Finite Group Databasebeta

Subgroups — #512514

Order 512.

510 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 218 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#33311	C₂	1	1	yes	maximal, characteristic
3	#13284	C₂	1	4	yes	maximal
4	#13281	C₂	1	4	yes	maximal
5	#12890	C₂	1	4	yes	maximal
6	C₄×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#499	—	2	1	no	—
8	C₂×#499	—	2	1	no	—
9	C₂×#499	C₂²	1	2	yes	—
10	C₂×#499	C₂²	1	2	yes	—
11	#2859	—	2	1	no	—
12	#2859	—	2	1	no	—
13	C₂×#346	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1846	—	2	1	no	—
17	#1846	—	2	1	no	—
18	#1846	C₂²	1	2	yes	—
19	#1846	C₂²	1	2	yes	—
20	#1845	C₂²	1	2	yes	—
21	#1845	C₂²	1	2	yes	—
22	#1844	—	2	1	no	—
23	#1844	—	2	1	no	—
24	C₄²×C₈	C₂²	1	1	yes	characteristic
25	#1380	C₂²	1	16	yes	—
26	C₂×#125	—	2	2	no	—
27	C₂×#125	—	2	2	no	—
28	C₂²×C₁₆	C₂³	1	4	yes	—
29	#499	—	2	8	no	—
30	#499	—	2	8	no	—
31	#499	—	4	4	no	—
32	#499	—	4	4	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	C₂³	1	4	yes	—
36	C₂×#107	C₂³	1	4	yes	—
37	C₂×#106	—	2	2	no	—
38	C₂×#106	—	2	2	no	—
39	C₂×#106	—	2	2	no	—
40	C₂×#106	—	2	2	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#389	—	4	4	no	—
48	#389	—	4	4	no	—
49	#384	—	2	2	no	—
50	#384	—	2	2	no	—
51	#382	—	2	2	no	—
52	#382	—	2	2	no	—
53	C₄³	D₄	1	1	yes	characteristic
54	#346	—	2	4	no	—
55	#346	Q₈	1	8	yes	—
56	C₄×C₁₆	—	2	4	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂⁴	1	1	yes	characteristic
61	#125	—	4	4	no	—
62	#125	—	4	4	no	—
63	#125	—	4	4	no	—
64	#125	—	4	4	no	—
65	C₂×C₄⋊C₄	—	2	4	no	—
66	C₂×C₄⋊C₄	—	2	4	no	—
67	C₂×C₄⋊C₄	—	4	4	no	—
68	C₂×C₄⋊C₄	—	4	4	no	—
69	C₂×C₄⋊C₄	—	4	4	no	—
70	C₂×C₄⋊C₄	—	4	4	no	—
71	C₂×C₄²	D₈	1	4	yes	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	C₂×Q₈	1	8	yes	—
78	#107	—	2	8	no	—
79	#107	—	2	8	no	—
80	#107	—	4	4	no	—
81	#107	—	4	4	no	—
82	#106	—	4	4	no	—
83	#106	—	4	4	no	—
84	#106	—	4	4	no	—
85	#106	—	4	4	no	—
86	C₄×C₈	SD₁₆	1	4	yes	—
87	C₄×C₈	SD₁₆	1	4	yes	—
88	C₄×C₈	Q₁₆	1	4	yes	—
89	C₄×C₈	C₂×Q₈	1	4	yes	—
90	C₄×C₈	#41	1	4	yes	—
91	C₄×C₈	#41	1	4	yes	—
92	#95	—	4	4	no	—
93	#95	—	4	4	no	—
94	C₂²×C₄	—	4	4	no	—
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	C₂×D₈	1	2	yes	—
97	C₂²×C₄	C₂×D₈	1	2	yes	—
98	C₂²×C₄	C₂×D₈	1	2	yes	—
99	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₈	#124	1	2	yes	—
101	C₂×C₈	#124	1	2	yes	—
102	C₂×C₈	C₄⋊Q₈	1	2	yes	—
103	C₂×C₈	C₂×SD₁₆	1	2	yes	—
104	C₂×C₈	C₂×SD₁₆	1	2	yes	—
105	C₂×C₈	C₂×SD₁₆	1	2	yes	—
106	C₂×C₈	C₂×SD₁₆	1	2	yes	—
107	C₂×C₈	C₂×Q₁₆	1	2	yes	—
108	C₂×C₈	C₂×Q₁₆	1	2	yes	—
109	C₂×C₈	C₂²×Q₈	1	2	yes	—
110	C₂×C₈	C₂×#41	1	2	yes	—
111	C₂×C₈	C₂×#41	1	2	yes	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	—	8	8	no	—
117	C₄⋊C₄	—	8	8	no	—
118	C₄⋊C₄	—	8	8	no	—
119	C₄⋊C₄	—	8	8	no	—
120	C₄²	—	4	4	no	—
121	C₄²	#124	1	2	yes	—
122	C₄²	#124	1	2	yes	—
123	C₄²	C₄⋊Q₈	1	2	yes	—
124	C₄²	C₂×SD₁₆	1	2	yes	—
125	C₄²	C₂×SD₁₆	1	2	yes	—
126	C₄²	C₂×Q₁₆	1	2	yes	—
127	C₁₆	—	2	8	no	—
128	C₂³	C₂²×D₈	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	8	8	no	—
135	C₂×C₄	—	8	8	no	—
136	C₂×C₄	#486	1	4	yes	—
137	C₂×C₄	#486	1	4	yes	—
138	C₂×C₄	#500	1	4	yes	—
139	C₂×C₄	C₂×#124	1	1	yes	characteristic
140	C₂×C₄	C₂×#124	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
145	C₈	#487	1	4	yes	—
146	C₈	#487	1	4	yes	—
147	C₈	#498	1	4	yes	—
148	C₈	#533	1	4	yes	—
149	C₂²	C₂×#486	1	1	yes	characteristic
150	C₂²	C₂×#486	1	1	yes	characteristic
151	C₂²	C₂×#500	1	1	yes	characteristic
152	C₂²	#3414	1	4	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	4	no	—
155	C₄	—	2	4	no	—
156	C₄	—	16	4	no	—
157	C₄	—	16	4	no	—
158	C₄	#2862	1	1	yes	characteristic
159	C₄	C₂×#487	1	1	yes	characteristic
160	C₄	C₂×#487	1	1	yes	characteristic
161	C₄	C₂×#498	1	1	yes	characteristic
162	C₂	#33289	1	1	yes	characteristic
163	C₂	#33701	1	2	yes	—
164	C₂	#33708	1	2	yes	—
165	C₂	#33732	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

Maximal subgroups

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

subgroup	index
#33311	2
#13284	2
#13281	2
#12890	2
C₄×#346	2