Finite Group Databasebeta

Subgroups — #513192

Order 512.

414 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 118 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	#33984	C₂	1	2	yes	maximal, complemented
2	#33860	C₂	1	2	yes	maximal, complemented
3	#13726	C₂	1	1	yes	maximal, characteristic, complemented
4	#13701	C₂	1	2	yes	maximal, complemented
5	#13693	C₂	1	2	yes	maximal, complemented
6	#13672	C₂	1	4	yes	maximal, complemented
7	#13655	C₂	1	1	yes	maximal, characteristic, complemented
8	#13647	C₂	1	1	yes	maximal, characteristic, complemented
9	#3417	C₂²	1	4	yes	—
10	#3293	—	2	2	no	—
11	#2264	—	2	4	no	—
12	#2264	C₂²	1	4	yes	—
13	#2262	—	2	2	no	—
14	#2262	C₂²	1	1	yes	characteristic
15	C₂×Q₆₄	—	2	2	no	—
16	C₂×Q₆₄	C₂²	1	2	yes	—
17	C₂×Q₆₄	C₂²	1	2	yes	—
18	#2258	C₂²	1	2	yes	—
19	#2247	C₂²	1	2	yes	—
20	#2222	C₂²	1	4	yes	—
21	#2214	—	2	2	no	—
22	#2195	—	2	4	no	—
23	#2178	C₂²	1	2	yes	—
24	#2170	C₂²	1	1	yes	characteristic
25	#1425	C₂²	1	1	yes	characteristic
26	#1422	C₂²	1	1	yes	characteristic
27	#1420	C₂²	1	4	yes	—
28	#1417	C₂²	1	2	yes	—
29	#1401	C₂²	1	2	yes	—
30	C₄×C₃₂	C₂²	1	1	yes	characteristic
31	#578	—	2	4	no	—
32	#578	—	4	2	no	—
33	#510	—	2	4	no	—
34	#510	—	4	4	no	—
35	#508	—	2	4	no	—
36	#508	C₂³	1	2	yes	—
37	C₂×Q₃₂	—	2	4	no	—
38	C₂×Q₃₂	—	2	2	no	—
39	C₂×Q₃₂	C₂³	1	4	yes	—
40	#504	C₂³	1	2	yes	—
41	#494	—	2	2	no	—
42	#473	—	2	4	no	—
43	#473	—	4	2	no	—
44	#456	—	4	4	no	—
45	#443	—	2	2	no	—
46	#443	D₄	1	1	yes	characteristic
47	Q₆₄	—	2	4	no	—
48	Q₆₄	—	2	2	no	—
49	Q₆₄	—	2	2	no	—
50	Q₆₄	—	4	2	no	—
51	Q₆₄	D₄	1	1	yes	characteristic
52	SD₆₄	—	2	4	no	—
53	SD₆₄	D₄	1	2	yes	—
54	D₃₂	D₄	1	1	yes	characteristic
55	M₆₄	—	2	2	no	—
56	M₆₄	C₂³	1	2	yes	—
57	C₂×C₃₂	—	2	2	no	—
58	C₂×C₃₂	C₂³	1	1	yes	characteristic
59	C₂×C₃₂	C₂³	1	1	yes	characteristic
60	#368	C₂³	1	2	yes	—
61	#364	D₄	1	1	yes	characteristic
62	#362	—	2	4	no	—
63	#359	—	2	2	no	—
64	#350	D₄	1	2	yes	—
65	C₄×C₁₆	C₂³	1	1	yes	characteristic
66	2₋¹⁺⁴	—	4	4	no	—
67	#137	—	4	4	no	—
68	#137	—	4	4	no	—
69	#137	—	8	4	no	—
70	#137	—	8	4	no	—
71	#135	—	2	2	no	—
72	#135	—	4	4	no	—
73	#135	—	8	2	no	—
74	#135	D₈	1	1	yes	characteristic
75	C₂×Q₁₆	—	2	4	no	—
76	C₂×Q₁₆	—	4	4	no	—
77	C₂×Q₁₆	—	4	2	no	—
78	C₂×Q₁₆	—	8	2	no	—
79	#131	—	4	2	no	—
80	#131	C₂×D₄	1	2	yes	—
81	C₄⋊Q₈	—	4	2	no	—
82	Q₃₂	—	2	4	no	—
83	Q₃₂	—	2	4	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	8	2	no	—
86	Q₃₂	C₂×D₄	1	2	yes	—
87	SD₃₂	—	2	2	no	—
88	SD₃₂	—	4	4	no	—
89	D₁₆	C₂×D₄	1	2	yes	—
90	M₃₂	—	2	2	no	—
91	M₃₂	C₂×D₄	1	2	yes	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
94	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
95	#108	—	2	2	no	—
96	#108	D₈	1	1	yes	characteristic
97	#104	—	4	2	no	—
98	#104	D₈	1	2	yes	—
99	#101	—	4	4	no	—
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₃₂	—	2	2	no	—
102	C₃₂	—	2	1	no	—
103	C₃₂	C₂×D₄	1	1	yes	characteristic
104	C₃₂	C₂×D₄	1	1	yes	characteristic
105	#41	—	4	2	no	—
106	#41	—	8	4	no	—
107	#41	—	8	4	no	—
108	#41	—	16	4	no	—
109	#41	C₂×D₈	1	2	yes	—
110	C₂×Q₈	—	4	4	no	—
111	C₂×Q₈	—	8	4	no	—
112	C₂×Q₈	—	8	4	no	—
113	Q₁₆	—	2	2	no	—
114	Q₁₆	—	4	4	no	—
115	Q₁₆	—	4	4	no	—
116	Q₁₆	—	8	4	no	—
117	Q₁₆	—	8	4	no	—
118	Q₁₆	—	8	2	no	—
119	Q₁₆	—	8	2	no	—
120	Q₁₆	D₁₆	1	1	yes	characteristic
121	SD₁₆	—	4	2	no	—
122	SD₁₆	—	8	4	no	—
123	SD₁₆	—	8	4	no	—
124	SD₁₆	D₁₆	1	2	yes	—
125	D₈	—	2	2	no	—
126	D₈	D₁₆	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	4	2	no	—
129	M₁₆	—	8	2	no	—
130	M₁₆	C₂×D₈	1	2	yes	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	8	2	no	—
133	C₂×C₈	C₂×D₈	1	1	yes	characteristic
134	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
135	C₄⋊C₄	—	16	2	no	—
136	C₄²	C₂×D₈	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	2	1	no	—
139	C₁₆	C₂²×D₄	1	1	yes	characteristic
140	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
141	Q₈	—	4	2	no	—
142	Q₈	—	8	4	no	—
143	Q₈	—	8	4	no	—
144	Q₈	—	16	4	no	—
145	Q₈	C₂×D₁₆	1	2	yes	—
146	D₄	—	2	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	8	2	no	—
149	D₄	—	16	4	no	—
150	D₄	C₂×D₁₆	1	2	yes	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	16	4	no	—
156	C₂×C₄	—	16	4	no	—
157	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	8	2	no	—
160	C₈	C₂×D₁₆	1	1	yes	characteristic
161	C₈	C₂×D₁₆	1	1	yes	characteristic
162	C₈	D₄²	1	1	yes	characteristic
163	C₈	#576	1	1	yes	characteristic
164	C₂²	—	2	2	no	—
165	C₂²	—	8	2	no	—
166	C₂²	#3415	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	—	8	2	no	—
170	C₄	—	16	4	no	—
171	C₄	D₄×D₈	1	1	yes	characteristic
172	C₄	C₂²×D₁₆	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	2	no	—
175	C₂	—	16	2	no	—
176	C₂	D₄×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	D₄×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170 ◁ 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
1. #33984	C₂	semidirect
2. #33860	C₂	semidirect
3. #13726	C₂	semidirect
4. #13701	C₂	semidirect
5. #13693	C₂	semidirect
6. #13672	C₂	semidirect
7. #13655	C₂	semidirect
8. #13647	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33984	2
#33860	2
#13726	2
#13701	2
#13693	2
#13672	2
#13655	2
#13647	2