Finite Group Databasebeta

Subgroups — #146259

Order 512.

273 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13211	C₂	1	1	yes	maximal, characteristic
2	#13208	C₂	1	1	yes	maximal, characteristic
3	#12904	C₂	1	1	yes	maximal, characteristic, complemented
4	#12889	C₂	1	1	yes	maximal, characteristic, complemented
5	#9756	C₂	1	1	yes	maximal, characteristic
6	#8793	C₂	1	1	yes	maximal, characteristic, complemented
7	#7141	C₂	1	1	yes	maximal, characteristic, complemented
8	#3147	C₂²	1	1	yes	characteristic
9	C₂×#365	—	2	2	no	—
10	C₂×#360	—	2	2	no	—
11	C₂×#359	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	1	yes	characteristic
15	#1846	C₂²	1	1	yes	characteristic
16	#1846	C₂²	1	1	yes	characteristic
17	#1517	—	2	2	no	—
18	#1450	C₂²	1	1	yes	characteristic
19	#1380	—	2	2	no	—
20	#1286	—	2	2	no	—
21	C₂×#135	—	2	1	no	—
22	C₂×#135	—	2	1	no	—
23	#532	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	2	yes	—
26	#495	—	2	1	no	—
27	#495	—	2	1	no	—
28	#494	Q₈	1	1	yes	characteristic
29	#493	Q₈	1	1	yes	characteristic
30	#492	—	2	1	no	—
31	#431	D₄	1	2	yes	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	C₂×#106	—	2	1	no	—
38	#423	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#382	—	2	1	no	—
41	#382	—	2	1	no	—
42	#365	—	4	4	no	—
43	#360	—	4	4	no	—
44	#359	—	4	4	no	—
45	#346	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	#327	—	4	4	no	—
50	#322	—	2	1	no	—
51	C₈²	—	2	1	no	—
52	C₂×#41	—	4	1	no	—
53	#135	—	4	2	no	—
54	#135	—	4	2	no	—
55	#135	—	4	2	no	—
56	#135	—	4	2	no	—
57	C₂×Q₁₆	—	2	1	no	—
58	C₂×Q₁₆	—	2	1	no	—
59	C₂×SD₁₆	—	4	1	no	—
60	C₂×SD₁₆	—	4	1	no	—
61	C₂×D₈	—	2	1	no	—
62	C₂×D₈	—	2	1	no	—
63	C₂×M₁₆	—	4	1	no	—
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	1	no	—
67	#127	—	2	1	no	—
68	#124	—	4	1	no	—
69	#121	—	4	2	no	—
70	C₄×D₄	—	4	2	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	M₃₂	SD₁₆	1	4	yes	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	#108	—	4	2	no	—
81	#108	—	4	2	no	—
82	#107	—	4	2	no	—
83	#107	—	4	2	no	—
84	#106	—	4	2	no	—
85	#106	—	4	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	#41	—	8	1	no	—
97	#41	—	8	1	no	—
98	#41	—	8	1	no	—
99	#41	—	8	1	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	8	1	no	—
104	C₂²×C₄	—	8	1	no	—
105	C₂²×C₄	—	8	1	no	—
106	C₂²×C₄	—	8	1	no	—
107	C₂²×C₄	#127	1	1	yes	characteristic
108	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
109	Q₁₆	—	4	2	no	—
110	Q₁₆	—	4	2	no	—
111	SD₁₆	—	8	2	no	—
112	SD₁₆	—	8	2	no	—
113	D₈	—	4	2	no	—
114	D₈	—	4	2	no	—
115	M₁₆	—	8	1	no	—
116	M₁₆	—	8	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	8	1	no	—
119	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₈	#123	1	1	yes	characteristic
122	C₂×C₈	#123	1	1	yes	characteristic
123	C₂×C₈	C₂×SD₁₆	1	2	yes	—
124	C₂×C₈	#135	1	2	yes	—
125	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	8	1	no	—
129	C₄⋊C₄	—	8	1	no	—
130	C₄⋊C₄	—	8	1	no	—
131	C₄⋊C₄	—	8	1	no	—
132	C₂²⋊C₄	—	8	2	no	—
133	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
134	C₄²	#123	1	1	yes	characteristic
135	C₄²	#135	1	2	yes	—
136	C₁₆	—	2	4	no	—
137	C₁₆	—	4	2	no	—
138	C₂³	—	8	1	no	—
139	C₂³	#496	1	1	yes	characteristic
140	Q₈	—	8	1	no	—
141	Q₈	—	8	1	no	—
142	D₄	—	8	1	no	—
143	D₄	—	8	1	no	—
144	D₄	—	16	2	no	—
145	D₄	—	16	1	no	—
146	D₄	—	16	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	16	1	no	—
150	C₂×C₄	—	16	1	no	—
151	C₂×C₄	—	16	2	no	—
152	C₂×C₄	—	16	2	no	—
153	C₂×C₄	—	16	1	no	—
154	C₂×C₄	—	16	1	no	—
155	C₂×C₄	#399	1	1	yes	characteristic
156	C₂×C₄	#478	1	2	yes	—
157	C₂×C₄	#482	1	2	yes	—
158	C₂×C₄	#492	1	1	yes	characteristic
159	C₂×C₄	#495	1	1	yes	characteristic
160	C₂×C₄	#495	1	1	yes	characteristic
161	C₂×C₄	#496	1	1	yes	characteristic
162	C₂×C₄	#496	1	1	yes	characteristic
163	C₈	—	8	2	no	—
164	C₈	—	16	1	no	—
165	C₈	#476	1	2	yes	—
166	C₈	#478	1	2	yes	—
167	C₈	#481	1	2	yes	—
168	C₈	#482	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	16	2	no	—
172	C₂²	#1715	1	1	yes	characteristic
173	C₂²	#1717	1	1	yes	characteristic
174	C₂²	#2092	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	16	1	no	—
178	C₄	—	32	1	no	—
179	C₄	—	32	1	no	—
180	C₄	#1708	1	1	yes	characteristic
181	C₄	#1717	1	1	yes	characteristic
182	C₄	#2092	1	1	yes	characteristic
183	C₄	#2094	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	—	32	1	no	—
187	C₂	#10677	1	1	yes	characteristic
188	C₂	#13525	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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. #12904	C₂	semidirect
4. #12889	C₂	semidirect
6. #8793	C₂	semidirect
7. #7141	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13211	2
#13208	2
#12904	2
#12889	2
#9756	2
#8793	2
#7141	2