Finite Group Databasebeta

Subgroups — #129128

Order 512.

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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	#12590	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic
3	#12008	C₂	1	1	yes	maximal, characteristic
4	#8267	C₂	1	1	yes	maximal, characteristic
5	#8266	C₂	1	1	yes	maximal, characteristic
6	#7155	C₂	1	1	yes	maximal, characteristic
7	#7155	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	#1564	C₂²	1	1	yes	characteristic
12	#1564	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1452	—	2	1	no	—
16	#1450	—	2	1	no	—
17	#1374	C₄	1	2	yes	—
18	#1374	C₄	1	2	yes	—
19	#1372	C₄	1	2	yes	complemented
20	#1372	C₄	1	2	yes	complemented
21	#1371	C₄	1	2	yes	complemented
22	#1371	C₄	1	2	yes	complemented
23	#1276	—	2	1	no	—
24	#1276	—	2	1	no	—
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	#431	—	2	1	no	—
29	#431	—	2	1	no	—
30	#431	—	2	1	no	—
31	#431	—	2	1	no	—
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	Q₈	1	2	yes	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄³	D₄	1	1	yes	characteristic
37	C₄×C₁₆	—	2	1	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	#335	C₂×C₄	1	2	yes	—
44	#335	C₂×C₄	1	2	yes	—
45	#335	C₂×C₄	1	2	yes	—
46	#335	C₂×C₄	1	2	yes	—
47	#334	—	2	1	no	—
48	#334	—	2	1	no	—
49	#334	—	2	1	no	—
50	#334	—	2	1	no	—
51	#322	—	4	1	no	—
52	#322	—	4	1	no	—
53	#322	—	4	1	no	—
54	C₈²	—	4	1	no	—
55	C₂²×C₈	C₄⋊C₄	1	2	yes	—
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄²	—	2	1	no	—
59	C₂×C₄²	—	2	1	no	—
60	C₂×C₄²	C₄⋊C₄	1	2	yes	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	C₄²	1	1	yes	characteristic
68	C₂×C₁₆	C₄²	1	1	yes	characteristic
69	C₂×C₁₆	C₄²	1	1	yes	characteristic
70	C₂×C₁₆	C₄²	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₂²⋊C₈	—	4	1	no	—
76	C₂²⋊C₈	—	4	1	no	—
77	C₂²⋊C₈	—	4	1	no	—
78	C₂²⋊C₈	—	4	1	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	4	2	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	D₈	1	1	yes	characteristic
87	C₄×C₈	D₈	1	1	yes	characteristic
88	C₄×C₈	Q₁₆	1	2	yes	—
89	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₄	#135	1	1	yes	characteristic
96	C₂²×C₄	#135	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	#107	1	2	yes	—
105	C₂×C₈	#107	1	2	yes	—
106	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₈	#117	1	1	yes	characteristic
108	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₈	#131	1	2	yes	—
111	C₂×C₈	#131	1	2	yes	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	4	2	no	—
116	C₄²	C₄×D₄	1	1	yes	characteristic
117	C₄²	C₄×D₄	1	1	yes	characteristic
118	C₄²	C₂×D₈	1	1	yes	characteristic
119	C₄²	C₂×Q₁₆	1	1	yes	characteristic
120	C₁₆	—	2	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	4	2	no	—
125	C₂³	#427	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	#424	1	2	yes	—
133	C₂×C₄	C₂×#107	1	1	yes	characteristic
134	C₂×C₄	#430	1	2	yes	—
135	C₂×C₄	C₄×D₈	1	1	yes	characteristic
136	C₂×C₄	C₄×D₈	1	1	yes	characteristic
137	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	2	no	—
146	C₈	—	8	2	no	—
147	C₈	#405	1	1	yes	characteristic
148	C₈	#405	1	1	yes	characteristic
149	C₈	#433	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#1572	1	2	yes	—
153	C₂²	C₄×#107	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	4	no	—
157	C₄	#1571	1	2	yes	—
158	C₄	#1768	1	1	yes	characteristic
159	C₄	#1778	1	1	yes	characteristic
160	C₄	#2178	1	2	yes	—
161	C₄	#2179	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	#8278	1	1	yes	characteristic
164	C₂	#12670	1	1	yes	characteristic
165	C₂	#12671	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ 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
19. #1372	C₄	semidirect
20. #1372	C₄	semidirect
21. #1371	C₄	semidirect
22. #1371	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12590	2
#12008	2
#8267	2
#8266	2
#7155	2