Finite Group Databasebeta

Subgroups — #141242

Order 512.

323 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 102 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#13278	C₂	1	2	yes	maximal
3	#12004	C₂	1	1	yes	maximal, characteristic
4	#11936	C₂	1	2	yes	maximal
5	C₈×#106	C₂	1	1	yes	maximal, characteristic
6	#3170	—	2	1	no	—
7	#3170	—	2	1	no	—
8	#3160	C₂²	1	2	yes	—
9	#2150	C₂²	1	2	yes	—
10	C₄×#106	C₂²	1	1	yes	characteristic
11	C₈×C₄⋊C₄	—	2	1	no	—
12	#1713	C₄	1	1	yes	characteristic, complemented
13	#1713	C₄	1	1	yes	characteristic, complemented
14	#1710	C₄	1	2	yes	complemented
15	#1707	—	2	1	no	—
16	#1707	—	2	1	no	—
17	C₂×#334	C₂²	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1386	—	2	4	no	—
20	#1370	—	2	1	no	—
21	#1370	—	2	1	no	—
22	#1363	—	2	4	no	—
23	#1279	—	2	1	no	—
24	#533	—	2	2	no	—
25	C₂×M₃₂	—	2	2	no	—
26	#500	—	2	1	no	—
27	#500	—	2	1	no	—
28	#500	C₂×C₄	1	2	yes	—
29	#500	C₂×C₄	1	2	yes	—
30	#499	—	2	2	no	—
31	#499	—	2	2	no	—
32	#498	—	2	2	no	—
33	#494	—	2	1	no	—
34	#494	—	2	1	no	—
35	#487	—	2	2	no	—
36	#487	—	2	2	no	—
37	#467	—	4	2	no	—
38	#467	—	4	1	no	—
39	#467	—	4	1	no	—
40	C₄×Q₁₆	—	4	2	no	—
41	C₄×Q₁₆	—	4	1	no	—
42	C₄×Q₁₆	—	4	1	no	—
43	#427	—	2	2	no	—
44	#427	—	2	2	no	—
45	C₂×#106	D₄	1	2	yes	—
46	C₂×C₄⋊C₈	—	2	1	no	—
47	C₂×C₄×C₈	—	4	1	no	—
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₁₆	—	2	2	no	—
53	C₄⋊C₁₆	—	2	2	no	—
54	#340	—	2	1	no	—
55	#336	—	4	1	no	—
56	#334	—	2	1	no	—
57	#334	D₄	1	2	yes	—
58	C₈²	C₂×C₄	1	1	yes	characteristic
59	C₈²	C₂×C₄	1	1	yes	characteristic
60	C₈²	D₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₂×Q₁₆	—	4	2	no	—
63	C₂×Q₁₆	—	4	1	no	—
64	C₂×Q₁₆	—	4	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	4	1	no	—
67	C₂²×C₈	D₈	1	1	yes	characteristic
68	C₂²×C₈	D₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	#41	1	1	yes	characteristic
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	#125	—	4	2	no	—
74	C₂²⋊Q₈	—	4	2	no	—
75	C₂²⋊Q₈	—	4	2	no	—
76	C₄×Q₈	—	4	2	no	—
77	C₄×Q₈	—	4	2	no	—
78	#117	—	4	2	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄²	—	4	1	no	—
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	M₃₂	—	4	4	no	—
83	C₂×C₁₆	—	4	2	no	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	4	2	no	—
89	#107	—	4	1	no	—
90	#107	—	4	1	no	—
91	#106	—	2	2	no	—
92	#106	—	4	2	no	—
93	#106	—	4	2	no	—
94	#106	Q₁₆	1	4	yes	complemented
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	4	1	no	—
97	#103	—	4	4	no	—
98	#103	—	4	2	no	—
99	#103	—	4	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	8	2	no	—
102	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈	SD₁₆	1	1	yes	characteristic
107	C₄×C₈	SD₁₆	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	#95	—	4	1	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	#121	1	1	yes	characteristic
117	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
118	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	Q₁₆	—	8	4	no	—
120	Q₁₆	—	8	2	no	—
121	Q₁₆	—	8	2	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	1	no	—
130	C₂×C₈	—	8	1	no	—
131	C₂×C₈	#102	1	1	yes	characteristic
132	C₂×C₈	#102	1	1	yes	characteristic
133	C₂×C₈	#102	1	1	yes	characteristic
134	C₂×C₈	#102	1	1	yes	characteristic
135	C₂×C₈	C₄×D₄	1	1	yes	characteristic
136	C₂×C₈	C₄×D₄	1	1	yes	characteristic
137	C₂×C₈	#124	1	1	yes	characteristic
138	C₂×C₈	#127	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₁₆	1	2	yes	—
140	C₂×C₈	#135	1	2	yes	—
141	C₄⋊C₄	—	2	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	2	no	—
148	C₄⋊C₄	—	8	2	no	—
149	C₂²⋊C₄	—	8	2	no	—
150	C₄²	—	8	1	no	—
151	C₄²	—	8	1	no	—
152	C₄²	—	8	2	no	—
153	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₄²	#121	1	1	yes	characteristic
155	C₄²	C₂×SD₁₆	1	1	yes	characteristic
156	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₁₆	—	8	2	no	—
158	C₂³	#459	1	1	yes	characteristic
159	Q₈	—	8	4	no	—
160	Q₈	—	8	2	no	—
161	Q₈	—	8	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	#390	1	1	yes	characteristic
171	C₂×C₄	C₂×#102	1	1	yes	characteristic
172	C₂×C₄	#418	1	1	yes	characteristic
173	C₂×C₄	#459	1	1	yes	characteristic
174	C₂×C₄	#460	1	1	yes	characteristic
175	C₂×C₄	#460	1	1	yes	characteristic
176	C₂×C₄	#467	1	2	yes	—
177	C₂×C₄	#473	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	4	1	no	—
185	C₈	—	8	2	no	—
186	C₈	#360	1	2	yes	—
187	C₈	C₄×Q₁₆	1	2	yes	—
188	C₈	#494	1	1	yes	characteristic
189	C₈	#495	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#1669	1	2	yes	—
193	C₂²	#1982	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	—	16	2	no	—
199	C₄	—	16	2	no	—
200	C₄	#1670	1	2	yes	—
201	C₄	#1969	1	1	yes	characteristic
202	C₄	#1979	1	1	yes	characteristic
203	C₄	C₂×#360	1	1	yes	characteristic
204	C₄	#2145	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	#9733	1	1	yes	characteristic
207	C₂	#13255	1	1	yes	characteristic
208	C₂	#13259	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
12. #1713	C₄	semidirect
13. #1713	C₄	semidirect
14. #1710	C₄	semidirect
94. #106	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#19604	2
#13278	2
#12004	2
#11936	2
C₈×#106	2