Finite Group Databasebeta

Subgroups — #513099

Order 512.

395 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 110 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	#33607	C₂	1	2	yes	maximal, complemented
2	#13713	C₂	1	4	yes	maximal, complemented
3	#13697	C₂	1	2	yes	maximal, complemented
4	#13694	C₂	1	2	yes	maximal, complemented
5	#13661	C₂	1	4	yes	maximal, complemented
6	#13554	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆	C₂²	1	2	yes	complemented
8	C₂×#504	C₂²	1	1	yes	characteristic
9	#3058	—	2	2	no	—
10	C₂×D₃₂	C₂²	1	2	yes	—
11	C₂×D₃₂	C₂²	1	2	yes	—
12	C₂×M₆₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃₂	C₂²	1	1	yes	characteristic, complemented
14	#2218	—	2	2	no	—
15	#2218	—	2	2	no	—
16	#2215	C₂²	1	4	yes	—
17	#2215	C₂²	1	4	yes	complemented
18	#2158	C₂²	1	2	yes	complemented
19	#2140	—	2	2	no	—
20	#1424	C₂²	1	2	yes	complemented
21	#1423	C₂²	1	2	yes	complemented
22	#1415	C₂²	1	4	yes	complemented
23	#1415	C₂²	1	4	yes	complemented
24	C₂²⋊C₃₂	C₂²	1	4	yes	complemented
25	C₂²×D₈	—	2	2	no	—
26	C₂×#131	D₄	1	1	yes	characteristic
27	#535	—	4	2	no	—
28	C₂×D₁₆	—	2	2	no	—
29	C₂×D₁₆	—	4	2	no	—
30	C₂×D₁₆	—	4	2	no	—
31	C₂×D₁₆	C₂³	1	4	yes	—
32	#504	—	4	1	no	—
33	#504	—	4	1	no	—
34	C₂×M₃₂	D₄	1	2	yes	—
35	C₂×M₃₂	D₄	1	1	yes	characteristic
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₂²×C₁₆	C₂³	1	1	yes	characteristic
38	#469	—	4	2	no	—
39	#469	—	4	2	no	—
40	#466	—	2	4	no	—
41	#466	—	2	4	no	—
42	#427	—	2	2	no	—
43	#418	—	4	2	no	—
44	D₃₂	—	4	2	no	—
45	D₃₂	—	4	2	no	—
46	M₆₄	—	4	1	no	—
47	C₂×C₃₂	—	4	1	no	—
48	C₂×C₃₂	C₂³	1	2	yes	—
49	C₂×C₃₂	C₂³	1	2	yes	—
50	#367	—	2	2	no	—
51	#366	C₂³	1	4	yes	—
52	#357	—	2	4	no	—
53	#357	—	2	4	no	—
54	C₂×#41	D₈	1	1	yes	characteristic
55	C₂²×D₄	—	4	2	no	—
56	C₂×D₈	—	2	4	no	—
57	C₂×D₈	—	4	4	no	—
58	C₂×D₈	—	4	4	no	—
59	C₂×D₈	—	4	2	no	—
60	#131	—	4	1	no	—
61	#131	—	4	1	no	—
62	C₂×M₁₆	D₈	1	2	yes	—
63	C₂×M₁₆	D₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	2	yes	—
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	#127	—	8	2	no	—
67	#124	—	8	2	no	—
68	#121	—	4	4	no	—
69	#121	—	4	4	no	—
70	#120	—	8	2	no	—
71	#120	—	8	2	no	—
72	#117	—	4	2	no	—
73	D₁₆	—	2	4	no	—
74	D₁₆	—	2	4	no	—
75	D₁₆	—	4	4	no	—
76	M₃₂	—	4	2	no	—
77	M₃₂	—	4	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂×D₄	1	2	yes	—
84	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
85	#107	—	2	4	no	—
86	#106	—	4	2	no	—
87	#102	—	4	4	no	—
88	#102	—	4	4	no	—
89	C₃₂	—	2	2	no	—
90	C₃₂	—	2	2	no	—
91	C₂⁴	—	8	2	no	—
92	#41	—	4	1	no	—
93	#41	—	4	1	no	—
94	C₂×Q₈	D₁₆	1	1	yes	characteristic
95	C₂×D₄	—	4	4	no	—
96	C₂×D₄	—	8	4	no	—
97	C₂×D₄	—	8	4	no	—
98	C₂×D₄	—	8	4	no	—
99	C₂×D₄	—	8	4	no	—
100	C₂×D₄	—	8	2	no	—
101	C₂×D₄	D₁₆	1	1	yes	characteristic
102	C₂×D₄	D₁₆	1	2	yes	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	D₈	—	4	4	no	—
106	D₈	—	4	4	no	—
107	D₈	—	8	4	no	—
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂×D₈	1	1	yes	characteristic
115	C₂×C₈	C₂×D₈	1	2	yes	—
116	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
117	C₄⋊C₄	—	4	4	no	—
118	C₂²⋊C₄	—	8	2	no	—
119	C₂²⋊C₄	—	8	4	no	—
120	C₂²⋊C₄	—	8	4	no	—
121	C₄²	—	8	2	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
125	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂³	—	8	4	no	—
127	C₂³	—	16	4	no	—
128	C₂³	—	32	4	no	—
129	C₂³	C₂×D₁₆	1	2	yes	—
130	C₂³	C₂×D₁₆	1	1	yes	characteristic
131	Q₈	—	4	1	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	1	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	D₄	—	16	4	no	—
137	D₄	—	32	4	no	—
138	D₄	—	32	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	C₂×D₁₆	1	2	yes	—
145	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	2	no	—
149	C₈	#534	1	1	yes	characteristic
150	C₈	#575	1	1	yes	characteristic
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	16	4	no	—
156	C₂²	—	16	4	no	—
157	C₂²	—	16	4	no	—
158	C₂²	—	32	8	no	—
159	C₂²	C₂²×D₁₆	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	16	4	no	—
163	C₄	#3186	1	1	yes	characteristic
164	C₄	#3414	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	4	2	no	—
167	C₂	—	32	4	no	—
168	C₂	#33752	1	1	yes	characteristic
169	C₂	#33982	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ 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
1. #33607	C₂	semidirect
2. #13713	C₂	semidirect
3. #13697	C₂	semidirect
4. #13694	C₂	semidirect
5. #13661	C₂	semidirect
6. #13554	C₂	semidirect
7. C₂²×D₁₆	C₂²	semidirect
12. C₂×M₆₄	C₂²	semidirect
13. C₂²×C₃₂	C₂²	semidirect
17. #2215	C₂²	semidirect
18. #2158	C₂²	semidirect
20. #1424	C₂²	semidirect
21. #1423	C₂²	semidirect
22. #1415	C₂²	semidirect
23. #1415	C₂²	semidirect
24. C₂²⋊C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33607	2
#13713	2
#13697	2
#13694	2
#13661	2
#13554	2