Finite Group Databasebeta

Subgroups — #512773

Order 512.

421 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 176 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	#33981	C₂	1	1	yes	maximal, characteristic
2	C₂×#2178	C₂	1	2	yes	maximal, complemented
3	#13656	C₂	1	4	yes	maximal, complemented
4	#13655	C₂	1	4	yes	maximal, complemented
5	#13636	C₂	1	1	yes	maximal, characteristic, complemented
6	#13620	C₂	1	2	yes	maximal, complemented
7	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	C₂²	1	2	yes	—
9	C₂×#443	—	2	2	no	—
10	#2264	C₄	1	4	yes	complemented
11	#2263	C₄	1	4	yes	complemented
12	#2262	C₂²	1	4	yes	—
13	C₂×M₆₄	C₂²	1	1	yes	characteristic
14	C₂×M₆₄	C₂²	1	1	yes	characteristic
15	#2178	—	2	2	no	—
16	#2178	—	2	2	no	—
17	#2178	C₂²	1	4	yes	complemented
18	#2178	C₂²	1	4	yes	complemented
19	C₂×#368	C₂²	1	2	yes	—
20	C₂×#359	—	2	2	no	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#1425	C₂²	1	4	yes	complemented
23	#1417	C₂²	1	8	yes	complemented
24	#1397	C₂²	1	2	yes	complemented
25	C₄×C₃₂	C₂²	1	2	yes	complemented
26	C₂×#135	—	2	2	no	—
27	C₂×#131	—	4	2	no	—
28	#508	—	2	2	no	—
29	#508	—	2	2	no	—
30	#508	C₂×C₄	1	4	yes	—
31	#508	C₂³	1	4	yes	—
32	C₂×Q₃₂	C₂×C₄	1	2	yes	—
33	C₂×SD₃₂	—	2	2	no	—
34	C₂×D₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#443	—	2	4	no	—
38	#443	—	2	4	no	—
39	#443	—	4	2	no	—
40	#443	—	4	2	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#104	—	4	2	no	—
43	C₂×C₄×C₈	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	M₆₄	C₂×C₄	1	4	yes	—
49	C₂×C₃₂	C₂³	1	2	yes	—
50	C₂×C₃₂	C₂³	1	2	yes	—
51	#368	—	2	2	no	—
52	#368	C₂³	1	4	yes	—
53	#359	—	2	4	no	—
54	#359	—	2	4	no	—
55	C₄×C₁₆	D₄	1	1	yes	characteristic
56	C₄×C₁₆	D₄	1	1	yes	characteristic
57	C₄×C₁₆	C₂³	1	1	yes	characteristic
58	C₄×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×#41	—	4	2	no	—
60	#135	—	2	4	no	—
61	#135	—	2	4	no	—
62	#135	—	4	2	no	—
63	#135	—	4	2	no	—
64	C₂×Q₁₆	—	2	2	no	—
65	C₂×SD₁₆	—	4	2	no	—
66	C₂×D₈	—	2	2	no	—
67	#131	—	4	4	no	—
68	#131	—	4	4	no	—
69	#131	—	8	2	no	—
70	#131	—	8	2	no	—
71	C₂×M₁₆	—	4	2	no	—
72	C₂×M₁₆	—	8	2	no	—
73	C₂²×C₈	—	8	2	no	—
74	C₂²×C₈	D₈	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	D₈	1	1	yes	characteristic
77	Q₃₂	—	2	2	no	—
78	Q₃₂	C₂²×C₄	1	4	yes	—
79	SD₃₂	—	2	4	no	—
80	SD₃₂	—	2	4	no	—
81	D₁₆	—	2	2	no	—
82	D₁₆	C₂²×C₄	1	4	yes	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	C₂²×C₄	1	2	yes	—
85	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
86	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
87	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₆	#41	1	2	yes	—
89	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
90	#108	—	2	4	no	—
91	#108	—	4	2	no	—
92	#104	—	4	4	no	—
93	#104	—	4	4	no	—
94	C₄×C₈	D₈	1	1	yes	characteristic
95	C₄×C₈	D₈	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂×D₄	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	C₂²×C₄	1	4	yes	—
100	#41	—	4	4	no	—
101	#41	—	4	4	no	—
102	#41	—	8	2	no	—
103	#41	—	8	2	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	8	2	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	8	2	no	—
109	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	4	2	no	—
112	SD₁₆	—	4	4	no	—
113	SD₁₆	—	4	4	no	—
114	D₈	—	2	4	no	—
115	D₈	—	4	2	no	—
116	M₁₆	—	4	4	no	—
117	M₁₆	—	8	2	no	—
118	M₁₆	—	8	4	no	—
119	M₁₆	—	8	4	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	—	8	4	no	—
122	C₂×C₈	—	8	4	no	—
123	C₂×C₈	D₁₆	1	2	yes	—
124	C₂×C₈	D₁₆	1	2	yes	—
125	C₂×C₈	C₄×D₄	1	2	yes	—
126	C₂×C₈	C₂×D₈	1	1	yes	characteristic
127	C₂×C₈	C₂×D₈	1	1	yes	characteristic
128	C₂×C₈	C₂×D₈	1	1	yes	characteristic
129	C₂×C₈	#135	1	2	yes	—
130	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
131	C₄²	—	2	1	no	—
132	C₄²	C₂×D₈	1	1	yes	characteristic
133	C₄²	C₂×D₈	1	1	yes	characteristic
134	C₁₆	—	2	1	no	—
135	C₁₆	—	2	1	no	—
136	C₁₆	C₄×D₄	1	2	yes	—
137	C₁₆	C₂³×C₄	1	1	yes	characteristic
138	C₁₆	C₂×#41	1	1	yes	characteristic
139	C₂³	—	8	2	no	—
140	C₂³	#508	1	1	yes	characteristic
141	Q₈	—	4	4	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	4	4	no	—
144	D₄	—	8	4	no	—
145	D₄	—	8	4	no	—
146	D₄	—	8	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₄×D₈	1	2	yes	—
154	C₂×C₄	C₂×D₁₆	1	2	yes	—
155	C₂×C₄	#508	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
157	C₈	—	8	4	no	—
158	C₈	C₄×D₈	1	2	yes	—
159	C₈	C₂×D₁₆	1	2	yes	—
160	C₈	C₂×D₁₆	1	1	yes	characteristic
161	C₈	C₂×D₁₆	1	1	yes	characteristic
162	C₈	C₂×C₄×D₄	1	1	yes	characteristic
163	C₈	C₂×#135	1	1	yes	characteristic
164	C₂²	—	4	1	no	—
165	C₂²	—	8	4	no	—
166	C₂²	—	16	4	no	—
167	C₂²	C₄×D₁₆	1	2	yes	—
168	C₂²	C₂×#508	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	1	no	—
171	C₄	—	8	4	no	—
172	C₄	C₄×D₁₆	1	2	yes	—
173	C₄	C₂×C₄×D₈	1	1	yes	characteristic
174	C₄	C₂²×D₁₆	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	2	no	—
177	C₂	—	16	4	no	—
178	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×C₄×D₁₆
prime radical	#33981	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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. C₂×#2178	C₂	semidirect
3. #13656	C₂	semidirect
4. #13655	C₂	semidirect
5. #13636	C₂	semidirect
6. #13620	C₂	semidirect
7. C₄×M₆₄	C₂	semidirect
10. #2264	C₄	semidirect
11. #2263	C₄	semidirect
17. #2178	C₂²	semidirect
18. #2178	C₂²	semidirect
22. #1425	C₂²	semidirect
23. #1417	C₂²	semidirect
24. #1397	C₂²	semidirect
25. C₄×C₃₂	C₂²	semidirect
44. Q₆₄	C₂×C₄	semidirect
45. SD₆₄	C₂×C₄	semidirect
46. D₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#33981	2
C₂×#2178	2
#13656	2
#13655	2
#13636	2
#13620	2
C₄×M₆₄	2