Finite Group Databasebeta

Subgroups — #507658

Order 512.

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

normal subgroups only

This group has 399 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	#33260	C₂	1	1	yes	maximal, characteristic
2	#21735	C₂	1	1	yes	maximal, characteristic
3	#13510	C₂	1	2	yes	maximal
4	#13454	C₂	1	2	yes	maximal
5	#13186	C₂	1	2	yes	maximal
6	#13180	C₂	1	2	yes	maximal
7	#13174	C₂	1	4	yes	maximal
8	#12660	C₂	1	1	yes	maximal, characteristic
9	#3162	C₂²	1	1	yes	characteristic
10	#2960	—	2	1	no	—
11	#2647	—	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	#1995	—	2	1	no	—
17	#1946	—	2	1	no	—
18	#1946	C₂²	1	2	yes	—
19	#1944	C₂²	1	2	yes	—
20	#1943	C₂²	1	2	yes	—
21	#1941	—	2	1	no	—
22	#1941	—	2	1	no	—
23	#1849	C₂²	1	1	yes	characteristic
24	#1846	C₂²	1	1	yes	characteristic
25	#1844	C₂²	1	1	yes	characteristic
26	#1836	—	2	1	no	—
27	#1772	—	2	1	no	—
28	#1772	C₂²	1	2	yes	—
29	#1726	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	2	no	—
37	C₈×Q₈	—	4	2	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	C₂³	1	2	yes	—
40	#427	—	2	1	no	—
41	#427	C₂³	1	2	yes	—
42	C₂×#107	C₂³	1	2	yes	—
43	C₂×#106	—	2	1	no	—
44	C₂×#106	—	2	1	no	—
45	C₂×#106	C₂³	1	2	yes	—
46	#424	—	2	2	no	—
47	#423	—	2	1	no	—
48	#405	—	2	2	no	—
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	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	#400	—	4	2	no	—
56	#398	—	4	2	no	—
57	#389	—	2	2	no	—
58	#389	—	2	2	no	—
59	#384	—	2	1	no	—
60	#382	—	2	1	no	—
61	#376	D₄	1	1	yes	characteristic
62	#367	—	2	4	no	—
63	#366	—	2	4	no	—
64	#365	—	4	4	no	—
65	#364	Q₈	1	8	yes	complemented
66	#347	—	4	2	no	—
67	#340	—	2	2	no	—
68	#340	—	2	2	no	—
69	#339	—	2	2	no	—
70	#339	—	2	2	no	—
71	C₂×M₁₆	—	2	2	no	—
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂⁴	1	1	yes	characteristic
76	C₄⋊Q₈	—	2	2	no	—
77	#125	—	2	2	no	—
78	C₂²⋊Q₈	—	4	2	no	—
79	C₄×Q₈	—	4	2	no	—
80	#117	—	2	2	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	—	4	2	no	—
84	C₂×C₄⋊C₄	—	4	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	M₃₂	—	2	4	no	—
90	M₃₂	C₂×Q₈	1	8	yes	—
91	C₂×C₁₆	—	2	4	no	—
92	#108	—	2	2	no	—
93	#108	—	2	2	no	—
94	#108	—	4	2	no	—
95	#107	—	2	2	no	—
96	#107	—	2	4	no	—
97	#107	#41	1	4	yes	—
98	#106	—	2	2	no	—
99	#106	—	2	2	no	—
100	#106	—	2	2	no	—
101	#106	—	2	2	no	—
102	#106	—	4	1	no	—
103	#106	—	4	1	no	—
104	#106	—	4	2	no	—
105	C₄⋊C₈	—	4	2	no	—
106	C₄⋊C₈	—	4	4	no	—
107	#97	—	2	2	no	—
108	#97	—	2	2	no	—
109	#97	—	2	1	no	—
110	#97	—	2	1	no	—
111	#97	—	4	2	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	4	2	no	—
116	C₄×C₈	C₂×Q₈	1	4	yes	—
117	#95	—	2	2	no	—
118	#95	—	4	2	no	—
119	#95	—	4	2	no	—
120	#95	—	4	2	no	—
121	C₂×Q₈	—	4	2	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
127	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
128	M₁₆	—	4	2	no	—
129	M₁₆	—	4	2	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₄⋊Q₈	1	2	yes	—
137	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₈	#137	1	2	yes	—
140	C₂×C₈	#137	1	2	yes	—
141	C₂×C₈	C₂²×Q₈	1	2	yes	—
142	C₂×C₈	C₂×#41	1	2	yes	—
143	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	16	2	no	—
153	C₄⋊C₄	—	16	2	no	—
154	C₄⋊C₄	—	16	2	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₄⋊Q₈	1	2	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	#525	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₂×C₄	#537	1	1	yes	characteristic
182	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×C₄	C₂×#137	1	1	yes	characteristic
184	C₂×C₄	C₂×#137	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	4	1	no	—
187	C₈	—	8	2	no	—
188	C₈	#501	1	2	yes	—
189	C₈	#501	1	2	yes	—
190	C₈	#554	1	2	yes	—
191	C₈	#557	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	#3109	1	1	yes	characteristic
194	C₂²	#3165	1	1	yes	characteristic
195	C₂²	#3254	1	1	yes	characteristic
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	8	1	no	—
199	C₄	—	8	4	no	—
200	C₄	—	16	2	no	—
201	C₄	—	16	2	no	—
202	C₄	#2584	1	1	yes	characteristic
203	C₄	#3113	1	1	yes	characteristic
204	C₄	C₂×#501	1	1	yes	characteristic
205	C₄	#3254	1	1	yes	characteristic
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#29319	1	1	yes	characteristic
209	C₂	#33691	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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
65. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33260	2
#21735	2
#13510	2
#13454	2
#13186	2
#13180	2
#13174	2
#12660	2