	C₄²×C₉	C₂³	1	3	yes	—
40	#2984	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
42	C₃×C₄×Q₈	C₃×C₄	1	3	yes	—
43	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
44	C₃×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
45	C₃×C₄×Q₈	C₂²×C₃	1	3	yes	—
46	C₂×C₃×C₄⋊C₄	C₃×C₄	1	3	yes	—
47	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
48	C₃×C₄⋊C₈	C₂²×C₃	1	6	yes	—
49	C₃×C₄⋊C₈	C₂²×C₃	1	6	yes	—
50	C₃×C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₃×C₂²⋊C₈	C₂²×C₃	1	3	yes	—
52	C₃×#97	C₂²×C₃	1	6	yes	—
53	C₃×C₄×C₈	C₂²×C₃	1	6	yes	—
54	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₉	—	2	6	no	—
56	Q₈×C₉	M₁₆	1	4	yes	—
57	C₂×C₄×C₉	—	2	6	no	—
58	C₂×C₄×C₉	—	2	6	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
62	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
63	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
64	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
65	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
66	C₈×C₉	—	2	2	no	—
67	C₈×C₉	—	2	6	no	—
68	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
69	#446	C₂×C₉	1	6	yes	complemented
70	C₈×Q₈	C₂×C₉	1	2	yes	complemented
71	#432	C₂×C₉	1	3	yes	—
72	#431	C₂×C₉	1	3	yes	—
73	C₂×C₃×Q₈	—	2	4	no	—
74	C₂×C₃×Q₈	C₂×C₃×C₄	1	3	yes	—
75	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
77	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
78	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
80	C₂×C₃×C₈	C₂³×C₃	1	6	yes	—
81	C₃×C₄⋊C₄	C₂×C₃×C₄	1	6	yes	—
82	C₃×C₄⋊C₄	C₂×C₃×C₄	1	3	yes	—
83	C₃×C₄⋊C₄	C₂×C₃×C₄	1	3	yes	—
84	C₃×C₄²	C₂×C₃×C₄	1	3	yes	—
85	C₃×C₄²	C₂×C₃×C₄	1	3	yes	—
86	C₃×C₄²	C₂³×C₃	1	3	yes	—
87	C₃×C₄²	C₂³×C₃	1	3	yes	—
88	C₂²×C₉	—	2	1	no	—
89	C₂²×C₉	—	2	1	no	—
90	C₂²×C₉	#131	1	2	yes	—
91	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₉	—	2	1	no	—
93	C₄×C₉	—	2	3	no	—
94	C₄×C₉	—	2	3	no	—
95	C₄×C₉	—	2	3	no	—
96	C₄×C₉	C₂×M₁₆	1	6	yes	—
97	C₄×C₉	2₋¹⁺⁴	1	2	yes	—
98	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
99	C₄×Q₈	C₄×C₉	1	3	yes	—
100	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
101	C₄×Q₈	C₂²×C₉	1	1	yes	characteristic
102	C₄×Q₈	C₂²×C₉	1	3	yes	—
103	C₂×C₄⋊C₄	C₄×C₉	1	3	yes	—
104	C₂×C₄²	C₂²×C₉	1	3	yes	—
105	C₄⋊C₈	C₂²×C₉	1	6	yes	—
106	C₄⋊C₈	C₂²×C₉	1	6	yes	—
107	C₂²⋊C₈	C₂²×C₉	1	1	yes	characteristic
108	C₂²⋊C₈	C₂²×C₉	1	3	yes	—
109	#97	C₂²×C₉	1	6	yes	—
110	C₄×C₈	C₂²×C₉	1	6	yes	—
111	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
112	C₃×Q₈	—	2	6	no	—
113	C₃×Q₈	C₃×M₁₆	1	4	yes	—
114	C₂×C₃×C₄	—	2	6	no	—
115	C₂×C₃×C₄	—	2	6	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
119	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
120	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
121	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
122	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	—	2	6	no	—
125	C₂×C₉	—	2	2	no	—
126	C₂×C₉	#519	1	1	yes	characteristic
127	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
128	C₂×C₉	C₂×#131	1	1	yes	characteristic
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	C₂×C₄×C₉	1	3	yes	—
131	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
133	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
134	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₉	1	2	yes	—
136	C₂×C₈	C₂³×C₉	1	6	yes	—
137	C₄⋊C₄	C₂×C₄×C₉	1	6	yes	—
138	C₄⋊C₄	C₂×C₄×C₉	1	3	yes	—
139	C₄⋊C₄	C₂×C₄×C₉	1	3	yes	—
140	C₄²	C₂×C₄×C₉	1	3	yes	—
141	C₄²	C₂×C₄×C₉	1	3	yes	—
142	C₄²	C₂³×C₉	1	3	yes	—
143	C₄²	C₂³×C₉	1	3	yes	—
144	C₂²×C₃	—	2	1	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	C₃×#131	1	2	yes	—
147	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
148	C₃×C₄	—	2	3	no	—
149	C₃×C₄	—	2	3	no	—
150	C₃×C₄	—	2	3	no	—
151	C₃×C₄	—	2	1	no	—
152	C₃×C₄	C₂×C₃×M₁₆	1	6	yes	—
153	C₃×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
154	C₉	#2984	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
156	Q₈	—	2	6	no	—
157	Q₈	C₉×M₁₆	1	4	yes	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
163	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
164	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
165	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
166	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	2	6	no	—
169	C₂×C₃	—	2	2	no	—
170	C₂×C₃	C₃×#519	1	1	yes	characteristic
171	C₂×C₃	C₂²×C₃×M₁₆	1	1	yes	characteristic
172	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₉×#131	1	2	yes	—
176	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
177	C₄	—	2	3	no	—
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	1	no	—
181	C₄	C₂×C₉×M₁₆	1	6	yes	—
182	C₄	C₉×2₋¹⁺⁴	1	2	yes	—
183	C₃	C₃×#2984	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	C₉×#519	1	1	yes	characteristic
186	C₂	C₂²×C₉×M₁₆	1	1	yes	characteristic
187	C₂	C₂×C₉×#131	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
3. C₈×Q₈×C₉	C₂	semidirect
40. #2984	C₉	direct (N × H)
69. #446	C₂×C₉	semidirect
70. C₈×Q₈	C₂×C₉	semidirect
154. C₉	#2984	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₉×#446	2
C₈×Q₈×C₉	2
C₉×#432	2
C₉×#431	2
C₃×#2984	3

Subgroups — C9×#2984 #11767705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2984 #11767705