Subgroups — C₃²×#1845 #11768894

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×#107	C₂	1	4	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#384	C₂	1	2	yes	maximal
4	C₃×#1845	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₈	C₂²	1	2	yes	—
6	C₂×C₃²×C₄⋊C₄	—	2	4	no	—
7	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₃²×#107	—	2	8	no	—
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	C₂×C₃×#107	C₂×C₃	1	16	yes	—
12	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
13	C₃×#384	C₂×C₃	1	8	yes	—
14	C₂²×C₃²×C₄	—	2	4	no	—
15	C₂²×C₃²×C₄	D₄	1	2	yes	—
16	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
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×C₄⋊C₄	—	2	8	no	—
21	C₃²×C₄⋊C₄	—	4	8	no	—
22	C₃²×C₄²	C₂×C₄	1	2	yes	—
23	C₃²×C₄²	Q₈	1	2	yes	—
24	#1845	C₃²	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
26	C₂×C₃×C₄⋊C₄	—	2	16	no	—
27	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	16	yes	—
28	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
29	C₃×#107	—	2	32	no	—
30	C₃×C₄×C₈	C₃×C₄	1	16	yes	complemented
31	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃²×C₄	—	4	4	no	—
33	C₂×C₃²×C₄	—	4	8	no	—
34	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
35	C₂×C₃²×C₄	D₈	1	4	yes	—
36	C₂×C₃²×C₄	Q₁₆	1	4	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×Q₈	1	2	yes	—
40	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₃²×C₈	C₄⋊C₄	1	8	yes	complemented
42	C₂×#107	C₂×C₃²	1	4	yes	—
43	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
44	#384	C₂×C₃²	1	2	yes	—
45	C₂²×C₃×C₄	—	2	16	no	—
46	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
47	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
48	C₂×C₃×C₈	C₂×C₃×C₄	1	16	yes	—
49	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
50	C₂×C₃×C₈	C₃×Q₈	1	16	yes	—
51	C₃×C₄⋊C₄	—	2	32	no	—
52	C₃×C₄⋊C₄	—	4	32	no	—
53	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
54	C₃×C₄²	C₃×Q₈	1	8	yes	—
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	—	8	4	no	—
61	C₃²×C₄	#107	1	8	yes	complemented
62	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
63	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₈	C₂²×C₃²	1	2	yes	—
65	C₂×C₄⋊C₄	—	2	4	no	—
66	C₂×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
67	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
68	#107	—	2	8	no	—
69	C₄×C₈	C₃²×C₄	1	4	yes	complemented
70	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
71	C₂×C₃×C₄	—	4	16	no	—
72	C₂×C₃×C₄	—	4	32	no	—
73	C₂×C₃×C₄	C₃×C₄⋊C₄	1	16	yes	—
74	C₂×C₃×C₄	C₃×D₈	1	16	yes	—
75	C₂×C₃×C₄	C₃×Q₁₆	1	16	yes	—
76	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
77	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
78	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
79	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
80	C₃×C₈	C₃×C₄⋊C₄	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	—	2	4	no	—
87	C₂²×C₄	C₃²×D₄	1	2	yes	—
88	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
89	C₂×C₈	C₂×C₃²×C₄	1	4	yes	—
90	C₂×C₈	C₃²×D₄	1	4	yes	—
91	C₂×C₈	C₃²×Q₈	1	4	yes	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	4	8	no	—
94	C₄²	C₂×C₃²×C₄	1	2	yes	—
95	C₄²	C₃²×Q₈	1	2	yes	—
96	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
97	C₂²×C₃	C₃×#127	1	4	yes	—
98	C₂²×C₃	C₃×C₄⋊Q₈	1	4	yes	—
99	C₂²×C₃	C₂×C₃×D₈	1	8	yes	—
100	C₂²×C₃	C₂×C₃×Q₁₆	1	8	yes	—
101	C₃×C₄	—	8	16	no	—
102	C₃×C₄	C₃×#107	1	32	yes	complemented
103	C₃×C₄	C₂×C₃×C₄⋊C₄	1	8	yes	—
104	C₃×C₄	C₃×C₄⋊Q₈	1	8	yes	—
105	C₃²	#1845	1	1	yes	characteristic, direct product factor, Sylow
106	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	4	8	no	—
109	C₂×C₄	C₃²×C₄⋊C₄	1	4	yes	—
110	C₂×C₄	C₃²×D₈	1	4	yes	—
111	C₂×C₄	C₃²×Q₁₆	1	4	yes	—
112	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
114	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
115	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
116	C₈	C₃²×C₄⋊C₄	1	8	yes	complemented
117	C₂×C₃	C₃×#384	1	4	yes	—
118	C₂×C₃	C₂×C₃×#107	1	8	yes	—
119	C₂×C₃	C₃×#493	1	4	yes	—
120	C₂×C₃	C₃×#494	1	4	yes	—
121	C₂×C₃	C₃×#500	1	8	yes	—
122	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²	C₃²×#127	1	1	yes	characteristic
124	C₂²	C₃²×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²	C₂×C₃²×D₈	1	2	yes	—
126	C₂²	C₂×C₃²×Q₁₆	1	2	yes	—
127	C₄	—	8	4	no	—
128	C₄	C₃²×#107	1	8	yes	complemented
129	C₄	C₂×C₃²×C₄⋊C₄	1	2	yes	—
130	C₄	C₃²×C₄⋊Q₈	1	2	yes	—
131	C₃	C₃×#1845	1	4	yes	direct product factor
132	C₂	C₃²×#384	1	1	yes	characteristic
133	C₂	C₂×C₃²×#107	1	2	yes	—
134	C₂	C₃²×#493	1	1	yes	characteristic
135	C₂	C₃²×#494	1	1	yes	characteristic
136	C₂	C₃²×#500	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃×#1845	C₃	direct (N × H)
10. C₃²×C₄×C₈	C₄	semidirect
24. #1845	C₃²	direct (N × H)
30. C₃×C₄×C₈	C₃×C₄	semidirect
41. C₃²×C₈	C₄⋊C₄	semidirect
61. C₃²×C₄	#107	semidirect
69. C₄×C₈	C₃²×C₄	semidirect
80. C₃×C₈	C₃×C₄⋊C₄	semidirect
102. C₃×C₄	C₃×#107	semidirect
105. C₃²	#1845	direct (N × H)
116. C₈	C₃²×C₄⋊C₄	semidirect
128. C₄	C₃²×#107	semidirect
131. C₃	C₃×#1845	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×#107	2
C₂×C₃²×C₄×C₈	2
C₃²×#384	2
C₃×#1845	3

Subgroups — C32×#1845 #11768894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1845 #11768894