Subgroups — C₂×#10592839 #11899381

Order 1152.

1104 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×#131	C₂	1	1	yes	maximal, characteristic
2	#10592839	C₂	1	8	yes	maximal, direct product factor
3	C₂×#63639	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	3	yes	maximal, complemented
5	C₂×#5140	—	3	4	no	maximal
6	C₂×C₃²×#41	C₄	1	1	yes	characteristic
7	C₃²×#131	C₄	1	4	yes	—
8	C₃²×#131	C₂²	1	4	yes	—
9	C₂×C₃²×M₁₆	C₄	1	3	yes	—
10	C₂²×C₃²×C₈	C₂²	1	3	yes	—
11	#63639	C₂²	1	12	yes	complemented
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	3	yes	—
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	12	yes	complemented
15	C₂×C₃×#131	D₃	1	4	yes	—
16	#5140	—	3	32	no	—
17	C₂×#812	—	3	12	no	—
18	C₂²×C₃⋊C₁₆	—	3	12	no	—
19	C₃²×#41	C₈	1	4	yes	—
20	C₃²×#41	C₂×C₄	1	4	yes	—
21	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
22	C₂×C₃²×D₄	C₈	1	3	yes	—
23	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
24	C₃²×M₁₆	C₂×C₄	1	12	yes	—
25	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
26	C₂×C₃²×C₈	C₂×C₄	1	6	yes	—
27	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂³	1	6	yes	—
29	C₃⋊(C₃⋊C₁₆)	C₂³	1	2	yes	—
30	C₃⋊(C₃⋊C₁₆)	C₂³	1	6	yes	—
31	C₂×#504	—	9	1	no	Sylow
32	C₂×C₃×#41	Dic₃	1	4	yes	—
33	C₃×#131	Dic₃	1	16	yes	—
34	C₃×#131	C₂×D₃	1	16	yes	—
35	C₂×C₃×M₁₆	Dic₃	1	12	yes	—
36	C₂²×C₃×C₈	C₂×D₃	1	12	yes	—
37	#812	—	3	48	no	—
38	C₂×C₃⋊C₁₆	—	3	48	no	—
39	C₂×C₃⋊C₁₆	—	3	4	no	—
40	C₂×C₃⋊C₁₆	—	3	12	no	—
41	C₂³×C₃²	C₂×C₈	1	3	yes	—
42	C₃²×Q₈	C₂×C₈	1	4	yes	—
43	C₃²×D₄	C₂×C₈	1	12	yes	—
44	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
45	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
46	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
48	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₃²×C₈	C₂²×C₄	1	6	yes	—
50	C₃²×C₈	C₂⁴	1	1	yes	characteristic
51	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
52	#504	—	9	8	no	—
53	C₂×M₃₂	—	9	3	no	—
54	C₂²×C₁₆	—	9	3	no	—
55	C₃×#41	C₃⋊C₈	1	16	yes	—
56	C₃×#41	C₂×Dic₃	1	16	yes	—
57	C₂×C₃×Q₈	C₃⋊C₈	1	4	yes	—
58	C₂×C₃×D₄	C₃⋊C₈	1	12	yes	—
59	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
60	C₃×M₁₆	C₂×Dic₃	1	48	yes	—
61	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
62	C₂×C₃×C₈	C₂×Dic₃	1	24	yes	—
63	C₂×C₃×C₈	C₂²×D₃	1	24	yes	—
64	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
65	C₃⋊C₁₆	—	3	8	no	—
66	C₃⋊C₁₆	—	3	24	no	—
67	C₂²×C₃²	—	2	6	no	—
68	C₂²×C₃²	C₂²×C₈	1	6	yes	—
69	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
70	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
71	C₃²×C₄	C₂²×C₈	1	6	yes	—
72	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
73	C₂×#41	C₃⋊Dic₃	1	1	yes	characteristic
74	#131	C₃⋊Dic₃	1	4	yes	—
75	#131	C₂×C₃⋊D₃	1	4	yes	—
76	C₂×M₁₆	C₃⋊Dic₃	1	3	yes	—
77	C₂²×C₈	C₂×C₃⋊D₃	1	3	yes	—
78	M₃₂	—	9	12	no	—
79	C₂×C₁₆	—	9	12	no	—
80	C₂×C₁₆	—	9	1	no	—
81	C₂×C₁₆	—	9	3	no	—
82	C₂³×C₃	C₂×C₃⋊C₈	1	12	yes	—
83	C₃×Q₈	C₂×C₃⋊C₈	1	16	yes	—
84	C₃×D₄	C₂×C₃⋊C₈	1	48	yes	—
85	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
86	C₂×C₃×C₄	C₂×C₃⋊C₈	1	24	yes	—
87	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
88	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
89	C₃×C₈	C₂²×Dic₃	1	4	yes	—
90	C₃×C₈	C₂²×Dic₃	1	24	yes	—
91	C₃×C₈	C₂³×D₃	1	4	yes	—
92	C₂×C₃²	—	2	6	no	—
93	C₂×C₃²	#504	1	2	yes	complemented
94	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
95	#41	C₃⋊(C₃⋊C₈)	1	4	yes	—
96	#41	C₂×C₃⋊Dic₃	1	4	yes	—
97	C₂×Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
98	C₂×D₄	C₃⋊(C₃⋊C₈)	1	3	yes	—
99	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
100	M₁₆	C₂×C₃⋊Dic₃	1	12	yes	—
101	C₂×C₈	C₂×C₃⋊Dic₃	1	3	yes	—
102	C₂×C₈	C₂×C₃⋊Dic₃	1	6	yes	—
103	C₂×C₈	C₂²×C₃⋊D₃	1	6	yes	—
104	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
105	C₁₆	—	9	2	no	—
106	C₁₆	—	9	6	no	—
107	C₂²×C₃	—	2	24	no	—
108	C₂²×C₃	C₂²×C₃⋊C₈	1	24	yes	—
109	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
110	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
111	C₃×C₄	C₂²×C₃⋊C₈	1	24	yes	—
112	C₃×C₄	C₂³×Dic₃	1	4	yes	—
113	C₃²	C₂×#504	1	1	yes	characteristic, complemented, Sylow
114	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
115	Q₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
116	D₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	—
117	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
118	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
119	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
121	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
122	C₈	C₂²×C₃⋊Dic₃	1	6	yes	—
123	C₈	C₂³×C₃⋊D₃	1	1	yes	characteristic
124	C₂×C₃	—	2	24	no	—
125	C₂×C₃	#5140	1	8	yes	complemented
126	C₂×C₃	C₂³×C₃⋊C₈	1	4	yes	—
127	C₂²	—	2	6	no	—
128	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
129	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
130	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
131	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
132	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
133	C₃	C₂×#5140	1	4	yes	complemented
134	C₂	—	2	6	no	—
135	C₂	#10592839	1	2	yes	direct product factor
136	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₈
Frattini	C₈	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃²×#41	C₄
cosocle	C₃²×C₈	C₂⁴
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	C₂×#504
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #10592839	C₂	direct (N × H)
3. C₂×#63639	C₂	semidirect
4. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
11. #63639	C₂²	semidirect
14. C₂×C₃⋊(C₃⋊C₁₆)	C₂²	semidirect
93. C₂×C₃²	#504	semidirect
113. C₃²	C₂×#504	semidirect
125. C₂×C₃	#5140	semidirect
133. C₃	C₂×#5140	semidirect
135. C₂	#10592839	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×#131	2
#10592839	2
C₂×#63639	2
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂×#5140	3

Subgroups — C2×#10592839 #11899381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10592839 #11899381