Finite Group Databasebeta

Subgroups — #140259

Order 512.

236 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 86 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	#13285	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#11939	C₂	1	1	yes	maximal, characteristic
4	#9749	C₂	1	1	yes	maximal, characteristic
5	#9627	C₂	1	1	yes	maximal, characteristic
6	#7276	C₂	1	1	yes	maximal, characteristic
7	#7155	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	—	2	1	no	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#1571	—	2	1	no	—
15	#1564	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	1	yes	characteristic
17	#1503	—	2	1	no	—
18	#1481	—	2	1	no	—
19	#1382	—	2	1	no	—
20	#1382	—	2	1	no	—
21	#1374	—	2	1	no	—
22	#1365	C₄	1	2	yes	complemented
23	#1364	C₄	1	2	yes	complemented
24	#1303	—	2	1	no	—
25	#533	—	2	1	no	—
26	C₂²×C₁₆	—	2	1	no	—
27	C₂²×C₁₆	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	D₄	1	1	yes	characteristic
29	#500	C₂×C₄	1	1	yes	characteristic
30	#500	C₂×C₄	1	1	yes	characteristic
31	#499	—	2	1	no	—
32	#499	—	2	1	no	—
33	#498	—	2	1	no	—
34	#431	—	2	1	no	—
35	#431	—	2	1	no	—
36	C₂×#108	—	2	1	no	—
37	C₂×#108	—	2	1	no	—
38	#427	—	2	1	no	—
39	#427	—	2	1	no	—
40	#423	—	2	1	no	—
41	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	1	no	—
46	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	4	2	no	—
50	#336	—	2	1	no	—
51	#335	—	2	1	no	—
52	#335	—	4	2	no	—
53	#335	C₂×C₄	1	2	yes	—
54	#334	—	2	1	no	—
55	#334	—	2	1	no	—
56	#328	—	4	2	no	—
57	Q₈⋊C₈	—	4	2	no	—
58	C₂×M₁₆	—	4	1	no	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	#41	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	1	no	—
63	C₄⋊Q₈	—	2	1	no	—
64	#125	—	4	1	no	—
65	C₂²⋊Q₈	—	4	1	no	—
66	C₂²⋊Q₈	—	4	1	no	—
67	C₄×Q₈	—	4	1	no	—
68	C₄×Q₈	—	4	1	no	—
69	#117	—	4	1	no	—
70	C₂×C₄⋊C₄	—	2	1	no	—
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	2	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	Q₁₆	1	2	yes	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#108	—	8	1	no	—
85	#107	—	2	1	no	—
86	#107	—	2	1	no	—
87	#107	—	2	1	no	—
88	#107	—	2	1	no	—
89	#106	—	4	1	no	—
90	#106	—	4	1	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄⋊C₈	—	4	1	no	—
95	C₄⋊C₈	—	4	1	no	—
96	C₄⋊C₈	D₈	1	2	yes	—
97	C₂²⋊C₈	—	4	1	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₄×C₈	#41	1	1	yes	characteristic
105	C₄×C₈	#41	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	#95	SD₁₆	1	2	yes	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	#121	1	1	yes	characteristic
112	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₄	#135	1	1	yes	characteristic
114	M₁₆	—	8	1	no	—
115	M₁₆	—	8	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	#117	1	1	yes	characteristic
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	#123	1	1	yes	characteristic
127	C₂×C₈	#126	1	1	yes	characteristic
128	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₈	#135	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈	#137	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	8	1	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₂²⋊C₄	—	8	1	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	—	8	1	no	—
143	C₄²	C₄×D₄	1	1	yes	characteristic
144	C₄²	#124	1	1	yes	characteristic
145	C₄²	C₂×D₈	1	1	yes	characteristic
146	C₄²	#135	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	—	4	2	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	#465	1	1	yes	characteristic
151	Q₈	—	8	2	no	—
152	Q₈	—	16	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	#388	1	1	yes	characteristic
161	C₂×C₄	C₄×D₈	1	1	yes	characteristic
162	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
163	C₂×C₄	#462	1	1	yes	characteristic
164	C₂×C₄	#463	1	1	yes	characteristic
165	C₂×C₄	#466	1	1	yes	characteristic
166	C₂×C₄	#471	1	1	yes	characteristic
167	C₂×C₄	#473	1	1	yes	characteristic
168	C₂×C₄	#486	1	1	yes	characteristic
169	C₂×C₄	#488	1	1	yes	characteristic
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	16	1	no	—
175	C₈	—	16	1	no	—
176	C₈	C₄×Q₁₆	1	1	yes	characteristic
177	C₈	#440	1	1	yes	characteristic
178	C₈	#484	1	1	yes	characteristic
179	C₈	#485	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	#1672	1	1	yes	characteristic
183	C₂²	#1679	1	1	yes	characteristic
184	C₂²	#1934	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	4	no	—
189	C₄	—	16	1	no	—
190	C₄	—	16	1	no	—
191	C₄	#1674	1	1	yes	characteristic
192	C₄	#1681	1	1	yes	characteristic
193	C₄	#1928	1	1	yes	characteristic
194	C₄	#1930	1	1	yes	characteristic
195	C₄	#2178	1	2	yes	—
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	#9566	1	1	yes	characteristic
199	C₂	#13146	1	1	yes	characteristic
200	C₂	#13151	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂³	#465
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ 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
22. #1365	C₄	semidirect
23. #1364	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13285	2
C₁₆×C₄⋊C₄	2
#11939	2
#9749	2
#9627	2
#7276	2
#7155	2