Subgroups — C₂×C₉×#384 #11767005

Order 1152.

1332 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₉×#384	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#384	C₃	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₉	C₂²	1	4	yes	—
6	C₂³×C₄×C₉	C₂²	1	3	yes	—
7	C₂×C₉×C₄⋊C₄	—	2	24	no	—
8	C₂×C₉×C₄⋊C₄	C₂²	1	24	yes	—
9	C₂×C₄²×C₉	C₄	1	8	yes	complemented
10	C₂×C₄²×C₉	C₂²	1	4	yes	—
11	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	6	yes	—
12	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
13	C₃×#384	C₂×C₃	1	8	yes	—
14	C₂⁴×C₉	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₉	—	2	24	no	—
16	C₂²×C₄×C₉	C₂×C₄	1	12	yes	—
17	C₂²×C₄×C₉	D₄	1	12	yes	—
18	C₂²×C₄×C₉	Q₈	1	12	yes	—
19	C₂²×C₄×C₉	C₂³	1	8	yes	—
20	C₂²×C₄×C₉	C₂³	1	6	yes	—
21	C₉×C₄⋊C₄	—	2	48	no	—
22	C₄²×C₉	C₂×C₄	1	16	yes	complemented
23	C₂×#384	C₉	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
25	C₂³×C₃×C₄	C₂²×C₃	1	3	yes	—
26	C₂×C₃×C₄⋊C₄	—	2	24	no	—
27	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	24	yes	—
28	C₂×C₃×C₄²	C₃×C₄	1	8	yes	—
29	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
30	C₂³×C₉	C₂²×C₄	1	2	yes	—
31	C₂³×C₉	C₂×D₄	1	6	yes	—
32	C₂³×C₉	C₂×Q₈	1	6	yes	—
33	C₂³×C₉	C₂⁴	1	1	yes	characteristic
34	C₂×C₄×C₉	—	2	24	no	—
35	C₂×C₄×C₉	—	4	16	no	—
36	C₂×C₄×C₉	C₄⋊C₄	1	48	yes	complemented
37	C₂×C₄×C₉	C₂²×C₄	1	12	yes	—
38	C₂×C₄×C₉	C₂×D₄	1	12	yes	—
39	C₂×C₄×C₉	C₂×Q₈	1	12	yes	—
40	C₂²×C₄⋊C₄	C₂×C₉	1	6	yes	—
41	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
42	#384	C₂×C₉	1	8	yes	direct product factor
43	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
44	C₂²×C₃×C₄	—	2	24	no	—
45	C₂²×C₃×C₄	C₂×C₃×C₄	1	12	yes	—
46	C₂²×C₃×C₄	C₃×D₄	1	12	yes	—
47	C₂²×C₃×C₄	C₃×Q₈	1	12	yes	—
48	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
49	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
50	C₃×C₄⋊C₄	—	2	48	no	—
51	C₃×C₄²	C₂×C₃×C₄	1	16	yes	—
52	C₂²×C₉	C₂×C₄⋊C₄	1	12	yes	—
53	C₂²×C₉	#127	1	4	yes	—
54	C₂²×C₉	C₄⋊Q₈	1	12	yes	—
55	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
56	C₂²×C₉	C₂²×D₄	1	3	yes	—
57	C₂²×C₉	C₂²×Q₈	1	3	yes	—
58	C₄×C₉	—	4	8	no	—
59	C₄×C₉	C₂×C₄⋊C₄	1	24	yes	complemented
60	C₂³×C₄	C₂²×C₉	1	4	yes	—
61	C₂³×C₄	C₂²×C₉	1	3	yes	—
62	C₂×C₄⋊C₄	—	2	24	no	—
63	C₂×C₄⋊C₄	C₂²×C₉	1	24	yes	—
64	C₂×C₄²	C₄×C₉	1	8	yes	complemented
65	C₂×C₄²	C₂²×C₉	1	4	yes	—
66	C₂³×C₃	C₂²×C₃×C₄	1	2	yes	—
67	C₂³×C₃	C₂×C₃×D₄	1	6	yes	—
68	C₂³×C₃	C₂×C₃×Q₈	1	6	yes	—
69	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
70	C₂×C₃×C₄	—	2	24	no	—
71	C₂×C₃×C₄	—	4	16	no	—
72	C₂×C₃×C₄	C₃×C₄⋊C₄	1	48	yes	—
73	C₂×C₃×C₄	C₂²×C₃×C₄	1	12	yes	—
74	C₂×C₃×C₄	C₂×C₃×D₄	1	12	yes	—
75	C₂×C₃×C₄	C₂×C₃×Q₈	1	12	yes	—
76	C₂×C₉	#384	1	8	yes	direct product factor
77	C₂×C₉	C₂²×C₄⋊C₄	1	3	yes	—
78	C₂×C₉	C₂×#127	1	1	yes	characteristic
79	C₂×C₉	C₂×C₄⋊Q₈	1	3	yes	—
80	C₂⁴	C₂³×C₉	1	1	yes	characteristic
81	C₂²×C₄	—	2	24	no	—
82	C₂²×C₄	C₂×C₄×C₉	1	12	yes	—
83	C₂²×C₄	D₄×C₉	1	12	yes	—
84	C₂²×C₄	Q₈×C₉	1	12	yes	—
85	C₂²×C₄	C₂³×C₉	1	8	yes	—
86	C₂²×C₄	C₂³×C₉	1	6	yes	—
87	C₄⋊C₄	—	2	48	no	—
88	C₄²	C₂×C₄×C₉	1	16	yes	complemented
89	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	12	yes	—
90	C₂²×C₃	C₃×#127	1	4	yes	—
91	C₂²×C₃	C₃×C₄⋊Q₈	1	12	yes	—
92	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
93	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
94	C₂²×C₃	C₂²×C₃×Q₈	1	3	yes	—
95	C₃×C₄	—	4	8	no	—
96	C₃×C₄	C₂×C₃×C₄⋊C₄	1	24	yes	—
97	C₉	C₂×#384	1	1	yes	characteristic, direct product factor, Sylow
98	C₂³	C₂²×C₄×C₉	1	2	yes	—
99	C₂³	C₂×D₄×C₉	1	6	yes	—
100	C₂³	C₂×Q₈×C₉	1	6	yes	—
101	C₂³	C₂⁴×C₉	1	1	yes	characteristic
102	C₂×C₄	—	2	24	no	—
103	C₂×C₄	—	4	16	no	—
104	C₂×C₄	C₉×C₄⋊C₄	1	48	yes	complemented
105	C₂×C₄	C₂²×C₄×C₉	1	12	yes	—
106	C₂×C₄	C₂×D₄×C₉	1	12	yes	—
107	C₂×C₄	C₂×Q₈×C₉	1	12	yes	—
108	C₂×C₃	C₃×#384	1	8	yes	—
109	C₂×C₃	C₂²×C₃×C₄⋊C₄	1	3	yes	—
110	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
111	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	3	yes	—
112	C₂²	C₂×C₉×C₄⋊C₄	1	12	yes	—
113	C₂²	C₉×#127	1	4	yes	—
114	C₂²	C₉×C₄⋊Q₈	1	12	yes	—
115	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
116	C₂²	C₂²×D₄×C₉	1	3	yes	—
117	C₂²	C₂²×Q₈×C₉	1	3	yes	—
118	C₄	—	4	8	no	—
119	C₄	C₂×C₉×C₄⋊C₄	1	24	yes	complemented
120	C₃	C₂×C₃×#384	1	1	yes	characteristic
121	C₂	C₉×#384	1	8	yes	direct product factor
122	C₂	C₂²×C₉×C₄⋊C₄	1	3	yes	—
123	C₂	C₂×C₉×#127	1	1	yes	characteristic
124	C₂	C₂×C₉×C₄⋊Q₈	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₉×#384	C₂	direct (N × H)
9. C₂×C₄²×C₉	C₄	semidirect
22. C₄²×C₉	C₂×C₄	semidirect
23. C₂×#384	C₉	direct (N × H)
36. C₂×C₄×C₉	C₄⋊C₄	semidirect
42. #384	C₂×C₉	direct (N × H)
59. C₄×C₉	C₂×C₄⋊C₄	semidirect
64. C₂×C₄²	C₄×C₉	semidirect
76. C₂×C₉	#384	direct (N × H)
88. C₄²	C₂×C₄×C₉	semidirect
97. C₉	C₂×#384	direct (N × H)
104. C₂×C₄	C₉×C₄⋊C₄	semidirect
119. C₄	C₂×C₉×C₄⋊C₄	semidirect
121. C₂	C₉×#384	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×C₄⋊C₄	2
C₂²×C₄²×C₉	2
C₉×#384	2
C₂×C₃×#384	3

Subgroups — C2×C9×#384 #11767005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#384 #11767005