Subgroups — #483341

Order 512.

518 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#504	C₂	1	1	yes	maximal, characteristic
2	#17789	C₂	1	2	yes	maximal
3	#12009	C₂	1	6	yes	maximal, complemented
4	#12008	C₂	1	3	yes	maximal, complemented
5	C₂×#1372	C₂	1	3	yes	maximal, complemented
6	C₂×#504	C₄	1	1	yes	characteristic
7	C₂×#504	C₄	1	1	yes	characteristic
8	#2988	—	2	2	no	—
9	C₄×#131	C₂²	1	1	yes	characteristic
10	#2108	C₄	1	3	yes	—
11	#2108	C₄	1	3	yes	—
12	C₄×M₃₂	C₂²	1	3	yes	—
13	C₂×C₄×C₁₆	C₂²	1	3	yes	—
14	#1565	—	2	6	no	—
15	#1564	C₂²	1	6	yes	—
16	C₂×#335	C₂²	1	6	yes	—
17	#1372	C₂²	1	3	yes	—
18	#1372	C₂²	1	6	yes	—
19	#1372	C₂²	1	1	yes	characteristic
20	#1371	C₂²	1	6	yes	—
21	C₂×#131	C₂×C₄	1	1	yes	characteristic
22	C₂×#131	Q₈	1	1	yes	characteristic
23	C₄×#41	D₄	1	1	yes	characteristic
24	#504	—	2	4	no	—
25	#504	C₈	1	8	yes	complemented
26	C₂×M₃₂	C₂×C₄	1	3	yes	—
27	C₂×M₃₂	C₂×C₄	1	3	yes	—
28	C₂²×C₁₆	C₂×C₄	1	3	yes	—
29	C₂²×C₁₆	C₂×C₄	1	3	yes	—
30	C₈×Q₈	—	4	2	no	—
31	C₈×D₄	—	4	6	no	—
32	#431	—	2	6	no	—
33	C₂×C₄⋊C₈	—	2	6	no	—
34	#405	C₂×C₄	1	3	yes	—
35	#405	Q₈	1	3	yes	—
36	C₄×M₁₆	D₄	1	3	yes	—
37	C₂×C₄×C₈	C₂³	1	3	yes	—
38	#346	C₂×C₄	1	6	yes	—
39	C₄×C₁₆	C₂×C₄	1	3	yes	—
40	C₄×C₁₆	C₂×C₄	1	3	yes	—
41	C₄×C₁₆	C₂³	1	1	yes	characteristic
42	C₄×C₁₆	C₂³	1	3	yes	—
43	#335	—	2	6	no	—
44	#335	C₂³	1	6	yes	—
45	#335	C₂³	1	2	yes	—
46	#334	—	2	6	no	—
47	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
48	#131	—	2	4	no	—
49	#131	C₂×C₈	1	4	yes	—
50	#131	M₁₆	1	4	yes	—
51	C₂×M₁₆	C₄⋊C₄	1	3	yes	—
52	C₂×M₁₆	C₄⋊C₄	1	3	yes	—
53	C₂²×C₈	—	4	6	no	—
54	C₂²×C₈	C₂²×C₄	1	3	yes	—
55	C₂²×C₈	C₂×Q₈	1	3	yes	—
56	C₄×Q₈	D₈	1	1	yes	characteristic
57	C₄×Q₈	Q₁₆	1	1	yes	characteristic
58	C₄×D₄	D₈	1	3	yes	—
59	C₄×D₄	Q₁₆	1	3	yes	—
60	#117	C₄⋊C₄	1	3	yes	—
61	C₂×C₄²	C₂×D₄	1	3	yes	—
62	M₃₂	—	2	6	no	—
63	M₃₂	C₂×C₈	1	12	yes	complemented
64	C₂×C₁₆	—	2	6	no	—
65	C₂×C₁₆	C₂×C₈	1	12	yes	complemented
66	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₁₆	C₂²×C₄	1	3	yes	—
68	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂²×C₄	1	3	yes	—
70	C₄⋊C₈	—	2	2	no	—
71	C₄⋊C₈	—	2	6	no	—
72	C₄⋊C₈	—	4	6	no	—
73	C₂²⋊C₈	—	4	6	no	—
74	#97	C₄⋊C₄	1	6	yes	—
75	C₄×C₈	—	4	6	no	—
76	C₄×C₈	C₂²×C₄	1	3	yes	—
77	C₄×C₈	C₂×D₄	1	3	yes	—
78	C₄×C₈	C₂×Q₈	1	3	yes	—
79	C₄×C₈	C₂⁴	1	1	yes	characteristic
80	#41	C₄⋊C₈	1	4	yes	—
81	#41	#108	1	4	yes	—
82	C₂×Q₈	#107	1	1	yes	characteristic
83	C₂×D₄	#107	1	3	yes	—
84	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
85	C₂²×C₄	C₂×D₈	1	3	yes	—
86	C₂²×C₄	C₂×Q₁₆	1	3	yes	—
87	M₁₆	—	2	6	no	—
88	M₁₆	C₄⋊C₈	1	12	yes	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	4	6	no	—
92	C₂×C₈	—	16	6	no	—
93	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
94	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×C₈	C₂²×C₈	1	6	yes	—
96	C₂×C₈	C₂×M₁₆	1	6	yes	—
97	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
99	C₄⋊C₄	#107	1	3	yes	—
100	C₄⋊C₄	#107	1	3	yes	—
101	C₂²⋊C₄	#107	1	6	yes	—
102	C₄²	C₂×C₄⋊C₄	1	3	yes	—
103	C₄²	C₂×D₈	1	3	yes	—
104	C₄²	C₂×Q₁₆	1	3	yes	—
105	C₄²	C₂²×D₄	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	2	3	no	—
108	C₁₆	C₂²×C₈	1	2	yes	—
109	C₁₆	C₂²×C₈	1	6	yes	—
110	C₂³	C₂×#107	1	3	yes	—
111	Q₈	#335	1	4	yes	—
112	D₄	#335	1	12	yes	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	C₂×C₄⋊C₈	1	6	yes	—
116	C₂×C₄	C₂×#107	1	3	yes	—
117	C₂×C₄	C₂×#107	1	3	yes	—
118	C₂×C₄	C₂×#107	1	3	yes	—
119	C₂×C₄	C₂×#108	1	6	yes	—
120	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
122	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	3	no	—
125	C₈	—	8	2	no	—
126	C₈	—	8	6	no	—
127	C₈	C₂×C₄⋊C₈	1	6	yes	—
128	C₈	C₂³×C₈	1	1	yes	characteristic
129	C₈	C₂²×M₁₆	1	1	yes	characteristic
130	C₂²	—	2	3	no	—
131	C₂²	—	2	3	no	—
132	C₂²	C₂×#335	1	6	yes	—
133	C₂²	C₂²×#107	1	1	yes	characteristic
134	C₄	—	2	3	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	3	no	—
137	C₄	C₂×#335	1	6	yes	—
138	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
139	C₄	C₂²×#108	1	1	yes	characteristic
140	C₄	#3416	1	2	yes	—
141	C₂	—	2	6	no	—
142	C₂	C₂²×#335	1	1	yes	characteristic
143	C₂	#33411	1	1	yes	characteristic
144	C₂	#33414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ 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. #12009	C₂	semidirect
4. #12008	C₂	semidirect
5. C₂×#1372	C₂	semidirect
25. #504	C₈	semidirect
63. M₃₂	C₂×C₈	semidirect
65. C₂×C₁₆	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#504	2
#17789	2
#12009	2
#12008	2
C₂×#1372	2