Finite Group Databasebeta

Subgroups — #129015

Order 512.

296 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 94 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	#17962	C₂	1	1	yes	maximal, characteristic
2	#11951	C₂	1	1	yes	maximal, characteristic
3	#11751	C₂	1	1	yes	maximal, characteristic
4	#11415	C₂	1	1	yes	maximal, characteristic, complemented
5	#8295	C₂	1	1	yes	maximal, characteristic, complemented
6	#7300	C₂	1	1	yes	maximal, characteristic, complemented
7	#7295	C₂	1	1	yes	maximal, characteristic, complemented
8	#3169	C₂²	1	1	yes	characteristic
9	#2931	—	2	1	no	—
10	#2163	C₂²	1	1	yes	characteristic
11	#2163	C₂²	1	1	yes	characteristic
12	C₄×#108	C₂²	1	1	yes	characteristic
13	#1594	—	2	1	no	—
14	#1594	C₄	1	2	yes	—
15	#1580	—	2	1	no	—
16	#1580	C₄	1	2	yes	—
17	#1566	C₂²	1	1	yes	characteristic
18	#1564	C₂²	1	1	yes	characteristic
19	#1492	—	2	1	no	—
20	#1451	—	2	1	no	—
21	C₂×#322	C₂²	1	1	yes	characteristic
22	#1364	C₄	1	2	yes	—
23	#1361	C₄	1	2	yes	—
24	#1349	C₄	1	4	yes	—
25	#1313	—	2	1	no	—
26	#1313	—	2	1	no	—
27	#1278	—	2	1	no	—
28	C₂×#131	—	4	1	no	—
29	#532	—	2	1	no	—
30	#500	C₂×C₄	1	1	yes	characteristic
31	#493	C₂×C₄	1	1	yes	characteristic
32	#486	—	2	1	no	—
33	#466	—	2	1	no	—
34	#466	C₂×C₄	1	2	yes	—
35	C₄×D₈	—	2	1	no	—
36	C₄×D₈	C₂×C₄	1	2	yes	—
37	#436	—	2	2	no	—
38	#436	—	4	1	no	—
39	C₈×D₄	—	2	2	no	—
40	C₈×D₄	—	4	1	no	—
41	#431	—	2	1	no	—
42	C₂×#108	—	2	1	no	—
43	#423	—	2	1	no	—
44	#408	—	4	1	no	—
45	C₄×M₁₆	—	2	1	no	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₂²⋊C₁₆	—	2	1	no	—
52	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
53	#346	—	2	1	no	—
54	#346	—	2	1	no	—
55	#341	C₂×C₄	1	2	yes	—
56	#335	C₂×C₄	1	2	yes	—
57	#334	—	2	1	no	—
58	#334	—	2	1	no	—
59	#334	C₂×C₄	1	2	yes	—
60	D₄⋊C₈	—	2	2	no	—
61	D₄⋊C₈	—	2	2	no	—
62	#322	—	4	2	no	—
63	#322	C₂×C₄	1	1	yes	characteristic
64	#322	C₂×C₄	1	1	yes	characteristic
65	#322	D₄	1	1	yes	characteristic
66	#322	D₄	1	1	yes	characteristic
67	C₂×#41	—	4	1	no	—
68	C₂×D₈	—	2	1	no	—
69	C₂×D₈	C₄²	1	2	yes	—
70	#131	—	8	4	no	—
71	C₂×M₁₆	—	2	2	no	—
72	C₂×M₁₆	—	4	2	no	—
73	C₂×M₁₆	—	4	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	#41	1	1	yes	characteristic
79	C₄⋊Q₈	—	2	1	no	—
80	#127	—	2	1	no	—
81	#124	—	4	1	no	—
82	#121	—	2	2	no	—
83	#121	—	4	1	no	—
84	C₄×D₄	—	2	2	no	—
85	C₄×D₄	—	4	1	no	—
86	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	#108	—	4	2	no	—
90	#107	—	2	1	no	—
91	#107	C₄²	1	2	yes	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	4	1	no	—
95	#102	—	2	2	no	—
96	#102	—	2	2	no	—
97	C₂²⋊C₈	—	4	2	no	—
98	C₂²⋊C₈	—	4	2	no	—
99	C₂²⋊C₈	—	4	1	no	—
100	#97	—	4	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	#41	—	8	2	no	—
114	#41	—	8	2	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	4	1	no	—
119	C₂×D₄	—	4	1	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂²×C₄	#137	1	1	yes	characteristic
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	M₁₆	—	4	2	no	—
127	M₁₆	—	4	2	no	—
128	M₁₆	—	8	4	no	—
129	M₁₆	—	8	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	8	1	no	—
141	C₂×C₈	—	8	4	no	—
142	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₈	#117	1	1	yes	characteristic
144	C₂×C₈	C₄×D₄	1	1	yes	characteristic
145	C₂×C₈	C₄×D₄	1	1	yes	characteristic
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₂²⋊C₄	—	4	2	no	—
149	C₂²⋊C₄	—	4	2	no	—
150	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₄²	C₄×D₄	1	1	yes	characteristic
152	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₁₆	—	4	2	no	—
155	C₁₆	—	4	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	#440	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	D₄	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	4	no	—
163	D₄	—	8	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₄	#417	1	1	yes	characteristic
173	C₂×C₄	#418	1	1	yes	characteristic
174	C₂×C₄	#440	1	1	yes	characteristic
175	C₂×C₄	#442	1	1	yes	characteristic
176	C₂×C₄	#442	1	1	yes	characteristic
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	1	no	—
184	C₈	—	8	2	no	—
185	C₈	—	8	2	no	—
186	C₈	—	8	2	no	—
187	C₈	#444	1	2	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	8	2	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	#1762	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	#1757	1	1	yes	characteristic
201	C₄	#1765	1	1	yes	characteristic
202	C₄	AGL(1,ℤ/16)	1	2	yes	—
203	C₂	—	4	1	no	—
204	C₂	—	16	2	no	—
205	C₂	#8253	1	1	yes	characteristic
206	C₂	#12647	1	1	yes	characteristic
207	C₂	#12651	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1762
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
4. #11415	C₂	semidirect
5. #8295	C₂	semidirect
6. #7300	C₂	semidirect
7. #7295	C₂	semidirect

Maximal subgroups

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

subgroup	index
#17962	2
#11951	2
#11751	2
#11415	2
#8295	2
#7300	2
#7295	2