Finite Group Databasebeta

Subgroups — #512448

Order 512.

730 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 234 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	C₂×#2245	C₂	1	4	yes	maximal
2	#33311	C₂	1	1	yes	maximal, characteristic
3	#33302	C₂	1	1	yes	maximal, characteristic
4	#13207	C₂	1	4	yes	maximal
5	#13203	C₂	1	4	yes	maximal
6	C₄×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#499	—	2	1	no	—
8	C₂×#499	C₂²	1	2	yes	—
9	C₂×#494	—	2	1	no	—
10	C₂×#494	—	2	1	no	—
11	C₂×#494	—	2	1	no	—
12	C₂×#494	C₂²	1	2	yes	—
13	C₂×#494	C₂²	1	2	yes	—
14	C₂×#494	C₂²	1	2	yes	—
15	#2868	—	2	1	no	—
16	#2859	—	2	1	no	—
17	#2245	—	2	16	no	—
18	C₂×#358	C₂²	1	16	yes	—
19	C₂×#346	C₂²	1	2	yes	—
20	C₂×#346	C₂²	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	2	yes	—
22	#1846	—	2	1	no	—
23	#1846	C₂²	1	2	yes	—
24	#1845	C₂²	1	2	yes	—
25	#1844	—	2	1	no	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	C₂²×Q₁₆	—	2	2	no	—
28	C₂²×Q₁₆	—	2	2	no	—
29	C₂²×Q₁₆	—	2	2	no	—
30	C₂²×Q₁₆	C₂³	1	4	yes	—
31	C₂×C₄⋊Q₈	—	2	2	no	—
32	C₂×C₄⋊Q₈	—	2	2	no	—
33	C₂×C₄⋊Q₈	—	2	2	no	—
34	C₂×#125	—	2	2	no	—
35	C₂²×C₁₆	C₂³	1	4	yes	—
36	#499	—	2	8	no	—
37	#499	—	4	4	no	—
38	#494	—	2	8	no	—
39	#494	—	2	8	no	—
40	#494	—	2	8	no	—
41	#494	—	4	4	no	—
42	#494	—	4	4	no	—
43	#494	—	4	4	no	—
44	C₂×#107	—	2	2	no	—
45	C₂×#107	C₂³	1	4	yes	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#389	—	4	4	no	—
55	#384	—	2	2	no	—
56	#384	—	4	4	no	—
57	#382	—	2	2	no	—
58	C₄³	D₄	1	1	yes	characteristic
59	#358	—	2	32	no	—
60	#346	—	2	4	no	—
61	#346	D₄	1	8	yes	—
62	C₄×C₁₆	—	2	4	no	—
63	C₂²×Q₈	—	2	4	no	—
64	C₂×Q₁₆	—	2	8	no	—
65	C₂×Q₁₆	—	2	8	no	—
66	C₂×Q₁₆	—	2	8	no	—
67	C₂×Q₁₆	—	4	4	no	—
68	C₂×Q₁₆	—	4	4	no	—
69	C₂×Q₁₆	—	4	4	no	—
70	C₂×Q₁₆	—	4	4	no	—
71	C₂×Q₁₆	—	4	4	no	—
72	C₂×Q₁₆	—	4	4	no	—
73	C₂×Q₁₆	—	4	4	no	—
74	C₂×Q₁₆	—	4	4	no	—
75	C₂×Q₁₆	—	4	4	no	—
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₈	C₂⁴	1	1	yes	characteristic
80	C₄⋊Q₈	—	4	4	no	—
81	C₄⋊Q₈	—	4	4	no	—
82	C₄⋊Q₈	—	4	4	no	—
83	C₄⋊Q₈	—	4	4	no	—
84	C₄⋊Q₈	—	4	4	no	—
85	C₄⋊Q₈	—	4	4	no	—
86	#125	—	4	4	no	—
87	#125	—	4	4	no	—
88	C₂×C₄⋊C₄	—	2	4	no	—
89	C₂×C₄⋊C₄	—	4	4	no	—
90	C₂×C₄⋊C₄	—	4	4	no	—
91	C₂×C₄⋊C₄	—	4	4	no	—
92	C₂×C₄⋊C₄	—	4	4	no	—
93	C₂×C₄⋊C₄	—	4	4	no	—
94	C₂×C₄²	D₈	1	4	yes	—
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	2	4	no	—
100	C₂×C₁₆	C₂×D₄	1	8	yes	—
101	#107	—	2	8	no	—
102	#107	—	4	4	no	—
103	#106	—	4	4	no	—
104	#106	—	4	4	no	—
105	C₄×C₈	D₈	1	4	yes	—
106	C₄×C₈	SD₁₆	1	4	yes	—
107	C₄×C₈	SD₁₆	1	4	yes	—
108	C₄×C₈	C₂×D₄	1	4	yes	—
109	C₄×C₈	#41	1	4	yes	—
110	C₄×C₈	#41	1	4	yes	—
111	#95	—	4	4	no	—
112	C₂×Q₈	—	4	4	no	—
113	C₂×Q₈	—	4	4	no	—
114	C₂×Q₈	—	4	4	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×Q₈	—	4	4	no	—
118	C₂²×C₄	—	4	4	no	—
119	C₂²×C₄	—	4	4	no	—
120	C₂²×C₄	C₂×D₈	1	2	yes	—
121	C₂²×C₄	C₂×D₈	1	2	yes	—
122	C₂²×C₄	C₂×D₈	1	2	yes	—
123	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
124	Q₁₆	—	4	16	no	—
125	Q₁₆	—	8	8	no	—
126	Q₁₆	—	8	8	no	—
127	Q₁₆	—	8	8	no	—
128	C₂×C₈	#124	1	2	yes	—
129	C₂×C₈	#124	1	2	yes	—
130	C₂×C₈	#127	1	2	yes	—
131	C₂×C₈	C₂×D₈	1	2	yes	—
132	C₂×C₈	C₂×D₈	1	2	yes	—
133	C₂×C₈	C₂×SD₁₆	1	2	yes	—
134	C₂×C₈	C₂×SD₁₆	1	2	yes	—
135	C₂×C₈	C₂×SD₁₆	1	2	yes	—
136	C₂×C₈	C₂×SD₁₆	1	2	yes	—
137	C₂×C₈	C₂²×D₄	1	2	yes	—
138	C₂×C₈	C₂×#41	1	2	yes	—
139	C₂×C₈	C₂×#41	1	2	yes	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	8	8	no	—
143	C₄⋊C₄	—	8	8	no	—
144	C₄⋊C₄	—	8	8	no	—
145	C₄⋊C₄	—	8	8	no	—
146	C₄⋊C₄	—	8	8	no	—
147	C₄²	—	2	8	no	—
148	C₄²	#124	1	2	yes	—
149	C₄²	#124	1	2	yes	—
150	C₄²	#127	1	2	yes	—
151	C₄²	C₂×D₈	1	2	yes	—
152	C₄²	C₂×SD₁₆	1	2	yes	—
153	C₄²	C₂×SD₁₆	1	2	yes	—
154	C₁₆	—	2	8	no	—
155	C₂³	C₂²×D₈	1	1	yes	characteristic
156	Q₈	—	8	4	no	—
157	Q₈	—	8	4	no	—
158	Q₈	—	8	4	no	—
159	Q₈	—	8	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	—	8	8	no	—
167	C₂×C₄	#486	1	4	yes	—
168	C₂×C₄	#486	1	4	yes	—
169	C₂×C₄	#493	1	4	yes	—
170	C₂×C₄	#510	1	8	yes	—
171	C₂×C₄	C₂×#124	1	1	yes	characteristic
172	C₂×C₄	C₂×#124	1	1	yes	characteristic
173	C₂×C₄	C₂×#127	1	1	yes	characteristic
174	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
175	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
177	C₈	#486	1	4	yes	—
178	C₈	#486	1	4	yes	—
179	C₈	#492	1	4	yes	—
180	C₈	#532	1	4	yes	—
181	C₂²	C₂×#486	1	1	yes	characteristic
182	C₂²	C₂×#486	1	1	yes	characteristic
183	C₂²	C₂×#493	1	1	yes	characteristic
184	C₂²	C₂×#510	1	4	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	16	4	no	—
188	C₄	—	16	4	no	—
189	C₄	#2245	1	8	yes	complemented
190	C₄	#2858	1	1	yes	characteristic
191	C₄	C₂×#486	1	1	yes	characteristic
192	C₄	C₂×#486	1	1	yes	characteristic
193	C₄	C₂×#492	1	1	yes	characteristic
194	C₂	#33224	1	1	yes	characteristic
195	C₂	C₂×#2245	1	2	yes	—
196	C₂	#33705	1	2	yes	—
197	C₂	#33718	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
189. C₄	#2245	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2245	2
#33311	2
#33302	2
#13207	2
#13203	2
C₄×#346	2