Finite Group Databasebeta

← #145060

Subgroups — #145060

Order 512.

285 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29154C211yesmaximal, characteristic
2#13182C211yesmaximal, characteristic, complemented
3#12822C211yesmaximal, characteristic
4#11956C211yesmaximal, characteristic, complemented
5C2×#1335C211yesmaximal, characteristic
6#9855C211yesmaximal, characteristic, complemented
7#7179C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2161—21no—
11C2×#364C2211yescharacteristic
12C2×#364C2211yescharacteristic
13#2143—21no—
14#2031—22no—
15#1976—21no—
16#1848C2211yescharacteristic
17#1833—21no—
18#1814—21no—
19#1812—21no—
20#1772C2211yescharacteristic
21#1493—22no—
22C2×C82C2211yescharacteristic
23#1389—21no—
24#1388—21no—
25#1366—22no—
26#1335—22no—
27C2×#135D411yescharacteristic
28C2×#131—21no—
29#533D412yes—
30C2×M32—21no—
31C2×M32—21no—
32#500—21no—
33#499—21no—
34#499—21no—
35#498—21no—
36#443—21no—
37#443—21no—
38#443—21no—
39#443—21no—
40C2×#108—21no—
41C2×#108—21no—
42C2×#108D411yescharacteristic
43C2×#108D411yescharacteristic
44#427—21no—
45#427—21no—
46#427D411yescharacteristic
47#423—21no—
48#421—22no—
49C2×#104—21no—
50#410—22no—
51#405—21no—
52C2×C4×C8—21no—
53C2×C4×C8C2311yescharacteristic
54#368—42no—
55#364—21no—
56#364—21no—
57#364—21no—
58#364D412yes—
59#359—42no—
60#344—22no—
61#339—21no—
62H(3,ℤ/4)—22no—
63Q8⋊C8—44no—
64C82—21no—
65C82—21no—
66C2×#41—21no—
67#135—21no—
68#135—21no—
69#135—21no—
70#135—21no—
71C2×Q16Q1611yescharacteristic
72C2×SD16—21no—
73C2×D8Q1611yescharacteristic
74#131—41no—
75#131—41no—
76#131—41no—
77#131—41no—
78C2×M16—21no—
79C2×M16—21no—
80C2×M16—41no—
81C2×M16—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4⋊Q8—22no—
87#125—22no—
88C22⋊Q8—42no—
89C4×Q8—42no—
90#117—22no—
91#117—21no—
92C2×C42C2×D411yescharacteristic
93M32—22no—
94M32—41no—
95M32—41no—
96M32—41no—
97C2×C16—41no—
98C2×C16—41no—
99#108—21no—
100#108—21no—
101#108—21no—
102#108—21no—
103#108—41no—
104#108—41no—
105#108—41no—
106#108—41no—
107#107—21no—
108#107—41no—
109#107—41no—
110#106—21no—
111#106—41no—
112#106—41no—
113C4⋊C8—42no—
114#104—41no—
115#104—41no—
116#104—82no—
117#99—82no—
118#97—41no—
119C4×C8—22no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125C4×C8#4111yescharacteristic
126#41—41no—
127#41—41no—
128#41—41no—
129#41—41no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×D4—21no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4#12011yescharacteristic
137Q16—21no—
138Q16—21no—
139SD16—41no—
140SD16—41no—
141D8—21no—
142D8—21no—
143M16—41no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—81no—
148M16—81no—
149M16—81no—
150M16—81no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8#12111yescharacteristic
159C2×C8#12111yescharacteristic
160C2×C8#12311yescharacteristic
161C2×C8#12411yescharacteristic
162C2×C8C2×Q1611yescharacteristic
163C2×C8#13511yescharacteristic
164C2×C8#13511yescharacteristic
165C2×C8#13511yescharacteristic
166C2×C8AGL(1,ℤ/8)11yescharacteristic
167C2×C8AGL(1,ℤ/8)11yescharacteristic
168C2×C8#13711yescharacteristic
169C2×C8#13711yescharacteristic
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—82no—
174C22⋊C4—41no—
175C22⋊C4—42no—
176C42—21no—
177C42—41no—
178C42—42no—
179C42#12111yescharacteristic
180C42#12311yescharacteristic
181C16—42no—
182C16—81no—
183C23—41no—
184C23#45511yescharacteristic
185Q8—41no—
186Q8—41no—
187Q8—84no—
188D4—41no—
189D4—41no—
190D4—81no—
191D4—81no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4#39411yescharacteristic
201C2×C4#44911yescharacteristic
202C2×C4#44911yescharacteristic
203C2×C4#45111yescharacteristic
204C2×C4#45211yescharacteristic
205C2×C4#45311yescharacteristic
206C8—44no—
207C8—81no—
208C8—81no—
209C8—161no—
210C8#46311yescharacteristic
211C8#46411yescharacteristic
212C8#46711yescharacteristic
213C8#46811yescharacteristic
214C8#48411yescharacteristic
215C8#48511yescharacteristic
216C8#48811yescharacteristic
217C8#48911yescharacteristic
218C22—41no—
219C22—81no—
220C22—161no—
221C22#164411yescharacteristic
222C22#164611yescharacteristic
223C22#203711yescharacteristic
224C4—22no—
225C4—41no—
226C4—81no—
227C4—161no—
228C4—162no—
229C4#162411yescharacteristic
230C4#163211yescharacteristic
231C4#203611yescharacteristic
232C4#204211yescharacteristic
233C2—21no—
234C2—21no—
235C2—21no—
236C2—161no—
237C2#1041311yescharacteristic
subgroupG/H
centerC4#1624
commutator, Frattini, cosocleC2×C4×C8C23
socleC2#10413
prime radicalC2×#135D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13182C2semidirect
4. #11956C2semidirect
6. #9855C2semidirect
7. #7179C2semidirect

Maximal subgroups

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

subgroupindex
#291542
#131822
#128222
#119562
C2×#13352
#98552
#71792