Finite Group Databasebeta

Subgroups — #512692

Order 512.

761 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 274 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×#3417	C₂	1	1	yes	maximal, characteristic
2	C₂×#2258	C₂	1	1	yes	maximal, characteristic, complemented
3	#33410	C₂	1	1	yes	maximal, characteristic, complemented
4	#13618	C₂	1	3	yes	maximal, complemented
5	#13616	C₂	1	3	yes	maximal, complemented
6	#13615	C₂	1	3	yes	maximal, complemented
7	C₂×#1416	C₂	1	3	yes	maximal, complemented
8	C₂×#578	—	2	1	no	—
9	#3417	—	2	4	no	—
10	#3417	C₄	1	8	yes	complemented
11	C₂×#510	—	2	3	no	—
12	C₂×#508	C₂²	1	3	yes	—
13	C₂²×Q₃₂	C₂²	1	3	yes	—
14	C₂×#504	C₂²	1	1	yes	characteristic
15	#2913	—	2	1	no	—
16	#2258	—	2	4	no	—
17	C₂×M₆₄	C₂²	1	3	yes	—
18	C₂²×C₃₂	C₂²	1	3	yes	—
19	#2159	—	2	3	no	—
20	#2158	C₂²	1	3	yes	—
21	C₂×#366	C₂²	1	3	yes	—
22	#1416	C₂²	1	3	yes	—
23	#1416	C₂²	1	6	yes	—
24	#1416	C₂²	1	3	yes	—
25	#1415	C₂²	1	1	yes	characteristic
26	#1415	C₂²	1	3	yes	—
27	C₂×2₋¹⁺⁴	—	4	1	no	—
28	#578	—	2	8	no	—
29	#578	—	4	4	no	—
30	C₂×#137	—	4	3	no	—
31	C₂×#135	—	2	3	no	—
32	C₂²×Q₁₆	—	2	3	no	—
33	C₂×#131	D₄	1	1	yes	characteristic
34	#520	—	4	1	no	—
35	#510	—	2	24	no	—
36	#508	—	2	6	no	—
37	#508	C₂×C₄	1	12	yes	complemented
38	C₂×Q₃₂	—	2	3	no	—
39	C₂×Q₃₂	—	2	6	no	—
40	C₂×Q₃₂	C₂×C₄	1	12	yes	complemented
41	C₂×Q₃₂	C₂³	1	3	yes	—
42	C₂×SD₃₂	—	2	3	no	—
43	C₂×D₁₆	C₂³	1	1	yes	characteristic
44	#504	C₂×C₄	1	4	yes	—
45	#504	D₄	1	4	yes	—
46	C₂×M₃₂	D₄	1	3	yes	—
47	C₂²×C₁₆	C₂³	1	3	yes	—
48	#428	—	4	3	no	—
49	#427	—	2	3	no	—
50	C₂×#107	—	2	3	no	—
51	M₆₄	—	2	6	no	—
52	C₂×C₃₂	—	2	6	no	—
53	C₂×C₃₂	C₂³	1	1	yes	characteristic
54	C₂×C₃₂	C₂³	1	3	yes	—
55	#367	—	2	3	no	—
56	#366	—	2	3	no	—
57	#366	C₂³	1	3	yes	—
58	#366	C₂³	1	1	yes	characteristic
59	2₋¹⁺⁴	—	4	8	no	—
60	2₋¹⁺⁴	—	8	4	no	—
61	C₂×#41	—	4	3	no	—
62	C₂×#41	—	8	3	no	—
63	C₂×#41	D₈	1	1	yes	characteristic
64	C₂²×Q₈	—	4	3	no	—
65	C₂²×Q₈	—	8	1	no	—
66	#137	—	4	24	no	—
67	#135	—	2	12	no	—
68	#135	—	4	6	no	—
69	C₂×Q₁₆	—	2	12	no	—
70	C₂×Q₁₆	—	2	3	no	—
71	C₂×Q₁₆	—	4	3	no	—
72	C₂×Q₁₆	—	4	6	no	—
73	C₂×SD₁₆	—	4	3	no	—
74	C₂×D₈	—	2	1	no	—
75	#131	C₂²⋊C₄	1	4	yes	—
76	#131	SD₁₆	1	4	yes	—
77	C₂×M₁₆	D₈	1	3	yes	—
78	C₂²×C₈	C₂×D₄	1	3	yes	—
79	C₄×Q₈	—	8	1	no	—
80	C₄×D₄	—	8	3	no	—
81	#117	—	4	3	no	—
82	C₂×C₄⋊C₄	—	4	3	no	—
83	Q₃₂	—	2	12	no	—
84	Q₃₂	—	2	3	no	—
85	Q₃₂	C₂²×C₄	1	6	yes	—
86	SD₃₂	—	2	12	no	—
87	D₁₆	—	2	1	no	—
88	D₁₆	C₂²×C₄	1	2	yes	—
89	M₃₂	C₂²⋊C₄	1	12	yes	—
90	C₂×C₁₆	C₂²×C₄	1	6	yes	—
91	C₂×C₁₆	C₂×D₄	1	3	yes	—
92	C₂×C₁₆	C₂×D₄	1	6	yes	—
93	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
94	#107	—	2	3	no	—
95	#107	—	2	1	no	—
96	#107	—	4	3	no	—
97	#106	—	4	3	no	—
98	C₃₂	—	2	1	no	—
99	C₃₂	—	2	3	no	—
100	#41	—	4	12	no	—
101	#41	—	8	24	no	—
102	#41	—	8	6	no	—
103	#41	#102	1	4	yes	—
104	#41	SD₃₂	1	4	yes	—
105	C₂×Q₈	—	4	3	no	—
106	C₂×Q₈	—	4	12	no	—
107	C₂×Q₈	—	8	8	no	—
108	C₂×Q₈	—	8	3	no	—
109	C₂×Q₈	—	8	6	no	—
110	C₂×Q₈	D₁₆	1	1	yes	characteristic
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	8	3	no	—
113	C₂×D₄	D₁₆	1	3	yes	—
114	C₂²×C₄	—	8	3	no	—
115	C₂²×C₄	—	8	3	no	—
116	C₂²×C₄	—	8	3	no	—
117	C₂²×C₄	C₂×D₈	1	3	yes	—
118	Q₁₆	—	2	6	no	—
119	Q₁₆	—	4	3	no	—
120	Q₁₆	—	4	12	no	—
121	SD₁₆	—	4	12	no	—
122	D₈	—	2	2	no	—
123	D₈	—	4	1	no	—
124	M₁₆	#102	1	12	yes	—
125	C₂×C₈	C₂×C₂²⋊C₄	1	6	yes	—
126	C₂×C₈	C₂×D₈	1	3	yes	—
127	C₂×C₈	C₂×SD₁₆	1	6	yes	—
128	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	3	no	—
131	C₄⋊C₄	—	8	3	no	—
132	C₂²⋊C₄	—	8	3	no	—
133	C₄²	—	8	3	no	—
134	C₁₆	C₂×C₂²⋊C₄	1	6	yes	—
135	C₁₆	C₂³×C₄	1	1	yes	characteristic
136	C₁₆	C₂²×D₄	1	1	yes	characteristic
137	C₂³	—	8	1	no	—
138	C₂³	C₂×D₁₆	1	3	yes	—
139	Q₈	—	4	6	no	—
140	Q₈	—	8	3	no	—
141	Q₈	—	8	12	no	—
142	Q₈	#357	1	4	yes	—
143	D₄	—	4	2	no	—
144	D₄	—	8	12	no	—
145	D₄	—	8	1	no	—
146	D₄	#357	1	12	yes	—
147	C₂×C₄	—	8	3	no	—
148	C₂×C₄	—	8	12	no	—
149	C₂×C₄	—	8	12	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	3	no	—
152	C₂×C₄	—	32	3	no	—
153	C₂×C₄	C₂×#102	1	6	yes	—
154	C₂×C₄	C₂×D₁₆	1	3	yes	—
155	C₂×C₄	C₂×SD₃₂	1	6	yes	—
156	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
157	C₈	C₂×#102	1	6	yes	—
158	C₈	C₂²×C₂²⋊C₄	1	1	yes	characteristic
159	C₈	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂²	—	2	3	no	—
161	C₂²	—	2	3	no	—
162	C₂²	—	8	2	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	C₂×#357	1	6	yes	—
166	C₂²	C₂²×D₁₆	1	1	yes	characteristic
167	C₄	—	8	6	no	—
168	C₄	—	16	3	no	—
169	C₄	—	16	1	no	—
170	C₄	C₂×#357	1	6	yes	—
171	C₄	C₂²×#102	1	1	yes	characteristic
172	C₄	C₂²×SD₃₂	1	1	yes	characteristic
173	C₂	—	2	6	no	—
174	C₂	—	16	2	no	—
175	C₂	C₂²×#357	1	1	yes	characteristic
176	C₂	#33983	1	1	yes	characteristic
177	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×#3417	C₂
LCS	G ▷ 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
2. C₂×#2258	C₂	semidirect
3. #33410	C₂	semidirect
4. #13618	C₂	semidirect
5. #13616	C₂	semidirect
6. #13615	C₂	semidirect
7. C₂×#1416	C₂	semidirect
10. #3417	C₄	semidirect
37. #508	C₂×C₄	semidirect
40. C₂×Q₃₂	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×#3417	2
C₂×#2258	2
#33410	2
#13618	2
#13616	2
#13615	2
C₂×#1416	2