Finite Group Databasebeta

Subgroups — #145880

Order 512.

210 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 74 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	#29230	C₂	1	1	yes	maximal, characteristic
2	#13187	C₂	1	1	yes	maximal, characteristic
3	#13180	C₂	1	1	yes	maximal, characteristic
4	#8793	C₂	1	1	yes	maximal, characteristic
5	#7270	C₂	1	1	yes	maximal, characteristic
6	#7181	C₂	1	1	yes	maximal, characteristic
7	#7181	C₂	1	1	yes	maximal, characteristic
8	#2159	—	2	1	no	—
9	C₂×#365	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	#2102	—	2	2	no	—
13	#1953	—	2	1	no	—
14	#1953	—	2	1	no	—
15	#1848	C₂²	1	1	yes	characteristic
16	#1846	C₂²	1	1	yes	characteristic
17	#1833	—	2	1	no	—
18	#1833	—	2	1	no	—
19	#1772	C₂²	1	1	yes	characteristic
20	#1772	C₂²	1	1	yes	characteristic
21	#1450	C₂²	1	1	yes	characteristic
22	#1387	—	2	1	no	—
23	#1386	—	2	1	no	—
24	#1286	—	2	2	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	#431	—	2	1	no	—
28	#430	—	2	2	no	—
29	C₂×#108	—	2	1	no	—
30	C₂×#108	—	2	1	no	—
31	C₂×#108	D₄	1	1	yes	characteristic
32	C₂×#108	D₄	1	1	yes	characteristic
33	#427	D₄	1	1	yes	characteristic
34	#427	D₄	1	1	yes	characteristic
35	C₂×#107	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	#423	D₄	1	2	yes	—
38	#405	—	2	1	no	—
39	#405	—	2	1	no	—
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	#382	—	2	1	no	—
42	#367	—	4	1	no	—
43	#366	—	4	1	no	—
44	#365	—	4	2	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	Q₈	1	2	yes	—
48	#364	Q₈	1	2	yes	—
49	#344	—	2	2	no	—
50	#344	—	2	2	no	—
51	#339	—	2	1	no	—
52	#339	—	2	1	no	—
53	H(3,ℤ/4)	—	2	2	no	—
54	#322	Q₈	1	2	yes	—
55	C₈²	—	2	1	no	—
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	—	2	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	#117	—	2	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
65	M₃₂	—	2	2	no	—
66	M₃₂	—	2	2	no	—
67	M₃₂	—	4	1	no	—
68	M₃₂	—	4	1	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₂×C₁₆	—	4	1	no	—
71	#108	—	2	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	2	1	no	—
75	#108	—	4	1	no	—
76	#108	—	4	1	no	—
77	#108	—	4	1	no	—
78	#108	—	4	1	no	—
79	#108	—	8	2	no	—
80	#107	—	2	1	no	—
81	#107	—	2	1	no	—
82	#107	—	4	1	no	—
83	#107	—	4	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	#106	—	4	1	no	—
87	#106	—	4	1	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₂²⋊C₈	—	4	1	no	—
91	#97	—	4	1	no	—
92	#97	—	4	1	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	4	1	no	—
95	C₄×C₈	—	4	1	no	—
96	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	#95	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	#120	1	1	yes	characteristic
104	M₁₆	—	4	1	no	—
105	M₁₆	—	4	2	no	—
106	M₁₆	—	4	1	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	1	no	—
109	M₁₆	—	4	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₈	#135	1	1	yes	characteristic
119	C₂×C₈	#135	1	1	yes	characteristic
120	C₂×C₈	#135	1	1	yes	characteristic
121	C₂×C₈	#135	1	1	yes	characteristic
122	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₈	#137	1	1	yes	characteristic
125	C₂×C₈	#137	1	1	yes	characteristic
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	8	1	no	—
129	C₄⋊C₄	—	8	1	no	—
130	C₂²⋊C₄	—	4	1	no	—
131	C₂²⋊C₄	—	4	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	4	1	no	—
134	C₄²	—	4	1	no	—
135	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
136	C₄²	C₄⋊Q₈	1	1	yes	characteristic
137	C₁₆	—	4	2	no	—
138	C₁₆	—	4	2	no	—
139	C₂³	#454	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	—	16	2	no	—
146	C₂×C₄	—	16	1	no	—
147	C₂×C₄	#395	1	1	yes	characteristic
148	C₂×C₄	#449	1	1	yes	characteristic
149	C₂×C₄	#449	1	1	yes	characteristic
150	C₂×C₄	#452	1	1	yes	characteristic
151	C₂×C₄	#452	1	1	yes	characteristic
152	C₂×C₄	#455	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₈	—	8	1	no	—
156	C₈	—	8	1	no	—
157	C₈	#478	1	1	yes	characteristic
158	C₈	#478	1	1	yes	characteristic
159	C₈	#479	1	1	yes	characteristic
160	C₈	#479	1	1	yes	characteristic
161	C₈	#499	1	1	yes	characteristic
162	C₈	#499	1	1	yes	characteristic
163	C₈	#501	1	1	yes	characteristic
164	C₈	#501	1	1	yes	characteristic
165	C₂²	—	4	1	no	—
166	C₂²	#1628	1	1	yes	characteristic
167	C₂²	#1638	1	1	yes	characteristic
168	C₂²	#2061	1	1	yes	characteristic
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	16	1	no	—
172	C₄	—	16	1	no	—
173	C₄	—	32	1	no	—
174	C₄	#1640	1	1	yes	characteristic
175	C₄	#1646	1	1	yes	characteristic
176	C₄	#2059	1	1	yes	characteristic
177	C₄	#2059	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#10578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#10578
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#454
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#29230	2
#13187	2
#13180	2
#8793	2
#7270	2
#7181	2