Finite Group Databasebeta

Subgroups — #152425

Order 512.

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

normal subgroups only

This group has 252 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	#33414	C₂	1	1	yes	maximal, characteristic
2	#13636	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1401	C₂	1	1	yes	maximal, characteristic
5	#12841	C₂	1	1	yes	maximal, characteristic
6	#7480	C₂	1	1	yes	maximal, characteristic, complemented
7	#7473	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#504	C₂²	1	1	yes	characteristic
9	#2916	—	2	1	no	—
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	#2161	—	2	1	no	—
13	#2161	—	2	2	no	—
14	C₂×#368	—	2	2	no	—
15	C₂×#368	C₂²	1	1	yes	characteristic
16	C₂×#368	C₂²	1	1	yes	characteristic
17	C₂×#368	C₂²	1	1	yes	characteristic
18	#2158	C₂²	1	1	yes	characteristic
19	#1814	—	2	1	no	—
20	#1425	—	2	2	no	—
21	#1425	—	2	2	no	—
22	#1401	C₄	1	4	yes	complemented
23	#1389	—	2	1	no	—
24	#568	—	4	1	no	—
25	C₂×#131	D₄	1	1	yes	characteristic
26	#504	—	2	1	no	—
27	#504	—	2	1	no	—
28	#504	C₂×C₄	1	2	yes	—
29	#504	Q₈	1	2	yes	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	—	2	1	no	—
33	C₂²×C₁₆	C₂³	1	1	yes	characteristic
34	#430	—	4	2	no	—
35	#430	—	4	1	no	—
36	C₂×#108	—	2	4	no	—
37	C₂×#108	—	2	1	no	—
38	C₂×#108	—	2	1	no	—
39	C₂×#108	—	2	1	no	—
40	#427	—	2	1	no	—
41	#421	—	4	1	no	—
42	M₆₄	—	2	1	no	—
43	M₆₄	C₂×C₄	1	2	yes	—
44	C₂×C₃₂	—	2	1	no	—
45	C₂×C₃₂	—	2	1	no	—
46	C₂×C₃₂	—	2	1	no	—
47	C₂×C₃₂	C₂×C₄	1	2	yes	—
48	#368	—	2	4	no	—
49	#368	—	2	1	no	—
50	#368	—	2	1	no	—
51	#368	—	2	1	no	—
52	#368	—	2	1	no	—
53	#368	D₄	1	2	yes	—
54	#368	D₄	1	2	yes	—
55	#367	—	2	1	no	—
56	#366	—	2	1	no	—
57	#344	—	4	1	no	—
58	C₂×#41	D₈	1	1	yes	characteristic
59	#131	—	2	1	no	—
60	#131	—	2	1	no	—
61	#131	—	8	2	no	—
62	#131	—	8	2	no	—
63	#131	C₄⋊C₄	1	2	yes	—
64	#131	Q₁₆	1	2	yes	—
65	C₂×M₁₆	—	2	1	no	—
66	C₂×M₁₆	—	4	4	no	—
67	C₂×M₁₆	—	4	1	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	D₈	1	1	yes	characteristic
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	#117	—	4	1	no	—
74	M₃₂	—	2	1	no	—
75	M₃₂	—	2	2	no	—
76	M₃₂	—	2	2	no	—
77	M₃₂	C₄⋊C₄	1	2	yes	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₁₆	#41	1	1	yes	characteristic
86	C₂×C₁₆	#41	1	1	yes	characteristic
87	#108	—	2	2	no	—
88	#108	—	2	2	no	—
89	#108	—	4	4	no	—
90	#108	—	4	1	no	—
91	#108	—	4	1	no	—
92	#108	—	4	1	no	—
93	#108	—	4	1	no	—
94	#107	—	4	1	no	—
95	#106	—	4	1	no	—
96	#104	—	8	2	no	—
97	C₃₂	—	2	2	no	—
98	C₃₂	—	2	2	no	—
99	#41	—	2	1	no	—
100	#41	—	2	1	no	—
101	#41	#107	1	2	yes	—
102	#41	D₁₆	1	2	yes	—
103	C₂×Q₈	Q₃₂	1	1	yes	characteristic
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	Q₃₂	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	8	4	no	—
114	M₁₆	—	8	1	no	—
115	M₁₆	—	8	1	no	—
116	M₁₆	—	8	1	no	—
117	M₁₆	—	8	1	no	—
118	M₁₆	#107	1	2	yes	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	8	4	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	—	8	1	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#120	1	1	yes	characteristic
128	C₂×C₈	#123	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈	#137	1	1	yes	characteristic
132	C₄⋊C₄	—	8	1	no	—
133	C₂²⋊C₄	—	8	1	no	—
134	C₄²	—	8	1	no	—
135	C₁₆	—	2	2	no	—
136	C₁₆	C₄×D₄	1	1	yes	characteristic
137	C₁₆	C₄×D₄	1	1	yes	characteristic
138	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
139	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂³	—	2	1	no	—
141	C₂³	#510	1	1	yes	characteristic
142	Q₈	—	2	1	no	—
143	Q₈	#366	1	2	yes	—
144	D₄	—	2	2	no	—
145	D₄	—	2	2	no	—
146	D₄	—	2	1	no	—
147	D₄	#366	1	2	yes	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	16	1	no	—
152	C₂×C₄	C₂×#107	1	1	yes	characteristic
153	C₂×C₄	#447	1	1	yes	characteristic
154	C₂×C₄	#480	1	1	yes	characteristic
155	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
156	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
157	C₂×C₄	#509	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	8	2	no	—
160	C₈	—	8	2	no	—
161	C₈	—	16	1	no	—
162	C₈	#385	1	1	yes	characteristic
163	C₈	#428	1	1	yes	characteristic
164	C₈	#451	1	1	yes	characteristic
165	C₈	#484	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	#2159	1	1	yes	characteristic
170	C₂²	#2188	1	1	yes	characteristic
171	C₂²	#2232	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	32	1	no	—
174	C₄	#1852	1	1	yes	characteristic
175	C₄	C₂×#366	1	1	yes	characteristic
176	C₄	#2184	1	1	yes	characteristic
177	C₄	#2236	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	#12919	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12919
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13636	C₂	semidirect
3. C₂×#1425	C₂	semidirect
6. #7480	C₂	semidirect
7. #7473	C₂	semidirect
22. #1401	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33414	2
#13636	2
C₂×#1425	2
C₂×#1401	2
#12841	2
#7480	2
#7473	2