Finite Group Databasebeta

Subgroups — #134697

Order 512.

269 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 100 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	#33420	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1372	C₂	1	1	yes	maximal, characteristic, complemented
4	#8605	C₂	1	2	yes	maximal
5	#7159	C₂	1	2	yes	maximal, complemented
6	C₄×#131	C₂²	1	1	yes	characteristic
7	#2164	—	2	1	no	—
8	#2164	—	2	1	no	—
9	#2162	—	2	2	no	—
10	#2151	C₄	1	2	yes	—
11	#2113	C₄	1	2	yes	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	2	yes	—
15	C₂×#335	C₂²	1	2	yes	—
16	#1477	—	2	2	no	—
17	#1372	—	2	2	no	—
18	#1372	—	2	1	no	—
19	#1371	—	2	1	no	—
20	#1283	—	2	2	no	—
21	C₂×#131	C₂×C₄	1	1	yes	characteristic
22	C₂×#131	Q₈	1	1	yes	characteristic
23	C₄×#41	D₄	1	1	yes	characteristic
24	C₂×M₃₂	C₂×C₄	1	2	yes	—
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	#431	—	2	2	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	#405	—	2	1	no	—
29	#405	—	2	1	no	—
30	#405	C₂×C₄	1	1	yes	characteristic
31	#405	Q₈	1	1	yes	characteristic
32	C₄×M₁₆	—	2	1	no	—
33	C₄×M₁₆	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄⋊C₁₆	—	2	2	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₂²⋊C₁₆	—	2	2	no	—
39	C₂²⋊C₁₆	—	2	2	no	—
40	#346	—	2	1	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	D₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	2	yes	—
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	#335	—	2	2	no	—
46	#335	—	2	2	no	—
47	#335	—	2	2	no	—
48	#334	—	2	2	no	—
49	Q₈⋊C₈	—	4	2	no	—
50	D₄⋊C₈	—	4	2	no	—
51	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
52	#131	—	2	4	no	—
53	#131	—	4	2	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	2	1	no	—
56	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
57	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	—	4	2	no	—
61	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₄×Q₈	D₈	1	1	yes	characteristic
64	C₄×Q₈	Q₁₆	1	1	yes	characteristic
65	C₄×D₄	—	2	1	no	—
66	C₄×D₄	—	2	1	no	—
67	C₄×D₄	D₈	1	1	yes	characteristic
68	C₄×D₄	Q₁₆	1	1	yes	characteristic
69	#117	—	2	1	no	—
70	#117	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	M₃₂	—	2	4	no	—
74	C₂×C₁₆	—	2	4	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
77	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₂²⋊C₈	—	4	2	no	—
82	#97	—	2	1	no	—
83	#97	—	2	1	no	—
84	#97	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	C₂×Q₈	#107	1	1	yes	characteristic
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	#107	1	1	yes	characteristic
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄	#135	1	1	yes	characteristic
103	C₂²×C₄	#135	1	1	yes	characteristic
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	—	8	2	no	—
115	C₂×C₈	—	16	2	no	—
116	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₈	#117	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₈	#131	1	4	yes	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	#107	1	1	yes	characteristic
124	C₄⋊C₄	#107	1	1	yes	characteristic
125	C₂²⋊C₄	—	2	1	no	—
126	C₂²⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₄²	#121	1	1	yes	characteristic
132	C₄²	#121	1	1	yes	characteristic
133	C₄²	C₂×D₈	1	1	yes	characteristic
134	C₄²	C₂×Q₁₆	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	—	2	2	no	—
137	C₁₆	—	4	2	no	—
138	C₂³	—	2	1	no	—
139	C₂³	#427	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	4	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	#380	1	1	yes	characteristic
153	C₂×C₄	#424	1	2	yes	—
154	C₂×C₄	C₂×#107	1	1	yes	characteristic
155	C₂×C₄	#430	1	2	yes	—
156	C₂×C₄	#466	1	1	yes	characteristic
157	C₂×C₄	#466	1	1	yes	characteristic
158	C₂×C₄	#467	1	1	yes	characteristic
159	C₂×C₄	#467	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	1	no	—
163	C₈	—	4	1	no	—
164	C₈	—	16	2	no	—
165	C₈	—	16	2	no	—
166	C₈	#408	1	2	yes	—
167	C₈	#433	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	#1572	1	2	yes	—
174	C₂²	#1808	1	1	yes	characteristic
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	#1571	1	2	yes	—
182	C₄	#1798	1	1	yes	characteristic
183	C₄	#1811	1	1	yes	characteristic
184	C₄	#2223	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	8	1	no	—
187	C₂	#8599	1	1	yes	characteristic
188	C₂	#12841	1	1	yes	characteristic
189	C₂	#12843	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #12008	C₂	semidirect
3. C₂×#1372	C₂	semidirect
5. #7159	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33420	2
#12008	2
C₂×#1372	2
#8605	2
#7159	2