Finite Group Databasebeta

Subgroups — #150879

Order 512.

263 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 68 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	#33607	C₂	1	1	yes	maximal, characteristic
2	#13556	C₂	1	1	yes	maximal, characteristic, complemented
3	#12839	C₂	1	1	yes	maximal, characteristic, complemented
4	#7422	C₂	1	4	yes	maximal, complemented
5	C₂²×D₁₆	C₄	1	1	yes	characteristic, complemented
6	C₂×#504	C₂²	1	1	yes	characteristic
7	#3058	—	2	1	no	—
8	C₂×M₆₄	—	2	1	no	—
9	#2218	C₄	1	2	yes	complemented
10	#2218	C₂²	1	2	yes	—
11	#2215	—	2	1	no	—
12	#2215	—	2	1	no	—
13	#2158	—	2	1	no	—
14	#2158	C₄	1	1	yes	characteristic, complemented
15	#2140	—	2	1	no	—
16	#1812	—	2	1	no	—
17	#1400	—	2	1	no	—
18	#1400	C₂²	1	2	yes	complemented
19	#1384	C₂²	1	2	yes	complemented
20	C₂²×D₈	—	2	1	no	—
21	C₂×#131	D₄	1	1	yes	characteristic
22	#535	—	4	1	no	—
23	C₂×D₁₆	—	2	1	no	—
24	C₂×D₁₆	—	4	1	no	—
25	C₂×D₁₆	—	4	1	no	—
26	C₂×D₁₆	C₂×C₄	1	2	yes	—
27	#504	—	4	1	no	—
28	#504	—	4	1	no	—
29	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	D₄	1	1	yes	characteristic
31	C₂×M₃₂	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	D₄	1	1	yes	characteristic
34	C₂²×C₁₆	D₄	1	1	yes	characteristic
35	#469	—	4	1	no	—
36	#469	—	4	1	no	—
37	#466	—	2	2	no	—
38	#466	—	2	2	no	—
39	#427	—	2	1	no	—
40	#427	—	2	2	no	—
41	#418	—	4	1	no	—
42	#410	—	4	1	no	—
43	M₆₄	—	2	2	no	—
44	M₆₄	—	4	1	no	—
45	C₂×C₃₂	—	4	1	no	—
46	#367	—	4	1	no	—
47	#367	C₂×C₄	1	2	yes	—
48	#366	—	2	1	no	—
49	#366	—	4	1	no	—
50	#357	—	2	2	no	—
51	#357	—	2	2	no	—
52	H(3,ℤ/4)	—	4	1	no	—
53	C₂×#41	D₈	1	1	yes	characteristic
54	C₂²×D₄	—	4	1	no	—
55	C₂×D₈	—	2	2	no	—
56	C₂×D₈	—	4	2	no	—
57	C₂×D₈	—	4	2	no	—
58	C₂×D₈	—	4	1	no	—
59	#131	—	4	1	no	—
60	#131	—	4	1	no	—
61	C₂×M₁₆	D₈	1	1	yes	characteristic
62	C₂×M₁₆	SD₁₆	1	1	yes	characteristic
63	C₂×M₁₆	SD₁₆	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	#127	—	8	1	no	—
68	#124	—	8	1	no	—
69	#121	—	4	2	no	—
70	#121	—	4	2	no	—
71	#120	—	8	1	no	—
72	#120	—	8	1	no	—
73	#117	—	4	1	no	—
74	#117	—	4	2	no	—
75	D₁₆	—	4	2	no	—
76	D₁₆	—	4	1	no	—
77	D₁₆	—	4	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	4	1	no	—
80	M₃₂	—	4	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
91	#107	—	2	2	no	—
92	#107	—	4	2	no	—
93	#106	—	4	1	no	—
94	#106	—	4	2	no	—
95	#102	—	4	2	no	—
96	#102	—	4	2	no	—
97	#99	—	8	1	no	—
98	#99	—	8	1	no	—
99	C₃₂	—	4	2	no	—
100	C₂⁴	—	8	1	no	—
101	#41	—	4	1	no	—
102	#41	—	4	1	no	—
103	C₂×Q₈	SD₃₂	1	1	yes	characteristic
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	8	2	no	—
106	C₂×D₄	—	8	2	no	—
107	C₂×D₄	—	8	2	no	—
108	C₂×D₄	—	8	2	no	—
109	C₂×D₄	—	8	1	no	—
110	C₂×D₄	D₁₆	1	1	yes	characteristic
111	C₂×D₄	D₁₆	1	1	yes	characteristic
112	C₂×D₄	SD₃₂	1	1	yes	characteristic
113	C₂²×C₄	#102	1	1	yes	characteristic
114	C₂²×C₄	#102	1	1	yes	characteristic
115	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
116	D₈	—	4	2	no	—
117	D₈	—	4	2	no	—
118	D₈	—	8	2	no	—
119	M₁₆	—	4	1	no	—
120	M₁₆	—	4	1	no	—
121	M₁₆	—	4	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	#102	1	1	yes	characteristic
129	C₂×C₈	#102	1	1	yes	characteristic
130	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	8	2	no	—
134	C₂²⋊C₄	—	8	1	no	—
135	C₂²⋊C₄	—	8	2	no	—
136	C₂²⋊C₄	—	8	2	no	—
137	C₂²⋊C₄	—	8	2	no	—
138	C₄²	—	8	2	no	—
139	C₄²	—	8	1	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	—	2	1	no	—
143	C₁₆	#99	1	1	yes	characteristic
144	C₁₆	#99	1	1	yes	characteristic
145	C₂³	—	8	2	no	—
146	C₂³	—	16	2	no	—
147	C₂³	—	32	2	no	—
148	C₂³	#357	1	1	yes	characteristic
149	C₂³	#357	1	1	yes	characteristic
150	C₂³	C₂×D₁₆	1	1	yes	characteristic
151	Q₈	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	1	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	D₄	—	16	2	no	—
158	D₄	—	32	2	no	—
159	D₄	—	32	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	16	2	no	—
168	C₂×C₄	#357	1	1	yes	characteristic
169	C₂×C₄	#357	1	1	yes	characteristic
170	C₂×C₄	C₂×#102	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	C₂×#99	1	1	yes	characteristic
176	C₈	#421	1	1	yes	characteristic
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	16	2	no	—
184	C₂²	—	16	2	no	—
185	C₂²	—	16	2	no	—
186	C₂²	—	32	4	no	—
187	C₂²	C₂×#357	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	16	2	no	—
192	C₄	—	32	2	no	—
193	C₄	#1519	1	1	yes	characteristic
194	C₄	#2145	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	4	1	no	—
197	C₂	—	4	1	no	—
198	C₂	—	4	1	no	—
199	C₂	—	32	2	no	—
200	C₂	#11886	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#11886
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	#33607	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13556	C₂	semidirect
3. #12839	C₂	semidirect
4. #7422	C₂	semidirect
5. C₂²×D₁₆	C₄	semidirect
9. #2218	C₄	semidirect
14. #2158	C₄	semidirect
18. #1400	C₂²	semidirect
19. #1384	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#33607	2
#13556	2
#12839	2
#7422	2