Finite Group Databasebeta

Subgroups — #140302

Order 512.

198 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13289	C₂	1	1	yes	maximal, characteristic
2	#13273	C₂	1	1	yes	maximal, characteristic
3	#13182	C₂	1	1	yes	maximal, characteristic
4	#12667	C₂	1	1	yes	maximal, characteristic
5	C₂×#1366	C₂	1	1	yes	maximal, characteristic
6	#7273	C₂	1	1	yes	maximal, characteristic
7	#7162	C₂	1	1	yes	maximal, characteristic
8	C₂×#499	C₂²	1	1	yes	characteristic
9	#2161	—	2	1	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	#2110	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	#1848	C₂²	1	1	yes	characteristic
16	#1772	C₂²	1	1	yes	characteristic
17	#1389	—	2	1	no	—
18	#1387	—	2	1	no	—
19	#1387	—	2	1	no	—
20	#1386	—	2	1	no	—
21	#1383	—	2	1	no	—
22	#1378	—	2	1	no	—
23	#1366	C₄	1	4	yes	complemented
24	C₂×#125	—	2	1	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂×M₃₂	D₄	1	1	yes	characteristic
29	C₂×M₃₂	D₄	1	1	yes	characteristic
30	#499	—	2	1	no	—
31	#499	—	2	1	no	—
32	#499	—	2	1	no	—
33	#499	C₂×C₄	1	2	yes	—
34	C₂×#108	—	2	1	no	—
35	C₂×#108	—	2	1	no	—
36	C₂×#108	D₄	1	1	yes	characteristic
37	#427	D₄	1	1	yes	characteristic
38	C₂×#107	—	2	1	no	—
39	C₂×#106	—	2	1	no	—
40	#423	—	2	1	no	—
41	#405	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#368	—	4	2	no	—
44	#364	—	2	1	no	—
45	#364	—	2	1	no	—
46	#364	—	2	1	no	—
47	#364	—	2	1	no	—
48	#364	C₂×C₄	1	2	yes	—
49	#364	C₂×C₄	1	2	yes	—
50	#347	—	4	2	no	—
51	#346	—	2	1	no	—
52	C₄×C₁₆	—	2	1	no	—
53	#339	—	2	1	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	4	1	no	—
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	#125	—	2	2	no	—
59	#125	—	4	1	no	—
60	#125	—	8	1	no	—
61	#117	—	2	1	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄⋊C₄	—	8	1	no	—
64	C₂×C₄²	#41	1	1	yes	characteristic
65	M₃₂	—	2	1	no	—
66	M₃₂	—	2	2	no	—
67	M₃₂	—	2	2	no	—
68	M₃₂	—	2	1	no	—
69	M₃₂	—	2	1	no	—
70	M₃₂	—	4	1	no	—
71	M₃₂	—	4	1	no	—
72	M₃₂	—	4	1	no	—
73	M₃₂	—	4	1	no	—
74	M₃₂	Q₁₆	1	2	yes	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	4	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	#108	—	2	1	no	—
81	#108	—	2	1	no	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#108	—	4	1	no	—
85	#108	—	4	1	no	—
86	#107	—	2	2	no	—
87	#107	—	2	1	no	—
88	#107	—	4	1	no	—
89	#106	—	2	2	no	—
90	#106	—	2	1	no	—
91	#106	—	4	1	no	—
92	C₄⋊C₈	—	8	1	no	—
93	#97	—	4	1	no	—
94	C₄×C₈	—	4	1	no	—
95	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	C₄×C₈	#41	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	#121	1	1	yes	characteristic
102	M₁₆	—	4	1	no	—
103	M₁₆	—	4	1	no	—
104	M₁₆	—	8	1	no	—
105	M₁₆	—	8	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	C₄×D₄	1	1	yes	characteristic
109	C₂×C₈	C₄×D₄	1	1	yes	characteristic
110	C₂×C₈	#123	1	1	yes	characteristic
111	C₂×C₈	#124	1	1	yes	characteristic
112	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₈	#135	1	1	yes	characteristic
114	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₈	#137	1	1	yes	characteristic
117	C₂×C₈	#137	1	1	yes	characteristic
118	C₂×C₈	#137	1	1	yes	characteristic
119	C₂×C₈	#137	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	4	1	no	—
128	C₄²	#117	1	1	yes	characteristic
129	C₄²	#126	1	1	yes	characteristic
130	C₁₆	—	2	2	no	—
131	C₁₆	—	4	2	no	—
132	C₁₆	—	4	2	no	—
133	C₁₆	—	4	1	no	—
134	C₁₆	—	8	1	no	—
135	C₂³	#471	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	16	2	no	—
142	C₂×C₄	#388	1	1	yes	characteristic
143	C₂×C₄	#462	1	1	yes	characteristic
144	C₂×C₄	#463	1	1	yes	characteristic
145	C₂×C₄	#468	1	1	yes	characteristic
146	C₂×C₄	#470	1	1	yes	characteristic
147	C₂×C₄	#472	1	1	yes	characteristic
148	C₈	—	8	1	no	—
149	C₈	—	16	1	no	—
150	C₈	C₄×Q₁₆	1	1	yes	characteristic
151	C₈	#440	1	1	yes	characteristic
152	C₈	#440	1	1	yes	characteristic
153	C₈	#441	1	1	yes	characteristic
154	C₈	#484	1	1	yes	characteristic
155	C₈	#485	1	1	yes	characteristic
156	C₈	#489	1	1	yes	characteristic
157	C₈	#491	1	1	yes	characteristic
158	C₂²	—	4	1	no	—
159	C₂²	#1692	1	1	yes	characteristic
160	C₂²	#1703	1	1	yes	characteristic
161	C₂²	#1931	1	1	yes	characteristic
162	C₄	—	4	1	no	—
163	C₄	—	4	1	no	—
164	C₄	—	16	1	no	—
165	C₄	—	16	2	no	—
166	C₄	#1696	1	1	yes	characteristic
167	C₄	#1697	1	1	yes	characteristic
168	C₄	#1928	1	1	yes	characteristic
169	C₄	#1935	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#9576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#9576
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×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
23. #1366	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13289	2
#13273	2
#13182	2
#12667	2
C₂×#1366	2
#7273	2
#7162	2