Finite Group Databasebeta

← #145057

Subgroups — #145057

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#13188C211yesmaximal, characteristic, complemented
3#12825C211yesmaximal, characteristic
4#11956C211yesmaximal, characteristic, complemented
5C2×#1335C211yesmaximal, characteristic
6#9855C211yesmaximal, characteristic, complemented
7#7179C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2155—21no—
11C2×#364C2211yescharacteristic
12C2×#364C2211yescharacteristic
13#2148—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#1388—21no—
24#1385—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#365—42no—
55#364—21no—
56#364—21no—
57#364—21no—
58#364D412yes—
59#362—41no—
60#361—41no—
61#344—22no—
62#339—21no—
63H(3,ℤ/4)—22no—
64Q8⋊C8—44no—
65C82—21no—
66C82—21no—
67C2×#41—21no—
68#135—21no—
69#135—21no—
70#135—21no—
71#135—21no—
72C2×Q16SD1611yescharacteristic
73C2×SD16—21no—
74C2×D8SD1611yescharacteristic
75#131—41no—
76#131—41no—
77#131—41no—
78#131—41no—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—41no—
82C2×M16—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—42no—
90C4×Q8—42no—
91#117—22no—
92#117—21no—
93C2×C42C2×D411yescharacteristic
94M32—22no—
95M32—41no—
96M32—41no—
97M32—41no—
98C2×C16—41no—
99C2×C16—41no—
100#108—21no—
101#108—21no—
102#108—21no—
103#108—21no—
104#108—41no—
105#108—41no—
106#108—41no—
107#108—41no—
108#107—21no—
109#107—41no—
110#107—41no—
111#106—21no—
112#106—41no—
113#106—41no—
114C4⋊C8—42no—
115#104—41no—
116#104—41no—
117#104—82no—
118#99—82no—
119#97—41no—
120C4×C8—22no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8C2×D411yescharacteristic
124C4×C8#4111yescharacteristic
125C4×C8#4111yescharacteristic
126C4×C8#4111yescharacteristic
127#41—41no—
128#41—41no—
129#41—41no—
130#41—41no—
131C2×Q8—21no—
132C2×Q8—42no—
133C2×D4—21no—
134C2×D4—41no—
135C22×C4—21no—
136C22×C4—41no—
137C22×C4#12011yescharacteristic
138Q16—21no—
139Q16—21no—
140SD16—41no—
141SD16—41no—
142D8—21no—
143D8—21no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—41no—
148M16—81no—
149M16—81no—
150M16—81no—
151M16—81no—
152C2×C8—42no—
153C2×C8—44no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8—81no—
159C2×C8#12111yescharacteristic
160C2×C8#12111yescharacteristic
161C2×C8#12311yescharacteristic
162C2×C8#12411yescharacteristic
163C2×C8C2×SD1611yescharacteristic
164C2×C8#13511yescharacteristic
165C2×C8#13511yescharacteristic
166C2×C8#13511yescharacteristic
167C2×C8AGL(1,ℤ/8)11yescharacteristic
168C2×C8#13711yescharacteristic
169C2×C8#13711yescharacteristic
170C2×C8#13711yescharacteristic
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—82no—
175C22⋊C4—41no—
176C22⋊C4—42no—
177C42—21no—
178C42—41no—
179C42—42no—
180C42#12111yescharacteristic
181C42#12311yescharacteristic
182C16—42no—
183C16—81no—
184C23—41no—
185C23#45511yescharacteristic
186Q8—41no—
187Q8—41no—
188Q8—84no—
189D4—41no—
190D4—41no—
191D4—81no—
192D4—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4#39411yescharacteristic
202C2×C4#44811yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#45211yescharacteristic
205C2×C4#45211yescharacteristic
206C2×C4#45311yescharacteristic
207C8—44no—
208C8—81no—
209C8—81no—
210C8—161no—
211C8#46311yescharacteristic
212C8#46411yescharacteristic
213C8#46511yescharacteristic
214C8#47011yescharacteristic
215C8#48311yescharacteristic
216C8#48511yescharacteristic
217C8#48811yescharacteristic
218C8#48911yescharacteristic
219C22—41no—
220C22—81no—
221C22—161no—
222C22#164311yescharacteristic
223C22#164611yescharacteristic
224C22#203711yescharacteristic
225C4—22no—
226C4—41no—
227C4—81no—
228C4—161no—
229C4—162no—
230C4#162211yescharacteristic
231C4#163411yescharacteristic
232C4#203511yescharacteristic
233C4#204211yescharacteristic
234C2—21no—
235C2—21no—
236C2—21no—
237C2—161no—
238C2#1041211yescharacteristic
subgroupG/H
centerC4#1634
commutator, Frattini, cosocleC2×C4×C8C23
socleC2#10412
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. #13188C2semidirect
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
#131882
#128252
#119562
C2×#13352
#98552
#71792