Subgroups — #11899411

Order 1152.

1080 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×#131	C₂	1	1	yes	maximal, characteristic
2	#10592839	C₂	1	8	yes	maximal, complemented
3	C₂×#63639	C₂	1	6	yes	maximal, complemented
4	#83780	—	3	4	no	maximal
5	C₂×C₃²×#41	C₄	1	1	yes	characteristic
6	C₃²×#131	C₄	1	4	yes	—
7	C₃²×#131	C₂²	1	4	yes	—
8	C₂×C₃²×M₁₆	C₄	1	3	yes	—
9	C₂²×C₃²×C₈	C₂²	1	3	yes	—
10	#63639	C₂²	1	4	yes	—
11	#63639	C₂²	1	12	yes	complemented
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	12	yes	complemented
13	C₂×C₃×#131	D₃	1	4	yes	—
14	#5140	—	3	32	no	—
15	C₂×#812	—	3	24	no	—
16	C₃²×#41	C₈	1	4	yes	—
17	C₃²×#41	C₂×C₄	1	4	yes	—
18	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
19	C₂×C₃²×D₄	C₈	1	3	yes	—
20	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
21	C₃²×M₁₆	C₂×C₄	1	12	yes	—
22	C₂×C₃²×C₈	C₂×C₄	1	3	yes	—
23	C₂×C₃²×C₈	C₂×C₄	1	6	yes	—
24	C₂×C₃²×C₈	C₂³	1	6	yes	—
25	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
26	C₃⋊(C₃⋊C₁₆)	C₂³	1	8	yes	—
27	#3407	—	9	1	no	Sylow
28	C₂×C₃×#41	Dic₃	1	4	yes	—
29	C₃×#131	Dic₃	1	16	yes	—
30	C₃×#131	C₂×D₃	1	16	yes	—
31	C₂×C₃×M₁₆	Dic₃	1	12	yes	—
32	C₂²×C₃×C₈	C₂×D₃	1	12	yes	—
33	#812	—	3	48	no	—
34	#812	—	3	16	no	—
35	C₂×C₃⋊C₁₆	—	3	48	no	—
36	C₂³×C₃²	C₂×C₈	1	3	yes	—
37	C₃²×Q₈	C₂×C₈	1	4	yes	—
38	C₃²×D₄	C₂×C₈	1	12	yes	—
39	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
40	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
41	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
43	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₃²×C₈	C₂²×C₄	1	6	yes	—
45	C₃²×C₈	C₂⁴	1	1	yes	characteristic
46	C₂×#131	C₃⋊D₃	1	1	yes	characteristic
47	#504	—	9	8	no	—
48	C₂×M₃₂	—	9	6	no	—
49	C₃×#41	C₃⋊C₈	1	16	yes	—
50	C₃×#41	C₂×Dic₃	1	16	yes	—
51	C₂×C₃×Q₈	C₃⋊C₈	1	4	yes	—
52	C₂×C₃×D₄	C₃⋊C₈	1	12	yes	—
53	C₂²×C₃×C₄	C₂×Dic₃	1	12	yes	—
54	C₃×M₁₆	C₂×Dic₃	1	48	yes	—
55	C₂×C₃×C₈	C₂×Dic₃	1	12	yes	—
56	C₂×C₃×C₈	C₂×Dic₃	1	24	yes	—
57	C₂×C₃×C₈	C₂²×D₃	1	24	yes	—
58	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
59	C₃⋊C₁₆	—	3	32	no	—
60	C₂²×C₃²	—	4	3	no	—
61	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
62	C₂²×C₃²	C₂²×C₈	1	6	yes	—
63	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
64	C₃²×C₄	C₂²×C₈	1	6	yes	—
65	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
66	C₂×#41	C₃⋊Dic₃	1	1	yes	characteristic
67	#131	C₃⋊Dic₃	1	4	yes	—
68	#131	C₂×C₃⋊D₃	1	4	yes	—
69	C₂×M₁₆	C₃⋊Dic₃	1	3	yes	—
70	C₂²×C₈	C₂×C₃⋊D₃	1	3	yes	—
71	M₃₂	—	9	4	no	—
72	M₃₂	—	9	12	no	—
73	C₂×C₁₆	—	9	12	no	—
74	C₂³×C₃	C₂×C₃⋊C₈	1	12	yes	—
75	C₃×Q₈	C₂×C₃⋊C₈	1	16	yes	—
76	C₃×D₄	C₂×C₃⋊C₈	1	48	yes	—
77	C₂×C₃×C₄	C₂×C₃⋊C₈	1	12	yes	—
78	C₂×C₃×C₄	C₂×C₃⋊C₈	1	24	yes	—
79	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
80	C₂×C₃×C₄	C₂²×Dic₃	1	24	yes	—
81	C₃×C₈	C₂²×Dic₃	1	4	yes	—
82	C₃×C₈	C₂²×Dic₃	1	24	yes	—
83	C₃×C₈	C₂³×D₃	1	4	yes	—
84	C₂×C₃²	—	2	1	no	—
85	C₂×C₃²	—	2	6	no	—
86	C₂×C₃²	C₂³×C₈	1	1	yes	characteristic
87	#41	C₃⋊(C₃⋊C₈)	1	4	yes	—
88	#41	C₂×C₃⋊Dic₃	1	4	yes	—
89	C₂×Q₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
90	C₂×D₄	C₃⋊(C₃⋊C₈)	1	3	yes	—
91	C₂²×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
92	M₁₆	C₂×C₃⋊Dic₃	1	12	yes	—
93	C₂×C₈	C₂×C₃⋊Dic₃	1	3	yes	—
94	C₂×C₈	C₂×C₃⋊Dic₃	1	6	yes	—
95	C₂×C₈	C₂²×C₃⋊D₃	1	6	yes	—
96	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
97	C₁₆	—	9	8	no	—
98	C₂²×C₃	—	4	12	no	—
99	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
100	C₂²×C₃	C₂²×C₃⋊C₈	1	24	yes	—
101	C₃×C₄	C₂²×C₃⋊C₈	1	4	yes	—
102	C₃×C₄	C₂²×C₃⋊C₈	1	24	yes	—
103	C₃×C₄	C₂³×Dic₃	1	4	yes	—
104	C₃²	#3407	1	1	yes	characteristic, complemented, Sylow
105	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
106	Q₈	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
107	D₄	C₂×C₃⋊(C₃⋊C₈)	1	12	yes	—
108	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	3	yes	—
109	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	—
110	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₃⋊Dic₃	1	6	yes	—
112	C₈	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
113	C₈	C₂²×C₃⋊Dic₃	1	6	yes	—
114	C₈	C₂³×C₃⋊D₃	1	1	yes	characteristic
115	C₂×C₃	—	2	4	no	—
116	C₂×C₃	—	2	24	no	—
117	C₂×C₃	C₂³×C₃⋊C₈	1	4	yes	—
118	C₂²	—	4	3	no	—
119	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
120	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
121	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
122	C₄	C₂²×C₃⋊(C₃⋊C₈)	1	6	yes	—
123	C₄	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
124	C₃	#83780	1	4	yes	complemented
125	C₂	—	2	1	no	—
126	C₂	—	2	6	no	—
127	C₂	C₂³×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×C₃⋊D₃
commutator, socle	C₂×C₃²	C₂³×C₈
Fitting	C₂×C₃²×#131	C₂
prime radical	C₂×C₃²×#41	C₄
cosocle	C₃²×C₈	C₂⁴
hypercenter	C₂×#131	C₃⋊D₃
hypocenter	C₃²	#3407
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #10592839	C₂	semidirect
3. C₂×#63639	C₂	semidirect
11. #63639	C₂²	semidirect
12. C₂×C₃⋊(C₃⋊C₁₆)	C₂²	semidirect
104. C₃²	#3407	semidirect
124. C₃	#83780	semidirect

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
#83780	3