Finite Group Databasebeta

Subgroups — #141246

Order 512.

379 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 102 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	#19603	C₂	1	1	yes	maximal, characteristic
2	#13251	C₂	1	2	yes	maximal, complemented
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	#11924	C₂	1	2	yes	maximal, complemented
5	C₈×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#3169	—	2	1	no	—
7	#3169	—	2	1	no	—
8	#3147	C₂²	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₂×#359	—	2	4	no	—
11	C₄×#108	C₂²	1	1	yes	characteristic
12	#1713	C₄	1	1	yes	characteristic
13	#1712	C₄	1	1	yes	characteristic
14	#1709	C₄	1	2	yes	—
15	#1706	—	2	1	no	—
16	#1706	—	2	1	no	—
17	#1564	C₂²	1	1	yes	characteristic
18	C₈×M₁₆	—	2	1	no	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1370	—	2	1	no	—
21	#1370	—	2	1	no	—
22	#1360	—	2	4	no	—
23	#1278	—	2	1	no	—
24	C₂×#135	—	2	2	no	—
25	C₂×#135	—	2	2	no	—
26	#532	—	2	2	no	—
27	C₂²×C₁₆	—	2	2	no	—
28	#500	—	2	1	no	—
29	#500	—	2	1	no	—
30	#495	—	2	2	no	—
31	#495	—	2	2	no	—
32	#494	C₂×C₄	1	2	yes	—
33	#493	—	2	1	no	—
34	#493	—	2	1	no	—
35	#493	C₂×C₄	1	2	yes	—
36	#492	—	2	2	no	—
37	#486	—	2	2	no	—
38	#486	—	2	2	no	—
39	#466	—	4	2	no	—
40	#466	—	4	1	no	—
41	#466	—	4	1	no	—
42	C₄×D₈	—	4	2	no	—
43	C₄×D₈	—	4	1	no	—
44	C₄×D₈	—	4	1	no	—
45	#431	—	2	1	no	—
46	C₂×#108	D₄	1	2	yes	—
47	C₄×M₁₆	—	2	1	no	—
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	2	no	—
52	C₄⋊C₁₆	—	2	2	no	—
53	#359	—	4	8	no	—
54	#341	—	2	1	no	—
55	#335	—	2	1	no	—
56	#334	D₄	1	2	yes	—
57	#322	—	4	1	no	—
58	C₈²	—	4	1	no	—
59	C₈²	C₂×C₄	1	1	yes	characteristic
60	C₈²	C₂×C₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₈²	D₄	1	1	yes	characteristic
63	C₂×#41	—	4	2	no	—
64	#135	—	4	4	no	—
65	#135	—	4	4	no	—
66	#135	—	8	4	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×SD₁₆	—	4	2	no	—
70	C₂×SD₁₆	—	4	2	no	—
71	C₂×D₈	—	2	2	no	—
72	C₂×D₈	—	2	2	no	—
73	C₂×D₈	—	4	2	no	—
74	C₂×D₈	—	4	1	no	—
75	C₂×D₈	—	4	1	no	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	SD₁₆	1	1	yes	characteristic
79	C₂²×C₈	SD₁₆	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₈	#41	1	1	yes	characteristic
82	C₄⋊Q₈	—	2	2	no	—
83	#127	—	2	2	no	—
84	#124	—	4	2	no	—
85	#121	—	4	2	no	—
86	#121	—	4	2	no	—
87	C₄×D₄	—	4	2	no	—
88	C₄×D₄	—	4	2	no	—
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₂×C₁₆	—	4	2	no	—
91	C₂×C₁₆	—	4	4	no	—
92	#108	—	2	2	no	—
93	#108	D₈	1	4	yes	complemented
94	#107	—	4	2	no	—
95	#107	—	4	1	no	—
96	#107	—	4	1	no	—
97	C₄⋊C₈	—	2	2	no	—
98	#102	—	4	4	no	—
99	#102	—	4	2	no	—
100	#102	—	4	2	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	#97	—	4	1	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₈	D₈	1	1	yes	characteristic
111	C₄×C₈	D₈	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₄×C₈	C₂×D₄	1	1	yes	characteristic
114	C₄×C₈	C₂×D₄	1	1	yes	characteristic
115	C₄×C₈	#41	1	1	yes	characteristic
116	#41	—	8	4	no	—
117	#41	—	8	4	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×D₄	—	4	2	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	8	2	no	—
123	C₂²×C₄	—	8	2	no	—
124	C₂²×C₄	#121	1	1	yes	characteristic
125	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
126	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
127	Q₁₆	—	4	4	no	—
128	Q₁₆	—	8	2	no	—
129	SD₁₆	—	8	4	no	—
130	SD₁₆	—	8	2	no	—
131	SD₁₆	—	8	2	no	—
132	D₈	—	4	4	no	—
133	D₈	—	8	2	no	—
134	D₈	—	8	4	no	—
135	D₈	—	8	2	no	—
136	D₈	—	8	2	no	—
137	M₁₆	—	2	4	no	—
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	#102	1	1	yes	characteristic
148	C₂×C₈	#102	1	1	yes	characteristic
149	C₂×C₈	#102	1	1	yes	characteristic
150	C₂×C₈	#102	1	1	yes	characteristic
151	C₂×C₈	C₄×D₄	1	1	yes	characteristic
152	C₂×C₈	C₄×D₄	1	1	yes	characteristic
153	C₂×C₈	#124	1	1	yes	characteristic
154	C₂×C₈	#127	1	1	yes	characteristic
155	C₂×C₈	C₂×D₈	1	2	yes	—
156	C₂×C₈	#135	1	2	yes	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	8	2	no	—
160	C₂²⋊C₄	—	8	2	no	—
161	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₄²	#121	1	1	yes	characteristic
163	C₄²	C₂×D₈	1	1	yes	characteristic
164	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₁₆	—	8	2	no	—
166	C₂³	—	8	2	no	—
167	C₂³	#460	1	1	yes	characteristic
168	Q₈	—	8	2	no	—
169	Q₈	—	8	2	no	—
170	D₄	—	8	4	no	—
171	D₄	—	8	2	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	2	no	—
175	D₄	—	16	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	16	4	no	—
181	C₂×C₄	#390	1	1	yes	characteristic
182	C₂×C₄	C₂×#102	1	1	yes	characteristic
183	C₂×C₄	#418	1	1	yes	characteristic
184	C₂×C₄	#459	1	1	yes	characteristic
185	C₂×C₄	#459	1	1	yes	characteristic
186	C₂×C₄	#460	1	1	yes	characteristic
187	C₂×C₄	#466	1	2	yes	—
188	C₂×C₄	#472	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	1	no	—
196	C₈	—	4	4	no	—
197	C₈	—	8	2	no	—
198	C₈	#359	1	2	yes	complemented
199	C₈	C₄×D₈	1	2	yes	—
200	C₈	#493	1	1	yes	characteristic
201	C₈	#495	1	1	yes	characteristic
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	16	2	no	—
205	C₂²	—	16	2	no	—
206	C₂²	—	16	2	no	—
207	C₂²	#1668	1	2	yes	—
208	C₂²	#1982	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	16	2	no	—
213	C₄	#1667	1	2	yes	—
214	C₄	#1970	1	1	yes	characteristic
215	C₄	#1978	1	1	yes	characteristic
216	C₄	C₂×#359	1	1	yes	characteristic
217	C₄	#2143	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	32	2	no	—
220	C₂	#9734	1	1	yes	characteristic
221	C₂	#13251	1	1	yes	characteristic
222	C₂	#13259	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1982
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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. #13251	C₂	semidirect
3. C₂×#1370	C₂	semidirect
4. #11924	C₂	semidirect
5. C₈×#108	C₂	semidirect
93. #108	D₈	semidirect
198. C₈	#359	semidirect

Maximal subgroups

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

subgroup	index
#19603	2
#13251	2
C₂×#1370	2
#11924	2
C₈×#108	2