Finite Group Databasebeta

Subgroups — #512992

Order 512.

352 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 128 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	#33608	C₂	1	1	yes	maximal, characteristic
2	#33414	C₂	1	1	yes	maximal, characteristic
3	#13718	C₂	1	8	yes	maximal, complemented
4	#13638	C₂	1	2	yes	maximal, complemented
5	C₂×#1420	C₂	1	2	yes	maximal, complemented
6	#13556	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
8	C₂×#504	C₂²	1	1	yes	characteristic
9	#3059	—	2	1	no	—
10	#2916	—	2	1	no	—
11	C₂×M₆₄	C₂²	1	2	yes	—
12	#2219	C₂²	1	4	yes	—
13	#2216	—	2	2	no	—
14	#2161	—	2	1	no	—
15	#2161	C₂²	1	4	yes	—
16	C₂×#368	—	2	2	no	—
17	C₂×#368	C₂²	1	2	yes	—
18	#2158	C₂²	1	1	yes	characteristic
19	#2141	—	2	1	no	—
20	#1426	C₂²	1	8	yes	—
21	#1420	C₂²	1	8	yes	—
22	#1400	C₂²	1	4	yes	—
23	C₂²×Q₁₆	—	2	1	no	—
24	#568	—	4	1	no	—
25	C₂×#131	D₄	1	1	yes	characteristic
26	#536	—	4	1	no	—
27	C₂×Q₃₂	—	2	1	no	—
28	C₂×Q₃₂	—	2	2	no	—
29	C₂×Q₃₂	—	2	2	no	—
30	C₂×Q₃₂	C₂³	1	2	yes	—
31	#504	—	2	2	no	—
32	#504	—	2	2	no	—
33	C₂×M₃₂	D₄	1	1	yes	characteristic
34	C₂×M₃₂	C₂³	1	2	yes	—
35	C₂²×C₁₆	D₄	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	1	yes	characteristic
37	#470	—	4	2	no	—
38	#467	—	2	4	no	—
39	#430	—	4	2	no	—
40	#430	—	4	1	no	—
41	C₂×#108	—	2	4	no	—
42	C₂×#108	—	2	2	no	—
43	#427	—	2	1	no	—
44	#419	—	4	1	no	—
45	M₆₄	—	2	2	no	—
46	M₆₄	C₂³	1	4	yes	—
47	C₂×C₃₂	—	2	2	no	—
48	#368	—	2	2	no	—
49	#368	—	2	4	no	—
50	#368	C₂³	1	4	yes	—
51	#367	C₂³	1	2	yes	—
52	#366	—	2	1	no	—
53	#358	—	2	4	no	—
54	C₂×#41	D₈	1	1	yes	characteristic
55	C₂²×Q₈	—	4	1	no	—
56	C₂×Q₁₆	—	2	2	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	#131	—	2	2	no	—
61	#131	—	2	2	no	—
62	#131	—	8	4	no	—
63	C₂×M₁₆	—	4	4	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂×M₁₆	D₈	1	2	yes	—
66	C₂×M₁₆	D₈	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	2	yes	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₄⋊Q₈	—	8	1	no	—
70	#124	—	8	1	no	—
71	#123	—	8	2	no	—
72	C₂²⋊Q₈	—	4	4	no	—
73	#117	—	4	1	no	—
74	Q₃₂	—	2	4	no	—
75	Q₃₂	—	4	1	no	—
76	Q₃₂	—	4	1	no	—
77	M₃₂	—	2	2	no	—
78	M₃₂	—	2	4	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	C₂×D₄	1	2	yes	—
82	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	#41	1	2	yes	—
84	C₂×C₁₆	#41	1	2	yes	—
85	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
86	#108	—	2	4	no	—
87	#108	—	4	4	no	—
88	#108	—	4	2	no	—
89	#107	—	2	2	no	—
90	#106	—	4	1	no	—
91	#103	—	4	4	no	—
92	C₃₂	—	2	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	2	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂×Q₈	SD₃₂	1	1	yes	characteristic
100	C₂×D₄	SD₃₂	1	2	yes	—
101	C₂×D₄	SD₃₂	1	1	yes	characteristic
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	Q₁₆	—	4	2	no	—
106	Q₁₆	—	4	2	no	—
107	Q₁₆	—	8	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	4	no	—
110	M₁₆	—	4	4	no	—
111	M₁₆	—	8	4	no	—
112	M₁₆	—	8	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	8	4	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	#123	1	2	yes	—
118	C₂×C₈	C₂×D₈	1	2	yes	—
119	C₂×C₈	C₂×D₈	1	1	yes	characteristic
120	C₂×C₈	#137	1	2	yes	—
121	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	8	4	no	—
124	C₂²⋊C₄	—	8	1	no	—
125	C₂²⋊C₄	—	8	4	no	—
126	C₄²	—	8	1	no	—
127	C₁₆	—	2	2	no	—
128	C₁₆	#123	1	1	yes	characteristic
129	C₁₆	#123	1	1	yes	characteristic
130	C₁₆	C₂×#41	1	2	yes	—
131	C₂³	C₂×SD₃₂	1	2	yes	—
132	C₂³	C₂×SD₃₂	1	1	yes	characteristic
133	Q₈	—	2	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	16	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	16	2	no	—
144	C₂×C₄	#480	1	2	yes	—
145	C₂×C₄	C₂×SD₃₂	1	2	yes	—
146	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
147	C₂×C₄	#509	1	2	yes	—
148	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	8	4	no	—
151	C₈	#480	1	2	yes	—
152	C₈	C₂×#123	1	1	yes	characteristic
153	C₈	C₂×#137	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	2	no	—
157	C₂²	#2233	1	2	yes	—
158	C₂²	C₂²×SD₃₂	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	16	2	no	—
161	C₄	—	16	2	no	—
162	C₄	#2233	1	2	yes	—
163	C₄	C₂×#480	1	1	yes	characteristic
164	C₄	C₂×#509	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	2	no	—
167	C₂	—	4	2	no	—
168	C₂	C₂×#2233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #13718	C₂	semidirect
4. #13638	C₂	semidirect
5. C₂×#1420	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33608	2
#33414	2
#13718	2
#13638	2
C₂×#1420	2
#13556	2