Finite Group Databasebeta

Subgroups — #507660

Order 512.

401 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 146 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	#33264	C₂	1	1	yes	maximal, characteristic
2	#21736	C₂	1	1	yes	maximal, characteristic
3	#13510	C₂	1	2	yes	maximal
4	#13454	C₂	1	2	yes	maximal
5	#13187	C₂	1	2	yes	maximal
6	#13180	C₂	1	2	yes	maximal
7	#13174	C₂	1	4	yes	maximal
8	#12659	C₂	1	1	yes	maximal, characteristic
9	#3162	C₂²	1	1	yes	characteristic
10	#2961	—	2	1	no	—
11	#2864	—	2	1	no	—
12	#2159	C₂²	1	4	yes	—
13	C₂×#365	C₂²	1	4	yes	—
14	C₂×#364	C₂²	1	2	yes	—
15	C₂×#347	C₂²	1	2	yes	—
16	#1994	—	2	1	no	—
17	#1946	C₂²	1	2	yes	—
18	#1945	—	2	1	no	—
19	#1944	—	2	1	no	—
20	#1944	C₂²	1	2	yes	—
21	#1943	—	2	1	no	—
22	#1943	C₂²	1	2	yes	—
23	#1849	C₂²	1	1	yes	characteristic
24	#1846	C₂²	1	1	yes	characteristic
25	#1846	C₂²	1	1	yes	characteristic
26	#1835	—	2	1	no	—
27	#1773	—	2	1	no	—
28	#1772	C₂²	1	2	yes	—
29	C₄×#97	C₂²	1	1	yes	characteristic
30	#1387	C₂²	1	4	yes	—
31	#1386	C₂²	1	4	yes	—
32	#1377	C₂²	1	2	yes	—
33	#533	—	2	1	no	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	#501	—	2	4	no	—
36	#446	—	4	4	no	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	C₂³	1	2	yes	—
39	#427	—	2	1	no	—
40	#427	C₂³	1	2	yes	—
41	C₂×#107	—	2	1	no	—
42	C₂×#107	C₂³	1	2	yes	—
43	C₂×#106	—	2	1	no	—
44	C₂×#106	C₂³	1	2	yes	—
45	#424	—	2	2	no	—
46	#423	—	2	1	no	—
47	#405	—	2	2	no	—
48	C₂×#97	D₄	1	1	yes	characteristic
49	C₂×#97	D₄	1	1	yes	characteristic
50	C₂×#97	C₂³	1	1	yes	characteristic
51	C₂×#97	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#389	—	2	2	no	—
55	#389	—	2	2	no	—
56	#387	—	4	4	no	—
57	#382	—	2	1	no	—
58	#382	—	2	1	no	—
59	C₄³	D₄	1	1	yes	characteristic
60	#367	—	2	4	no	—
61	#366	—	2	4	no	—
62	#365	—	4	4	no	—
63	#364	Q₈	1	8	yes	complemented
64	#347	—	4	2	no	—
65	#340	—	2	2	no	—
66	#340	—	2	2	no	—
67	#339	—	2	2	no	—
68	#339	—	2	2	no	—
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂⁴	1	1	yes	characteristic
74	C₄⋊Q₈	—	2	2	no	—
75	#125	—	2	2	no	—
76	C₂²⋊Q₈	—	4	2	no	—
77	C₄×Q₈	—	4	2	no	—
78	#117	—	2	2	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	—	2	2	no	—
81	C₂×C₄⋊C₄	—	4	2	no	—
82	C₂×C₄⋊C₄	—	4	2	no	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
87	M₃₂	—	2	4	no	—
88	M₃₂	C₂×Q₈	1	8	yes	—
89	C₂×C₁₆	—	2	4	no	—
90	#108	—	2	2	no	—
91	#108	—	2	2	no	—
92	#108	—	4	2	no	—
93	#107	—	2	2	no	—
94	#107	—	2	4	no	—
95	#107	—	4	2	no	—
96	#107	#41	1	4	yes	—
97	#106	—	2	2	no	—
98	#106	—	2	2	no	—
99	#106	—	2	2	no	—
100	#106	—	2	2	no	—
101	#106	—	4	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	4	4	no	—
104	#97	—	2	2	no	—
105	#97	—	2	2	no	—
106	#97	—	2	2	no	—
107	#97	—	2	1	no	—
108	#97	—	2	1	no	—
109	#97	—	4	2	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	4	2	no	—
112	C₄×C₈	C₂×Q₈	1	4	yes	—
113	#95	—	4	2	no	—
114	#95	—	4	2	no	—
115	#95	—	4	2	no	—
116	#95	—	4	2	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	C₄⋊Q₈	1	2	yes	—
133	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₈	#137	1	2	yes	—
135	C₂×C₈	#137	1	2	yes	—
136	C₂×C₈	C₂²×Q₈	1	2	yes	—
137	C₂×C₈	C₂×#41	1	2	yes	—
138	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	8	4	no	—
148	C₄⋊C₄	—	8	4	no	—
149	C₂²⋊C₄	—	4	2	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	2	no	—
156	C₄²	—	8	2	no	—
157	C₄²	C₄⋊Q₈	1	2	yes	—
158	C₁₆	—	2	8	no	—
159	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	Q₈	—	16	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	#525	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	#537	1	1	yes	characteristic
177	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×C₄	C₂×#137	1	1	yes	characteristic
179	C₂×C₄	C₂×#137	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	4	1	no	—
182	C₈	—	8	2	no	—
183	C₈	#501	1	4	yes	—
184	C₈	#554	1	2	yes	—
185	C₈	#557	1	2	yes	—
186	C₂²	—	2	2	no	—
187	C₂²	#3109	1	1	yes	characteristic
188	C₂²	#3165	1	1	yes	characteristic
189	C₂²	#3254	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	1	no	—
193	C₄	—	8	4	no	—
194	C₄	—	16	2	no	—
195	C₄	—	16	2	no	—
196	C₄	#2584	1	1	yes	characteristic
197	C₄	#3113	1	1	yes	characteristic
198	C₄	C₂×#501	1	1	yes	characteristic
199	C₄	#3254	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	#29319	1	1	yes	characteristic
203	C₂	#33690	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
63. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33264	2
#21736	2
#13510	2
#13454	2
#13187	2
#13180	2
#13174	2
#12659	2