Finite Group Databasebeta

Subgroups — #140307

Order 512.

228 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 70 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	#12036	C₂	1	1	yes	maximal, characteristic
2	#11950	C₂	1	1	yes	maximal, characteristic
3	#11650	C₂	1	1	yes	maximal, characteristic
4	#9845	C₂	1	1	yes	maximal, characteristic
5	#8266	C₂	1	1	yes	maximal, characteristic
6	#7291	C₂	1	1	yes	maximal, characteristic
7	#7148	C₂	1	1	yes	maximal, characteristic
8	#3170	C₂²	1	1	yes	characteristic
9	#2162	C₂²	1	1	yes	characteristic
10	#2162	C₂²	1	1	yes	characteristic
11	#1994	—	2	1	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1567	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1564	C₂²	1	1	yes	characteristic
16	#1493	—	2	1	no	—
17	#1450	—	2	1	no	—
18	#1372	—	2	1	no	—
19	#1371	—	2	1	no	—
20	#1364	C₄	1	1	yes	characteristic, complemented
21	#1364	C₄	1	1	yes	characteristic, complemented
22	#1362	C₄	1	1	yes	characteristic, complemented
23	#1362	C₄	1	1	yes	characteristic, complemented
24	#1332	—	2	1	no	—
25	#1332	—	2	1	no	—
26	#1276	—	2	1	no	—
27	#533	—	2	1	no	—
28	#500	C₂×C₄	1	1	yes	characteristic
29	#494	C₂×C₄	1	1	yes	characteristic
30	#487	—	2	1	no	—
31	#467	—	2	1	no	—
32	#467	—	2	1	no	—
33	C₄×Q₁₆	—	2	1	no	—
34	C₄×Q₁₆	—	2	1	no	—
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#423	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄³	D₄	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	1	no	—
44	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₁₆	—	2	1	no	—
47	C₂²⋊C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	#339	—	4	2	no	—
51	#335	—	2	1	no	—
52	#335	—	2	1	no	—
53	#335	C₂×C₄	1	1	yes	characteristic
54	#335	C₂×C₄	1	1	yes	characteristic
55	#335	D₄	1	1	yes	characteristic
56	#335	D₄	1	1	yes	characteristic
57	#334	—	2	1	no	—
58	#334	—	2	1	no	—
59	Q₈⋊C₈	—	4	1	no	—
60	Q₈⋊C₈	—	4	1	no	—
61	#322	—	4	1	no	—
62	C₈²	—	4	1	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×Q₁₆	—	2	1	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	#41	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	—	4	1	no	—
73	C₂²⋊Q₈	—	4	1	no	—
74	C₄×Q₈	—	4	1	no	—
75	C₄×Q₈	—	4	1	no	—
76	#117	—	4	1	no	—
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	—	2	1	no	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	#107	—	2	1	no	—
85	#107	—	2	1	no	—
86	C₄⋊C₈	—	2	1	no	—
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	#103	—	4	1	no	—
93	#103	—	4	1	no	—
94	C₂²⋊C₈	—	4	1	no	—
95	C₂²⋊C₈	—	4	1	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	4	2	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	D₈	1	1	yes	characteristic
103	C₄×C₈	D₈	1	1	yes	characteristic
104	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	C₄×C₈	#41	1	1	yes	characteristic
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	C₂²×C₄	—	4	1	no	—
111	C₂²×C₄	#123	1	1	yes	characteristic
112	C₂²×C₄	#135	1	1	yes	characteristic
113	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₁₆	—	4	1	no	—
115	Q₁₆	—	4	1	no	—
116	Q₁₆	—	8	1	no	—
117	M₁₆	—	8	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	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	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	8	1	no	—
130	C₂×C₈	#117	1	1	yes	characteristic
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	#124	1	1	yes	characteristic
133	C₂×C₈	#126	1	1	yes	characteristic
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	8	1	no	—
139	C₄⋊C₄	—	8	1	no	—
140	C₂²⋊C₄	—	8	1	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	4	1	no	—
145	C₄²	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	8	1	no	—
148	C₄²	C₄×D₄	1	1	yes	characteristic
149	C₄²	#121	1	1	yes	characteristic
150	C₄²	C₂×D₈	1	1	yes	characteristic
151	C₄²	#137	1	1	yes	characteristic
152	C₁₆	—	4	1	no	—
153	C₁₆	—	4	1	no	—
154	C₁₆	—	8	1	no	—
155	C₂³	#482	1	1	yes	characteristic
156	Q₈	—	8	1	no	—
157	Q₈	—	8	1	no	—
158	Q₈	—	8	1	no	—
159	Q₈	—	8	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	#388	1	1	yes	characteristic
172	C₂×C₄	C₄×D₈	1	1	yes	characteristic
173	C₂×C₄	#440	1	1	yes	characteristic
174	C₂×C₄	#459	1	1	yes	characteristic
175	C₂×C₄	#464	1	1	yes	characteristic
176	C₂×C₄	#480	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	8	1	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	1	no	—
185	C₈	—	8	1	no	—
186	C₈	—	16	1	no	—
187	C₈	#443	1	1	yes	characteristic
188	C₈	#443	1	1	yes	characteristic
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#1925	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	4	4	no	—
197	C₄	—	16	1	no	—
198	C₄	—	16	1	no	—
199	C₄	#1932	1	1	yes	characteristic
200	C₄	#1937	1	1	yes	characteristic
201	C₄	#2178	1	1	yes	characteristic
202	C₄	#2179	1	1	yes	characteristic
203	C₄	#2212	1	1	yes	characteristic
204	C₄	#2214	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	#9577	1	1	yes	characteristic
207	C₂	#13134	1	1	yes	characteristic
208	C₂	#13135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1925
prime radical	C₂³	#482
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×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
20. #1364	C₄	semidirect
21. #1364	C₄	semidirect
22. #1362	C₄	semidirect
23. #1362	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12036	2
#11950	2
#11650	2
#9845	2
#8266	2
#7291	2
#7148	2