Finite Group Databasebeta

Subgroups — #140263

Order 512.

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

normal subgroups only

This group has 213 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	#13284	C₂	1	1	yes	maximal, characteristic
2	#13273	C₂	1	1	yes	maximal, characteristic
3	#13181	C₂	1	1	yes	maximal, characteristic
4	#12665	C₂	1	1	yes	maximal, characteristic
5	C₂×#1366	C₂	1	1	yes	maximal, characteristic
6	#7269	C₂	1	1	yes	maximal, characteristic
7	#7161	C₂	1	1	yes	maximal, characteristic
8	C₂×#499	C₂²	1	1	yes	characteristic
9	#2159	—	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	#2108	—	2	1	no	—
14	C₂×#346	C₂²	1	1	yes	characteristic
15	#1845	C₂²	1	1	yes	characteristic
16	#1772	C₂²	1	1	yes	characteristic
17	#1387	—	2	1	no	—
18	#1386	—	2	1	no	—
19	#1383	—	2	1	no	—
20	#1380	—	2	1	no	—
21	#1380	—	2	1	no	—
22	#1379	—	2	1	no	—
23	#1366	C₄	1	4	yes	complemented
24	C₂×#125	—	2	1	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂²×C₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₁₆	D₄	1	1	yes	characteristic
30	#499	—	2	1	no	—
31	#499	—	2	1	no	—
32	#499	—	2	1	no	—
33	#499	C₂×C₄	1	2	yes	—
34	C₂×#108	D₄	1	1	yes	characteristic
35	#427	D₄	1	1	yes	characteristic
36	C₂×#107	—	2	1	no	—
37	C₂×#107	—	2	1	no	—
38	C₂×#107	—	2	1	no	—
39	C₂×#106	—	2	1	no	—
40	#405	—	2	1	no	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#384	—	2	1	no	—
43	#367	—	4	1	no	—
44	#366	—	4	1	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	—	2	1	no	—
49	#364	C₂×C₄	1	2	yes	—
50	#364	C₂×C₄	1	2	yes	—
51	#346	—	2	1	no	—
52	#346	—	2	1	no	—
53	#346	—	4	1	no	—
54	C₄×C₁₆	—	4	1	no	—
55	#339	—	2	1	no	—
56	C₂×M₁₆	—	2	1	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	#125	—	2	2	no	—
60	#125	—	4	1	no	—
61	#125	—	8	1	no	—
62	#117	—	2	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	8	1	no	—
66	C₂×C₄⋊C₄	—	8	1	no	—
67	C₂×C₄²	#41	1	1	yes	characteristic
68	M₃₂	—	2	2	no	—
69	M₃₂	—	2	2	no	—
70	M₃₂	—	4	1	no	—
71	M₃₂	—	4	1	no	—
72	M₃₂	—	4	1	no	—
73	M₃₂	—	4	1	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	Q₁₆	1	2	yes	—
83	#108	—	2	1	no	—
84	#108	Q₁₆	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	—	4	1	no	—
87	#107	—	4	1	no	—
88	#107	—	4	1	no	—
89	#107	—	4	1	no	—
90	#107	—	4	1	no	—
91	#107	SD₁₆	1	1	yes	characteristic
92	#107	SD₁₆	1	1	yes	characteristic
93	#106	—	2	1	no	—
94	#106	—	2	2	no	—
95	#106	—	4	1	no	—
96	#97	—	4	1	no	—
97	C₄×C₈	—	4	1	no	—
98	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₄×C₈	#41	1	1	yes	characteristic
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	8	1	no	—
104	C₂²×C₄	—	8	1	no	—
105	C₂²×C₄	#121	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	C₄×D₄	1	1	yes	characteristic
110	C₂×C₈	C₄×D₄	1	1	yes	characteristic
111	C₂×C₈	#123	1	1	yes	characteristic
112	C₂×C₈	#124	1	1	yes	characteristic
113	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₈	#135	1	1	yes	characteristic
117	C₂×C₈	#135	1	1	yes	characteristic
118	C₂×C₈	#135	1	1	yes	characteristic
119	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₈	#137	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₄⋊C₄	—	8	1	no	—
128	C₄⋊C₄	—	8	1	no	—
129	C₄⋊C₄	—	16	2	no	—
130	C₂²⋊C₄	—	4	1	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	4	1	no	—
133	C₄²	#117	1	1	yes	characteristic
134	C₄²	#126	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	—	4	2	no	—
137	C₁₆	—	4	2	no	—
138	C₁₆	—	4	1	no	—
139	C₁₆	—	8	1	no	—
140	C₂³	#472	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	16	1	no	—
149	C₂×C₄	—	16	1	no	—
150	C₂×C₄	—	16	2	no	—
151	C₂×C₄	—	16	1	no	—
152	C₂×C₄	#388	1	1	yes	characteristic
153	C₂×C₄	#462	1	1	yes	characteristic
154	C₂×C₄	#463	1	1	yes	characteristic
155	C₂×C₄	#465	1	1	yes	characteristic
156	C₂×C₄	#467	1	1	yes	characteristic
157	C₂×C₄	#471	1	1	yes	characteristic
158	C₈	—	4	2	no	—
159	C₈	C₄×SD₁₆	1	1	yes	characteristic
160	C₈	C₄×Q₁₆	1	1	yes	characteristic
161	C₈	C₄×Q₁₆	1	1	yes	characteristic
162	C₈	#440	1	1	yes	characteristic
163	C₈	#484	1	1	yes	characteristic
164	C₈	#485	1	1	yes	characteristic
165	C₈	#487	1	1	yes	characteristic
166	C₈	#488	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#1676	1	1	yes	characteristic
170	C₂²	#1677	1	1	yes	characteristic
171	C₂²	#1931	1	1	yes	characteristic
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	16	2	no	—
177	C₄	—	32	1	no	—
178	C₄	#1672	1	1	yes	characteristic
179	C₄	#1683	1	1	yes	characteristic
180	C₄	#1928	1	1	yes	characteristic
181	C₄	#1933	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#9568	1	1	yes	characteristic
185	C₂	#12465	1	1	yes	characteristic
186	C₂	#12466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #1366	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13284	2
#13273	2
#13181	2
#12665	2
C₂×#1366	2
#7269	2
#7161	2