Finite Group Databasebeta

Subgroups — #152988

Order 512.

324 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 96 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	C₂×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1417	C₂	1	4	yes	maximal, complemented
4	C₄×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#508	—	2	1	no	—
6	C₂×#508	C₂²	1	2	yes	—
7	#3147	—	2	1	no	—
8	C₂²×C₃₂	C₂²	1	2	yes	—
9	#2249	—	2	1	no	—
10	#2249	C₄	1	2	yes	—
11	#2248	—	2	1	no	—
12	#2247	C₄	1	1	yes	characteristic
13	#2246	C₄	1	1	yes	characteristic
14	C₂×#368	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₄×#108	—	2	1	no	—
17	C₄⋊C₃₂	—	2	1	no	—
18	C₄⋊C₃₂	—	2	1	no	—
19	#1417	—	2	8	no	—
20	#1381	—	2	1	no	—
21	C₂×#135	—	2	2	no	—
22	C₂×#135	—	4	1	no	—
23	#532	—	4	1	no	—
24	#508	—	2	4	no	—
25	#508	—	2	4	no	—
26	#508	—	4	4	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	D₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	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×#108	—	2	2	no	—
41	C₄×M₁₆	—	4	1	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃₂	—	2	4	no	—
44	C₂×C₃₂	—	2	2	no	—
45	#368	—	2	2	no	—
46	#368	D₄	1	4	yes	complemented
47	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₄×C₁₆	D₄	1	1	yes	characteristic
50	C₄×C₁₆	D₄	1	1	yes	characteristic
51	#341	—	4	1	no	—
52	C₂×#41	—	4	2	no	—
53	#135	—	4	4	no	—
54	#135	—	4	4	no	—
55	#135	—	8	4	no	—
56	C₂×Q₁₆	—	2	2	no	—
57	C₂×Q₁₆	—	4	1	no	—
58	C₂×SD₁₆	—	4	2	no	—
59	C₂×SD₁₆	—	4	2	no	—
60	C₂×D₈	—	2	2	no	—
61	C₂×D₈	—	4	1	no	—
62	C₂×M₁₆	—	4	2	no	—
63	C₂²×C₈	D₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Q₈	—	4	1	no	—
66	#127	—	4	1	no	—
67	#124	—	4	2	no	—
68	#121	—	8	1	no	—
69	#121	—	8	1	no	—
70	C₄×D₄	—	8	1	no	—
71	C₄×D₄	—	8	1	no	—
72	C₂×C₄²	D₈	1	1	yes	characteristic
73	Q₃₂	—	2	4	no	—
74	Q₃₂	—	4	2	no	—
75	SD₃₂	—	4	4	no	—
76	SD₃₂	—	4	2	no	—
77	SD₃₂	—	4	2	no	—
78	D₁₆	—	2	4	no	—
79	D₁₆	—	4	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₁₆	C₂×D₄	1	2	yes	—
85	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
86	C₂×C₁₆	#41	1	2	yes	—
87	#108	—	2	4	no	—
88	#108	—	4	2	no	—
89	#97	—	8	1	no	—
90	C₄×C₈	—	8	1	no	—
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	SD₁₆	1	1	yes	characteristic
94	C₄×C₈	SD₁₆	1	1	yes	characteristic
95	C₃₂	—	4	2	no	—
96	#41	—	8	4	no	—
97	#41	—	8	4	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	8	2	no	—
101	C₂×D₄	—	8	1	no	—
102	C₂×D₄	—	8	1	no	—
103	C₂²×C₄	—	8	2	no	—
104	C₂²×C₄	SD₃₂	1	1	yes	characteristic
105	C₂²×C₄	SD₃₂	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	Q₁₆	—	4	2	no	—
108	Q₁₆	—	4	2	no	—
109	Q₁₆	—	8	2	no	—
110	SD₁₆	—	8	4	no	—
111	SD₁₆	—	8	4	no	—
112	D₈	—	4	2	no	—
113	D₈	—	4	2	no	—
114	D₈	—	8	2	no	—
115	M₁₆	—	4	4	no	—
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	8	2	no	—
119	C₂×C₈	#102	1	1	yes	characteristic
120	C₂×C₈	#102	1	1	yes	characteristic
121	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₈	#121	1	2	yes	—
123	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₂²⋊C₄	—	8	2	no	—
128	C₂²⋊C₄	—	8	2	no	—
129	C₄²	#102	1	1	yes	characteristic
130	C₄²	#102	1	1	yes	characteristic
131	C₄²	D₁₆	1	1	yes	characteristic
132	C₄²	D₁₆	1	1	yes	characteristic
133	C₁₆	—	2	1	no	—
134	C₁₆	—	2	1	no	—
135	C₁₆	C₄×D₄	1	2	yes	—
136	C₁₆	#124	1	1	yes	characteristic
137	C₁₆	#127	1	1	yes	characteristic
138	C₂³	—	8	2	no	—
139	C₂³	C₂×SD₃₂	1	1	yes	characteristic
140	Q₈	—	8	2	no	—
141	Q₈	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	16	2	no	—
145	D₄	—	16	4	no	—
146	D₄	—	16	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	16	4	no	—
151	C₂×C₄	#357	1	1	yes	characteristic
152	C₂×C₄	#357	1	1	yes	characteristic
153	C₂×C₄	#357	1	1	yes	characteristic
154	C₂×C₄	#357	1	1	yes	characteristic
155	C₂×C₄	C₂×#102	1	1	yes	characteristic
156	C₂×C₄	#459	1	2	yes	—
157	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
158	C₂×C₄	#509	1	2	yes	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	4	no	—
162	C₈	#390	1	1	yes	characteristic
163	C₈	#418	1	1	yes	characteristic
164	C₈	#460	1	2	yes	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	16	2	no	—
168	C₂²	—	16	2	no	—
169	C₂²	—	16	2	no	—
170	C₂²	C₂×#357	1	1	yes	characteristic
171	C₂²	#2207	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	16	2	no	—
176	C₄	#1417	1	2	yes	complemented
177	C₄	#1982	1	1	yes	characteristic
178	C₄	#2140	1	1	yes	characteristic
179	C₄	#2206	1	2	yes	—
180	C₂	—	2	2	no	—
181	C₂	—	32	2	no	—
182	C₂	#13252	1	1	yes	characteristic
183	C₂	C₂×#1417	1	1	yes	characteristic
184	C₂	#13620	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ 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. C₂×C₄⋊C₃₂	C₂	semidirect
3. C₂×#1417	C₂	semidirect
4. C₄×#368	C₂	semidirect
46. #368	D₄	semidirect
176. C₄	#1417	semidirect

Maximal subgroups

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

subgroup	index
#33713	2
C₂×C₄⋊C₃₂	2
C₂×#1417	2
C₄×#368	2