Subgroups — C₃²×#1848 #11768897

Order 1152.

660 conjugacy classes of subgroups.

The subgroup lattice has 129 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	4	yes	maximal
2	C₃²×#423	C₂	1	2	yes	maximal
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#1848	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₃²×C₈	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₃²×#108	—	2	8	no	—
9	C₃²×C₄⋊C₈	—	2	4	no	—
10	C₃²×C₄×C₈	C₄	1	2	yes	—
11	C₃²×C₄×C₈	C₄	1	2	yes	—
12	C₂×C₃×#108	C₂×C₃	1	16	yes	—
13	C₃×#423	C₂×C₃	1	8	yes	—
14	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
15	C₂²×C₃²×C₄	Q₈	1	2	yes	—
16	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
17	C₃²×M₁₆	—	2	8	no	—
18	C₂×C₃²×C₈	—	2	4	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
20	C₂×C₃²×C₈	D₄	1	4	yes	—
21	C₂×C₃²×C₈	Q₈	1	4	yes	—
22	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃²×C₄²	D₄	1	2	yes	—
25	#1848	C₃²	1	1	yes	characteristic, direct product factor, Sylow
26	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	—
27	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
28	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
29	C₃×#108	—	2	32	no	—
30	C₃×C₄⋊C₈	—	2	16	no	—
31	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
32	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
33	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₃²×C₄	—	2	4	no	—
35	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
36	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
37	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
39	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
41	C₃²×C₈	—	4	4	no	—
42	C₃²×C₈	C₄⋊C₄	1	8	yes	—
43	C₂×#108	C₂×C₃²	1	4	yes	—
44	#423	C₂×C₃²	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₃×Q₈	1	8	yes	—
47	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
48	C₃×M₁₆	—	2	32	no	—
49	C₂×C₃×C₈	—	2	16	no	—
50	C₂×C₃×C₈	C₂×C₃×C₄	1	16	yes	—
51	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
52	C₂×C₃×C₈	C₃×Q₈	1	16	yes	—
53	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
54	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
55	C₃×C₄²	C₃×D₄	1	8	yes	—
56	C₂²×C₃²	—	2	2	no	—
57	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
59	C₃²×C₄	—	2	2	no	—
60	C₃²×C₄	#108	1	4	yes	complemented
61	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
62	C₃²×C₄	#127	1	1	yes	characteristic
63	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×M₁₆	C₂²×C₃²	1	4	yes	—
65	C₂²×C₈	C₂²×C₃²	1	2	yes	—
66	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
67	#108	—	2	8	no	—
68	C₄⋊C₈	—	2	4	no	—
69	C₄×C₈	C₃²×C₄	1	2	yes	—
70	C₄×C₈	C₃²×C₄	1	2	yes	—
71	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
72	C₂×C₃×C₄	—	2	16	no	—
73	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
74	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
75	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
77	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
78	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
79	C₃×C₈	—	4	16	no	—
80	C₃×C₈	C₃×C₄⋊C₄	1	32	yes	—
81	C₂×C₃²	—	2	2	no	—
82	C₂×C₃²	#384	1	1	yes	characteristic
83	C₂×C₃²	C₂×#108	1	2	yes	—
84	C₂²×C₄	C₃²×Q₈	1	2	yes	—
85	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
86	M₁₆	—	2	8	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	C₂×C₃²×C₄	1	4	yes	—
89	C₂×C₈	C₃²×D₄	1	4	yes	—
90	C₂×C₈	C₃²×Q₈	1	4	yes	—
91	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
92	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
93	C₄²	C₃²×D₄	1	2	yes	—
94	C₂²×C₃	—	2	8	no	—
95	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
96	C₂²×C₃	C₃×C₄⋊Q₈	1	8	yes	—
97	C₃×C₄	—	2	8	no	—
98	C₃×C₄	C₃×#108	1	16	yes	complemented
99	C₃×C₄	C₂×C₃×C₄⋊C₄	1	8	yes	—
100	C₃×C₄	C₃×#127	1	4	yes	—
101	C₃×C₄	C₃×C₄⋊Q₈	1	4	yes	—
102	C₃²	#1848	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³	C₂×C₃²×Q₈	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
106	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
107	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
108	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
109	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
111	C₈	—	4	4	no	—
112	C₈	C₃²×C₄⋊C₄	1	8	yes	—
113	C₂×C₃	—	2	8	no	—
114	C₂×C₃	C₃×#384	1	4	yes	—
115	C₂×C₃	C₂×C₃×#108	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²	C₃²×C₄⋊Q₈	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₃²×#108	1	4	yes	complemented
121	C₄	C₂×C₃²×C₄⋊C₄	1	2	yes	—
122	C₄	C₃²×#127	1	1	yes	characteristic
123	C₄	C₃²×C₄⋊Q₈	1	1	yes	characteristic
124	C₃	C₃×#1848	1	4	yes	direct product factor
125	C₂	—	2	2	no	—
126	C₂	C₃²×#384	1	1	yes	characteristic
127	C₂	C₂×C₃²×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1848	C₃	direct (N × H)
25. #1848	C₃²	direct (N × H)
60. C₃²×C₄	#108	semidirect
98. C₃×C₄	C₃×#108	semidirect
102. C₃²	#1848	direct (N × H)
120. C₄	C₃²×#108	semidirect
124. C₃	C₃×#1848	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×C₃²×#108	2
C₃²×#423	2
C₂×C₃²×C₄×C₈	2
C₃×#1848	3

Subgroups — C32×#1848 #11768897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1848 #11768897