Subgroups — #11786030

Order 1152.

744 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#10592634	C₂	1	2	yes	maximal
3	#10589417	C₂	1	4	yes	maximal
4	#69762	—	3	4	no	maximal
5	C₂²×C₃²×C₈	C₂²	1	2	yes	—
6	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
8	C₂×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
12	#4935	—	3	8	no	—
13	#4551	—	3	16	no	—
14	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
15	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
17	C₂²×C₃⋊Dic₃	—	2	4	no	—
18	C₂×C₃²×C₈	—	2	4	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
20	C₃²×C₄²	C₂×C₄	1	2	yes	—
21	C₃²×C₄²	Q₈	1	2	yes	—
22	C₄⋊(C₃⋊Dic₃)	—	2	8	no	—
23	C₄⋊(C₃⋊Dic₃)	—	4	4	no	—
24	C₄⋊(C₃⋊Dic₃)	—	4	4	no	—
25	#1834	—	9	1	no	Sylow
26	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
27	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
28	C₂×C₄⋊Dic₃	—	3	16	no	—
29	C₂×C₄⋊Dic₃	—	6	8	no	—
30	C₂×C₄⋊Dic₃	—	6	8	no	—
31	C₃×C₄×C₈	Dic₃	1	16	yes	complemented
32	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
34	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
35	C₂×C₃²×C₄	D₈	1	2	yes	—
36	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
37	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
38	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₃²×C₄	#41	1	4	yes	—
41	C₂×C₃⋊Dic₃	—	4	4	no	—
42	C₂×C₃⋊Dic₃	—	4	4	no	—
43	C₂×C₃⋊Dic₃	—	4	4	no	—
44	C₃²×C₈	—	2	4	no	—
45	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
46	#384	—	9	2	no	—
47	#340	—	9	4	no	—
48	C₂²×C₃×C₄	D₁₂	1	4	yes	—
49	C₂²×C₃×C₄	D₁₂	1	4	yes	—
50	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
51	C₂²×Dic₃	—	6	16	no	—
52	C₂×C₃×C₈	—	2	16	no	—
53	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
54	C₃×C₄²	Dic₆	1	8	yes	—
55	C₃×C₄²	C₂×Dic₃	1	8	yes	—
56	C₄⋊Dic₃	—	6	32	no	—
57	C₄⋊Dic₃	—	12	16	no	—
58	C₄⋊Dic₃	—	12	16	no	—
59	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃²	#124	1	2	yes	—
61	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
62	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
63	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
64	C₃²×C₄	#106	1	4	yes	—
65	C₃²×C₄	#107	1	4	yes	—
66	C₃²×C₄	#117	1	2	yes	—
67	C₃²×C₄	#125	1	2	yes	—
68	C₃⋊Dic₃	—	8	4	no	—
69	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
70	C₂×C₄⋊C₄	—	9	4	no	—
71	C₂×C₄⋊C₄	—	18	2	no	—
72	C₂×C₄⋊C₄	—	18	2	no	—
73	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
74	C₄×C₈	C₃⋊Dic₃	1	4	yes	complemented
75	C₂³×C₃	C₂×D₁₂	1	4	yes	—
76	C₂×C₃×C₄	#204	1	16	yes	—
77	C₂×C₃×C₄	D₂₄	1	8	yes	—
78	C₂×C₃×C₄	Dic₁₂	1	8	yes	—
79	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
80	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
81	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
82	C₂×C₃×C₄	#235	1	16	yes	—
83	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
84	C₂×Dic₃	—	12	16	no	—
85	C₂×Dic₃	—	12	16	no	—
86	C₂×Dic₃	—	12	16	no	—
87	C₃×C₈	—	2	16	no	—
88	C₂×C₃²	#382	1	1	yes	characteristic
89	C₂×C₃²	C₂×#106	1	1	yes	characteristic
90	C₂×C₃²	C₂×#107	1	1	yes	characteristic
91	C₂×C₃²	#486	1	2	yes	—
92	C₂×C₃²	#487	1	2	yes	—
93	C₂²×C₄	—	18	4	no	—
94	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
95	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
99	C₄⋊C₄	—	18	8	no	—
100	C₄⋊C₄	—	36	4	no	—
101	C₄⋊C₄	—	36	4	no	—
102	C₄²	#634	1	2	yes	—
103	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
104	C₂²×C₃	#875	1	8	yes	—
105	C₂²×C₃	C₂×#204	1	8	yes	—
106	C₂²×C₃	C₂×D₂₄	1	4	yes	—
107	C₂²×C₃	C₂×Dic₁₂	1	4	yes	—
108	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
109	C₃×C₄	#817	1	16	yes	—
110	C₃×C₄	#818	1	16	yes	—
111	C₃×C₄	#870	1	8	yes	—
112	C₃×C₄	#926	1	8	yes	—
113	Dic₃	—	24	16	no	—
114	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
116	C₂×C₄	—	36	4	no	—
117	C₂×C₄	—	36	4	no	—
118	C₂×C₄	—	36	4	no	—
119	C₂×C₄	#3744	1	4	yes	—
120	C₂×C₄	#3745	1	2	yes	—
121	C₂×C₄	#3746	1	2	yes	—
122	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
123	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
124	C₂×C₄	C₂×#634	1	1	yes	characteristic
125	C₂×C₄	#3828	1	4	yes	—
126	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂×C₃	#4681	1	8	yes	—
129	C₂×C₃	#4694	1	8	yes	—
130	C₂×C₃	#4936	1	4	yes	—
131	C₂×C₃	C₂×#817	1	4	yes	—
132	C₂×C₃	C₂×#818	1	4	yes	—
133	C₂²	#64084	1	2	yes	—
134	C₂²	C₂×#3744	1	2	yes	—
135	C₂²	C₂×#3745	1	1	yes	characteristic
136	C₂²	C₂×#3746	1	1	yes	characteristic
137	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
138	C₄	—	72	4	no	—
139	C₄	#63644	1	4	yes	—
140	C₄	#63645	1	4	yes	—
141	C₄	#64079	1	2	yes	—
142	C₄	#64135	1	2	yes	—
143	C₃	#69762	1	4	yes	complemented
144	C₂	#10592380	1	2	yes	—
145	C₂	#10592393	1	2	yes	—
146	C₂	#10592635	1	1	yes	characteristic
147	C₂	C₂×#63644	1	1	yes	characteristic
148	C₂	C₂×#63645	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊(C₃⋊D₃)
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#1834
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₃²×C₄×C₈	C₄	semidirect
31. C₃×C₄×C₈	Dic₃	semidirect
74. C₄×C₈	C₃⋊Dic₃	semidirect
114. C₃²	#1834	semidirect
143. C₃	#69762	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
#10592634	2
#10589417	2
#69762	3