Finite Group Databasebeta

Subgroups — #150939

Order 512.

288 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 92 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	#33713	C₂	1	1	yes	maximal, characteristic
2	#13627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
4	#7445	C₂	1	2	yes	maximal, complemented
5	#7444	C₂	1	2	yes	maximal, complemented
6	C₂×#508	—	2	1	no	—
7	C₂×#508	C₄	1	2	yes	—
8	#3147	—	2	1	no	—
9	C₂×M₆₄	—	2	1	no	—
10	#2249	—	2	1	no	—
11	#2249	C₄	1	2	yes	—
12	#2248	—	2	1	no	—
13	#2247	C₂²	1	1	yes	characteristic
14	#2246	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×#335	—	2	1	no	—
17	C₄⋊C₃₂	—	2	1	no	—
18	C₄⋊C₃₂	C₂²	1	2	yes	—
19	#1371	—	2	1	no	—
20	#1371	C₂²	1	2	yes	—
21	C₂×#135	—	2	2	no	—
22	C₂×#135	—	4	1	no	—
23	#532	—	4	1	no	—
24	#508	—	4	4	no	—
25	#508	—	4	2	no	—
26	#508	—	4	2	no	—
27	C₂×Q₃₂	—	2	1	no	—
28	C₂×Q₃₂	C₂×C₄	1	2	yes	—
29	C₂×SD₃₂	—	2	2	no	—
30	C₂×SD₃₂	—	2	2	no	—
31	C₂×D₁₆	—	2	1	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	—	4	1	no	—
37	#494	—	2	1	no	—
38	#493	—	2	1	no	—
39	#492	—	4	1	no	—
40	C₂×C₄⋊C₈	—	4	1	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	M₆₄	—	4	2	no	—
43	C₂×C₃₂	—	2	2	no	—
44	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₄×C₁₆	D₄	1	1	yes	characteristic
46	C₄×C₁₆	D₄	1	1	yes	characteristic
47	C₄×C₁₆	C₂³	1	1	yes	characteristic
48	#335	—	2	2	no	—
49	#335	—	4	1	no	—
50	C₂×#41	—	4	2	no	—
51	#135	—	4	4	no	—
52	#135	—	4	4	no	—
53	#135	—	8	4	no	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×SD₁₆	—	4	2	no	—
57	C₂×SD₁₆	—	4	2	no	—
58	C₂×D₈	—	2	2	no	—
59	C₂×D₈	—	4	1	no	—
60	C₂²×C₈	—	8	1	no	—
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	SD₁₆	1	1	yes	characteristic
63	C₄⋊Q₈	—	4	1	no	—
64	#127	—	4	1	no	—
65	#124	—	4	2	no	—
66	#121	—	8	1	no	—
67	#121	—	8	1	no	—
68	C₄×D₄	—	8	1	no	—
69	C₄×D₄	—	8	1	no	—
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	Q₃₂	—	4	2	no	—
72	Q₃₂	—	4	1	no	—
73	Q₃₂	—	4	1	no	—
74	SD₃₂	—	4	4	no	—
75	SD₃₂	—	4	4	no	—
76	D₁₆	—	4	2	no	—
77	D₁₆	—	4	1	no	—
78	D₁₆	—	4	1	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₁₆	D₈	1	2	yes	—
83	C₂×C₁₆	SD₁₆	1	2	yes	—
84	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	8	1	no	—
88	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₈	D₈	1	1	yes	characteristic
90	C₄×C₈	SD₁₆	1	1	yes	characteristic
91	C₄×C₈	C₂×D₄	1	1	yes	characteristic
92	C₃₂	—	4	2	no	—
93	#41	—	8	4	no	—
94	#41	—	8	4	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	8	2	no	—
98	C₂×D₄	—	8	1	no	—
99	C₂×D₄	—	8	1	no	—
100	C₂²×C₄	—	8	2	no	—
101	C₂²×C₄	#102	1	1	yes	characteristic
102	C₂²×C₄	D₁₆	1	1	yes	characteristic
103	C₂²×C₄	SD₃₂	1	1	yes	characteristic
104	Q₁₆	—	4	2	no	—
105	Q₁₆	—	4	2	no	—
106	Q₁₆	—	8	2	no	—
107	SD₁₆	—	8	4	no	—
108	SD₁₆	—	8	4	no	—
109	D₈	—	4	2	no	—
110	D₈	—	4	2	no	—
111	D₈	—	8	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	—	16	2	no	—
115	C₂×C₈	#100	1	2	yes	—
116	C₂×C₈	#102	1	1	yes	characteristic
117	C₂×C₈	#102	1	1	yes	characteristic
118	C₂×C₈	#102	1	2	yes	—
119	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₂²⋊C₄	—	8	2	no	—
124	C₂²⋊C₄	—	8	2	no	—
125	C₄²	#102	1	1	yes	characteristic
126	C₄²	D₁₆	1	1	yes	characteristic
127	C₄²	SD₃₂	1	1	yes	characteristic
128	C₄²	C₂×D₈	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	#102	1	2	yes	—
132	C₁₆	C₂×D₈	1	1	yes	characteristic
133	C₁₆	C₂×SD₁₆	1	1	yes	characteristic
134	C₂³	—	8	2	no	—
135	C₂³	#357	1	1	yes	characteristic
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	D₄	—	16	2	no	—
141	D₄	—	16	4	no	—
142	D₄	—	16	1	no	—
143	D₄	—	16	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	16	4	no	—
149	C₂×C₄	#331	1	2	yes	—
150	C₂×C₄	#357	1	1	yes	characteristic
151	C₂×C₄	#357	1	1	yes	characteristic
152	C₂×C₄	#357	1	1	yes	characteristic
153	C₂×C₄	#361	1	2	yes	—
154	C₂×C₄	C₂×#102	1	1	yes	characteristic
155	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
156	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	16	2	no	—
160	C₈	#331	1	2	yes	—
161	C₈	C₂×#100	1	1	yes	characteristic
162	C₈	C₂×#102	1	1	yes	characteristic
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	16	2	no	—
166	C₂²	—	16	2	no	—
167	C₂²	—	16	2	no	—
168	C₂²	#1361	1	2	yes	—
169	C₂²	C₂×#357	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	16	2	no	—
174	C₄	#1361	1	2	yes	—
175	C₄	C₂×#331	1	1	yes	characteristic
176	C₄	C₂×#361	1	1	yes	characteristic
177	C₄	#2263	1	1	yes	characteristic
178	C₄	#2264	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	32	2	no	—
181	C₂	C₂×#1361	1	1	yes	characteristic
182	C₂	#13616	1	1	yes	characteristic
183	C₂	#13625	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#33713	C₂
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. #13627	C₂	semidirect
3. C₂×#1371	C₂	semidirect
4. #7445	C₂	semidirect
5. #7444	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33713	2
#13627	2
C₂×#1371	2
#7445	2
#7444	2