Finite Group Databasebeta

Subgroups — #129016

Order 512.

262 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 94 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	#17961	C₂	1	1	yes	maximal, characteristic
2	#11953	C₂	1	1	yes	maximal, characteristic
3	#11752	C₂	1	1	yes	maximal, characteristic
4	#11415	C₂	1	1	yes	maximal, characteristic
5	#8295	C₂	1	1	yes	maximal, characteristic
6	#7300	C₂	1	1	yes	maximal, characteristic
7	#7295	C₂	1	1	yes	maximal, characteristic
8	#3170	C₂²	1	1	yes	characteristic
9	#2960	—	2	1	no	—
10	#2163	C₂²	1	1	yes	characteristic
11	#2163	C₂²	1	1	yes	characteristic
12	C₄×#108	C₂²	1	1	yes	characteristic
13	#1593	—	2	1	no	—
14	#1593	C₄	1	2	yes	—
15	#1579	—	2	1	no	—
16	#1579	C₄	1	2	yes	—
17	#1566	C₂²	1	1	yes	characteristic
18	#1564	C₂²	1	1	yes	characteristic
19	#1493	—	2	1	no	—
20	#1451	—	2	1	no	—
21	C₂×#322	C₂²	1	1	yes	characteristic
22	#1365	C₄	1	2	yes	—
23	#1362	C₄	1	2	yes	—
24	#1350	C₄	1	4	yes	—
25	#1313	—	2	1	no	—
26	#1313	—	2	1	no	—
27	#1278	—	2	1	no	—
28	#533	—	2	1	no	—
29	#500	C₂×C₄	1	1	yes	characteristic
30	#494	C₂×C₄	1	1	yes	characteristic
31	#487	—	2	1	no	—
32	#467	—	2	1	no	—
33	#467	C₂×C₄	1	2	yes	—
34	#446	—	2	2	no	—
35	#446	—	4	1	no	—
36	C₈×Q₈	—	2	2	no	—
37	C₈×Q₈	—	4	1	no	—
38	C₄×Q₁₆	—	2	1	no	—
39	C₄×Q₁₆	C₂×C₄	1	2	yes	—
40	#431	—	2	1	no	—
41	C₂×#108	—	2	1	no	—
42	#424	—	4	1	no	—
43	#423	—	2	1	no	—
44	#405	—	4	1	no	—
45	C₄×M₁₆	—	2	1	no	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₂²⋊C₁₆	—	2	1	no	—
52	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
53	#346	—	2	1	no	—
54	#346	—	2	1	no	—
55	#341	C₂×C₄	1	2	yes	—
56	#335	C₂×C₄	1	2	yes	—
57	#334	—	2	1	no	—
58	#334	—	2	1	no	—
59	#334	C₂×C₄	1	2	yes	—
60	Q₈⋊C₈	—	2	2	no	—
61	Q₈⋊C₈	—	2	2	no	—
62	#322	—	4	2	no	—
63	#322	C₂×C₄	1	1	yes	characteristic
64	#322	C₂×C₄	1	1	yes	characteristic
65	#322	D₄	1	1	yes	characteristic
66	#322	D₄	1	1	yes	characteristic
67	C₂×Q₁₆	—	2	1	no	—
68	C₂×Q₁₆	C₄²	1	2	yes	—
69	C₂×M₁₆	—	2	2	no	—
70	C₂×M₁₆	—	4	1	no	—
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	#41	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	—	2	1	no	—
77	#125	—	4	1	no	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₂²⋊Q₈	—	4	1	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	4	1	no	—
82	#117	—	4	1	no	—
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	2	no	—
86	#108	—	4	2	no	—
87	#107	—	2	1	no	—
88	#107	C₄²	1	2	yes	—
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	4	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	4	1	no	—
94	#103	—	2	2	no	—
95	#103	—	2	2	no	—
96	C₂²⋊C₈	—	4	1	no	—
97	#97	—	4	2	no	—
98	#97	—	4	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	4	2	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	—	4	1	no	—
114	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂²×C₄	#137	1	1	yes	characteristic
117	Q₁₆	—	4	2	no	—
118	Q₁₆	—	4	2	no	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	8	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	8	1	no	—
133	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₈	#117	1	1	yes	characteristic
135	C₂×C₈	C₄×D₄	1	1	yes	characteristic
136	C₂×C₈	C₄×D₄	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	—	4	2	no	—
145	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₄²	C₄×D₄	1	1	yes	characteristic
147	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	C₁₆	—	4	2	no	—
151	C₂³	#440	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	Q₈	—	4	2	no	—
154	Q₈	—	8	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#417	1	1	yes	characteristic
163	C₂×C₄	#418	1	1	yes	characteristic
164	C₂×C₄	#440	1	1	yes	characteristic
165	C₂×C₄	#442	1	1	yes	characteristic
166	C₂×C₄	#442	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	4	1	no	—
171	C₈	—	4	1	no	—
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	8	2	no	—
175	C₈	—	8	2	no	—
176	C₈	—	8	2	no	—
177	C₈	#444	1	2	yes	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#1762	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	2	no	—
187	C₄	—	8	2	no	—
188	C₄	#1757	1	1	yes	characteristic
189	C₄	#1765	1	1	yes	characteristic
190	C₄	#2180	1	2	yes	—
191	C₂	—	4	1	no	—
192	C₂	#8253	1	1	yes	characteristic
193	C₂	#12648	1	1	yes	characteristic
194	C₂	#12651	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1762
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#440
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#17961	2
#11953	2
#11752	2
#11415	2
#8295	2
#7300	2
#7295	2