Subgroups — C₉×#554 #10589986

Order 576.

342 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#125	C₂	1	2	yes	maximal, complemented
3	C₉×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
4	C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄×C₉	C₂	1	2	yes	maximal
7	C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₉×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃×#554	C₃	1	1	yes	maximal, characteristic
10	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₉	C₂²	1	2	yes	—
12	C₂×D₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₉	C₂²	1	4	yes	—
15	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₄	C₂²	1	4	yes	—
17	C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₉×C₄⋊C₄	C₂²	1	2	yes	—
20	C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₉×C₄⋊C₄	C₂²	1	4	yes	—
22	C₉×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₉×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₄²×C₉	C₂²	1	2	yes	—
25	C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
27	C₃×#125	C₂×C₃	1	2	yes	—
28	C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
29	C₃×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
30	C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
31	C₃×C₄×D₄	C₂×C₃	1	2	yes	—
32	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₄	C₂×C₃	1	2	yes	—
34	C₂³×C₉	C₂³	1	2	yes	—
35	Q₈×C₉	—	2	2	no	—
36	Q₈×C₉	—	4	2	no	—
37	D₄×C₉	Q₈	1	4	yes	complemented
38	C₂×C₄×C₉	—	2	4	no	—
39	C₂×C₄×C₉	—	4	4	no	—
40	C₂×C₄×C₉	C₂³	1	2	yes	—
41	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₄×C₉	C₂³	1	2	yes	—
44	C₂×C₄×C₉	C₂³	1	4	yes	—
45	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
47	#554	C₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
50	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
52	C₂²×C₃×C₄	C₂²×C₃	1	4	yes	—
53	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
54	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
59	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
60	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
61	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
62	C₃×C₄²	C₂²×C₃	1	2	yes	—
63	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₉	—	2	2	no	—
65	C₂²×C₉	—	2	2	no	—
66	C₂²×C₉	C₂×Q₈	1	4	yes	—
67	C₂²×C₉	C₂⁴	1	1	yes	characteristic
68	C₄×C₉	—	2	2	no	—
69	C₄×C₉	—	2	2	no	—
70	C₄×C₉	—	2	2	no	—
71	C₄×C₉	—	2	4	no	—
72	C₄×C₉	—	2	1	no	—
73	C₄×C₉	C₂×Q₈	1	2	yes	—
74	C₄×C₉	#41	1	2	yes	—
75	C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
76	#125	C₂×C₉	1	2	yes	complemented
77	C₂²⋊Q₈	C₂×C₉	1	4	yes	complemented
78	C₂²⋊Q₈	C₂×C₉	1	2	yes	complemented
79	C₄×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
80	C₄×D₄	C₂×C₉	1	2	yes	—
81	C₄×D₄	C₂×C₉	1	1	yes	characteristic
82	C₂×C₄⋊C₄	C₂×C₉	1	2	yes	complemented
83	C₂³×C₃	C₂³×C₃	1	2	yes	—
84	C₃×Q₈	—	2	2	no	—
85	C₃×Q₈	—	4	2	no	—
86	C₃×D₄	C₃×Q₈	1	4	yes	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	4	4	no	—
89	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
90	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
92	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
93	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
94	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₉	—	2	4	no	—
97	C₂×C₉	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₉	C₂×#41	1	1	yes	characteristic
99	C₂×C₉	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×Q₈	C₂²×C₉	1	1	yes	characteristic
101	C₂×Q₈	C₂²×C₉	1	2	yes	—
102	C₂×D₄	C₂²×C₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×C₉	1	2	yes	—
104	C₂²×C₄	C₂²×C₉	1	4	yes	—
105	C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
106	C₄⋊C₄	C₂²×C₉	1	4	yes	—
107	C₄⋊C₄	C₂²×C₉	1	2	yes	—
108	C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
109	C₄⋊C₄	C₂²×C₉	1	2	yes	—
110	C₄⋊C₄	C₂²×C₉	1	2	yes	—
111	C₄⋊C₄	C₂²×C₉	1	4	yes	—
112	C₂²⋊C₄	C₂²×C₉	1	4	yes	—
113	C₂²⋊C₄	C₂²×C₉	1	2	yes	—
114	C₄²	C₂²×C₉	1	2	yes	—
115	C₄²	C₂²×C₉	1	1	yes	characteristic
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	C₂×C₃×Q₈	1	4	yes	—
119	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
120	C₃×C₄	—	2	2	no	—
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	2	2	no	—
123	C₃×C₄	—	2	4	no	—
124	C₃×C₄	—	2	1	no	—
125	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₃×C₄	C₃×#41	1	2	yes	—
127	C₉	#554	1	1	yes	characteristic, direct product factor, Sylow
128	C₂³	C₂³×C₉	1	2	yes	—
129	Q₈	—	2	2	no	—
130	Q₈	—	4	2	no	—
131	D₄	Q₈×C₉	1	4	yes	complemented
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	C₂³×C₉	1	2	yes	—
135	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×C₉	1	2	yes	—
137	C₂×C₄	C₂³×C₉	1	2	yes	—
138	C₂×C₄	C₂³×C₉	1	4	yes	—
139	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
140	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
141	C₂×C₃	—	2	4	no	—
142	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
143	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
144	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₂×Q₈×C₉	1	4	yes	—
148	C₂²	C₂⁴×C₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	4	no	—
153	C₄	—	2	1	no	—
154	C₄	C₂×Q₈×C₉	1	2	yes	—
155	C₄	C₉×#41	1	2	yes	—
156	C₃	C₃×#554	1	1	yes	characteristic
157	C₂	—	2	4	no	—
158	C₂	C₂²×Q₈×C₉	1	1	yes	characteristic
159	C₂	C₂×C₉×#41	1	1	yes	characteristic
160	C₂	C₉×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂⁴
commutator	C₂²	C₂⁴×C₉
Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×C₄⋊Q₈	C₂	semidirect
2. C₉×#125	C₂	semidirect
3. C₉×C₂²⋊Q₈	C₂	semidirect
4. C₉×C₂²⋊Q₈	C₂	semidirect
5. C₄×Q₈×C₉	C₂	semidirect
8. C₂×C₉×C₄⋊C₄	C₂	semidirect
37. D₄×C₉	Q₈	semidirect
47. #554	C₉	direct (N × H)
75. C₄⋊Q₈	C₂×C₉	semidirect
76. #125	C₂×C₉	semidirect
77. C₂²⋊Q₈	C₂×C₉	semidirect
78. C₂²⋊Q₈	C₂×C₉	semidirect
79. C₄×Q₈	C₂×C₉	semidirect
82. C₂×C₄⋊C₄	C₂×C₉	semidirect
127. C₉	#554	direct (N × H)
131. D₄	Q₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₄⋊Q₈	2
C₉×#125	2
C₉×C₂²⋊Q₈	2
C₄×Q₈×C₉	2
C₄×D₄×C₉	2
C₂×C₉×C₄⋊C₄	2
C₃×#554	3

Subgroups — C9×#554 #10589986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#554 #10589986