Finite Group Databasebeta

Subgroups — #153246

Order 512.

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2221	C₂	1	1	yes	maximal, characteristic
2	#13636	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1419	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1419	C₂	1	1	yes	maximal, characteristic, complemented
5	#13558	C₂	1	1	yes	maximal, characteristic, complemented
6	#12840	C₂	1	1	yes	maximal, characteristic, complemented
7	#7479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₆	C₂²	1	1	yes	characteristic
9	C₂×#504	C₂²	1	1	yes	characteristic
10	C₂×#472	—	2	1	no	—
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	C₂×M₆₄	C₂²	1	1	yes	characteristic
13	#2221	—	2	2	no	—
14	#2221	—	2	2	no	—
15	C₂×#368	C₂²	1	1	yes	characteristic
16	C₂×#368	C₂²	1	1	yes	characteristic
17	C₂×#368	C₂²	1	1	yes	characteristic
18	C₂×#361	—	2	1	no	—
19	#1813	—	2	1	no	—
20	#1425	—	2	2	no	—
21	#1419	—	2	2	no	—
22	#1419	—	2	2	no	—
23	#1401	—	2	2	no	—
24	#1385	—	2	1	no	—
25	C₂×AGL(1,ℤ/8)	—	4	1	no	—
26	C₂²×D₈	—	2	1	no	—
27	C₂×#131	D₄	1	1	yes	characteristic
28	C₂×D₁₆	—	2	1	no	—
29	C₂×D₁₆	—	2	1	no	—
30	C₂×D₁₆	—	2	1	no	—
31	C₂×D₁₆	—	2	1	no	—
32	C₂×D₁₆	—	2	1	no	—
33	C₂×D₁₆	D₄	1	2	yes	—
34	#504	—	2	1	no	—
35	#504	—	2	1	no	—
36	#504	—	2	1	no	—
37	#504	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	D₄	1	1	yes	characteristic
40	C₂²×C₁₆	—	2	1	no	—
41	C₂²×C₁₆	C₂³	1	1	yes	characteristic
42	#472	—	2	4	no	—
43	#472	—	4	2	no	—
44	C₂×#108	—	2	1	no	—
45	C₂×#108	—	2	1	no	—
46	C₂×#108	—	2	1	no	—
47	#413	—	4	1	no	—
48	C₂×#100	—	4	1	no	—
49	M₆₄	—	2	1	no	—
50	M₆₄	—	2	1	no	—
51	M₆₄	—	2	1	no	—
52	M₆₄	D₄	1	2	yes	—
53	C₂×C₃₂	—	2	1	no	—
54	C₂×C₃₂	—	2	1	no	—
55	#368	—	2	1	no	—
56	#368	—	2	1	no	—
57	#368	—	2	1	no	—
58	#368	—	2	1	no	—
59	#368	—	2	1	no	—
60	#368	D₄	1	2	yes	—
61	#361	—	2	4	no	—
62	#338	—	4	1	no	—
63	C₂×#41	D₈	1	1	yes	characteristic
64	C₂²×D₄	—	4	1	no	—
65	AGL(1,ℤ/8)	—	4	4	no	—
66	AGL(1,ℤ/8)	—	8	2	no	—
67	C₂×SD₁₆	—	8	1	no	—
68	C₂×D₈	—	2	2	no	—
69	C₂×D₈	—	4	1	no	—
70	C₂×D₈	—	4	1	no	—
71	C₂×D₈	—	4	1	no	—
72	C₂×D₈	—	4	1	no	—
73	C₂×D₈	—	4	1	no	—
74	C₂×D₈	—	8	1	no	—
75	#131	—	2	1	no	—
76	#131	—	2	1	no	—
77	#131	—	2	1	no	—
78	#131	—	2	1	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂×M₁₆	—	4	1	no	—
82	C₂×M₁₆	—	4	1	no	—
83	C₂×M₁₆	D₈	1	1	yes	characteristic
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	D₁₆	—	2	2	no	—
87	D₁₆	—	2	2	no	—
88	D₁₆	—	4	1	no	—
89	D₁₆	—	4	1	no	—
90	M₃₂	—	2	1	no	—
91	M₃₂	—	2	1	no	—
92	M₃₂	—	4	1	no	—
93	M₃₂	—	4	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	—	4	1	no	—
97	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
98	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
99	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
100	C₂×C₁₆	#41	1	1	yes	characteristic
101	C₂×C₁₆	#41	1	1	yes	characteristic
102	C₂×C₁₆	#41	1	1	yes	characteristic
103	#108	—	2	2	no	—
104	#108	—	4	1	no	—
105	#108	—	4	1	no	—
106	#108	—	4	1	no	—
107	#108	—	4	1	no	—
108	#108	—	4	1	no	—
109	#101	—	8	1	no	—
110	#100	—	4	4	no	—
111	#100	—	8	1	no	—
112	C₃₂	—	2	2	no	—
113	C₃₂	—	4	1	no	—
114	C₂⁴	—	8	1	no	—
115	#41	—	2	1	no	—
116	#41	—	2	1	no	—
117	#41	—	2	1	no	—
118	#41	—	2	1	no	—
119	C₂×Q₈	D₁₆	1	1	yes	characteristic
120	C₂×D₄	—	2	1	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	8	1	no	—
123	C₂×D₄	—	8	1	no	—
124	C₂×D₄	—	8	1	no	—
125	C₂×D₄	—	8	1	no	—
126	C₂×D₄	—	8	1	no	—
127	C₂×D₄	D₁₆	1	1	yes	characteristic
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
130	SD₁₆	—	8	4	no	—
131	D₈	—	4	2	no	—
132	D₈	—	4	2	no	—
133	D₈	—	8	4	no	—
134	D₈	—	8	1	no	—
135	D₈	—	8	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	M₁₆	—	4	2	no	—
141	M₁₆	—	8	1	no	—
142	M₁₆	—	8	1	no	—
143	M₁₆	—	8	1	no	—
144	M₁₆	—	8	1	no	—
145	M₁₆	—	8	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	8	1	no	—
150	C₂×C₈	—	8	1	no	—
151	C₂×C₈	—	8	1	no	—
152	C₂×C₈	#120	1	1	yes	characteristic
153	C₂×C₈	#121	1	1	yes	characteristic
154	C₂×C₈	#123	1	1	yes	characteristic
155	C₂×C₈	#135	1	1	yes	characteristic
156	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₈	#137	1	1	yes	characteristic
158	C₁₆	—	4	1	no	—
159	C₁₆	#121	1	1	yes	characteristic
160	C₁₆	#121	1	1	yes	characteristic
161	C₁₆	#123	1	1	yes	characteristic
162	C₁₆	#124	1	1	yes	characteristic
163	C₂³	—	2	1	no	—
164	C₂³	—	8	2	no	—
165	C₂³	—	16	1	no	—
166	C₂³	—	16	1	no	—
167	C₂³	—	32	2	no	—
168	C₂³	#510	1	1	yes	characteristic
169	Q₈	—	2	1	no	—
170	Q₈	—	2	1	no	—
171	D₄	—	2	1	no	—
172	D₄	—	2	1	no	—
173	D₄	—	4	1	no	—
174	D₄	—	4	1	no	—
175	D₄	—	8	2	no	—
176	D₄	—	8	2	no	—
177	D₄	—	16	1	no	—
178	D₄	—	16	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#447	1	1	yes	characteristic
183	C₂×C₄	#466	1	1	yes	characteristic
184	C₂×C₄	#480	1	1	yes	characteristic
185	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
186	C₂×C₄	#508	1	1	yes	characteristic
187	C₂×C₄	#509	1	1	yes	characteristic
188	C₈	—	4	1	no	—
189	C₈	—	8	2	no	—
190	C₈	—	16	1	no	—
191	C₈	—	16	1	no	—
192	C₈	#394	1	1	yes	characteristic
193	C₈	#452	1	1	yes	characteristic
194	C₈	#468	1	1	yes	characteristic
195	C₈	#482	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	8	1	no	—
199	C₂²	—	16	2	no	—
200	C₂²	—	32	2	no	—
201	C₂²	—	32	2	no	—
202	C₂²	—	32	2	no	—
203	C₂²	#2189	1	1	yes	characteristic
204	C₂²	#2220	1	1	yes	characteristic
205	C₂²	#2232	1	1	yes	characteristic
206	C₄	—	4	1	no	—
207	C₄	#2033	1	1	yes	characteristic
208	C₄	#2184	1	1	yes	characteristic
209	C₄	#2215	1	1	yes	characteristic
210	C₄	#2234	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	—	8	1	no	—
215	C₂	—	32	2	no	—
216	C₂	#13380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#13380
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13636	C₂	semidirect
3. C₂×#1419	C₂	semidirect
4. C₂×#1419	C₂	semidirect
5. #13558	C₂	semidirect
6. #12840	C₂	semidirect
7. #7479	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2221	2
#13636	2
C₂×#1419	2
#13558	2
#12840	2
#7479	2