Finite Group Databasebeta

Subgroups — #140288

Order 512.

238 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13262	C₂	1	1	yes	maximal, characteristic
2	#13212	C₂	1	1	yes	maximal, characteristic
3	#13188	C₂	1	1	yes	maximal, characteristic
4	#12667	C₂	1	1	yes	maximal, characteristic
5	C₂×#1358	C₂	1	1	yes	maximal, characteristic
6	#7270	C₂	1	1	yes	maximal, characteristic
7	#7162	C₂	1	1	yes	maximal, characteristic
8	C₂×#494	C₂²	1	1	yes	characteristic
9	#2155	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#362	—	2	1	no	—
14	C₂×#362	—	2	1	no	—
15	C₂×#362	—	2	1	no	—
16	#2141	—	2	1	no	—
17	#2110	—	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	#1358	C₄	1	4	yes	complemented
24	C₂²×Q₁₆	—	2	1	no	—
25	C₂²×Q₁₆	—	2	1	no	—
26	C₂×C₄⋊Q₈	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	—	2	1	no	—
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	#494	—	2	1	no	—
33	#494	—	2	1	no	—
34	#494	—	2	1	no	—
35	#494	C₂×C₄	1	2	yes	—
36	C₂×#108	—	2	1	no	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	D₄	1	1	yes	characteristic
39	#427	D₄	1	1	yes	characteristic
40	#423	—	2	1	no	—
41	#405	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#365	—	4	2	no	—
44	#364	—	2	1	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	C₂×C₄	1	2	yes	—
49	#364	C₂×C₄	1	2	yes	—
50	#362	—	4	2	no	—
51	#362	—	4	2	no	—
52	#362	—	4	2	no	—
53	#358	—	4	2	no	—
54	#347	—	4	2	no	—
55	#346	—	2	1	no	—
56	C₄×C₁₆	—	2	1	no	—
57	#339	—	2	1	no	—
58	C₂²×Q₈	—	4	1	no	—
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×Q₁₆	—	2	2	no	—
61	C₂×Q₁₆	—	4	1	no	—
62	C₂×Q₁₆	—	4	1	no	—
63	C₂×Q₁₆	—	4	1	no	—
64	C₂×Q₁₆	—	4	1	no	—
65	C₂×Q₁₆	—	4	1	no	—
66	C₂×Q₁₆	—	4	1	no	—
67	C₂×Q₁₆	—	4	1	no	—
68	C₂×Q₁₆	—	4	1	no	—
69	C₂×Q₁₆	—	4	1	no	—
70	C₂×Q₁₆	—	4	1	no	—
71	C₂×M₁₆	—	2	1	no	—
72	C₂×M₁₆	—	4	1	no	—
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	8	1	no	—
78	#117	—	2	1	no	—
79	C₂×C₄⋊C₄	—	8	1	no	—
80	C₂×C₄²	#41	1	1	yes	characteristic
81	M₃₂	—	2	1	no	—
82	M₃₂	—	2	2	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	SD₁₆	1	2	yes	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	4	1	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	#108	—	2	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	#107	—	2	1	no	—
103	#106	—	2	1	no	—
104	C₄⋊C₈	—	8	1	no	—
105	#97	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₄×C₈	#41	1	1	yes	characteristic
109	C₄×C₈	#41	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	8	1	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	1	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	8	1	no	—
119	C₂²×C₄	#121	1	1	yes	characteristic
120	Q₁₆	—	4	2	no	—
121	Q₁₆	—	4	2	no	—
122	Q₁₆	—	4	2	no	—
123	Q₁₆	—	4	2	no	—
124	Q₁₆	—	8	1	no	—
125	Q₁₆	—	8	1	no	—
126	Q₁₆	—	8	1	no	—
127	Q₁₆	—	8	1	no	—
128	M₁₆	—	4	1	no	—
129	M₁₆	—	4	1	no	—
130	M₁₆	—	8	1	no	—
131	M₁₆	—	8	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	8	1	no	—
134	C₂×C₈	C₄×D₄	1	1	yes	characteristic
135	C₂×C₈	C₄×D₄	1	1	yes	characteristic
136	C₂×C₈	#123	1	1	yes	characteristic
137	C₂×C₈	#124	1	1	yes	characteristic
138	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₈	#135	1	1	yes	characteristic
140	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₈	#137	1	1	yes	characteristic
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	8	2	no	—
149	C₂²⋊C₄	—	4	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	4	1	no	—
152	C₄²	#117	1	1	yes	characteristic
153	C₄²	#126	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	—	4	2	no	—
156	C₁₆	—	4	2	no	—
157	C₁₆	—	4	1	no	—
158	C₁₆	—	8	1	no	—
159	C₂³	#471	1	1	yes	characteristic
160	Q₈	—	8	2	no	—
161	Q₈	—	8	2	no	—
162	Q₈	—	16	1	no	—
163	Q₈	—	16	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	16	1	no	—
170	C₂×C₄	—	16	1	no	—
171	C₂×C₄	#388	1	1	yes	characteristic
172	C₂×C₄	#460	1	1	yes	characteristic
173	C₂×C₄	#463	1	1	yes	characteristic
174	C₂×C₄	#468	1	1	yes	characteristic
175	C₂×C₄	#469	1	1	yes	characteristic
176	C₂×C₄	#472	1	1	yes	characteristic
177	C₈	—	8	1	no	—
178	C₈	—	16	1	no	—
179	C₈	C₄×SD₁₆	1	1	yes	characteristic
180	C₈	#440	1	1	yes	characteristic
181	C₈	#442	1	1	yes	characteristic
182	C₈	#442	1	1	yes	characteristic
183	C₈	#481	1	1	yes	characteristic
184	C₈	#482	1	1	yes	characteristic
185	C₈	#489	1	1	yes	characteristic
186	C₈	#490	1	1	yes	characteristic
187	C₂²	—	4	1	no	—
188	C₂²	#1686	1	1	yes	characteristic
189	C₂²	#1701	1	1	yes	characteristic
190	C₂²	#1931	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	16	1	no	—
194	C₄	—	16	2	no	—
195	C₄	#1688	1	1	yes	characteristic
196	C₄	#1697	1	1	yes	characteristic
197	C₄	#1926	1	1	yes	characteristic
198	C₄	#1936	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	#9572	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#9572
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ 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
23. #1358	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13262	2
#13212	2
#13188	2
#12667	2
C₂×#1358	2
#7270	2
#7162	2