Subgroups — C₉×#2399 #11767120

Order 1152.

756 conjugacy classes of subgroups.

The subgroup lattice has 165 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	2	yes	maximal
2	C₉×#398	C₂	1	4	yes	maximal
3	C₉×#395	C₂	1	4	yes	maximal
4	C₄×C₉×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₉×#376	C₂	1	1	yes	maximal, characteristic
6	C₃×#2399	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₄×Q₈×C₉	—	2	8	no	—
9	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₉	C₂²	1	4	yes	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉	C₂²	1	2	yes	—
15	C₉×#95	C₂²	1	4	yes	—
16	C₉×#95	C₂²	1	4	yes	—
17	C₉×#95	C₂²	1	4	yes	—
18	C₂×C₃×C₄×Q₈	C₂×C₃	1	2	yes	—
19	C₃×#398	C₂×C₃	1	4	yes	—
20	C₃×#395	C₂×C₃	1	4	yes	—
21	C₃×C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
22	C₃×#376	C₂×C₃	1	1	yes	characteristic
23	C₂×Q₈×C₉	—	2	4	no	—
24	C₂×Q₈×C₉	—	2	8	no	—
25	C₂²×C₄×C₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₉	C₂³	1	4	yes	—
28	C₂²×C₄×C₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₉×C₄⋊C₄	—	2	8	no	—
31	C₉×C₄⋊C₄	—	2	4	no	—
32	C₉×C₄⋊C₄	—	2	16	no	—
33	C₄²×C₉	—	2	8	no	—
34	C₄²×C₉	D₄	1	4	yes	—
35	C₄²×C₉	Q₈	1	8	yes	complemented
36	#2399	C₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
38	C₃×C₄×Q₈	—	2	8	no	—
39	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
40	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
41	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
42	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
43	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
45	C₃×#95	C₂²×C₃	1	4	yes	—
46	C₃×#95	C₂²×C₃	1	4	yes	—
47	C₃×#95	C₂²×C₃	1	4	yes	—
48	C₂³×C₉	C₂⁴	1	1	yes	characteristic
49	Q₈×C₉	—	2	16	no	—
50	C₂×C₄×C₉	—	2	4	no	—
51	C₂×C₄×C₉	—	2	8	no	—
52	C₂×C₄×C₉	—	2	4	no	—
53	C₂×C₄×C₉	—	2	4	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	2	8	no	—
56	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₉	C₂×Q₈	1	8	yes	—
59	C₂×C₄×C₉	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₉	#41	1	8	yes	—
61	C₂×C₄×Q₈	C₂×C₉	1	2	yes	—
62	#398	C₂×C₉	1	4	yes	—
63	#395	C₂×C₉	1	4	yes	—
64	C₄×C₄⋊C₄	C₂×C₉	1	4	yes	—
65	#376	C₂×C₉	1	1	yes	characteristic
66	C₂×C₃×Q₈	—	2	4	no	—
67	C₂×C₃×Q₈	—	2	8	no	—
68	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
69	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
70	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
71	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
72	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
73	C₃×C₄⋊C₄	—	2	8	no	—
74	C₃×C₄⋊C₄	—	2	4	no	—
75	C₃×C₄⋊C₄	—	2	16	no	—
76	C₃×C₄²	—	2	8	no	—
77	C₃×C₄²	C₃×D₄	1	4	yes	—
78	C₃×C₄²	C₃×Q₈	1	8	yes	—
79	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₉	C₂²×Q₈	1	2	yes	—
81	C₂²×C₉	C₂×#41	1	4	yes	—
82	C₄×C₉	—	2	4	no	—
83	C₄×C₉	—	2	8	no	—
84	C₄×C₉	—	2	8	no	—
85	C₄×C₉	—	4	4	no	—
86	C₄×C₉	C₄⋊Q₈	1	4	yes	—
87	C₂²×Q₈	C₂²×C₉	1	2	yes	—
88	C₄×Q₈	—	2	8	no	—
89	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
90	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
92	C₂×C₄²	C₂²×C₉	1	4	yes	—
93	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₉	1	2	yes	—
95	#95	C₂²×C₉	1	4	yes	—
96	#95	C₂²×C₉	1	4	yes	—
97	#95	C₂²×C₉	1	4	yes	—
98	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
99	C₃×Q₈	—	2	16	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	—	2	8	no	—
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	8	no	—
106	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
107	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
108	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
109	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
110	C₂×C₃×C₄	C₃×#41	1	8	yes	—
111	C₂×C₉	#525	1	2	yes	—
112	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₉	#533	1	4	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	—	2	8	no	—
116	C₂²×C₄	C₂³×C₉	1	4	yes	—
117	C₂²×C₄	C₂³×C₉	1	2	yes	—
118	C₂²×C₄	C₂³×C₉	1	4	yes	—
119	C₂²×C₄	C₂³×C₉	1	4	yes	—
120	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
121	C₄⋊C₄	—	2	8	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	16	no	—
124	C₄²	—	2	8	no	—
125	C₄²	D₄×C₉	1	4	yes	—
126	C₄²	Q₈×C₉	1	8	yes	complemented
127	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
128	C₂²×C₃	C₂²×C₃×Q₈	1	2	yes	—
129	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	—	2	8	no	—
132	C₃×C₄	—	2	8	no	—
133	C₃×C₄	—	4	4	no	—
134	C₃×C₄	C₃×C₄⋊Q₈	1	4	yes	—
135	C₉	#2399	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	C₂⁴×C₉	1	1	yes	characteristic
137	Q₈	—	2	16	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
145	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
146	C₂×C₄	C₂×Q₈×C₉	1	8	yes	—
147	C₂×C₄	C₂×Q₈×C₉	1	4	yes	—
148	C₂×C₄	C₉×#41	1	8	yes	—
149	C₂×C₃	C₃×#525	1	2	yes	—
150	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₃	C₃×#533	1	4	yes	—
152	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
153	C₂²	C₂²×Q₈×C₉	1	2	yes	—
154	C₂²	C₂×C₉×#41	1	4	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	8	no	—
157	C₄	—	2	8	no	—
158	C₄	—	4	4	no	—
159	C₄	C₉×C₄⋊Q₈	1	4	yes	—
160	C₃	C₃×#2399	1	1	yes	characteristic
161	C₂	C₉×#525	1	2	yes	—
162	C₂	C₂×C₉×C₄⋊Q₈	1	1	yes	characteristic
163	C₂	C₉×#533	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
35. C₄²×C₉	Q₈	semidirect
36. #2399	C₉	direct (N × H)
126. C₄²	Q₈×C₉	semidirect
135. C₉	#2399	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×#398	2
C₉×#395	2
C₄×C₉×C₄⋊C₄	2
C₉×#376	2
C₃×#2399	3

Subgroups — C9×#2399 #11767120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2399 #11767120