Finite Group Databasebeta

Subgroups — #140580

Order 512.

385 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 104 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	#21708	C₂	1	1	yes	maximal, characteristic
2	#12901	C₂	1	1	yes	maximal, characteristic, complemented
3	#11602	C₂	1	4	yes	maximal, complemented
4	#8137	C₂	1	1	yes	maximal, characteristic, complemented
5	#3150	C₄	1	1	yes	characteristic
6	C₂×#443	—	2	1	no	—
7	C₂×#443	C₂²	1	2	yes	—
8	#2931	—	2	1	no	—
9	C₂×#364	—	2	1	no	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	#1975	—	2	1	no	—
13	#1975	C₄	1	2	yes	—
14	#1963	—	2	1	no	—
15	#1849	C₄	1	1	yes	characteristic
16	#1726	C₂²	1	1	yes	characteristic
17	#1450	C₂²	1	2	yes	—
18	#1335	—	2	8	no	—
19	#1275	—	2	1	no	—
20	#1274	—	2	1	no	—
21	C₂×#135	—	2	1	no	—
22	C₂×#135	C₂×C₄	1	2	yes	—
23	C₂×#131	—	2	2	no	—
24	#532	—	2	1	no	—
25	C₂×M₃₂	—	2	2	no	—
26	#497	—	2	2	no	—
27	#496	—	2	2	no	—
28	#443	—	2	4	no	—
29	#443	—	2	4	no	—
30	#443	—	4	2	no	—
31	#443	—	4	2	no	—
32	#436	—	4	2	no	—
33	C₈×D₄	—	4	2	no	—
34	#431	—	2	2	no	—
35	C₂×#108	—	2	1	no	—
36	C₂×#108	C₂×C₄	1	2	yes	—
37	#423	—	2	1	no	—
38	C₂×#104	—	2	2	no	—
39	C₂×#104	—	2	2	no	—
40	#408	—	2	2	no	—
41	C₂×#97	C₂×C₄	1	1	yes	characteristic
42	C₂×#97	C₂×C₄	1	1	yes	characteristic
43	C₂×#97	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#376	D₄	1	1	yes	characteristic
48	#364	—	2	4	no	—
49	#364	—	4	2	no	—
50	C₄⋊C₁₆	—	4	2	no	—
51	#322	—	2	2	no	—
52	C₈²	D₄	1	4	yes	complemented
53	C₂×#41	—	2	2	no	—
54	#135	—	2	4	no	—
55	#135	—	2	4	no	—
56	#135	—	4	2	no	—
57	#135	—	4	2	no	—
58	C₂×Q₁₆	—	2	1	no	—
59	C₂×Q₁₆	M₁₆	1	2	yes	—
60	C₂×SD₁₆	—	2	2	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	2	1	no	—
63	C₂×D₈	M₁₆	1	2	yes	—
64	#131	—	4	2	no	—
65	#131	—	4	2	no	—
66	#131	—	4	2	no	—
67	#131	—	4	2	no	—
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	—	4	2	no	—
70	C₂²×C₈	—	4	2	no	—
71	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	1	no	—
76	#127	—	2	1	no	—
77	#124	—	2	2	no	—
78	#121	—	4	2	no	—
79	C₄×D₄	—	4	2	no	—
80	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	M₃₂	—	4	2	no	—
84	M₃₂	—	4	2	no	—
85	C₂×C₁₆	—	4	2	no	—
86	#108	—	2	4	no	—
87	#108	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	2	no	—
90	#104	—	4	4	no	—
91	#104	—	4	4	no	—
92	C₂²⋊C₈	—	4	2	no	—
93	C₂²⋊C₈	—	4	4	no	—
94	#97	—	2	2	no	—
95	#97	—	2	2	no	—
96	#97	—	4	1	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	D₈	1	2	yes	—
99	C₄×C₈	D₈	1	2	yes	—
100	C₄×C₈	SD₁₆	1	2	yes	—
101	C₄×C₈	SD₁₆	1	2	yes	—
102	C₄×C₈	C₂×D₄	1	2	yes	—
103	C₄×C₈	#41	1	2	yes	—
104	#95	—	4	1	no	—
105	#41	—	4	2	no	—
106	#41	—	4	2	no	—
107	#41	—	4	2	no	—
108	#41	—	4	2	no	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	4	2	no	—
112	C₂×D₄	—	4	2	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	Q₁₆	—	2	4	no	—
119	Q₁₆	—	4	2	no	—
120	SD₁₆	—	4	4	no	—
121	SD₁₆	—	4	4	no	—
122	D₈	—	2	4	no	—
123	D₈	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	M₁₆	—	8	2	no	—
127	M₁₆	—	8	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	#102	1	2	yes	—
137	C₂×C₈	#102	1	2	yes	—
138	C₂×C₈	C₄×D₄	1	2	yes	—
139	C₂×C₈	#124	1	1	yes	characteristic
140	C₂×C₈	#127	1	1	yes	characteristic
141	C₂×C₈	C₂×M₁₆	1	2	yes	—
142	C₂×C₈	#131	1	2	yes	—
143	C₂×C₈	C₂×D₈	1	1	yes	characteristic
144	C₂×C₈	C₂×D₈	1	1	yes	characteristic
145	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
147	C₄⋊C₄	—	4	2	no	—
148	C₂²⋊C₄	—	4	4	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	#102	1	2	yes	—
153	C₄²	#102	1	2	yes	—
154	C₄²	#121	1	2	yes	—
155	C₁₆	—	8	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	#421	1	1	yes	characteristic
158	Q₈	—	4	2	no	—
159	Q₈	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	2	no	—
162	D₄	—	8	4	no	—
163	D₄	—	8	2	no	—
164	D₄	—	8	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	8	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	8	1	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	#390	1	1	yes	characteristic
175	C₂×C₄	#407	1	1	yes	characteristic
176	C₂×C₄	#408	1	1	yes	characteristic
177	C₂×C₄	C₂×#102	1	1	yes	characteristic
178	C₂×C₄	C₂×#102	1	1	yes	characteristic
179	C₂×C₄	#421	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	8	4	no	—
183	C₈	—	8	2	no	—
184	C₈	#435	1	4	yes	—
185	C₈	#486	1	1	yes	characteristic
186	C₈	#486	1	1	yes	characteristic
187	C₈	#492	1	1	yes	characteristic
188	C₈	#493	1	1	yes	characteristic
189	C₂²	—	4	1	no	—
190	C₂²	—	8	4	no	—
191	C₂²	—	8	2	no	—
192	C₂²	#1480	1	1	yes	characteristic
193	C₂²	#1486	1	1	yes	characteristic
194	C₂²	#1963	1	1	yes	characteristic
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	8	2	no	—
199	C₄	#1480	1	1	yes	characteristic
200	C₄	#1486	1	1	yes	characteristic
201	C₄	#1955	1	1	yes	characteristic
202	C₄	#1957	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	—	16	2	no	—
207	C₂	#9644	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9644
prime radical	#3150	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ 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
2. #12901	C₂	semidirect
3. #11602	C₂	semidirect
4. #8137	C₂	semidirect
52. C₈²	D₄	semidirect

Maximal subgroups

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

subgroup	index
#21708	2
#12901	2
#11602	2
#8137	2