Subgroups — #11786228

Order 1152.

840 conjugacy classes of subgroups.

The subgroup lattice has 138 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	C₂×#63645	C₂	1	4	yes	maximal
3	#10592634	C₂	1	2	yes	maximal
4	#69765	—	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	4	no	—
8	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
9	C₃²×C₄×C₈	C₄	1	4	yes	complemented
10	#63645	—	2	8	no	—
11	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
12	C₂×#818	—	3	16	no	—
13	#4935	—	3	8	no	—
14	C₂²×C₃²×C₄	D₄	1	2	yes	—
15	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
16	C₂²×C₃⋊Dic₃	—	2	4	no	—
17	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
18	C₂×C₃²×C₈	D₄	1	4	yes	—
19	C₂×C₃²×C₈	Q₈	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	8	no	—
24	#1845	—	9	1	no	Sylow
25	C₂²×C₃×C₈	C₂×D₃	1	8	yes	—
26	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
27	C₂×C₄⋊Dic₃	—	3	16	no	—
28	C₂×C₄⋊Dic₃	—	6	16	no	—
29	C₃×C₄×C₈	Dic₃	1	16	yes	complemented
30	#818	—	6	32	no	—
31	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
33	C₂×C₃²×C₄	D₈	1	4	yes	—
34	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
35	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
37	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
38	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃⋊Dic₃	—	4	4	no	—
40	C₂×C₃⋊Dic₃	—	4	8	no	—
41	C₃²×C₈	C₄⋊C₄	1	8	yes	complemented
42	C₂×#107	—	9	4	no	—
43	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
44	#384	—	9	2	no	—
45	C₂²×C₃×C₄	D₁₂	1	8	yes	—
46	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
47	C₂²×Dic₃	—	6	16	no	—
48	C₂×C₃×C₈	Dic₆	1	16	yes	—
49	C₂×C₃×C₈	D₁₂	1	16	yes	—
50	C₂×C₃×C₈	C₂×Dic₃	1	16	yes	—
51	C₃×C₄²	Dic₆	1	8	yes	—
52	C₃×C₄²	C₂×Dic₃	1	8	yes	—
53	C₄⋊Dic₃	—	6	32	no	—
54	C₄⋊Dic₃	—	12	32	no	—
55	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃²	#127	1	1	yes	characteristic
57	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₃²	C₂×D₈	1	2	yes	—
59	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
60	C₃²×C₄	#107	1	8	yes	complemented
61	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
62	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
63	C₃⋊Dic₃	—	8	4	no	—
64	C₂²×C₈	C₂×C₃⋊D₃	1	2	yes	—
65	C₂×C₄⋊C₄	—	9	4	no	—
66	C₂×C₄⋊C₄	—	18	4	no	—
67	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
68	#107	—	18	8	no	—
69	C₄×C₈	C₃⋊Dic₃	1	4	yes	complemented
70	C₂³×C₃	C₂×D₁₂	1	4	yes	—
71	C₂×C₃×C₄	D₂₄	1	16	yes	—
72	C₂×C₃×C₄	Dic₁₂	1	16	yes	—
73	C₂×C₃×C₄	C₄⋊Dic₃	1	16	yes	—
74	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
75	C₂×C₃×C₄	C₂×Dic₆	1	8	yes	—
76	C₂×C₃×C₄	C₂×D₁₂	1	8	yes	—
77	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
78	C₂×Dic₃	—	12	16	no	—
79	C₂×Dic₃	—	12	32	no	—
80	C₃×C₈	C₄⋊Dic₃	1	32	yes	complemented
81	C₂×C₃²	#384	1	1	yes	characteristic
82	C₂×C₃²	C₂×#107	1	2	yes	—
83	C₂×C₃²	#493	1	1	yes	characteristic
84	C₂×C₃²	#494	1	1	yes	characteristic
85	C₂×C₃²	#500	1	2	yes	—
86	C₂²×C₄	—	18	4	no	—
87	C₂²×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
88	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
89	C₂×C₈	#634	1	4	yes	—
90	C₂×C₈	C₄⋊(C₃⋊D₃)	1	4	yes	—
91	C₂×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
92	C₄⋊C₄	—	18	8	no	—
93	C₄⋊C₄	—	36	8	no	—
94	C₄²	#634	1	2	yes	—
95	C₄²	C₂×C₃⋊Dic₃	1	2	yes	—
96	C₂²×C₃	#869	1	4	yes	—
97	C₂²×C₃	#874	1	4	yes	—
98	C₂²×C₃	C₂×D₂₄	1	8	yes	—
99	C₂²×C₃	C₂×Dic₁₂	1	8	yes	—
100	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
101	C₃×C₄	#818	1	32	yes	complemented
102	C₃×C₄	#869	1	8	yes	—
103	C₃×C₄	C₂×C₄⋊Dic₃	1	8	yes	—
104	Dic₃	—	24	16	no	—
105	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
106	C₂³	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
107	C₂×C₄	—	36	4	no	—
108	C₂×C₄	—	36	8	no	—
109	C₂×C₄	#3745	1	4	yes	—
110	C₂×C₄	#3746	1	4	yes	—
111	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	4	yes	—
112	C₂×C₄	C₂×#634	1	1	yes	characteristic
113	C₂×C₄	C₂×#634	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	2	yes	—
115	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
116	C₈	C₄⋊(C₃⋊Dic₃)	1	8	yes	complemented
117	C₂×C₃	#4682	1	8	yes	—
118	C₂×C₃	#4695	1	4	yes	—
119	C₂×C₃	#4699	1	4	yes	—
120	C₂×C₃	#4935	1	4	yes	—
121	C₂×C₃	C₂×#818	1	8	yes	—
122	C₂²	#64078	1	1	yes	characteristic
123	C₂²	#64083	1	1	yes	characteristic
124	C₂²	C₂×#3745	1	2	yes	—
125	C₂²	C₂×#3746	1	2	yes	—
126	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
127	C₄	—	72	4	no	—
128	C₄	#63645	1	8	yes	complemented
129	C₄	#64078	1	2	yes	—
130	C₄	C₂×C₄⋊(C₃⋊Dic₃)	1	2	yes	—
131	C₃	#69765	1	4	yes	complemented
132	C₂	#10592381	1	2	yes	—
133	C₂	#10592394	1	1	yes	characteristic
134	C₂	#10592398	1	1	yes	characteristic
135	C₂	#10592634	1	1	yes	characteristic
136	C₂	C₂×#63645	1	2	yes	—

	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	#1845
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
9. C₃²×C₄×C₈	C₄	semidirect
29. C₃×C₄×C₈	Dic₃	semidirect
41. C₃²×C₈	C₄⋊C₄	semidirect
60. C₃²×C₄	#107	semidirect
69. C₄×C₈	C₃⋊Dic₃	semidirect
80. C₃×C₈	C₄⋊Dic₃	semidirect
101. C₃×C₄	#818	semidirect
105. C₃²	#1845	semidirect
116. C₈	C₄⋊(C₃⋊Dic₃)	semidirect
128. C₄	#63645	semidirect
131. C₃	#69765	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
C₂×#63645	2
#10592634	2
#69765	3