Finite Group Databasebeta

Subgroups — #491439

Order 512.

636 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 182 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	#33726	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#2246	C₂	1	1	yes	maximal, characteristic
3	#33704	C₂	1	2	yes	maximal, complemented
4	#19652	C₂	1	2	yes	maximal, complemented
5	#12506	C₂	1	4	yes	maximal, complemented
6	#12503	C₂	1	4	yes	maximal, complemented
7	#11399	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₆	—	2	2	no	—
9	#3208	—	2	4	no	—
10	#3160	C₂²	1	2	yes	—
11	C₂×#493	C₂²	1	2	yes	—
12	#2254	C₂²	1	2	yes	complemented
13	#2253	—	2	2	no	—
14	#2252	C₂²	1	2	yes	complemented
15	#2246	—	2	2	no	—
16	#2246	C₂²	1	2	yes	—
17	#2246	C₂²	1	2	yes	—
18	#2244	—	2	4	no	—
19	#2240	C₂²	1	8	yes	complemented
20	#2158	—	2	2	no	—
21	#2140	—	2	4	no	—
22	C₄×M₃₂	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	#1715	—	2	4	no	—
25	#1706	C₂²	1	8	yes	complemented
26	#1450	C₂²	1	1	yes	characteristic
27	#1311	C₂²	1	2	yes	complemented
28	C₈×C₁₆	C₂²	1	2	yes	complemented
29	C₂²×D₈	—	2	2	no	—
30	C₂²×D₈	—	2	2	no	—
31	#533	—	2	2	no	—
32	C₂×#127	—	2	2	no	—
33	C₂×D₁₆	—	2	4	no	—
34	C₂×D₁₆	—	2	4	no	—
35	C₂×D₁₆	—	4	8	no	—
36	C₂×M₃₂	D₄	1	2	yes	—
37	C₂²×C₁₆	D₄	1	2	yes	—
38	#500	C₂³	1	4	yes	—
39	#499	—	2	2	no	—
40	#499	—	2	2	no	—
41	#498	—	2	2	no	—
42	#493	—	2	2	no	—
43	#493	—	2	2	no	—
44	#493	—	2	2	no	—
45	#493	C₂³	1	4	yes	—
46	#492	—	2	8	no	—
47	#486	—	2	8	no	—
48	#460	—	4	8	no	—
49	#450	—	4	8	no	—
50	#431	D₄	1	2	yes	—
51	#427	—	2	2	no	—
52	#427	—	2	2	no	—
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	#367	—	2	4	no	—
55	#366	—	2	4	no	—
56	#357	—	2	16	no	—
57	#346	D₄	1	2	yes	—
58	C₄×C₁₆	D₄	1	2	yes	—
59	C₄×C₁₆	C₂³	1	1	yes	characteristic
60	C₄×C₁₆	C₂³	1	2	yes	—
61	C₄×C₁₆	C₂³	1	1	yes	characteristic
62	#322	D₄	1	2	yes	—
63	C₈²	C₂³	1	2	yes	—
64	C₂²×D₄	—	4	2	no	—
65	C₂²×D₄	—	4	4	no	—
66	C₂×SD₁₆	—	4	16	no	—
67	C₂×D₈	—	2	4	no	—
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	2	no	—
70	C₂×D₈	—	4	2	no	—
71	C₂×D₈	—	4	8	no	—
72	C₂×D₈	—	4	8	no	—
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	4	no	—
76	#127	—	2	4	no	—
77	#127	—	4	2	no	—
78	#127	—	4	4	no	—
79	#125	—	4	2	no	—
80	C₂²⋊Q₈	—	4	4	no	—
81	C₄×Q₈	—	4	4	no	—
82	#117	—	4	2	no	—
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	D₁₆	—	4	8	no	—
85	D₁₆	—	4	8	no	—
86	M₃₂	—	2	4	no	—
87	C₂×C₁₆	SD₁₆	1	8	yes	complemented
88	C₂×C₁₆	C₂×D₄	1	4	yes	—
89	C₂×C₁₆	C₂×D₄	1	2	yes	—
90	C₂×C₁₆	C₂×D₄	1	2	yes	—
91	#107	—	2	4	no	—
92	#107	—	2	4	no	—
93	#106	—	4	2	no	—
94	#106	—	4	2	no	—
95	C₄⋊C₈	D₈	1	4	yes	—
96	#102	—	4	16	no	—
97	C₂²⋊C₈	D₈	1	4	yes	—
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	4	yes	—
102	C₄×C₈	C₂⁴	1	1	yes	characteristic
103	C₂⁴	—	8	2	no	—
104	C₂×Q₈	—	4	4	no	—
105	C₂×D₄	—	4	4	no	—
106	C₂×D₄	—	4	8	no	—
107	C₂×D₄	—	8	4	no	—
108	C₂×D₄	—	8	4	no	—
109	C₂×D₄	—	8	2	no	—
110	C₂×D₄	—	8	4	no	—
111	C₂×D₄	—	16	8	no	—
112	C₂²×C₄	#127	1	1	yes	characteristic
113	C₂²×C₄	C₂×D₈	1	2	yes	—
114	SD₁₆	—	16	16	no	—
115	D₈	—	4	8	no	—
116	D₈	—	4	8	no	—
117	D₈	—	8	4	no	—
118	D₈	—	8	4	no	—
119	C₂×C₈	#127	1	1	yes	characteristic
120	C₂×C₈	#127	1	1	yes	characteristic
121	C₂×C₈	C₂×D₈	1	8	yes	—
122	C₂×C₈	C₂×SD₁₆	1	4	yes	—
123	C₂×C₈	#135	1	4	yes	—
124	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₂²⋊C₄	—	8	2	no	—
130	C₄²	—	8	2	no	—
131	C₄²	#127	1	1	yes	characteristic
132	C₄²	C₂×D₈	1	2	yes	—
133	C₄²	C₂²×D₄	1	1	yes	characteristic
134	C₁₆	—	2	4	no	—
135	C₁₆	C₂×SD₁₆	1	4	yes	—
136	C₁₆	C₂×SD₁₆	1	4	yes	—
137	C₂³	—	8	4	no	—
138	C₂³	—	16	8	no	—
139	C₂³	—	16	4	no	—
140	C₂³	#493	1	1	yes	characteristic
141	Q₈	—	8	8	no	—
142	D₄	—	8	4	no	—
143	D₄	—	8	4	no	—
144	D₄	—	8	16	no	—
145	D₄	—	16	8	no	—
146	D₄	—	16	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	#492	1	2	yes	—
150	C₂×C₄	#493	1	2	yes	—
151	C₂×C₄	#493	1	1	yes	characteristic
152	C₂×C₄	#495	1	2	yes	—
153	C₂×C₄	C₂×#127	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₈	1	2	yes	—
155	C₈	—	2	8	no	—
156	C₈	#492	1	2	yes	—
157	C₈	#495	1	2	yes	—
158	C₈	C₂²×SD₁₆	1	2	yes	—
159	C₈	C₂×#135	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	16	4	no	—
162	C₂²	—	16	8	no	—
163	C₂²	—	32	8	no	—
164	C₂²	#1706	1	2	yes	—
165	C₂²	C₂×#493	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	16	4	no	—
168	C₄	#1706	1	2	yes	—
169	C₄	C₂×#492	1	1	yes	characteristic
170	C₄	C₂×#495	1	1	yes	characteristic
171	C₄	#3415	1	4	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	32	4	no	—
174	C₂	C₂×#1706	1	1	yes	characteristic
175	C₂	#33720	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #33726	C₂	semidirect
3. #33704	C₂	semidirect
4. #19652	C₂	semidirect
5. #12506	C₂	semidirect
6. #12503	C₂	semidirect
7. #11399	C₂	semidirect
12. #2254	C₂²	semidirect
14. #2252	C₂²	semidirect
19. #2240	C₂²	semidirect
25. #1706	C₂²	semidirect
27. #1311	C₂²	semidirect
28. C₈×C₁₆	C₂²	semidirect
87. C₂×C₁₆	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#33726	2
C₂×#2246	2
#33704	2
#19652	2
#12506	2
#12503	2
#11399	2