Subgroups — C₃²×#1844 #11768893

Order 1152.

840 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×#106	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×#1844	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×#106	—	2	8	no	—
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	C₂×C₃×#106	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	#1844	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×#106	—	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	SD₁₆	1	8	yes	—
36	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
38	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
39	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₃²×C₈	C₄⋊C₄	1	8	yes	complemented
41	C₂×#106	C₂×C₃²	1	4	yes	—
42	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
43	#384	C₂×C₃²	1	2	yes	—
44	C₂²×C₃×C₄	—	2	16	no	—
45	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
46	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
47	C₂×C₃×C₈	C₂×C₃×C₄	1	16	yes	—
48	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
49	C₂×C₃×C₈	C₃×Q₈	1	16	yes	—
50	C₃×C₄⋊C₄	—	2	32	no	—
51	C₃×C₄⋊C₄	—	4	32	no	—
52	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
53	C₃×C₄²	C₃×Q₈	1	8	yes	—
54	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₃²	#127	1	1	yes	characteristic
56	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
58	C₃²×C₄	—	8	4	no	—
59	C₃²×C₄	#106	1	8	yes	complemented
60	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
61	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
62	C₂²×C₈	C₂²×C₃²	1	2	yes	—
63	C₂×C₄⋊C₄	—	2	4	no	—
64	C₂×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
65	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
66	#106	—	2	8	no	—
67	C₄×C₈	C₃²×C₄	1	4	yes	complemented
68	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
69	C₂×C₃×C₄	—	4	16	no	—
70	C₂×C₃×C₄	—	4	32	no	—
71	C₂×C₃×C₄	C₃×C₄⋊C₄	1	16	yes	—
72	C₂×C₃×C₄	C₃×SD₁₆	1	32	yes	—
73	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
74	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
75	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
76	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
77	C₃×C₈	C₃×C₄⋊C₄	1	32	yes	complemented
78	C₂×C₃²	#384	1	1	yes	characteristic
79	C₂×C₃²	C₂×#106	1	2	yes	—
80	C₂×C₃²	#492	1	2	yes	—
81	C₂×C₃²	#498	1	2	yes	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	C₃²×D₄	1	2	yes	—
84	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
85	C₂×C₈	C₂×C₃²×C₄	1	4	yes	—
86	C₂×C₈	C₃²×D₄	1	4	yes	—
87	C₂×C₈	C₃²×Q₈	1	4	yes	—
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	4	8	no	—
90	C₄²	C₂×C₃²×C₄	1	2	yes	—
91	C₄²	C₃²×Q₈	1	2	yes	—
92	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
93	C₂²×C₃	C₃×#127	1	4	yes	—
94	C₂²×C₃	C₃×C₄⋊Q₈	1	4	yes	—
95	C₂²×C₃	C₂×C₃×SD₁₆	1	16	yes	—
96	C₃×C₄	—	8	16	no	—
97	C₃×C₄	C₃×#106	1	32	yes	complemented
98	C₃×C₄	C₂×C₃×C₄⋊C₄	1	8	yes	—
99	C₃×C₄	C₃×C₄⋊Q₈	1	8	yes	—
100	C₃²	#1844	1	1	yes	characteristic, direct product factor, Sylow
101	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	8	no	—
104	C₂×C₄	C₃²×C₄⋊C₄	1	4	yes	—
105	C₂×C₄	C₃²×SD₁₆	1	8	yes	—
106	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
107	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
108	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
109	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
110	C₈	C₃²×C₄⋊C₄	1	8	yes	complemented
111	C₂×C₃	C₃×#384	1	4	yes	—
112	C₂×C₃	C₂×C₃×#106	1	8	yes	—
113	C₂×C₃	C₃×#492	1	8	yes	—
114	C₂×C₃	C₃×#498	1	8	yes	—
115	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²	C₃²×#127	1	1	yes	characteristic
117	C₂²	C₃²×C₄⋊Q₈	1	1	yes	characteristic
118	C₂²	C₂×C₃²×SD₁₆	1	4	yes	—
119	C₄	—	8	4	no	—
120	C₄	C₃²×#106	1	8	yes	complemented
121	C₄	C₂×C₃²×C₄⋊C₄	1	2	yes	—
122	C₄	C₃²×C₄⋊Q₈	1	2	yes	—
123	C₃	C₃×#1844	1	4	yes	direct product factor
124	C₂	C₃²×#384	1	1	yes	characteristic
125	C₂	C₂×C₃²×#106	1	2	yes	—
126	C₂	C₃²×#492	1	2	yes	—
127	C₂	C₃²×#498	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×#1844	C₃	direct (N × H)
10. C₃²×C₄×C₈	C₄	semidirect
24. #1844	C₃²	direct (N × H)
30. C₃×C₄×C₈	C₃×C₄	semidirect
40. C₃²×C₈	C₄⋊C₄	semidirect
59. C₃²×C₄	#106	semidirect
67. C₄×C₈	C₃²×C₄	semidirect
77. C₃×C₈	C₃×C₄⋊C₄	semidirect
97. C₃×C₄	C₃×#106	semidirect
100. C₃²	#1844	direct (N × H)
110. C₈	C₃²×C₄⋊C₄	semidirect
120. C₄	C₃²×#106	semidirect
123. C₃	C₃×#1844	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×#106	2
C₂×C₃²×C₄×C₈	2
C₃²×#384	2
C₃×#1844	3

Subgroups — C32×#1844 #11768893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1844 #11768893