Finite Group Databasebeta

Subgroups — #135151

Order 512.

230 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 76 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	#17972	C₂	1	1	yes	maximal, characteristic
2	#12005	C₂	1	1	yes	maximal, characteristic
3	#11987	C₂	1	1	yes	maximal, characteristic
4	#11984	C₂	1	1	yes	maximal, characteristic
5	#8796	C₂	1	1	yes	maximal, characteristic
6	#7246	C₂	1	1	yes	maximal, characteristic
7	#7242	C₂	1	1	yes	maximal, characteristic
8	#3170	C₂²	1	1	yes	characteristic
9	#2985	—	2	1	no	—
10	#2150	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	#1593	C₄	1	2	yes	—
13	#1591	—	2	1	no	—
14	#1579	C₄	1	2	yes	—
15	C₈×Q₁₆	—	2	1	no	—
16	#1568	C₂²	1	1	yes	characteristic
17	#1568	C₂²	1	1	yes	characteristic
18	#1568	C₂²	1	1	yes	characteristic
19	#1497	—	2	1	no	—
20	#1450	C₂²	1	1	yes	characteristic
21	#1370	—	2	1	no	—
22	#1370	—	2	1	no	—
23	#1365	—	2	1	no	—
24	#1364	—	2	1	no	—
25	#1362	—	2	1	no	—
26	#1362	—	2	1	no	—
27	#1287	—	2	1	no	—
28	#1287	—	2	1	no	—
29	#533	—	2	1	no	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂²×C₁₆	—	2	1	no	—
32	#500	D₄	1	1	yes	characteristic
33	#494	D₄	1	1	yes	characteristic
34	#487	—	2	1	no	—
35	#467	—	2	1	no	—
36	#467	Q₈	1	2	yes	—
37	#446	—	2	2	no	—
38	#446	—	4	2	no	—
39	C₈×Q₈	—	2	2	no	—
40	C₄×Q₁₆	—	2	1	no	—
41	C₄×Q₁₆	C₂×C₄	1	2	yes	—
42	#433	—	4	2	no	—
43	#432	—	2	1	no	—
44	#432	—	2	1	no	—
45	#432	—	2	1	no	—
46	#431	D₄	1	1	yes	characteristic
47	#431	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	—	2	1	no	—
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	—	2	1	no	—
53	#335	—	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	C₂×C₄	1	2	yes	—
59	Q₈⋊C₈	—	2	2	no	—
60	Q₈⋊C₈	—	2	2	no	—
61	#322	C₂×C₄	1	1	yes	characteristic
62	#322	C₂×C₄	1	1	yes	characteristic
63	C₈²	D₄	1	1	yes	characteristic
64	C₈²	D₄	1	1	yes	characteristic
65	C₂×Q₁₆	—	2	1	no	—
66	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
67	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₈	#41	1	1	yes	characteristic
69	C₄⋊Q₈	—	2	1	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	#125	—	4	1	no	—
72	C₂²⋊Q₈	—	2	2	no	—
73	C₂²⋊Q₈	—	4	1	no	—
74	C₄×Q₈	—	2	2	no	—
75	C₄×Q₈	—	4	1	no	—
76	#117	—	4	1	no	—
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	M₃₂	—	8	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	8	1	no	—
82	#107	—	2	1	no	—
83	#107	C₄⋊C₄	1	2	yes	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄⋊C₈	—	4	1	no	—
91	#103	—	2	2	no	—
92	#103	—	2	2	no	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₂²⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	—	4	1	no	—
96	C₂²⋊C₈	—	4	1	no	—
97	C₂²⋊C₈	—	4	1	no	—
98	#97	—	4	2	no	—
99	#97	—	4	1	no	—
100	#97	—	4	1	no	—
101	#97	—	4	1	no	—
102	C₄×C₈	—	4	2	no	—
103	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂²×C₄	#135	1	1	yes	characteristic
115	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
116	Q₁₆	—	4	2	no	—
117	Q₁₆	—	4	1	no	—
118	Q₁₆	—	4	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#117	1	1	yes	characteristic
131	C₂×C₈	#121	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	4	2	no	—
141	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₄²	#121	1	1	yes	characteristic
143	C₄²	#135	1	1	yes	characteristic
144	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₁₆	—	8	1	no	—
146	C₁₆	—	8	1	no	—
147	C₂³	#478	1	1	yes	characteristic
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	16	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	#380	1	1	yes	characteristic
158	C₂×C₄	#416	1	1	yes	characteristic
159	C₂×C₄	#418	1	1	yes	characteristic
160	C₂×C₄	#463	1	1	yes	characteristic
161	C₂×C₄	#463	1	1	yes	characteristic
162	C₂×C₄	#478	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	8	2	no	—
171	C₈	—	16	1	no	—
172	C₈	—	16	1	no	—
173	C₈	#472	1	1	yes	characteristic
174	C₈	#473	1	1	yes	characteristic
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#1804	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	8	2	no	—
185	C₄	—	8	2	no	—
186	C₄	#1813	1	1	yes	characteristic
187	C₄	#1816	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	#8688	1	1	yes	characteristic
190	C₂	#12827	1	1	yes	characteristic
191	C₂	#12829	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#478
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#17972	2
#12005	2
#11987	2
#11984	2
#8796	2
#7246	2
#7242	2