Finite Group Databasebeta

Subgroups — #129171

Order 512.

255 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 132 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	C₄×#365	C₂	1	1	yes	maximal, characteristic
2	#12009	C₂	1	1	yes	maximal, characteristic
3	#11404	C₂	1	1	yes	maximal, characteristic
4	#8294	C₂	1	1	yes	maximal, characteristic
5	#8285	C₂	1	1	yes	maximal, characteristic
6	#7176	C₂	1	1	yes	maximal, characteristic
7	#7175	C₂	1	1	yes	maximal, characteristic
8	C₂×#365	C₄	1	2	yes	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	C₄×#108	C₂²	1	1	yes	characteristic
12	C₄×#106	C₂²	1	1	yes	characteristic
13	C₈×C₄⋊C₄	—	2	1	no	—
14	#1564	C₂²	1	1	yes	characteristic
15	C₂×#335	C₂²	1	1	yes	characteristic
16	C₈×M₁₆	—	2	1	no	—
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1383	C₄	1	2	yes	—
19	#1372	C₄	1	2	yes	complemented
20	#1371	C₄	1	2	yes	complemented
21	#1313	C₄	1	4	yes	complemented
22	#1279	—	2	1	no	—
23	#1278	—	2	1	no	—
24	C₂×M₃₂	—	2	1	no	—
25	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
26	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
27	#431	—	2	1	no	—
28	C₂×#108	C₂×C₄	1	2	yes	—
29	C₂×#106	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₄×M₁₆	—	2	1	no	—
32	C₂×C₄×C₈	—	4	1	no	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₄⋊C₄	—	2	1	no	—
37	#365	C₈	1	8	yes	complemented
38	#346	C₂×C₄	1	2	yes	—
39	#346	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	—	2	1	no	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	#341	—	2	1	no	—
43	#340	—	2	1	no	—
44	#336	—	4	1	no	—
45	#335	—	2	1	no	—
46	#335	C₂×C₄	1	2	yes	—
47	#335	C₂×C₄	1	1	yes	characteristic
48	#335	C₂×C₄	1	1	yes	characteristic
49	#334	—	2	1	no	—
50	#322	—	4	1	no	—
51	#322	C₂×C₄	1	2	yes	—
52	#322	Q₈	1	2	yes	—
53	C₈²	—	4	1	no	—
54	C₂×M₁₆	—	2	2	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	4	1	no	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	#41	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄²	—	4	1	no	—
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	M₃₂	—	2	2	no	—
64	M₃₂	—	4	2	no	—
65	M₃₂	C₂×C₈	1	4	yes	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	C₄²	1	2	yes	—
68	C₂×C₁₆	C₄²	1	2	yes	—
69	#108	—	2	2	no	—
70	#108	C₂×C₈	1	4	yes	—
71	#106	—	2	2	no	—
72	#106	C₂×C₈	1	4	yes	—
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	4	1	no	—
77	C₂²⋊C₈	—	4	1	no	—
78	#97	—	4	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	4	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	8	2	no	—
84	C₄×C₈	C₄⋊C₄	1	2	yes	—
85	C₄×C₈	C₄⋊C₄	1	2	yes	—
86	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	#95	—	4	1	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂²×C₄	#137	1	1	yes	characteristic
95	M₁₆	—	2	4	no	—
96	M₁₆	—	4	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	—	8	1	no	—
109	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₈	#117	1	1	yes	characteristic
111	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂²×C₈	1	2	yes	—
114	C₂×C₈	#131	1	2	yes	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	—	8	1	no	—
118	C₄²	—	8	1	no	—
119	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₄²	C₄×D₄	1	1	yes	characteristic
121	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₄²	#137	1	1	yes	characteristic
123	C₁₆	—	2	2	no	—
124	C₁₆	—	4	2	no	—
125	C₁₆	C₄×C₈	1	4	yes	complemented
126	C₂³	#440	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#428	1	1	yes	characteristic
135	C₂×C₄	#428	1	1	yes	characteristic
136	C₂×C₄	C₈×D₄	1	2	yes	—
137	C₂×C₄	#440	1	1	yes	characteristic
138	C₂×C₄	#441	1	1	yes	characteristic
139	C₂×C₄	#442	1	1	yes	characteristic
140	C₂×C₄	#444	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	2	no	—
144	C₈	—	4	2	no	—
145	C₈	—	4	4	no	—
146	C₈	—	8	1	no	—
147	C₈	—	8	1	no	—
148	C₈	—	8	2	no	—
149	C₈	C₂×C₄×C₈	1	1	yes	characteristic
150	C₈	#405	1	1	yes	characteristic
151	C₈	C₈×Q₈	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	SD₁₆⋊C₈	1	2	yes	—
155	C₂²	#1776	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	8	4	no	—
160	C₄	SD₁₆⋊C₈	1	2	yes	—
161	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
162	C₄	#1773	1	1	yes	characteristic
163	C₄	#2180	1	1	yes	characteristic
164	C₄	AGL(1,ℤ/16)	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	#8285	1	1	yes	characteristic
167	C₂	#12677	1	1	yes	characteristic
168	C₂	#12678	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#440
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
19. #1372	C₄	semidirect
20. #1371	C₄	semidirect
21. #1313	C₄	semidirect
37. #365	C₈	semidirect
125. C₁₆	C₄×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#365	2
#12009	2
#11404	2
#8294	2
#8285	2
#7176	2
#7175	2