Finite Group Databasebeta

Subgroups — #512944

Order 512.

432 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 138 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, complemented
2	C₂×#2258	C₂	1	1	yes	maximal, characteristic, complemented
3	#33610	C₂	1	2	yes	maximal, complemented
4	#13699	C₂	1	4	yes	maximal, complemented
5	#13695	C₂	1	2	yes	maximal, complemented
6	#13694	C₂	1	2	yes	maximal, complemented
7	C₂×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	#13616	C₂	1	2	yes	maximal, complemented
9	C₂×#508	C₂²	1	2	yes	—
10	C₂×#504	C₂²	1	1	yes	characteristic
11	#3061	—	2	2	no	—
12	#2262	—	2	4	no	—
13	C₂×Q₆₄	C₂²	1	1	yes	characteristic
14	C₂×SD₆₄	C₂²	1	2	yes	—
15	C₂×D₃₂	C₂²	1	1	yes	characteristic
16	#2258	—	2	4	no	—
17	C₂×M₆₄	C₂²	1	1	yes	characteristic
18	C₂×M₆₄	C₂²	1	2	yes	—
19	C₂²×C₃₂	C₂²	1	2	yes	—
20	C₂²×C₃₂	C₂²	1	1	yes	characteristic
21	#2220	—	2	4	no	—
22	#2216	C₂²	1	4	yes	—
23	#2215	C₂²	1	4	yes	—
24	C₂×#366	C₂²	1	2	yes	—
25	#2139	—	2	2	no	—
26	#1423	C₂²	1	2	yes	—
27	#1423	C₂²	1	2	yes	—
28	#1416	C₂²	1	4	yes	—
29	#1415	C₂²	1	4	yes	—
30	C₂×#135	—	2	2	no	—
31	C₂×#131	D₄	1	1	yes	characteristic
32	#537	—	4	2	no	—
33	#508	—	2	4	no	—
34	#508	—	2	4	no	—
35	C₂×Q₃₂	C₂³	1	2	yes	—
36	C₂×SD₃₂	—	2	2	no	—
37	C₂×D₁₆	C₂³	1	2	yes	—
38	#504	—	2	2	no	—
39	#504	—	2	2	no	—
40	C₂×M₃₂	D₄	1	1	yes	characteristic
41	C₂×M₃₂	D₄	1	2	yes	—
42	C₂²×C₁₆	C₂³	1	2	yes	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	#468	—	4	4	no	—
45	#467	—	2	4	no	—
46	#466	—	2	4	no	—
47	C₂×#107	—	2	2	no	—
48	#417	—	4	2	no	—
49	Q₆₄	—	2	2	no	—
50	SD₆₄	—	2	4	no	—
51	D₃₂	—	2	2	no	—
52	M₆₄	—	2	2	no	—
53	M₆₄	—	2	4	no	—
54	C₂×C₃₂	—	2	4	no	—
55	C₂×C₃₂	D₄	1	4	yes	complemented
56	C₂×C₃₂	C₂³	1	2	yes	—
57	C₂×C₃₂	C₂³	1	1	yes	characteristic
58	C₂×C₃₂	C₂³	1	1	yes	characteristic
59	#366	—	2	2	no	—
60	#366	C₂³	1	4	yes	—
61	#358	—	2	4	no	—
62	#357	—	2	4	no	—
63	C₂×#41	—	4	2	no	—
64	C₂×#41	D₈	1	1	yes	characteristic
65	#135	—	4	4	no	—
66	#135	—	4	4	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	4	2	no	—
69	C₂×D₈	—	2	2	no	—
70	#131	—	2	2	no	—
71	#131	—	2	2	no	—
72	C₂×M₁₆	D₈	1	1	yes	characteristic
73	C₂×M₁₆	D₈	1	2	yes	—
74	C₂²×C₈	C₂×D₄	1	2	yes	—
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²⋊Q₈	—	4	4	no	—
77	C₂²⋊Q₈	—	8	2	no	—
78	#121	—	4	4	no	—
79	#121	—	8	2	no	—
80	#121	—	8	4	no	—
81	C₂×C₄⋊C₄	—	4	2	no	—
82	Q₃₂	—	2	2	no	—
83	Q₃₂	—	2	2	no	—
84	SD₃₂	—	4	4	no	—
85	D₁₆	—	2	2	no	—
86	D₁₆	—	2	2	no	—
87	M₃₂	—	2	2	no	—
88	M₃₂	—	2	4	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	C₂×D₄	1	2	yes	—
92	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
93	C₂×C₁₆	C₂×D₄	1	2	yes	—
94	C₂×C₁₆	#41	1	2	yes	—
95	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
96	#107	—	2	4	no	—
97	#107	—	4	2	no	—
98	#103	—	4	4	no	—
99	#102	—	4	4	no	—
100	C₃₂	—	2	2	no	—
101	C₃₂	C₂×D₄	1	2	yes	—
102	C₃₂	C₂×D₄	1	2	yes	—
103	#41	—	2	2	no	—
104	#41	—	2	2	no	—
105	#41	—	8	4	no	—
106	#41	—	8	4	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	D₁₆	1	1	yes	characteristic
109	C₂×D₄	—	4	2	no	—
110	C₂×D₄	—	8	2	no	—
111	C₂×D₄	—	8	4	no	—
112	C₂×D₄	D₁₆	1	2	yes	—
113	C₂×D₄	D₁₆	1	1	yes	characteristic
114	C₂²×C₄	—	8	2	no	—
115	C₂²×C₄	—	8	2	no	—
116	C₂²×C₄	C₂×D₈	1	2	yes	—
117	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
118	Q₁₆	—	4	2	no	—
119	Q₁₆	—	4	2	no	—
120	SD₁₆	—	8	4	no	—
121	D₈	—	4	2	no	—
122	D₈	—	4	2	no	—
123	M₁₆	—	2	2	no	—
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	#121	1	2	yes	—
128	C₂×C₈	C₂×D₈	1	2	yes	—
129	C₂×C₈	C₂×D₈	1	1	yes	characteristic
130	C₂×C₈	#135	1	2	yes	—
131	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
132	C₄⋊C₄	—	4	4	no	—
133	C₄⋊C₄	—	8	2	no	—
134	C₄⋊C₄	—	8	4	no	—
135	C₂²⋊C₄	—	8	4	no	—
136	C₂²⋊C₄	—	8	4	no	—
137	C₁₆	—	2	2	no	—
138	C₁₆	#121	1	2	yes	—
139	C₁₆	C₂²×D₄	1	1	yes	characteristic
140	C₁₆	C₂×#41	1	1	yes	characteristic
141	C₂³	—	8	2	no	—
142	C₂³	C₂×D₁₆	1	2	yes	—
143	C₂³	C₂×D₁₆	1	1	yes	characteristic
144	Q₈	—	2	2	no	—
145	Q₈	—	8	2	no	—
146	Q₈	—	8	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	2	2	no	—
149	D₄	—	8	2	no	—
150	D₄	—	8	2	no	—
151	D₄	—	16	4	no	—
152	D₄	—	32	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	16	4	no	—
158	C₂×C₄	—	32	2	no	—
159	C₂×C₄	#466	1	2	yes	—
160	C₂×C₄	C₂×D₁₆	1	2	yes	—
161	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
162	C₂×C₄	#508	1	2	yes	—
163	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	#466	1	2	yes	—
166	C₈	C₂×#121	1	1	yes	characteristic
167	C₈	C₂×#135	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	—	16	2	no	—
176	C₂²	#2215	1	2	yes	—
177	C₂²	C₂²×D₁₆	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	16	2	no	—
180	C₄	—	16	4	no	—
181	C₄	#2215	1	2	yes	—
182	C₄	C₂×#466	1	1	yes	characteristic
183	C₄	C₂×#508	1	1	yes	characteristic
184	C₂	—	2	2	no	—
185	C₂	—	4	2	no	—
186	C₂	—	32	2	no	—
187	C₂	C₂×#2215	1	1	yes	characteristic
188	C₂	#33982	1	1	yes	characteristic
189	C₂	#33984	1	1	yes	characteristic

	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. C₂×#2262	C₂	semidirect
2. C₂×#2258	C₂	semidirect
3. #33610	C₂	semidirect
4. #13699	C₂	semidirect
5. #13695	C₂	semidirect
6. #13694	C₂	semidirect
7. C₂×#1423	C₂	semidirect
8. #13616	C₂	semidirect
55. C₂×C₃₂	D₄	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
C₂×#2258	2
#33610	2
#13699	2
#13695	2
#13694	2
C₂×#1423	2
#13616	2