Finite Group Databasebeta

Subgroups — #512769

Order 512.

432 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2262	C₂	1	1	yes	maximal, characteristic
2	#33459	C₂	1	2	yes	maximal, complemented
3	#13652	C₂	1	4	yes	maximal, complemented
4	C₄×Q₆₄	C₂	1	2	yes	maximal, complemented
5	C₄×D₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×#1423	C₂	1	1	yes	maximal, characteristic, complemented
7	#13616	C₂	1	2	yes	maximal, complemented
8	#13549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₂²	1	2	yes	—
10	#2951	—	2	2	no	—
11	#2262	C₄	1	8	yes	complemented
12	C₂×Q₆₄	C₂²	1	1	yes	characteristic
13	C₂×SD₆₄	C₂²	1	2	yes	—
14	C₂×D₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₆₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃₂	C₂²	1	1	yes	characteristic
17	#2175	—	2	4	no	—
18	C₄×Q₃₂	C₂²	1	4	yes	—
19	C₄×D₁₆	C₂²	1	4	yes	—
20	C₂×#366	C₂²	1	2	yes	—
21	#2139	—	2	2	no	—
22	#2108	C₂²	1	1	yes	characteristic
23	#1423	C₂²	1	2	yes	—
24	#1423	C₂²	1	2	yes	—
25	#1416	C₂²	1	4	yes	—
26	#1415	C₂²	1	4	yes	—
27	#1397	C₂²	1	2	yes	—
28	C₄×C₃₂	C₂²	1	2	yes	—
29	C₂×#135	—	2	2	no	—
30	#520	—	4	2	no	—
31	#508	—	2	4	no	—
32	#508	C₂×C₄	1	8	yes	complemented
33	C₂×Q₃₂	C₂³	1	2	yes	—
34	C₂×SD₃₂	—	2	2	no	—
35	C₂×D₁₆	C₂³	1	2	yes	—
36	C₂×M₃₂	D₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂³	1	1	yes	characteristic
38	#440	—	4	4	no	—
39	C₄×Q₁₆	—	2	4	no	—
40	C₄×D₈	—	2	4	no	—
41	C₂×#107	—	2	2	no	—
42	#417	—	4	2	no	—
43	#405	D₄	1	1	yes	characteristic
44	Q₆₄	C₂×C₄	1	4	yes	complemented
45	SD₆₄	C₂×C₄	1	8	yes	complemented
46	D₃₂	C₂×C₄	1	4	yes	complemented
47	M₆₄	—	2	2	no	—
48	C₂×C₃₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₃₂	C₂³	1	2	yes	—
50	C₂×C₃₂	C₂³	1	1	yes	characteristic
51	C₂×C₃₂	C₂³	1	1	yes	characteristic
52	#366	—	2	2	no	—
53	#366	C₂³	1	4	yes	—
54	#358	—	2	4	no	—
55	#357	—	2	4	no	—
56	#346	D₄	1	2	yes	—
57	C₄×C₁₆	C₂³	1	2	yes	—
58	C₂×#41	—	4	2	no	—
59	#135	—	2	8	no	—
60	#135	—	4	4	no	—
61	C₂×Q₁₆	—	2	2	no	—
62	C₂×SD₁₆	—	4	2	no	—
63	C₂×D₈	—	2	2	no	—
64	C₂×M₁₆	D₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄×Q₈	—	4	4	no	—
67	C₄×D₄	—	4	4	no	—
68	C₄×D₄	—	8	4	no	—
69	#117	—	8	2	no	—
70	#117	D₈	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄⋊C₄	—	8	2	no	—
73	Q₃₂	—	2	2	no	—
74	Q₃₂	C₂²×C₄	1	4	yes	—
75	SD₃₂	—	2	8	no	—
76	D₁₆	—	2	2	no	—
77	D₁₆	C₂²×C₄	1	4	yes	—
78	M₃₂	—	2	2	no	—
79	C₂×C₁₆	C₂²×C₄	1	2	yes	—
80	C₂×C₁₆	C₂×D₄	1	2	yes	—
81	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₆	#41	1	2	yes	—
83	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
84	#107	—	2	4	no	—
85	#107	—	4	2	no	—
86	#103	—	4	4	no	—
87	#102	—	4	4	no	—
88	#97	D₈	1	2	yes	—
89	C₄×C₈	C₂×D₄	1	2	yes	—
90	C₃₂	—	2	2	no	—
91	C₃₂	C₂²×C₄	1	2	yes	—
92	C₃₂	C₂²×C₄	1	2	yes	—
93	#41	—	4	8	no	—
94	#41	—	8	4	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	8	2	no	—
98	C₂²×C₄	—	8	2	no	—
99	C₂²×C₄	—	8	2	no	—
100	C₂²×C₄	—	8	4	no	—
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	Q₁₆	—	2	4	no	—
103	Q₁₆	—	4	2	no	—
104	SD₁₆	—	4	8	no	—
105	D₈	—	2	4	no	—
106	D₈	—	4	2	no	—
107	M₁₆	—	2	2	no	—
108	C₂×C₈	C₄×D₄	1	2	yes	—
109	C₂×C₈	C₂×D₈	1	2	yes	—
110	C₂×C₈	C₂×D₈	1	1	yes	characteristic
111	C₂×C₈	#135	1	2	yes	—
112	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₄⋊C₄	D₁₆	1	1	yes	characteristic
117	C₄⋊C₄	D₁₆	1	1	yes	characteristic
118	C₂²⋊C₄	—	8	4	no	—
119	C₂²⋊C₄	D₁₆	1	2	yes	—
120	C₄²	—	8	4	no	—
121	C₄²	C₂×D₈	1	2	yes	—
122	C₁₆	—	2	2	no	—
123	C₁₆	C₄×D₄	1	2	yes	—
124	C₁₆	C₂³×C₄	1	1	yes	characteristic
125	C₁₆	C₂×#41	1	1	yes	characteristic
126	C₂³	—	8	2	no	—
127	C₂³	C₂×D₁₆	1	1	yes	characteristic
128	Q₈	—	4	4	no	—
129	Q₈	—	8	2	no	—
130	D₄	—	4	4	no	—
131	D₄	—	8	8	no	—
132	D₄	—	8	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	8	8	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	32	4	no	—
137	C₂×C₄	—	32	2	no	—
138	C₂×C₄	C₄×D₈	1	2	yes	—
139	C₂×C₄	C₂×D₁₆	1	2	yes	—
140	C₂×C₄	C₂×D₁₆	1	2	yes	—
141	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
142	C₂×C₄	#508	1	2	yes	—
143	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	C₄×D₈	1	2	yes	—
146	C₈	C₂×C₄×D₄	1	1	yes	characteristic
147	C₈	C₂×#135	1	1	yes	characteristic
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	16	2	no	—
152	C₂²	—	16	2	no	—
153	C₂²	C₄×D₁₆	1	2	yes	—
154	C₂²	C₂²×D₁₆	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	8	4	no	—
158	C₄	—	16	4	no	—
159	C₄	C₄×D₁₆	1	2	yes	—
160	C₄	C₂×C₄×D₈	1	1	yes	characteristic
161	C₄	C₂×#508	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	16	4	no	—
164	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic
165	C₂	#33982	1	1	yes	characteristic
166	C₂	#33984	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #33459	C₂	semidirect
3. #13652	C₂	semidirect
4. C₄×Q₆₄	C₂	semidirect
5. C₄×D₃₂	C₂	semidirect
6. C₂×#1423	C₂	semidirect
7. #13616	C₂	semidirect
8. #13549	C₂	semidirect
11. #2262	C₄	semidirect
32. #508	C₂×C₄	semidirect
44. Q₆₄	C₂×C₄	semidirect
45. SD₆₄	C₂×C₄	semidirect
46. D₃₂	C₂×C₄	semidirect
48. C₂×C₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2262	2
#33459	2
#13652	2
C₄×Q₆₄	2
C₄×D₃₂	2
C₂×#1423	2
#13616	2
#13549	2