Finite Group Databasebeta

Subgroups — #152825

Order 512.

209 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₂×#1426	C₂	1	1	yes	maximal, characteristic
2	C₂×#1420	C₂	1	1	yes	maximal, characteristic
3	C₂×#1420	C₂	1	1	yes	maximal, characteristic
4	#13240	C₂	1	1	yes	maximal, characteristic
5	#12676	C₂	1	1	yes	maximal, characteristic
6	#7474	C₂	1	1	yes	maximal, characteristic
7	#7468	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	1	yes	characteristic
12	C₂×#368	C₂²	1	1	yes	characteristic
13	C₂×#367	C₂²	1	1	yes	characteristic
14	C₂×#358	—	2	1	no	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	#1969	—	2	1	no	—
17	C₄×#108	—	2	1	no	—
18	#1426	—	2	2	no	—
19	#1420	—	2	2	no	—
20	#1420	C₄	1	4	yes	complemented
21	#1381	—	2	1	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂×Q₃₂	—	2	1	no	—
24	C₂×Q₃₂	—	2	1	no	—
25	C₂×Q₃₂	—	2	1	no	—
26	C₂×Q₃₂	—	2	1	no	—
27	C₂×Q₃₂	—	2	1	no	—
28	C₂×Q₃₂	C₂×C₄	1	2	yes	—
29	C₂²×C₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#107	—	2	1	no	—
34	C₂×#103	—	4	1	no	—
35	C₄×M₁₆	—	4	1	no	—
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	#391	—	4	1	no	—
38	M₆₄	—	2	1	no	—
39	M₆₄	—	2	1	no	—
40	M₆₄	—	2	1	no	—
41	M₆₄	C₂×C₄	1	2	yes	—
42	C₂×C₃₂	—	2	1	no	—
43	C₂×C₃₂	—	2	1	no	—
44	#368	—	2	1	no	—
45	#368	—	2	1	no	—
46	#368	C₂×C₄	1	2	yes	—
47	#368	D₄	1	2	yes	—
48	#367	—	2	1	no	—
49	#367	—	2	1	no	—
50	#358	—	4	2	no	—
51	#346	—	2	1	no	—
52	#346	—	2	1	no	—
53	#341	—	4	1	no	—
54	C₂²×Q₈	—	4	1	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	4	1	no	—
57	C₂×Q₁₆	—	4	1	no	—
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×Q₁₆	—	4	1	no	—
60	C₂×Q₁₆	—	4	1	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂²×C₈	D₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	4	1	no	—
66	C₂×C₄²	D₈	1	1	yes	characteristic
67	Q₃₂	—	2	2	no	—
68	Q₃₂	—	2	2	no	—
69	Q₃₂	—	4	1	no	—
70	Q₃₂	—	4	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	#41	1	1	yes	characteristic
77	C₂×C₁₆	#41	1	1	yes	characteristic
78	C₂×C₁₆	#41	1	1	yes	characteristic
79	C₂×C₁₆	#41	1	1	yes	characteristic
80	#108	—	2	2	no	—
81	#108	—	2	2	no	—
82	#108	—	4	1	no	—
83	#108	—	4	1	no	—
84	#107	—	4	1	no	—
85	#107	—	4	1	no	—
86	#103	—	8	2	no	—
87	#97	—	8	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	8	1	no	—
91	#95	—	8	2	no	—
92	C₃₂	—	2	2	no	—
93	C₃₂	—	4	1	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂×Q₈	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	—	8	1	no	—
102	C₂²×C₄	SD₃₂	1	2	yes	—
103	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
104	Q₁₆	—	4	2	no	—
105	Q₁₆	—	4	2	no	—
106	Q₁₆	—	8	1	no	—
107	Q₁₆	—	8	1	no	—
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	8	1	no	—
111	M₁₆	—	8	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	8	1	no	—
116	C₂×C₈	—	8	1	no	—
117	C₂×C₈	C₄×D₄	1	1	yes	characteristic
118	C₂×C₈	#121	1	1	yes	characteristic
119	C₂×C₈	#123	1	1	yes	characteristic
120	C₂×C₈	#135	1	1	yes	characteristic
121	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₈	#137	1	1	yes	characteristic
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₁₆	—	4	1	no	—
128	C₁₆	#117	1	1	yes	characteristic
129	C₁₆	C₄×D₄	1	1	yes	characteristic
130	C₁₆	#124	1	1	yes	characteristic
131	C₁₆	#126	1	1	yes	characteristic
132	C₂³	C₂×SD₃₂	1	1	yes	characteristic
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	Q₈	—	16	1	no	—
136	Q₈	—	16	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	—	16	1	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	C₄×D₈	1	1	yes	characteristic
145	C₂×C₄	#459	1	1	yes	characteristic
146	C₂×C₄	#480	1	1	yes	characteristic
147	C₂×C₄	#508	1	1	yes	characteristic
148	C₂×C₄	#509	1	1	yes	characteristic
149	C₂×C₄	#509	1	1	yes	characteristic
150	C₈	—	4	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	2	no	—
153	C₈	#388	1	1	yes	characteristic
154	C₈	#440	1	1	yes	characteristic
155	C₈	#464	1	1	yes	characteristic
156	C₈	#482	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₄×SD₃₂	1	1	yes	characteristic
159	C₂²	#2207	1	1	yes	characteristic
160	C₂²	#2233	1	1	yes	characteristic
161	C₄	—	4	2	no	—
162	C₄	—	16	2	no	—
163	C₄	—	32	1	no	—
164	C₄	#1925	1	1	yes	characteristic
165	C₄	#2177	1	1	yes	characteristic
166	C₄	#2210	1	1	yes	characteristic
167	C₄	#2234	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#13131	1	1	yes	characteristic
171	C₂	#13718	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#2233
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ 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
20. #1420	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1426	2
C₂×#1420	2
#13240	2
#12676	2
#7474	2
#7468	2