Finite Group Databasebeta

Subgroups — #153381

Order 512.

187 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 80 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×#1424	C₂	1	1	yes	maximal, characteristic
2	C₂×#1423	C₂	1	1	yes	maximal, characteristic
3	#13165	C₂	1	1	yes	maximal, characteristic
4	#12907	C₂	1	1	yes	maximal, characteristic
5	#7471	C₂	1	1	yes	maximal, characteristic
6	#7471	C₂	1	1	yes	maximal, characteristic
7	#7465	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₃₂	C₂²	1	1	yes	characteristic
10	C₂×#367	—	2	1	no	—
11	C₂×#366	C₂²	1	1	yes	characteristic
12	C₂×#366	C₂²	1	1	yes	characteristic
13	C₂×#366	C₂²	1	1	yes	characteristic
14	C₂×#366	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1942	—	2	1	no	—
17	#1846	—	2	1	no	—
18	#1424	—	2	2	no	—
19	#1423	—	2	2	no	—
20	#1380	—	2	1	no	—
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×#107	—	2	1	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#107	—	2	1	no	—
27	C₂×#106	—	4	1	no	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	#389	—	4	1	no	—
30	#382	—	4	1	no	—
31	C₂×C₃₂	—	2	1	no	—
32	C₂×C₃₂	—	2	1	no	—
33	C₂×C₃₂	—	2	1	no	—
34	C₂×C₃₂	—	2	1	no	—
35	C₂×C₃₂	Q₈	1	2	yes	—
36	C₂×C₃₂	Q₈	1	2	yes	—
37	#367	—	4	2	no	—
38	#366	—	2	1	no	—
39	#366	—	2	1	no	—
40	#366	—	2	1	no	—
41	#366	—	2	1	no	—
42	#366	—	2	1	no	—
43	#366	—	2	1	no	—
44	#366	—	2	1	no	—
45	#366	D₄	1	2	yes	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	#340	—	4	1	no	—
49	C₂²×C₈	D₈	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄	—	4	1	no	—
52	C₂×C₄⋊C₄	—	4	1	no	—
53	C₂×C₄⋊C₄	—	4	1	no	—
54	C₂×C₄⋊C₄	—	4	1	no	—
55	C₂×C₄⋊C₄	—	8	1	no	—
56	C₂×C₄²	D₈	1	1	yes	characteristic
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	Q₁₆	1	2	yes	—
60	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
61	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₁₆	#41	1	1	yes	characteristic
64	C₂×C₁₆	#41	1	1	yes	characteristic
65	C₂×C₁₆	#41	1	1	yes	characteristic
66	#107	—	2	2	no	—
67	#107	—	4	1	no	—
68	#107	—	4	1	no	—
69	#107	—	4	1	no	—
70	#107	—	4	1	no	—
71	#107	—	4	1	no	—
72	#107	—	4	1	no	—
73	#107	—	4	1	no	—
74	#106	—	8	2	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	#95	—	8	1	no	—
78	#95	—	8	1	no	—
79	#95	—	8	1	no	—
80	C₃₂	—	2	2	no	—
81	C₃₂	—	2	2	no	—
82	C₂²×C₄	—	8	1	no	—
83	C₂²×C₄	—	8	1	no	—
84	C₂²×C₄	—	8	1	no	—
85	C₂²×C₄	—	8	1	no	—
86	C₂²×C₄	D₁₆	1	1	yes	characteristic
87	C₂²×C₄	D₁₆	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	Q₃₂	1	2	yes	—
92	C₂×C₈	#121	1	1	yes	characteristic
93	C₂×C₈	#123	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₈	#137	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	8	1	no	—
104	C₄⋊C₄	—	8	1	no	—
105	C₄⋊C₄	—	8	1	no	—
106	C₄⋊C₄	—	16	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
111	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
112	C₁₆	#125	1	1	yes	characteristic
113	C₁₆	#125	1	1	yes	characteristic
114	C₂³	C₂×D₁₆	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	—	16	1	no	—
123	C₂×C₄	—	16	1	no	—
124	C₂×C₄	—	16	1	no	—
125	C₂×C₄	—	16	1	no	—
126	C₂×C₄	—	16	1	no	—
127	C₂×C₄	—	16	1	no	—
128	C₂×C₄	—	16	1	no	—
129	C₂×C₄	—	16	1	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	16	1	no	—
133	C₂×C₄	#459	1	1	yes	characteristic
134	C₂×C₄	#480	1	1	yes	characteristic
135	C₂×C₄	#500	1	1	yes	characteristic
136	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
137	C₂×C₄	#509	1	1	yes	characteristic
138	C₂×C₄	#510	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	#398	1	1	yes	characteristic
141	C₈	#462	1	1	yes	characteristic
142	C₈	#484	1	1	yes	characteristic
143	C₈	#501	1	1	yes	characteristic
144	C₂²	#2206	1	1	yes	characteristic
145	C₂²	#2232	1	1	yes	characteristic
146	C₂²	#2252	1	1	yes	characteristic
147	C₂²	#2262	1	1	yes	characteristic
148	C₂²	#2262	1	1	yes	characteristic
149	C₂²	#2263	1	1	yes	characteristic
150	C₂²	#2264	1	1	yes	characteristic
151	C₄	—	4	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	16	2	no	—
154	C₄	—	32	1	no	—
155	C₄	—	32	1	no	—
156	C₄	—	32	1	no	—
157	C₄	#2083	1	1	yes	characteristic
158	C₄	#2209	1	1	yes	characteristic
159	C₄	#2236	1	1	yes	characteristic
160	C₄	#2255	1	1	yes	characteristic
161	C₂	#13505	1	1	yes	characteristic
162	C₂	#13689	1	1	yes	characteristic
163	C₂	#13690	1	1	yes	characteristic
164	C₂	#13713	1	1	yes	characteristic
165	C₂	#13716	1	1	yes	characteristic
166	C₂	#13732	1	1	yes	characteristic
167	C₂	#13734	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#1424	2
C₂×#1423	2
#13165	2
#12907	2
#7471	2
#7465	2