Finite Group Databasebeta

← #145062

Subgroups — #145062

Order 512.

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

normal subgroups only

This group has 278 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#13180C211yesmaximal, characteristic, complemented
3#12824C211yesmaximal, characteristic
4#11956C211yesmaximal, characteristic, complemented
5#11602C211yesmaximal, characteristic
6#9755C211yesmaximal, characteristic, complemented
7#7181C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2159—21no—
11C2×#364C2211yescharacteristic
12C2×#364C2211yescharacteristic
13C2×#360—21no—
14#2031—22no—
15#1976—21no—
16#1846C2211yescharacteristic
17#1833—21no—
18#1814—21no—
19#1812—21no—
20#1772C2211yescharacteristic
21#1518—22no—
22#1450C2211yescharacteristic
23#1387—21no—
24#1383—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—
40#431—21no—
41C2×#108D411yescharacteristic
42C2×#108D411yescharacteristic
43#427—21no—
44#427—21no—
45#427D411yescharacteristic
46C2×#107—21no—
47C2×#106—21no—
48#421—22no—
49C2×#104—21no—
50#410—22no—
51#405—21no—
52C2×C4×C8C2311yescharacteristic
53#382—21no—
54#367—41no—
55#366—41no—
56#364—21no—
57#364—21no—
58#364—21no—
59#364D412yes—
60#360—42no—
61#344—22no—
62#339—21no—
63H(3,ℤ/4)—22no—
64#328—44no—
65#322—21no—
66C82—21no—
67C2×#41—21no—
68#135—21no—
69#135—21no—
70#135—21no—
71#135—21no—
72C2×Q16Q1611yescharacteristic
73C2×SD16—21no—
74C2×D8Q1611yescharacteristic
75#131—41no—
76#131—41no—
77#131—41no—
78#131—41no—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—41no—
82C22×C8—41no—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—22no—
86#125—22no—
87C22⋊Q8—42no—
88C4×Q8—42no—
89#117—22no—
90#117—21no—
91C2×C4⋊C4—41no—
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#107—21no—
104#107—41no—
105#107—41no—
106#107—41no—
107#107—41no—
108#106—21no—
109#106—41no—
110#106—41no—
111#106—41no—
112#106—41no—
113C4⋊C8—41no—
114#104—41no—
115#104—41no—
116#104—82no—
117#99—82no—
118C22⋊C8—22no—
119#97—41no—
120C4×C8—41no—
121C4×C8—41no—
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125C4×C8#4111yescharacteristic
126#95—42no—
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—81no—
138C22×C4#12011yescharacteristic
139Q16—21no—
140Q16—21no—
141SD16—41no—
142SD16—41no—
143D8—21no—
144D8—21no—
145M16—41no—
146M16—41no—
147M16—41no—
148M16—41no—
149M16—81no—
150M16—81no—
151C2×C8—42no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8#12111yescharacteristic
157C2×C8#12111yescharacteristic
158C2×C8#12311yescharacteristic
159C2×C8#12411yescharacteristic
160C2×C8C2×Q1611yescharacteristic
161C2×C8#13511yescharacteristic
162C2×C8#13511yescharacteristic
163C2×C8#13511yescharacteristic
164C2×C8AGL(1,ℤ/8)11yescharacteristic
165C2×C8AGL(1,ℤ/8)11yescharacteristic
166C2×C8#13711yescharacteristic
167C2×C8#13711yescharacteristic
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C22⋊C4—41no—
175C22⋊C4—42no—
176C42—21no—
177C42—41no—
178C42—42no—
179C42#12111yescharacteristic
180C42#12311yescharacteristic
181C16—42no—
182C16—81no—
183C23—41no—
184C23#45311yescharacteristic
185Q8—41no—
186Q8—41no—
187Q8—162no—
188D4—41no—
189D4—41no—
190D4—81no—
191D4—81no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—162no—
201C2×C4—161no—
202C2×C4#39411yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#44911yescharacteristic
205C2×C4#45111yescharacteristic
206C2×C4#45211yescharacteristic
207C2×C4#45511yescharacteristic
208C8—82no—
209C8—81no—
210C8—81no—
211C8#46311yescharacteristic
212C8#46411yescharacteristic
213C8#46711yescharacteristic
214C8#46811yescharacteristic
215C8#48411yescharacteristic
216C8#48511yescharacteristic
217C8#48811yescharacteristic
218C8#48911yescharacteristic
219C22—41no—
220C22—81no—
221C22—161no—
222C22#162411yescharacteristic
223C22#163211yescharacteristic
224C22#203711yescharacteristic
225C4—41no—
226C4—41no—
227C4—81no—
228C4—161no—
229C4—162no—
230C4—321no—
231C4#164411yescharacteristic
232C4#164611yescharacteristic
233C4#203611yescharacteristic
234C4#204211yescharacteristic
235C2—21no—
236C2—21no—
237C2—21no—
238C2—161no—
239C2#1041311yescharacteristic
subgroupG/H
center, socleC2#10413
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ 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. #13180C2semidirect
4. #11956C2semidirect
6. #9755C2semidirect
7. #7181C2semidirect

Maximal subgroups

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

subgroupindex
#291542
#131802
#128242
#119562
#116022
#97552
#71812