Finite Group Databasebeta

Subgroups — #145954

Order 512.

264 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 74 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	#13261	C₂	1	1	yes	maximal, characteristic
2	#13213	C₂	1	1	yes	maximal, characteristic
3	#13188	C₂	1	1	yes	maximal, characteristic, complemented
4	#13164	C₂	1	1	yes	maximal, characteristic, complemented
5	#12002	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#1357	C₂	1	1	yes	maximal, characteristic
7	#7270	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#493	C₂²	1	1	yes	characteristic
9	#2155	—	2	1	no	—
10	#2153	—	2	1	no	—
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#364	C₂²	1	1	yes	characteristic
14	C₂×#361	—	2	1	no	—
15	C₂×#361	—	2	1	no	—
16	C₂×#361	—	2	1	no	—
17	#2140	—	2	1	no	—
18	C₄×M₃₂	C₂²	1	1	yes	characteristic
19	#1848	C₂²	1	1	yes	characteristic
20	#1772	C₂²	1	1	yes	characteristic
21	#1388	—	2	1	no	—
22	#1378	—	2	1	no	—
23	#1369	—	2	2	no	—
24	#1357	—	2	2	no	—
25	C₂²×D₈	—	2	1	no	—
26	C₂²×D₈	—	2	1	no	—
27	C₂×#127	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	—	2	1	no	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	C₂×M₃₂	D₄	1	1	yes	characteristic
33	#493	—	2	1	no	—
34	#493	—	2	1	no	—
35	#493	—	2	1	no	—
36	#493	Q₈	1	2	yes	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	D₄	1	1	yes	characteristic
40	#427	D₄	1	1	yes	characteristic
41	#423	—	2	1	no	—
42	#405	—	2	1	no	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#365	—	4	2	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	—	2	1	no	—
49	#364	—	2	1	no	—
50	#364	—	4	2	no	—
51	#364	D₄	1	2	yes	—
52	#361	—	4	2	no	—
53	#361	—	4	2	no	—
54	#361	—	4	2	no	—
55	#357	—	4	2	no	—
56	#346	—	2	1	no	—
57	C₄×C₁₆	—	2	1	no	—
58	#339	—	2	1	no	—
59	C₂²×D₄	—	4	1	no	—
60	C₂²×D₄	—	8	1	no	—
61	C₂×D₈	—	2	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	4	1	no	—
64	C₂×D₈	—	4	1	no	—
65	C₂×D₈	—	4	1	no	—
66	C₂×D₈	—	4	1	no	—
67	C₂×D₈	—	4	1	no	—
68	C₂×D₈	—	4	1	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×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	4	1	no	—
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	#127	—	2	2	no	—
78	#127	—	4	1	no	—
79	#127	—	8	1	no	—
80	#117	—	2	1	no	—
81	C₂×C₄²	#41	1	1	yes	characteristic
82	M₃₂	—	2	1	no	—
83	M₃₂	—	2	2	no	—
84	M₃₂	—	2	1	no	—
85	M₃₂	—	2	1	no	—
86	M₃₂	—	4	1	no	—
87	M₃₂	—	4	1	no	—
88	M₃₂	—	4	1	no	—
89	M₃₂	—	4	1	no	—
90	M₃₂	—	4	1	no	—
91	M₃₂	SD₁₆	1	2	yes	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	—	4	1	no	—
97	#108	—	2	1	no	—
98	#108	—	4	1	no	—
99	#108	—	4	1	no	—
100	#108	—	4	1	no	—
101	#108	—	4	1	no	—
102	#108	SD₁₆	1	2	yes	—
103	#107	—	2	1	no	—
104	#106	—	2	1	no	—
105	C₄⋊C₈	—	8	1	no	—
106	#97	—	4	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	C₂×D₄	1	1	yes	characteristic
109	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₂⁴	—	8	1	no	—
113	C₂×D₄	—	4	2	no	—
114	C₂×D₄	—	8	2	no	—
115	C₂×D₄	—	8	2	no	—
116	C₂×D₄	—	8	1	no	—
117	C₂×D₄	—	8	1	no	—
118	C₂×D₄	—	8	1	no	—
119	C₂×D₄	—	8	1	no	—
120	C₂×D₄	—	8	1	no	—
121	C₂×D₄	—	16	4	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#121	1	1	yes	characteristic
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	4	2	no	—
127	D₈	—	4	2	no	—
128	D₈	—	8	1	no	—
129	D₈	—	8	1	no	—
130	D₈	—	8	1	no	—
131	D₈	—	8	1	no	—
132	M₁₆	—	2	2	no	—
133	M₁₆	—	4	1	no	—
134	M₁₆	—	8	1	no	—
135	M₁₆	—	8	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	8	1	no	—
138	C₂×C₈	#121	1	1	yes	characteristic
139	C₂×C₈	#121	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₈	#135	1	1	yes	characteristic
145	C₂×C₈	#135	1	1	yes	characteristic
146	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄⋊C₄	—	4	1	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
155	C₄²	#126	1	1	yes	characteristic
156	C₁₆	—	2	2	no	—
157	C₁₆	—	4	2	no	—
158	C₁₆	—	4	1	no	—
159	C₁₆	—	8	1	no	—
160	C₁₆	—	8	1	no	—
161	C₂³	—	8	2	no	—
162	C₂³	—	16	4	no	—
163	C₂³	—	16	1	no	—
164	C₂³	—	16	1	no	—
165	C₂³	#471	1	1	yes	characteristic
166	D₄	—	8	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	16	4	no	—
169	D₄	—	16	4	no	—
170	D₄	—	16	1	no	—
171	D₄	—	16	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	#396	1	1	yes	characteristic
177	C₂×C₄	#460	1	1	yes	characteristic
178	C₂×C₄	#463	1	1	yes	characteristic
179	C₂×C₄	#465	1	1	yes	characteristic
180	C₂×C₄	#469	1	1	yes	characteristic
181	C₂×C₄	#472	1	1	yes	characteristic
182	C₈	—	4	2	no	—
183	C₈	—	16	1	no	—
184	C₈	#460	1	1	yes	characteristic
185	C₈	#463	1	1	yes	characteristic
186	C₈	#465	1	1	yes	characteristic
187	C₈	#469	1	1	yes	characteristic
188	C₈	#476	1	1	yes	characteristic
189	C₈	#476	1	1	yes	characteristic
190	C₈	#478	1	1	yes	characteristic
191	C₈	#478	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	—	16	4	no	—
194	C₂²	—	16	2	no	—
195	C₂²	—	32	2	no	—
196	C₂²	—	32	2	no	—
197	C₂²	#1666	1	1	yes	characteristic
198	C₂²	#1701	1	1	yes	characteristic
199	C₂²	#2063	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	16	1	no	—
203	C₄	#1677	1	1	yes	characteristic
204	C₄	#1688	1	1	yes	characteristic
205	C₄	#2065	1	1	yes	characteristic
206	C₄	#2065	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	—	32	2	no	—
211	C₂	#10595	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#10595
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #13188	C₂	semidirect
4. #13164	C₂	semidirect
5. #12002	C₂	semidirect
7. #7270	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13261	2
#13213	2
#13188	2
#13164	2
#12002	2
C₂×#1357	2
#7270	2