Finite Group Databasebeta

Subgroups — #140331

Order 512.

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

normal subgroups only

This group has 208 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	#11953	C₂	1	1	yes	maximal, characteristic
2	#11681	C₂	1	1	yes	maximal, characteristic
3	#11414	C₂	1	1	yes	maximal, characteristic
4	#9720	C₂	1	1	yes	maximal, characteristic
5	#9635	C₂	1	1	yes	maximal, characteristic
6	#7302	C₂	1	1	yes	maximal, characteristic
7	#7175	C₂	1	1	yes	maximal, characteristic
8	#3170	C₂²	1	1	yes	characteristic
9	#2163	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	#1986	—	2	1	no	—
12	C₄×#106	C₂²	1	1	yes	characteristic
13	#1574	—	2	1	no	—
14	#1566	C₂²	1	1	yes	characteristic
15	#1564	C₂²	1	1	yes	characteristic
16	#1503	—	2	1	no	—
17	#1451	C₂²	1	1	yes	characteristic
18	#1365	C₄	1	1	yes	characteristic, complemented
19	#1365	C₄	1	1	yes	characteristic, complemented
20	#1362	C₄	1	1	yes	characteristic, complemented
21	#1362	C₄	1	1	yes	characteristic, complemented
22	#1334	—	2	1	no	—
23	#1334	—	2	1	no	—
24	#1313	—	2	1	no	—
25	#1313	—	2	1	no	—
26	#1299	—	2	1	no	—
27	#533	—	2	1	no	—
28	C₂×M₃₂	—	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	—	2	1	no	—
34	C₄×Q₁₆	—	2	1	no	—
35	C₄×Q₁₆	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	D₄	1	1	yes	characteristic
38	C₂×#106	—	2	1	no	—
39	#423	—	2	1	no	—
40	C₄×M₁₆	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄×C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂²⋊C₁₆	—	2	1	no	—
46	#346	—	2	1	no	—
47	#346	D₄	1	1	yes	characteristic
48	#346	D₄	1	1	yes	characteristic
49	C₄×C₁₆	—	2	1	no	—
50	#340	—	2	1	no	—
51	#339	—	4	2	no	—
52	#335	—	4	1	no	—
53	#335	C₂×C₄	1	1	yes	characteristic
54	#335	C₂×C₄	1	1	yes	characteristic
55	#334	—	2	1	no	—
56	#334	—	2	1	no	—
57	#334	—	2	1	no	—
58	#334	—	4	1	no	—
59	#328	—	4	2	no	—
60	#322	—	2	1	no	—
61	#322	—	2	1	no	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×M₁₆	—	2	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂²×C₈	#41	1	1	yes	characteristic
67	C₂²×C₈	#41	1	1	yes	characteristic
68	C₄⋊Q₈	—	2	1	no	—
69	C₄⋊Q₈	—	2	1	no	—
70	#125	—	4	1	no	—
71	C₂²⋊Q₈	—	4	1	no	—
72	C₂²⋊Q₈	—	4	1	no	—
73	C₄×Q₈	—	4	1	no	—
74	C₄×Q₈	—	4	1	no	—
75	#117	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄²	—	4	1	no	—
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	M₃₂	—	4	2	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	#107	—	2	1	no	—
85	#107	—	2	1	no	—
86	#106	—	2	2	no	—
87	#106	—	4	1	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	#103	—	4	1	no	—
94	#103	—	4	1	no	—
95	C₂²⋊C₈	—	2	1	no	—
96	C₂²⋊C₈	—	4	1	no	—
97	#97	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₄×C₈	C₂×D₄	1	1	yes	characteristic
103	C₄×C₈	#41	1	1	yes	characteristic
104	C₄×C₈	#41	1	1	yes	characteristic
105	#95	—	4	1	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	#124	1	1	yes	characteristic
110	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
112	Q₁₆	—	4	1	no	—
113	Q₁₆	—	4	1	no	—
114	Q₁₆	—	8	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	M₁₆	—	16	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	#117	1	1	yes	characteristic
127	C₂×C₈	C₄×D₄	1	1	yes	characteristic
128	C₂×C₈	#123	1	1	yes	characteristic
129	C₂×C₈	#126	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	8	1	no	—
136	C₄⋊C₄	—	8	1	no	—
137	C₄⋊C₄	—	8	1	no	—
138	C₂²⋊C₄	—	8	1	no	—
139	C₄²	—	8	1	no	—
140	C₄²	—	8	1	no	—
141	C₄²	—	8	1	no	—
142	C₄²	C₄×D₄	1	1	yes	characteristic
143	C₄²	#121	1	1	yes	characteristic
144	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₄²	#137	1	1	yes	characteristic
146	C₁₆	—	2	2	no	—
147	C₁₆	—	4	1	no	—
148	C₁₆	—	4	1	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	#490	1	1	yes	characteristic
151	Q₈	—	8	1	no	—
152	Q₈	—	8	1	no	—
153	Q₈	—	16	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	16	1	no	—
163	C₂×C₄	—	16	1	no	—
164	C₂×C₄	#388	1	1	yes	characteristic
165	C₂×C₄	#440	1	1	yes	characteristic
166	C₂×C₄	#442	1	1	yes	characteristic
167	C₂×C₄	#468	1	1	yes	characteristic
168	C₂×C₄	#469	1	1	yes	characteristic
169	C₂×C₄	#489	1	1	yes	characteristic
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	8	1	no	—
174	C₈	—	8	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	8	1	no	—
177	C₈	—	16	1	no	—
178	C₈	#443	1	1	yes	characteristic
179	C₈	#444	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#1936	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	16	2	no	—
189	C₄	—	16	1	no	—
190	C₄	—	16	1	no	—
191	C₄	#1929	1	1	yes	characteristic
192	C₄	#1932	1	1	yes	characteristic
193	C₄	#2180	1	1	yes	characteristic
194	C₄	#2180	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	#9592	1	1	yes	characteristic
197	C₂	#13154	1	1	yes	characteristic
198	C₂	#13156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#490
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ 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
18. #1365	C₄	semidirect
19. #1365	C₄	semidirect
20. #1362	C₄	semidirect
21. #1362	C₄	semidirect

Maximal subgroups

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

subgroup	index
#11953	2
#11681	2
#11414	2
#9720	2
#9635	2
#7302	2
#7175	2