Finite Group Databasebeta

Subgroups — #140149

Order 512.

243 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 82 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	#13197	C₂	1	1	yes	maximal, characteristic
2	#12667	C₂	1	1	yes	maximal, characteristic, complemented
3	#12646	C₂	1	1	yes	maximal, characteristic
4	#12002	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#1335	C₂	1	1	yes	maximal, characteristic
6	#7179	C₂	1	1	yes	maximal, characteristic, complemented
7	#7138	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#443	C₂²	1	1	yes	characteristic
9	C₂×#364	C₂²	1	1	yes	characteristic
10	C₂×#364	C₂²	1	1	yes	characteristic
11	#2148	C₄	1	1	yes	characteristic
12	#2145	C₄	1	1	yes	characteristic
13	#2117	—	2	1	no	—
14	#2115	—	2	1	no	—
15	#2110	—	2	1	no	—
16	C₄×M₃₂	C₂²	1	1	yes	characteristic
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	#1772	C₂²	1	1	yes	characteristic
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1389	C₄	1	1	yes	characteristic
21	#1388	C₄	1	1	yes	characteristic
22	#1378	—	2	1	no	—
23	#1369	—	2	2	no	—
24	#1335	—	2	2	no	—
25	C₂×#135	C₂×C₄	1	1	yes	characteristic
26	C₂×#131	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	C₂×M₃₂	D₄	1	1	yes	characteristic
33	#443	—	2	1	no	—
34	#443	—	2	1	no	—
35	#443	—	2	1	no	—
36	#443	—	2	1	no	—
37	C₂×#108	C₂×C₄	1	1	yes	characteristic
38	C₂×#108	C₂×C₄	1	1	yes	characteristic
39	#427	C₂×C₄	1	1	yes	characteristic
40	C₂×#104	—	2	1	no	—
41	#405	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#364	—	2	1	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	D₄	1	2	yes	—
48	#362	—	2	1	no	—
49	#361	—	2	1	no	—
50	#360	—	2	2	no	—
51	#350	—	4	2	no	—
52	#349	—	4	2	no	—
53	#347	—	4	2	no	—
54	#346	—	2	1	no	—
55	#346	—	2	1	no	—
56	C₄×C₁₆	—	2	1	no	—
57	C₄×C₁₆	—	2	1	no	—
58	#339	—	2	1	no	—
59	C₈²	—	2	1	no	—
60	C₈²	—	2	1	no	—
61	C₂×#41	—	2	1	no	—
62	#135	—	2	1	no	—
63	#135	—	2	1	no	—
64	#135	—	2	1	no	—
65	#135	—	2	1	no	—
66	C₂×Q₁₆	M₁₆	1	1	yes	characteristic
67	C₂×SD₁₆	—	2	1	no	—
68	C₂×D₈	M₁₆	1	1	yes	characteristic
69	#131	—	4	1	no	—
70	#131	—	4	1	no	—
71	#131	—	4	1	no	—
72	#131	—	4	1	no	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	—	4	1	no	—
76	C₂²×C₈	—	4	1	no	—
77	C₂²×C₈	—	4	1	no	—
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	#117	—	2	1	no	—
81	C₂×C₄²	#41	1	1	yes	characteristic
82	M₃₂	—	2	1	no	—
83	M₃₂	—	2	1	no	—
84	M₃₂	—	2	2	no	—
85	M₃₂	—	2	1	no	—
86	M₃₂	—	2	1	no	—
87	M₃₂	—	2	1	no	—
88	M₃₂	—	2	1	no	—
89	M₃₂	—	4	1	no	—
90	M₃₂	—	4	1	no	—
91	M₃₂	—	4	1	no	—
92	M₃₂	M₁₆	1	2	yes	—
93	M₃₂	SD₁₆	1	2	yes	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	4	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	2	1	no	—
101	#108	—	2	1	no	—
102	#108	—	2	1	no	—
103	#108	—	2	1	no	—
104	#107	—	2	1	no	—
105	#106	—	2	1	no	—
106	#104	—	4	1	no	—
107	#104	—	4	1	no	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	4	1	no	—
111	C₄×C₈	—	4	1	no	—
112	C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	C₄×C₈	#41	1	1	yes	characteristic
115	C₄×C₈	#41	1	1	yes	characteristic
116	#41	—	4	1	no	—
117	#41	—	4	1	no	—
118	#41	—	4	1	no	—
119	#41	—	4	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×D₄	—	2	1	no	—
122	C₂×D₄	—	4	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
126	Q₁₆	—	2	1	no	—
127	Q₁₆	—	2	1	no	—
128	SD₁₆	—	4	1	no	—
129	SD₁₆	—	4	1	no	—
130	D₈	—	2	1	no	—
131	D₈	—	2	1	no	—
132	M₁₆	—	4	1	no	—
133	M₁₆	—	4	1	no	—
134	M₁₆	—	4	1	no	—
135	M₁₆	—	4	1	no	—
136	M₁₆	—	8	1	no	—
137	M₁₆	—	8	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	8	1	no	—
143	C₂×C₈	—	8	1	no	—
144	C₂×C₈	#117	1	1	yes	characteristic
145	C₂×C₈	C₄×D₄	1	1	yes	characteristic
146	C₂×C₈	#121	1	1	yes	characteristic
147	C₂×C₈	#123	1	1	yes	characteristic
148	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
149	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
150	C₂×C₈	#131	1	1	yes	characteristic
151	C₂×C₈	#131	1	1	yes	characteristic
152	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₈	#135	1	1	yes	characteristic
154	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₄⋊C₄	—	4	1	no	—
157	C₂²⋊C₄	—	4	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	#124	1	1	yes	characteristic
161	C₄²	#126	1	1	yes	characteristic
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₁₆	—	4	2	no	—
166	C₁₆	—	4	1	no	—
167	C₁₆	—	8	1	no	—
168	C₂³	—	4	1	no	—
169	C₂³	#444	1	1	yes	characteristic
170	Q₈	—	4	1	no	—
171	Q₈	—	4	1	no	—
172	D₄	—	4	1	no	—
173	D₄	—	4	1	no	—
174	D₄	—	8	1	no	—
175	D₄	—	8	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	#388	1	1	yes	characteristic
184	C₂×C₄	#435	1	1	yes	characteristic
185	C₂×C₄	#436	1	1	yes	characteristic
186	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
187	C₂×C₄	#442	1	1	yes	characteristic
188	C₂×C₄	#443	1	1	yes	characteristic
189	C₈	—	4	4	no	—
190	C₈	—	8	1	no	—
191	C₈	—	8	1	no	—
192	C₈	#432	1	1	yes	characteristic
193	C₈	#433	1	1	yes	characteristic
194	C₈	#435	1	1	yes	characteristic
195	C₈	#435	1	1	yes	characteristic
196	C₈	#460	1	1	yes	characteristic
197	C₈	#463	1	1	yes	characteristic
198	C₈	#481	1	1	yes	characteristic
199	C₈	#482	1	1	yes	characteristic
200	C₂²	—	4	1	no	—
201	C₂²	—	8	1	no	—
202	C₂²	—	16	1	no	—
203	C₂²	#1582	1	1	yes	characteristic
204	C₂²	#1594	1	1	yes	characteristic
205	C₂²	#1929	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	1	no	—
209	C₄	—	16	1	no	—
210	C₄	#1586	1	1	yes	characteristic
211	C₄	#1590	1	1	yes	characteristic
212	C₄	#1924	1	1	yes	characteristic
213	C₄	#1926	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	16	1	no	—
218	C₂	#9546	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9546
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ 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. #12667	C₂	semidirect
4. #12002	C₂	semidirect
6. #7179	C₂	semidirect
7. #7138	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13197	2
#12667	2
#12646	2
#12002	2
C₂×#1335	2
#7179	2
#7138	2