Finite Group Databasebeta

Subgroups — #152826

Order 512.

207 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 72 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	#13638	C₂	1	1	yes	maximal, characteristic, complemented
2	#13625	C₂	1	1	yes	maximal, characteristic
3	#13622	C₂	1	1	yes	maximal, characteristic
4	#13242	C₂	1	1	yes	maximal, characteristic
5	#12671	C₂	1	1	yes	maximal, characteristic, complemented
6	#7476	C₂	1	1	yes	maximal, characteristic, complemented
7	#7468	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	C₂×#368	C₂²	1	1	yes	characteristic
13	#2158	C₂²	1	1	yes	characteristic
14	C₂×#359	—	2	1	no	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	#1974	—	2	1	no	—
17	#1772	—	2	1	no	—
18	#1426	—	2	2	no	—
19	#1420	C₄	1	2	yes	complemented
20	#1419	C₄	1	2	yes	complemented
21	#1418	—	2	2	no	—
22	#1382	—	2	1	no	—
23	C₂×#135	—	2	1	no	—
24	#508	—	2	1	no	—
25	#508	—	2	1	no	—
26	#508	—	2	1	no	—
27	#508	—	2	1	no	—
28	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
29	C₂×SD₃₂	—	2	1	no	—
30	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	C₂×#108	—	2	1	no	—
34	C₂×#108	—	2	1	no	—
35	#427	—	2	1	no	—
36	C₂×#104	—	4	1	no	—
37	#408	—	4	1	no	—
38	#405	—	4	1	no	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	M₆₄	—	2	1	no	—
41	M₆₄	—	2	1	no	—
42	M₆₄	—	2	1	no	—
43	M₆₄	C₂×C₄	1	2	yes	—
44	C₂×C₃₂	—	2	1	no	—
45	C₂×C₃₂	—	2	1	no	—
46	#368	—	2	1	no	—
47	#368	—	2	1	no	—
48	#368	—	2	1	no	—
49	#368	C₂×C₄	1	2	yes	—
50	#367	D₄	1	2	yes	—
51	#366	—	2	1	no	—
52	#359	—	4	2	no	—
53	#346	—	2	1	no	—
54	#346	—	2	1	no	—
55	#339	—	4	1	no	—
56	C₂×#41	—	4	1	no	—
57	#135	—	4	1	no	—
58	#135	—	4	1	no	—
59	#135	—	4	1	no	—
60	#135	—	4	1	no	—
61	C₂×Q₁₆	—	2	1	no	—
62	C₂×SD₁₆	—	4	1	no	—
63	C₂×D₈	—	2	1	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂²×C₈	D₈	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	#117	—	4	1	no	—
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	Q₃₂	—	2	1	no	—
71	Q₃₂	—	2	1	no	—
72	SD₃₂	—	4	1	no	—
73	SD₃₂	—	4	1	no	—
74	D₁₆	—	2	1	no	—
75	D₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	#41	1	1	yes	characteristic
82	C₂×C₁₆	#41	1	1	yes	characteristic
83	C₂×C₁₆	#41	1	1	yes	characteristic
84	C₂×C₁₆	#41	1	1	yes	characteristic
85	#108	—	2	2	no	—
86	#108	—	4	1	no	—
87	#108	—	4	1	no	—
88	#108	—	4	1	no	—
89	#107	—	2	2	no	—
90	#106	—	4	1	no	—
91	#104	—	8	2	no	—
92	C₂²⋊C₈	—	8	2	no	—
93	#97	—	8	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	8	1	no	—
97	C₃₂	—	2	2	no	—
98	C₃₂	—	4	1	no	—
99	#41	—	8	1	no	—
100	#41	—	8	1	no	—
101	#41	—	8	1	no	—
102	#41	—	8	1	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×D₄	—	4	1	no	—
105	C₂×D₄	—	8	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
109	Q₁₆	—	4	1	no	—
110	Q₁₆	—	4	1	no	—
111	SD₁₆	—	8	1	no	—
112	SD₁₆	—	8	1	no	—
113	D₈	—	4	1	no	—
114	D₈	—	4	1	no	—
115	M₁₆	—	4	2	no	—
116	M₁₆	—	8	1	no	—
117	M₁₆	—	8	1	no	—
118	M₁₆	—	8	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	8	1	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	SD₃₂	1	2	yes	—
124	C₂×C₈	C₄×D₄	1	1	yes	characteristic
125	C₂×C₈	#121	1	1	yes	characteristic
126	C₂×C₈	#123	1	1	yes	characteristic
127	C₂×C₈	#135	1	1	yes	characteristic
128	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₈	#137	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₂²⋊C₄	—	8	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	8	1	no	—
135	C₁₆	—	4	1	no	—
136	C₁₆	#117	1	1	yes	characteristic
137	C₁₆	C₄×D₄	1	1	yes	characteristic
138	C₁₆	#124	1	1	yes	characteristic
139	C₁₆	#126	1	1	yes	characteristic
140	C₂³	—	8	1	no	—
141	C₂³	#508	1	1	yes	characteristic
142	Q₈	—	8	1	no	—
143	Q₈	—	8	1	no	—
144	D₄	—	8	1	no	—
145	D₄	—	8	1	no	—
146	D₄	—	16	1	no	—
147	D₄	—	16	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	16	1	no	—
153	C₂×C₄	—	16	1	no	—
154	C₂×C₄	C₄×D₈	1	1	yes	characteristic
155	C₂×C₄	#459	1	1	yes	characteristic
156	C₂×C₄	#480	1	1	yes	characteristic
157	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
158	C₂×C₄	#509	1	1	yes	characteristic
159	C₂×C₄	#509	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	8	2	no	—
162	C₈	—	16	1	no	—
163	C₈	#388	1	1	yes	characteristic
164	C₈	#440	1	1	yes	characteristic
165	C₈	#464	1	1	yes	characteristic
166	C₈	#482	1	1	yes	characteristic
167	C₂²	—	4	1	no	—
168	C₂²	—	16	2	no	—
169	C₂²	—	16	1	no	—
170	C₂²	C₄×SD₃₂	1	1	yes	characteristic
171	C₂²	#2210	1	1	yes	characteristic
172	C₂²	#2234	1	1	yes	characteristic
173	C₄	—	8	1	no	—
174	C₄	—	16	2	no	—
175	C₄	—	16	1	no	—
176	C₄	#1925	1	1	yes	characteristic
177	C₄	#2177	1	1	yes	characteristic
178	C₄	#2207	1	1	yes	characteristic
179	C₄	#2233	1	1	yes	characteristic
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	32	1	no	—
184	C₂	#13131	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2233
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13131
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #13638	C₂	semidirect
5. #12671	C₂	semidirect
6. #7476	C₂	semidirect
7. #7468	C₂	semidirect
19. #1420	C₄	semidirect
20. #1419	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13638	2
#13625	2
#13622	2
#13242	2
#12671	2
#7476	2
#7468	2