Subgroups — C₂²×#1372 #483333

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 127 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#335	C₂	1	2	yes	maximal
3	C₂×#1372	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊C₈	—	2	2	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₄	1	6	yes	—
9	C₂×C₄×C₁₆	C₂²	1	6	yes	—
10	C₂×#335	—	2	6	no	—
11	C₂×#335	C₂²	1	12	yes	—
12	#1372	C₂²	1	16	yes	direct product factor
13	C₂³×C₈	—	4	2	no	—
14	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₈	Q₈	1	1	yes	characteristic
16	C₂²×C₄²	D₄	1	1	yes	characteristic
17	C₂²×C₁₆	—	2	4	no	—
18	C₂²×C₁₆	C₈	1	8	yes	complemented
19	C₂²×C₁₆	C₂×C₄	1	6	yes	—
20	C₂²×C₁₆	C₂×C₄	1	6	yes	—
21	C₂×C₄⋊C₈	—	2	12	no	—
22	C₂×C₄⋊C₈	—	4	6	no	—
23	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
24	C₂×C₄×C₈	D₄	1	3	yes	—
25	C₂×C₄×C₈	Q₈	1	3	yes	—
26	C₂×C₄×C₈	C₂³	1	3	yes	—
27	C₄×C₁₆	C₂×C₄	1	12	yes	—
28	C₄×C₁₆	C₂³	1	4	yes	—
29	#335	—	2	12	no	—
30	#335	C₂³	1	8	yes	—
31	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
32	C₂³×C₄	D₈	1	1	yes	characteristic
33	C₂³×C₄	Q₁₆	1	1	yes	characteristic
34	C₂²×C₈	—	2	4	no	—
35	C₂²×C₈	—	4	12	no	—
36	C₂²×C₈	—	8	8	no	—
37	C₂²×C₈	C₄⋊C₄	1	3	yes	—
38	C₂²×C₈	C₄⋊C₄	1	3	yes	—
39	C₂²×C₈	C₂×C₈	1	4	yes	—
40	C₂²×C₈	M₁₆	1	4	yes	—
41	C₂²×C₈	C₂²×C₄	1	3	yes	—
42	C₂²×C₈	C₂×Q₈	1	3	yes	—
43	C₂×C₄²	C₄⋊C₄	1	3	yes	—
44	C₂×C₄²	D₈	1	3	yes	—
45	C₂×C₄²	Q₁₆	1	3	yes	—
46	C₂×C₄²	C₂×D₄	1	3	yes	—
47	C₂×C₁₆	—	2	12	no	—
48	C₂×C₁₆	C₂×C₈	1	24	yes	complemented
49	C₂×C₁₆	C₂²×C₄	1	4	yes	—
50	C₂×C₁₆	C₂²×C₄	1	4	yes	—
51	C₄⋊C₈	—	2	8	no	—
52	C₄⋊C₈	—	4	12	no	—
53	C₄×C₈	C₄⋊C₄	1	6	yes	—
54	C₄×C₈	C₂²×C₄	1	3	yes	—
55	C₄×C₈	C₂×D₄	1	3	yes	—
56	C₄×C₈	C₂×Q₈	1	3	yes	—
57	C₄×C₈	C₂⁴	1	1	yes	characteristic
58	C₂⁴	#107	1	1	yes	characteristic
59	C₂²×C₄	—	2	4	no	—
60	C₂²×C₄	C₄⋊C₈	1	4	yes	—
61	C₂²×C₄	#107	1	3	yes	—
62	C₂²×C₄	#107	1	3	yes	—
63	C₂²×C₄	#107	1	3	yes	—
64	C₂²×C₄	#108	1	4	yes	—
65	C₂²×C₄	SD₃₂	1	8	yes	—
66	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
67	C₂²×C₄	C₂×D₈	1	3	yes	—
68	C₂²×C₄	C₂×Q₁₆	1	3	yes	—
69	C₂×C₈	—	2	6	no	—
70	C₂×C₈	—	2	6	no	—
71	C₂×C₈	—	4	8	no	—
72	C₂×C₈	—	8	24	no	—
73	C₂×C₈	C₄⋊C₈	1	12	yes	—
74	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₈	C₂²×C₈	1	6	yes	—
77	C₂×C₈	C₂×M₁₆	1	6	yes	—
78	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
80	C₄²	#107	1	6	yes	—
81	C₄²	C₂×C₄⋊C₄	1	3	yes	—
82	C₄²	C₂×D₈	1	3	yes	—
83	C₄²	C₂×Q₁₆	1	3	yes	—
84	C₄²	C₂²×D₄	1	1	yes	characteristic
85	C₁₆	—	2	4	no	—
86	C₁₆	C₂²×C₈	1	8	yes	complemented
87	C₂³	#335	1	4	yes	—
88	C₂³	#367	1	4	yes	—
89	C₂³	#368	1	4	yes	—
90	C₂³	C₂×#107	1	3	yes	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	#335	1	12	yes	—
94	C₂×C₄	#367	1	12	yes	—
95	C₂×C₄	C₂×C₄⋊C₈	1	6	yes	—
96	C₂×C₄	C₂×#107	1	3	yes	—
97	C₂×C₄	C₂×#107	1	3	yes	—
98	C₂×C₄	C₂×#107	1	3	yes	—
99	C₂×C₄	C₂×#108	1	6	yes	—
100	C₂×C₄	C₂×SD₃₂	1	12	yes	—
101	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
103	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
104	C₈	—	2	3	no	—
105	C₈	—	2	1	no	—
106	C₈	—	8	8	no	—
107	C₈	C₂×C₄⋊C₈	1	6	yes	—
108	C₈	C₂³×C₈	1	1	yes	characteristic
109	C₈	C₂²×M₁₆	1	1	yes	characteristic
110	C₂²	#1372	1	16	yes	direct product factor
111	C₂²	C₂×#335	1	6	yes	—
112	C₂²	C₂×#367	1	6	yes	—
113	C₂²	C₂×#368	1	6	yes	—
114	C₂²	C₂²×#107	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	C₂×#335	1	6	yes	—
118	C₄	C₂×#367	1	6	yes	—
119	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
120	C₄	C₂²×#108	1	1	yes	characteristic
121	C₄	C₂²×SD₃₂	1	2	yes	—
122	C₂	C₂×#1372	1	12	yes	direct product factor
123	C₂	C₂²×#335	1	1	yes	characteristic
124	C₂	C₂²×#367	1	1	yes	characteristic
125	C₂	C₂²×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle, prime radical	C₂⁴	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#1372	C₂	direct (N × H)
12. #1372	C₂²	direct (N × H)
18. C₂²×C₁₆	C₈	semidirect
48. C₂×C₁₆	C₂×C₈	semidirect
86. C₁₆	C₂²×C₈	semidirect
110. C₂²	#1372	direct (N × H)
122. C₂	C₂×#1372	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₁₆	2
C₂²×#335	2
C₂×#1372	2

Subgroups — C22×#1372 #483333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1372 #483333