Subgroups — C₂×C₈×Q₈×C₉ #11767679

Order 1152.

660 conjugacy classes of subgroups.

The subgroup lattice has 156 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	C₈×Q₈×C₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₉×C₄⋊C₈	C₂	1	3	yes	maximal
4	C₂×C₄×C₈×C₉	C₂	1	3	yes	maximal
5	C₂×C₃×C₈×Q₈	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₉	C₂²	1	3	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₉	C₄	1	4	yes	—
10	C₄×Q₈×C₉	C₂²	1	4	yes	—
11	C₂×C₉×C₄⋊C₄	C₄	1	3	yes	—
12	C₂×C₄²×C₉	C₂²	1	3	yes	—
13	C₉×C₄⋊C₈	C₂²	1	12	yes	—
14	C₄×C₈×C₉	C₂²	1	12	yes	—
15	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
16	C₃×C₈×Q₈	C₂×C₃	1	8	yes	—
17	C₂×C₃×C₄⋊C₈	C₂×C₃	1	3	yes	—
18	C₂×C₃×C₄×C₈	C₂×C₃	1	3	yes	—
19	C₂×Q₈×C₉	C₈	1	8	yes	direct product factor
20	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₉	—	2	6	no	—
25	C₂×C₈×C₉	Q₈	1	4	yes	direct product factor
26	C₂×C₈×C₉	C₂³	1	6	yes	—
27	C₂×C₈×C₉	C₂³	1	2	yes	—
28	C₉×C₄⋊C₄	C₂×C₄	1	12	yes	—
29	C₄²×C₉	C₂×C₄	1	6	yes	—
30	C₄²×C₉	C₂³	1	6	yes	—
31	C₂×C₈×Q₈	C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂²×C₃	1	3	yes	—
34	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
35	C₃×C₄×Q₈	C₃×C₄	1	4	yes	—
36	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
37	C₂×C₃×C₄⋊C₄	C₃×C₄	1	3	yes	—
38	C₂×C₃×C₄²	C₂²×C₃	1	3	yes	—
39	C₃×C₄⋊C₈	C₂²×C₃	1	12	yes	—
40	C₃×C₄×C₈	C₂²×C₃	1	12	yes	—
41	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₉	C₂×C₈	1	16	yes	direct product factor
43	C₂×C₄×C₉	—	2	6	no	—
44	C₂×C₄×C₉	C₂×C₈	1	12	yes	—
45	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₉	C₂²×C₄	1	6	yes	—
48	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₉	#41	1	2	yes	—
50	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
51	C₈×C₉	—	2	6	no	—
52	C₈×C₉	C₂×Q₈	1	4	yes	direct product factor
53	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
54	C₈×Q₈	C₂×C₉	1	8	yes	direct product factor
55	C₂×C₄⋊C₈	C₂×C₉	1	3	yes	—
56	C₂×C₄×C₈	C₂×C₉	1	3	yes	—
57	C₂×C₃×Q₈	C₃×C₈	1	8	yes	—
58	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
59	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
60	C₂²×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
61	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₃×C₈	—	2	6	no	—
63	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
64	C₂×C₃×C₈	C₂³×C₃	1	6	yes	—
65	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
66	C₃×C₄⋊C₄	C₂×C₃×C₄	1	12	yes	—
67	C₃×C₄²	C₂×C₃×C₄	1	6	yes	—
68	C₃×C₄²	C₂³×C₃	1	6	yes	—
69	C₂²×C₉	C₄×Q₈	1	2	yes	—
70	C₂²×C₉	C₂²×C₈	1	2	yes	—
71	C₂²×C₉	#131	1	2	yes	—
72	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₉	—	2	6	no	—
74	C₄×C₉	C₄×Q₈	1	2	yes	—
75	C₄×C₉	C₂²×C₈	1	12	yes	—
76	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₉	C₂×#41	1	1	yes	characteristic
78	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₉	1	3	yes	—
80	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
81	C₄×Q₈	C₄×C₉	1	4	yes	—
82	C₄×Q₈	C₂²×C₉	1	4	yes	—
83	C₂×C₄⋊C₄	C₄×C₉	1	3	yes	—
84	C₂×C₄²	C₂²×C₉	1	3	yes	—
85	C₄⋊C₈	C₂²×C₉	1	12	yes	—
86	C₄×C₈	C₂²×C₉	1	12	yes	—
87	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
88	C₃×Q₈	C₂×C₃×C₈	1	16	yes	—
89	C₂×C₃×C₄	—	2	6	no	—
90	C₂×C₃×C₄	C₂×C₃×C₈	1	12	yes	—
91	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
92	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
94	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
95	C₂×C₃×C₄	C₃×#41	1	2	yes	—
96	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
97	C₃×C₈	—	2	6	no	—
98	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
99	C₂×C₉	C₈×Q₈	1	4	yes	direct product factor
100	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₉	C₂³×C₈	1	1	yes	characteristic
102	C₂×C₉	C₂×#131	1	1	yes	characteristic
103	C₂×Q₈	C₈×C₉	1	8	yes	direct product factor
104	C₂×Q₈	C₂×C₄×C₉	1	4	yes	—
105	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
106	C₂²×C₄	C₂×C₄×C₉	1	3	yes	—
107	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
108	C₂×C₈	—	2	6	no	—
109	C₂×C₈	Q₈×C₉	1	4	yes	direct product factor
110	C₂×C₈	C₂³×C₉	1	6	yes	—
111	C₂×C₈	C₂³×C₉	1	2	yes	—
112	C₄⋊C₄	C₂×C₄×C₉	1	12	yes	—
113	C₄²	C₂×C₄×C₉	1	6	yes	—
114	C₄²	C₂³×C₉	1	6	yes	—
115	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
116	C₂²×C₃	C₂²×C₃×C₈	1	2	yes	—
117	C₂²×C₃	C₃×#131	1	2	yes	—
118	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
119	C₃×C₄	—	2	6	no	—
120	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
121	C₃×C₄	C₂²×C₃×C₈	1	12	yes	—
122	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
123	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
124	C₉	C₂×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
125	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
126	Q₈	C₂×C₈×C₉	1	16	yes	direct product factor
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	C₂×C₈×C₉	1	12	yes	—
129	C₂×C₄	C₂²×C₄×C₉	1	6	yes	—
130	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₄×C₉	1	6	yes	—
132	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
133	C₂×C₄	C₉×#41	1	2	yes	—
134	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
135	C₈	—	2	6	no	—
136	C₈	C₂×Q₈×C₉	1	4	yes	direct product factor
137	C₂×C₃	C₃×C₈×Q₈	1	4	yes	—
138	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
139	C₂×C₃	C₂³×C₃×C₈	1	1	yes	characteristic
140	C₂×C₃	C₂×C₃×#131	1	1	yes	characteristic
141	C₂²	C₄×Q₈×C₉	1	2	yes	—
142	C₂²	C₂²×C₈×C₉	1	2	yes	—
143	C₂²	C₉×#131	1	2	yes	—
144	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	C₄×Q₈×C₉	1	2	yes	—
147	C₄	C₂²×C₈×C₉	1	12	yes	—
148	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
149	C₄	C₂×C₉×#41	1	1	yes	characteristic
150	C₃	C₂×C₃×C₈×Q₈	1	1	yes	characteristic
151	C₂	C₈×Q₈×C₉	1	4	yes	direct product factor
152	C₂	C₂×C₄×Q₈×C₉	1	1	yes	characteristic
153	C₂	C₂³×C₈×C₉	1	1	yes	characteristic
154	C₂	C₂×C₉×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₉	C₂²
commutator	C₂	C₂³×C₈×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle, prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₉	C₂	direct (N × H)
19. C₂×Q₈×C₉	C₈	direct (N × H)
25. C₂×C₈×C₉	Q₈	direct (N × H)
31. C₂×C₈×Q₈	C₉	direct (N × H)
42. Q₈×C₉	C₂×C₈	direct (N × H)
52. C₈×C₉	C₂×Q₈	direct (N × H)
54. C₈×Q₈	C₂×C₉	direct (N × H)
99. C₂×C₉	C₈×Q₈	direct (N × H)
103. C₂×Q₈	C₈×C₉	direct (N × H)
109. C₂×C₈	Q₈×C₉	direct (N × H)
124. C₉	C₂×C₈×Q₈	direct (N × H)
126. Q₈	C₂×C₈×C₉	direct (N × H)
136. C₈	C₂×Q₈×C₉	direct (N × H)
151. C₂	C₈×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
C₂×C₄×Q₈×C₉	2
C₈×Q₈×C₉	2
C₂×C₉×C₄⋊C₈	2
C₂×C₄×C₈×C₉	2
C₂×C₃×C₈×Q₈	3

Subgroups — C2×C8×Q8×C9 #11767679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Q₈×C₉ #11767679