Finite Group Databasebeta

Subgroups — #150986

Order 512.

280 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 76 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×#2249	C₂	1	1	yes	maximal, characteristic
2	#13639	C₂	1	1	yes	maximal, characteristic, complemented
3	#12024	C₂	1	1	yes	maximal, characteristic, complemented
4	#7446	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₃₂	C₄	1	1	yes	characteristic
6	C₂²×SD₃₂	—	2	1	no	—
7	C₂²×D₁₆	C₄	1	1	yes	characteristic
8	C₂×#495	—	2	1	no	—
9	#2249	—	2	1	no	—
10	#2249	—	2	1	no	—
11	#2249	C₄	1	2	yes	—
12	#2249	C₂²	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1568	—	2	1	no	—
15	C₄⋊C₃₂	C₂²	1	2	yes	complemented
16	C₂²⋊C₃₂	—	2	1	no	—
17	C₄×C₃₂	—	2	1	no	—
18	#1372	C₂²	1	2	yes	complemented
19	#1371	—	2	1	no	—
20	C₂²×Q₁₆	—	2	1	no	—
21	C₂²×SD₁₆	—	4	1	no	—
22	C₂²×D₈	—	2	1	no	—
23	C₂×#124	—	4	1	no	—
24	C₂×Q₃₂	—	2	1	no	—
25	C₂×Q₃₂	—	4	2	no	—
26	C₂×Q₃₂	C₂×C₄	1	2	yes	—
27	C₂×SD₃₂	—	2	2	no	—
28	C₂×SD₃₂	—	2	2	no	—
29	C₂×SD₃₂	—	4	4	no	—
30	C₂×D₁₆	—	2	1	no	—
31	C₂×D₁₆	—	4	2	no	—
32	C₂×D₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	D₄	1	1	yes	characteristic
35	#495	—	2	2	no	—
36	#495	—	2	2	no	—
37	#495	—	4	1	no	—
38	#495	—	4	1	no	—
39	#432	—	4	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃₂	—	2	2	no	—
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	C₄×C₁₆	C₂³	1	1	yes	characteristic
46	#335	—	2	2	no	—
47	#334	—	4	1	no	—
48	C₂²×Q₈	—	4	1	no	—
49	C₂²×D₄	—	4	1	no	—
50	C₂×Q₁₆	—	2	2	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	1	no	—
54	C₂×SD₁₆	—	4	2	no	—
55	C₂×SD₁₆	—	4	2	no	—
56	C₂×SD₁₆	—	8	4	no	—
57	C₂×D₈	—	2	2	no	—
58	C₂×D₈	—	4	2	no	—
59	C₂×D₈	—	4	2	no	—
60	C₂×D₈	—	4	1	no	—
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	SD₁₆	1	1	yes	characteristic
63	#124	—	4	2	no	—
64	#124	—	4	2	no	—
65	#124	—	8	1	no	—
66	#124	—	8	1	no	—
67	C₂×C₂²⋊C₄	—	8	1	no	—
68	C₂×C₂²⋊C₄	—	8	1	no	—
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	Q₃₂	—	4	2	no	—
71	Q₃₂	—	4	2	no	—
72	SD₃₂	—	4	4	no	—
73	SD₃₂	—	4	4	no	—
74	D₁₆	—	4	2	no	—
75	D₁₆	—	4	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₈	—	4	2	no	—
83	C₂²⋊C₈	—	8	1	no	—
84	#97	—	8	1	no	—
85	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×C₈	D₈	1	1	yes	characteristic
87	C₄×C₈	SD₁₆	1	1	yes	characteristic
88	C₄×C₈	C₂×D₄	1	1	yes	characteristic
89	C₃₂	—	4	2	no	—
90	C₂⁴	—	8	1	no	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	1	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	8	2	no	—
97	C₂×D₄	—	8	2	no	—
98	C₂×D₄	—	8	1	no	—
99	C₂²×C₄	—	8	1	no	—
100	C₂²×C₄	#102	1	1	yes	characteristic
101	C₂²×C₄	D₁₆	1	1	yes	characteristic
102	C₂²×C₄	SD₃₂	1	1	yes	characteristic
103	Q₁₆	—	4	2	no	—
104	Q₁₆	—	4	2	no	—
105	Q₁₆	—	8	2	no	—
106	SD₁₆	—	8	4	no	—
107	SD₁₆	—	8	4	no	—
108	D₈	—	4	2	no	—
109	D₈	—	4	2	no	—
110	D₈	—	8	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	#102	1	1	yes	characteristic
116	C₂×C₈	#102	1	1	yes	characteristic
117	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²⋊C₄	—	8	2	no	—
120	C₂²⋊C₄	—	8	2	no	—
121	C₂²⋊C₄	—	8	2	no	—
122	C₂²⋊C₄	—	8	2	no	—
123	C₄²	#102	1	1	yes	characteristic
124	C₄²	D₁₆	1	1	yes	characteristic
125	C₄²	SD₃₂	1	1	yes	characteristic
126	C₄²	C₂×D₈	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	2	1	no	—
129	C₁₆	#100	1	2	yes	—
130	C₁₆	AGL(1,ℤ/8)	1	2	yes	—
131	C₂³	—	8	2	no	—
132	C₂³	—	16	2	no	—
133	C₂³	—	16	2	no	—
134	C₂³	—	16	2	no	—
135	C₂³	#357	1	1	yes	characteristic
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	Q₈	—	16	2	no	—
139	D₄	—	8	2	no	—
140	D₄	—	8	2	no	—
141	D₄	—	16	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	16	2	no	—
148	C₂×C₄	#357	1	1	yes	characteristic
149	C₂×C₄	#357	1	1	yes	characteristic
150	C₂×C₄	#357	1	1	yes	characteristic
151	C₂×C₄	C₂×#102	1	1	yes	characteristic
152	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
153	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	16	2	no	—
157	C₈	#361	1	2	yes	—
158	C₈	C₂×#100	1	1	yes	characteristic
159	C₈	#418	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	16	2	no	—
163	C₂²	—	16	2	no	—
164	C₂²	—	16	2	no	—
165	C₂²	—	32	4	no	—
166	C₂²	#1417	1	2	yes	—
167	C₂²	C₂×#357	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	16	2	no	—
174	C₄	#1545	1	1	yes	characteristic
175	C₄	C₂×#361	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	32	2	no	—
178	C₂	#11961	1	1	yes	characteristic
179	C₂	C₂×#1417	1	1	yes	characteristic
180	C₂	#13622	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13639	C₂	semidirect
3. #12024	C₂	semidirect
4. #7446	C₂	semidirect
15. C₄⋊C₃₂	C₂²	semidirect
18. #1372	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2249	2
#13639	2
#12024	2
#7446	2