Finite Group Databasebeta

Subgroups — #145262

Order 512.

245 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₂×#1696	C₂	1	1	yes	maximal, characteristic
2	#12028	C₂	1	1	yes	maximal, characteristic
3	#11974	C₂	1	1	yes	maximal, characteristic
4	#11974	C₂	1	1	yes	maximal, characteristic
5	#11644	C₂	1	1	yes	maximal, characteristic
6	#8625	C₂	1	1	yes	maximal, characteristic
7	#7294	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#477	—	2	1	no	—
10	C₂×#446	C₂²	1	1	yes	characteristic
11	#2163	C₂²	1	1	yes	characteristic
12	#2163	C₂²	1	1	yes	characteristic
13	#1696	—	2	2	no	—
14	#1696	—	2	2	no	—
15	#1568	C₂²	1	1	yes	characteristic
16	#1568	C₂²	1	1	yes	characteristic
17	C₂×#334	C₂²	1	1	yes	characteristic
18	C₂×#333	—	2	1	no	—
19	#1467	—	2	1	no	—
20	#1368	—	2	2	no	—
21	#1364	—	2	2	no	—
22	#1364	—	2	2	no	—
23	#1338	—	2	2	no	—
24	#1290	—	2	1	no	—
25	C₂×C₄⋊Q₈	—	2	1	no	—
26	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
27	#500	—	2	1	no	—
28	#500	—	2	1	no	—
29	#500	—	2	1	no	—
30	#500	D₄	1	2	yes	—
31	#477	—	2	4	no	—
32	#477	—	4	2	no	—
33	#446	—	2	1	no	—
34	#446	—	2	1	no	—
35	#446	—	2	1	no	—
36	#446	—	2	1	no	—
37	#432	—	2	1	no	—
38	#432	—	2	1	no	—
39	C₂×#107	—	2	1	no	—
40	C₂×#107	—	2	1	no	—
41	C₂×#107	—	4	1	no	—
42	C₂×C₄⋊C₈	—	2	1	no	—
43	C₂×C₄⋊C₈	—	2	1	no	—
44	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
45	C₂×#103	—	4	1	no	—
46	C₂×#97	—	2	1	no	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	1	no	—
49	C₄⋊C₁₆	D₄	1	2	yes	—
50	C₂²⋊C₁₆	—	2	1	no	—
51	C₂²⋊C₁₆	—	2	1	no	—
52	#346	—	2	1	no	—
53	#346	—	2	1	no	—
54	#335	—	2	1	no	—
55	#335	—	2	1	no	—
56	#334	—	2	1	no	—
57	#334	—	2	1	no	—
58	#334	—	2	1	no	—
59	#334	D₄	1	2	yes	—
60	#333	—	2	4	no	—
61	#324	—	4	2	no	—
62	C₂²×Q₈	—	4	1	no	—
63	C₂²×Q₈	SD₁₆	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	4	1	no	—
66	C₂²×C₈	#41	1	1	yes	characteristic
67	C₂²×C₈	#41	1	1	yes	characteristic
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	—	4	1	no	—
70	C₄⋊Q₈	—	4	1	no	—
71	C₄⋊Q₈	—	4	1	no	—
72	C₄×Q₈	—	2	1	no	—
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	—	2	1	no	—
75	C₄×Q₈	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	4	1	no	—
78	C₂×C₄⋊C₄	—	4	1	no	—
79	C₂×C₄⋊C₄	SD₁₆	1	1	yes	characteristic
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	4	1	no	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	4	4	no	—
87	#107	—	4	1	no	—
88	#107	—	4	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄⋊C₈	—	4	1	no	—
95	C₄⋊C₈	—	4	1	no	—
96	C₄⋊C₈	—	4	1	no	—
97	#103	—	4	4	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	C₂²⋊C₈	—	4	1	no	—
100	#97	—	4	1	no	—
101	#97	—	4	1	no	—
102	#97	—	4	1	no	—
103	#97	—	4	1	no	—
104	C₄×C₈	C₂×D₄	1	1	yes	characteristic
105	C₄×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄×C₈	#41	1	1	yes	characteristic
108	C₂×Q₈	—	2	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	8	1	no	—
113	C₂×Q₈	—	16	2	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	8	1	no	—
117	C₂²×C₄	#123	1	1	yes	characteristic
118	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	8	2	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	—	8	1	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	#121	1	1	yes	characteristic
129	C₂×C₈	#121	1	1	yes	characteristic
130	C₂×C₈	#123	1	1	yes	characteristic
131	C₂×C₈	#124	1	1	yes	characteristic
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	#120	1	1	yes	characteristic
143	C₄²	#121	1	1	yes	characteristic
144	C₄²	#135	1	1	yes	characteristic
145	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₁₆	—	4	2	no	—
147	C₁₆	—	8	1	no	—
148	C₂³	#481	1	1	yes	characteristic
149	Q₈	—	4	2	no	—
150	Q₈	—	8	1	no	—
151	Q₈	—	16	2	no	—
152	Q₈	—	16	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	16	2	no	—
163	C₂×C₄	#394	1	1	yes	characteristic
164	C₂×C₄	#449	1	1	yes	characteristic
165	C₂×C₄	#450	1	1	yes	characteristic
166	C₂×C₄	#465	1	1	yes	characteristic
167	C₂×C₄	#469	1	1	yes	characteristic
168	C₂×C₄	#482	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	8	2	no	—
173	C₈	—	8	1	no	—
174	C₈	—	16	1	no	—
175	C₈	—	16	1	no	—
176	C₈	#473	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#2034	1	1	yes	characteristic
180	C₂²	#2239	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	1	no	—
187	C₄	—	8	1	no	—
188	C₄	—	16	2	no	—
189	C₄	#2047	1	1	yes	characteristic
190	C₄	#2049	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	#10447	1	1	yes	characteristic
193	C₂	#13399	1	1	yes	characteristic
194	C₂	#13404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#481
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ 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
C₂×#1696	2
#12028	2
#11974	2
#11644	2
#8625	2
#7294	2