Finite Group Databasebeta

Subgroups — #134775

Order 512.

299 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 84 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	#33434	C₂	1	1	yes	maximal, characteristic
2	#12839	C₂	1	2	yes	maximal, complemented
3	#12807	C₂	1	2	yes	maximal
4	#7228	C₂	1	2	yes	maximal, complemented
5	C₂×#504	C₄	1	1	yes	characteristic, complemented
6	C₂×#504	C₂²	1	1	yes	characteristic
7	#2931	C₂²	1	1	yes	characteristic
8	#2169	—	2	2	no	—
9	D₄×C₁₆	—	2	2	no	—
10	#2158	—	2	2	no	—
11	#2150	C₄	1	1	yes	characteristic, complemented
12	#2126	—	2	2	no	—
13	#2113	C₄	1	1	yes	characteristic, complemented
14	#2113	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic, complemented
16	#1812	C₂²	1	4	yes	—
17	#1391	—	2	2	no	—
18	#1384	—	2	2	no	—
19	C₂×#131	C₂×C₄	1	1	yes	characteristic
20	C₂×#131	C₂³	1	1	yes	characteristic
21	#532	D₄	1	1	yes	characteristic
22	#504	—	4	1	no	—
23	#504	—	4	1	no	—
24	#504	—	4	1	no	—
25	#504	—	4	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
28	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
29	C₂×M₃₂	D₄	1	2	yes	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	D₄	1	2	yes	—
34	#436	—	2	2	no	—
35	C₈×D₄	—	2	2	no	—
36	#427	—	2	4	no	—
37	#423	Q₈	1	1	yes	characteristic
38	#410	—	2	4	no	—
39	#408	D₄	1	1	yes	characteristic
40	#408	Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	#367	—	4	2	no	—
43	#366	—	4	2	no	—
44	C₄⋊C₁₆	—	2	2	no	—
45	#354	—	4	2	no	—
46	#353	—	4	2	no	—
47	C₂²⋊C₁₆	—	2	2	no	—
48	C₂²⋊C₁₆	—	2	2	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	H(3,ℤ/4)	—	2	4	no	—
52	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#41	C₂×D₄	1	1	yes	characteristic
54	#131	—	4	1	no	—
55	#131	—	4	1	no	—
56	#131	—	4	1	no	—
57	#131	—	4	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
60	C₂×M₁₆	C₂×Q₈	1	1	yes	characteristic
61	C₂×M₁₆	#41	1	2	yes	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	2	yes	—
66	C₄⋊Q₈	Q₁₆	1	1	yes	characteristic
67	#127	D₈	1	1	yes	characteristic
68	#124	D₈	1	1	yes	characteristic
69	#124	Q₁₆	1	1	yes	characteristic
70	#121	—	2	2	no	—
71	C₄×D₄	—	2	2	no	—
72	#117	—	4	4	no	—
73	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
74	M₃₂	—	4	1	no	—
75	M₃₂	—	4	2	no	—
76	M₃₂	—	4	2	no	—
77	M₃₂	—	4	1	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	4	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
89	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
90	#107	—	4	4	no	—
91	#106	—	4	4	no	—
92	C₄⋊C₈	—	2	2	no	—
93	#99	—	4	4	no	—
94	#99	—	4	4	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	#41	—	4	1	no	—
102	#41	—	4	1	no	—
103	C₂×Q₈	#107	1	1	yes	characteristic
104	C₂×Q₈	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	#107	1	1	yes	characteristic
108	C₂×D₄	C₂×D₈	1	1	yes	characteristic
109	C₂×D₄	#135	1	2	yes	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	#121	1	2	yes	—
114	M₁₆	—	4	1	no	—
115	M₁₆	—	4	2	no	—
116	M₁₆	—	4	2	no	—
117	M₁₆	—	4	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₈	#117	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	8	4	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	8	4	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	8	4	no	—
137	C₄²	#107	1	1	yes	characteristic
138	C₄²	#107	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	1	no	—
141	C₁₆	—	4	1	no	—
142	C₁₆	—	4	1	no	—
143	C₁₆	#99	1	1	yes	characteristic
144	C₁₆	#99	1	1	yes	characteristic
145	C₂³	—	2	1	no	—
146	C₂³	#427	1	1	yes	characteristic
147	C₂³	#466	1	2	yes	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	4	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	16	4	no	—
166	C₂×C₄	#380	1	1	yes	characteristic
167	C₂×C₄	C₂×#107	1	1	yes	characteristic
168	C₂×C₄	#467	1	2	yes	—
169	C₈	—	2	1	no	—
170	C₈	—	2	2	no	—
171	C₈	—	4	1	no	—
172	C₈	—	4	1	no	—
173	C₈	C₂×#99	1	1	yes	characteristic
174	C₈	#410	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	#1808	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	32	4	no	—
188	C₄	#1799	1	1	yes	characteristic
189	C₄	#1812	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	4	2	no	—
192	C₂	—	4	1	no	—
193	C₂	—	8	1	no	—
194	C₂	#8609	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8609
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12839	C₂	semidirect
4. #7228	C₂	semidirect
5. C₂×#504	C₄	semidirect
11. #2150	C₄	semidirect
13. #2113	C₄	semidirect
15. C₂×C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#33434	2
#12839	2
#12807	2
#7228	2