Finite Group Databasebeta

Subgroups — #141346

Order 512.

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

normal subgroups only

This group has 337 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	#19600	C₂	1	1	yes	maximal, characteristic
2	#11857	C₂	1	2	yes	maximal, complemented
3	#11856	C₂	1	2	yes	maximal
4	#11411	C₂	1	1	yes	maximal, characteristic, complemented
5	#8814	C₂	1	1	yes	maximal, characteristic, complemented
6	#3170	C₂²	1	1	yes	characteristic
7	#3169	C₂²	1	1	yes	characteristic
8	#3167	—	2	1	no	—
9	#3131	—	2	1	no	—
10	#2162	C₂²	1	2	yes	—
11	#1845	—	2	1	no	—
12	#1844	—	2	1	no	—
13	#1834	C₂²	1	2	yes	—
14	#1711	—	2	1	no	—
15	#1710	C₄	1	1	yes	characteristic, complemented
16	#1709	C₄	1	1	yes	characteristic, complemented
17	#1708	—	2	1	no	—
18	#1707	C₄	1	1	yes	characteristic, complemented
19	#1706	C₄	1	1	yes	characteristic, complemented
20	C₂×C₈²	C₂²	1	1	yes	characteristic
21	#1348	—	2	4	no	—
22	#1347	—	2	4	no	—
23	#1312	—	2	1	no	—
24	#1311	—	2	1	no	—
25	#533	—	2	1	no	—
26	#532	—	2	1	no	—
27	#500	C₂×C₄	1	1	yes	characteristic
28	#500	C₂×C₄	1	1	yes	characteristic
29	#498	—	2	1	no	—
30	#494	C₂×C₄	1	1	yes	characteristic
31	#493	C₂×C₄	1	1	yes	characteristic
32	#492	—	2	1	no	—
33	#488	—	4	2	no	—
34	#487	—	2	1	no	—
35	#487	—	2	1	no	—
36	#487	—	2	1	no	—
37	#487	—	2	1	no	—
38	#486	—	2	1	no	—
39	#486	—	2	1	no	—
40	#486	—	2	1	no	—
41	#486	—	2	1	no	—
42	#467	—	2	2	no	—
43	#466	—	2	2	no	—
44	#465	—	4	2	no	—
45	C₄×Q₁₆	—	2	2	no	—
46	C₄×SD₁₆	—	4	2	no	—
47	C₄×D₈	—	2	2	no	—
48	C₂×#107	—	4	2	no	—
49	C₂×#106	—	4	2	no	—
50	#416	—	4	2	no	—
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#384	—	2	2	no	—
55	C₄⋊C₁₆	—	2	2	no	—
56	C₂²⋊C₁₆	D₄	1	4	yes	—
57	C₄×C₁₆	—	2	2	no	—
58	#340	—	2	2	no	—
59	#340	—	2	2	no	—
60	C₈²	C₂×C₄	1	1	yes	characteristic
61	C₈²	C₂×C₄	1	1	yes	characteristic
62	C₈²	D₄	1	1	yes	characteristic
63	C₈²	D₄	1	1	yes	characteristic
64	C₂×#41	—	4	1	no	—
65	C₂×Q₁₆	—	2	2	no	—
66	C₂×SD₁₆	—	4	2	no	—
67	C₂×D₈	—	2	2	no	—
68	C₂²×C₈	D₈	1	2	yes	—
69	C₂²×C₈	D₈	1	2	yes	—
70	C₂²×C₈	C₂×D₄	1	2	yes	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	1	no	—
73	C₄⋊Q₈	—	2	1	no	—
74	#127	—	2	1	no	—
75	#125	—	4	1	no	—
76	#124	—	4	1	no	—
77	C₂²⋊Q₈	—	4	1	no	—
78	C₂²⋊Q₈	—	4	1	no	—
79	#121	—	4	1	no	—
80	#121	—	4	1	no	—
81	C₄×Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	C₄×D₄	—	4	1	no	—
84	C₄×D₄	—	4	1	no	—
85	#117	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	2	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₄⋊C₄	—	4	2	no	—
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#107	—	8	4	no	—
94	#106	—	4	2	no	—
95	#106	—	8	4	no	—
96	#103	—	4	2	no	—
97	#103	—	4	2	no	—
98	#103	—	4	2	no	—
99	#103	—	4	2	no	—
100	#102	—	4	2	no	—
101	#102	—	4	2	no	—
102	#102	—	4	2	no	—
103	#102	—	4	2	no	—
104	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈	SD₁₆	1	2	yes	—
109	C₄×C₈	SD₁₆	1	2	yes	—
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄×C₈	#41	1	2	yes	—
113	#41	—	8	2	no	—
114	#41	—	8	2	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×D₄	—	4	1	no	—
119	C₂×D₄	—	4	1	no	—
120	C₂×D₄	—	4	1	no	—
121	C₂×D₄	—	8	1	no	—
122	C₂²×C₄	—	8	1	no	—
123	C₂²×C₄	—	8	2	no	—
124	C₂²×C₄	#127	1	1	yes	characteristic
125	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
127	Q₁₆	—	8	2	no	—
128	SD₁₆	—	8	4	no	—
129	D₈	—	8	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	#102	1	2	yes	—
135	C₂×C₈	#102	1	2	yes	—
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	#121	1	2	yes	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₄⋊C₄	—	8	4	no	—
148	C₄⋊C₄	—	8	1	no	—
149	C₄⋊C₄	—	8	1	no	—
150	C₄⋊C₄	—	8	4	no	—
151	C₂²⋊C₄	—	8	1	no	—
152	C₂²⋊C₄	—	8	1	no	—
153	C₂²⋊C₄	—	8	1	no	—
154	C₄²	—	8	1	no	—
155	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₄²	#124	1	1	yes	characteristic
157	C₄²	C₂×SD₁₆	1	1	yes	characteristic
158	C₄²	C₂×SD₁₆	1	1	yes	characteristic
159	C₁₆	—	4	4	no	—
160	C₂³	—	8	1	no	—
161	C₂³	#493	1	1	yes	characteristic
162	Q₈	—	8	2	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	16	1	no	—
169	D₄	—	16	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	16	2	no	—
176	C₂×C₄	—	16	2	no	—
177	C₂×C₄	—	16	2	no	—
178	C₂×C₄	—	16	2	no	—
179	C₂×C₄	#390	1	1	yes	characteristic
180	C₂×C₄	C₂×#102	1	1	yes	characteristic
181	C₂×C₄	C₂×#102	1	1	yes	characteristic
182	C₂×C₄	#486	1	1	yes	characteristic
183	C₂×C₄	#486	1	1	yes	characteristic
184	C₂×C₄	#492	1	1	yes	characteristic
185	C₂×C₄	#509	1	2	yes	—
186	C₂×C₄	#510	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	#360	1	4	yes	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	16	1	no	—
196	C₂²	—	16	1	no	—
197	C₂²	—	16	1	no	—
198	C₂²	#1957	1	1	yes	characteristic
199	C₂²	#2250	1	1	yes	characteristic
200	C₂²	#2251	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	16	1	no	—
204	C₄	—	16	1	no	—
205	C₄	—	16	1	no	—
206	C₄	—	32	2	no	—
207	C₄	#1976	1	2	yes	—
208	C₄	#2139	1	2	yes	—
209	C₄	C₂×#360	1	2	yes	—
210	C₄	#2243	1	2	yes	—
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	—	32	1	no	—
214	C₂	#9756	1	1	yes	characteristic
215	C₂	#13205	1	1	yes	characteristic
216	C₂	#13210	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #11857	C₂	semidirect
4. #11411	C₂	semidirect
5. #8814	C₂	semidirect
15. #1710	C₄	semidirect
16. #1709	C₄	semidirect
18. #1707	C₄	semidirect
19. #1706	C₄	semidirect

Maximal subgroups

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

subgroup	index
#19600	2
#11857	2
#11856	2
#11411	2
#8814	2