Subgroups — Q₈×C₉×M₁₆ #11767684

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 201 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×#423	C₂	1	3	yes	maximal
5	C₄×C₉×M₁₆	C₂	1	3	yes	maximal
6	C₃×Q₈×M₁₆	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	3	yes	—
9	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉	C₄	1	3	yes	—
11	C₄×Q₈×C₉	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₉	C₂²	1	3	yes	—
14	C₂×C₉×C₄⋊C₄	C₄	1	3	yes	—
15	C₂×C₄²×C₉	C₂²	1	3	yes	—
16	C₉×C₄⋊C₈	C₂²	1	6	yes	—
17	C₉×C₄⋊C₈	C₂²	1	6	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×#423	C₂×C₃	1	3	yes	—
24	C₃×C₄×M₁₆	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	C₉×M₁₆	—	2	6	no	—
32	C₉×M₁₆	Q₈	1	4	yes	direct product factor
33	C₂×C₈×C₉	C₂³	1	6	yes	—
34	C₂×C₈×C₉	C₂³	1	2	yes	—
35	C₉×C₄⋊C₄	C₂×C₄	1	6	yes	—
36	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
37	C₉×C₄⋊C₄	C₂×C₄	1	3	yes	—
38	C₄²×C₉	C₂×C₄	1	3	yes	—
39	C₄²×C₉	C₂×C₄	1	3	yes	—
40	C₄²×C₉	C₂³	1	3	yes	—
41	C₄²×C₉	C₂³	1	3	yes	—
42	Q₈×M₁₆	C₉	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
44	C₂×C₃×M₁₆	C₂²×C₃	1	3	yes	—
45	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₃×C₄×Q₈	C₃×C₄	1	3	yes	—
47	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
48	C₃×C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄×Q₈	C₂²×C₃	1	3	yes	—
50	C₂×C₃×C₄⋊C₄	C₃×C₄	1	3	yes	—
51	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
52	C₃×C₄⋊C₈	C₂²×C₃	1	6	yes	—
53	C₃×C₄⋊C₈	C₂²×C₃	1	6	yes	—
54	C₃×#97	C₂²×C₃	1	6	yes	—
55	C₃×C₄×C₈	C₂²×C₃	1	6	yes	—
56	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
57	Q₈×C₉	—	2	6	no	—
58	Q₈×C₉	M₁₆	1	4	yes	direct product factor
59	C₂×C₄×C₉	—	2	6	no	—
60	C₂×C₄×C₉	—	4	3	no	—
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	1	yes	characteristic
65	C₂×C₄×C₉	C₂²×C₄	1	3	yes	—
66	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₉	#41	1	2	yes	—
68	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
69	C₈×C₉	—	2	6	no	—
70	C₈×C₉	C₂×Q₈	1	4	yes	complemented
71	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
72	#446	C₂×C₉	1	6	yes	complemented
73	C₈×Q₈	C₂×C₉	1	2	yes	complemented
74	#423	C₂×C₉	1	3	yes	—
75	C₄×M₁₆	C₂×C₉	1	3	yes	—
76	C₂×C₃×Q₈	—	2	4	no	—
77	C₂×C₃×Q₈	C₂×C₃×C₄	1	3	yes	—
78	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₃×M₁₆	—	2	6	no	—
83	C₃×M₁₆	C₃×Q₈	1	4	yes	—
84	C₂×C₃×C₈	C₂³×C₃	1	6	yes	—
85	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
86	C₃×C₄⋊C₄	C₂×C₃×C₄	1	6	yes	—
87	C₃×C₄⋊C₄	C₂×C₃×C₄	1	3	yes	—
88	C₃×C₄⋊C₄	C₂×C₃×C₄	1	3	yes	—
89	C₃×C₄²	C₂×C₃×C₄	1	3	yes	—
90	C₃×C₄²	C₂×C₃×C₄	1	3	yes	—
91	C₃×C₄²	C₂³×C₃	1	3	yes	—
92	C₃×C₄²	C₂³×C₃	1	3	yes	—
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	—	2	1	no	—
95	C₂²×C₉	C₄×Q₈	1	2	yes	—
96	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₉	—	2	3	no	—
98	C₄×C₉	—	2	3	no	—
99	C₄×C₉	—	2	3	no	—
100	C₄×C₉	C₄×Q₈	1	2	yes	—
101	C₄×C₉	C₂×M₁₆	1	6	yes	—
102	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
103	C₄×C₉	C₂×#41	1	1	yes	characteristic
104	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
105	C₂×M₁₆	C₂²×C₉	1	3	yes	—
106	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
107	C₄×Q₈	C₄×C₉	1	3	yes	—
108	C₄×Q₈	C₄×C₉	1	1	yes	characteristic
109	C₄×Q₈	C₂²×C₉	1	1	yes	characteristic
110	C₄×Q₈	C₂²×C₉	1	3	yes	—
111	C₂×C₄⋊C₄	C₄×C₉	1	3	yes	—
112	C₂×C₄²	C₂²×C₉	1	3	yes	—
113	C₄⋊C₈	C₂²×C₉	1	6	yes	—
114	C₄⋊C₈	C₂²×C₉	1	6	yes	—
115	#97	C₂²×C₉	1	6	yes	—
116	C₄×C₈	C₂²×C₉	1	6	yes	—
117	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₃×Q₈	—	2	6	no	—
119	C₃×Q₈	C₃×M₁₆	1	4	yes	—
120	C₂×C₃×C₄	—	2	6	no	—
121	C₂×C₃×C₄	—	4	3	no	—
122	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
124	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
125	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×C₃×C₄	1	3	yes	—
127	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
128	C₂×C₃×C₄	C₃×#41	1	2	yes	—
129	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
130	C₃×C₈	—	2	6	no	—
131	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
132	C₂×C₉	—	2	2	no	—
133	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂×C₉	C₂²×M₁₆	1	1	yes	characteristic
135	C₂×C₉	#568	1	1	yes	characteristic
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	C₂×C₄×C₉	1	3	yes	—
138	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
140	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
141	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
142	M₁₆	—	2	6	no	—
143	M₁₆	Q₈×C₉	1	4	yes	direct product factor
144	C₂×C₈	C₂³×C₉	1	6	yes	—
145	C₂×C₈	C₂³×C₉	1	2	yes	—
146	C₄⋊C₄	C₂×C₄×C₉	1	6	yes	—
147	C₄⋊C₄	C₂×C₄×C₉	1	3	yes	—
148	C₄⋊C₄	C₂×C₄×C₉	1	3	yes	—
149	C₄²	C₂×C₄×C₉	1	3	yes	—
150	C₄²	C₂×C₄×C₉	1	3	yes	—
151	C₄²	C₂³×C₉	1	3	yes	—
152	C₄²	C₂³×C₉	1	3	yes	—
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
156	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
157	C₃×C₄	—	2	3	no	—
158	C₃×C₄	—	2	3	no	—
159	C₃×C₄	—	2	3	no	—
160	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
161	C₃×C₄	C₂×C₃×M₁₆	1	6	yes	—
162	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
163	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
164	C₉	Q₈×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
166	Q₈	—	2	6	no	—
167	Q₈	C₉×M₁₆	1	4	yes	direct product factor
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	—	4	3	no	—
170	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
171	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
172	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
173	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₄×C₉	1	3	yes	—
175	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
176	C₂×C₄	C₉×#41	1	2	yes	—
177	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
178	C₈	—	2	6	no	—
179	C₈	C₂×Q₈×C₉	1	4	yes	complemented
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
182	C₂×C₃	C₂²×C₃×M₁₆	1	1	yes	characteristic
183	C₂×C₃	C₃×#568	1	1	yes	characteristic
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	C₄×Q₈×C₉	1	2	yes	—
187	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	3	no	—
191	C₄	C₄×Q₈×C₉	1	2	yes	—
192	C₄	C₂×C₉×M₁₆	1	6	yes	—
193	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
194	C₄	C₂×C₉×#41	1	1	yes	characteristic
195	C₃	C₃×Q₈×M₁₆	1	1	yes	characteristic
196	C₂	—	2	2	no	—
197	C₂	C₂×C₄×Q₈×C₉	1	1	yes	characteristic
198	C₂	C₂²×C₉×M₁₆	1	1	yes	characteristic
199	C₂	C₉×#568	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
32. C₉×M₁₆	Q₈	direct (N × H)
42. Q₈×M₁₆	C₉	direct (N × H)
58. Q₈×C₉	M₁₆	direct (N × H)
70. C₈×C₉	C₂×Q₈	semidirect
72. #446	C₂×C₉	semidirect
73. C₈×Q₈	C₂×C₉	semidirect
143. M₁₆	Q₈×C₉	direct (N × H)
164. C₉	Q₈×M₁₆	direct (N × H)
167. Q₈	C₉×M₁₆	direct (N × H)
179. C₈	C₂×Q₈×C₉	semidirect

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×#423	2
C₄×C₉×M₁₆	2
C₃×Q₈×M₁₆	3

Subgroups — Q8×C9×M16 #11767684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉×M₁₆ #11767684