Finite Group Databasebeta

← #136361

Subgroups — #136361

Order 512.

371 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 96 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#22689C211yesmaximal, characteristic
2#13186C211yesmaximal, characteristic, complemented
3#12823C211yesmaximal, characteristic
4C2×#1335C211yesmaximal, characteristic
5#8813C211yesmaximal, characteristic, complemented
6#7268C211yesmaximal, characteristic, complemented
7#7179C211yesmaximal, characteristic, complemented
8C2×#443C2211yescharacteristic
9#2952—21no—
10#2912—21no—
11C2×#365—21no—
12C2×#364C2211yescharacteristic
13C2×#364C2211yescharacteristic
14C2×#360—21no—
15#1951—21no—
16#1859—22no—
17#1844—22no—
18#1844C2211yescharacteristic
19#1844C2211yescharacteristic
20#1815—21no—
21#1772C2211yescharacteristic
22C2×C82C2211yescharacteristic
23#1386—21no—
24#1383—21no—
25#1335C414yescomplemented
26C2×#135D411yescharacteristic
27C2×#131—21no—
28#520—22no—
29C2×M32—21no—
30C2×M32—21no—
31#443—21no—
32#443—21no—
33#443C2×C412yes—
34#443Q812yes—
35#442—41no—
36#441—41no—
37C4×SD16—42no—
38C2×#108D411yescharacteristic
39C2×#108D411yescharacteristic
40#428—22no—
41#428—42no—
42#427—41no—
43#427D411yescharacteristic
44C2×#106—21no—
45C2×#106—21no—
46C2×#106—21no—
47C2×#106—21no—
48C2×#106—41no—
49C2×#106—44no—
50C2×#104—21no—
51#417—22no—
52#405—21no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55#384—21no—
56#384D412yes—
57#365—42no—
58#364—21no—
59#364—21no—
60#364—21no—
61#364C2×C412yescomplemented
62#360—42no—
63#343—22no—
64#339—21no—
65C82—21no—
66C82C2×C412yescomplemented
67C2×#41—21no—
68#135—21no—
69#135—21no—
70#135C4⋊C412yes—
71#135SD1612yes—
72C2×Q16SD1611yescharacteristic
73C2×SD16—21no—
74C2×D8SD1611yescharacteristic
75#131—22no—
76#131—22no—
77#131—41no—
78#131—41no—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—41no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4×Q8—42no—
87C4×D4—42no—
88C4×D4—82no—
89#117—21no—
90#117—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C42C2×D411yescharacteristic
97M32—22no—
98M32—41no—
99M32—41no—
100M32—41no—
101C2×C16—41no—
102C2×C16—41no—
103#108—21no—
104#108—21no—
105#108—21no—
106#108C4⋊C412yes—
107#107—42no—
108#107SD1612yes—
109#106—21no—
110#106—42no—
111#106—41no—
112#106—41no—
113#106—41no—
114#106—41no—
115#106—41no—
116#106—41no—
117#106—41no—
118#106—41no—
119#106—88no—
120#104—22no—
121#104—22no—
122#103—42no—
123#102—42no—
124#97—41no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—41no—
128C4×C8C22×C411yescharacteristic
129C4×C8C2×Q811yescharacteristic
130C4×C8#4111yescharacteristic
131C4×C8#4111yescharacteristic
132#41—22no—
133#41—22no—
134#41—41no—
135#41—41no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—41no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—82no—
143C22×C4—81no—
144C22×C4#12011yescharacteristic
145Q16—21no—
146Q16#10612yescomplemented
147SD16—22no—
148SD16—22no—
149D8—21no—
150D8#10612yescomplemented
151M16—22no—
152M16—41no—
153M16—42no—
154M16—42no—
155M16—41no—
156M16—41no—
157C2×C8—24no—
158C2×C8—28no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8C2×C4⋊C411yescharacteristic
164C2×C8C4×D411yescharacteristic
165C2×C8C22⋊Q811yescharacteristic
166C2×C8#12311yescharacteristic
167C2×C8C2×SD1611yescharacteristic
168C2×C8C2×SD1611yescharacteristic
169C2×C8C2×SD1611yescharacteristic
170C2×C8#13511yescharacteristic
171C2×C8AGL(1,ℤ/8)11yescharacteristic
172C2×C8AGL(1,ℤ/8)11yescharacteristic
173C2×C8AGL(1,ℤ/8)11yescharacteristic
174C2×C8#13711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—82no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C22⋊C4—41no—
184C22⋊C4—82no—
185C42—21no—
186C42—41no—
187C42—82no—
188C42C4×D411yescharacteristic
189C42C22⋊Q811yescharacteristic
190C16—42no—
191C16—81no—
192C23—41no—
193C23#45311yescharacteristic
194Q8—22no—
195Q8—41no—
196D4—22no—
197D4—42no—
198D4—42no—
199D4—41no—
200C2×C4—24no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—162no—
209C2×C4—162no—
210C2×C4—161no—
211C2×C4—322no—
212C2×C4#38511yescharacteristic
213C2×C4#44811yescharacteristic
214C2×C4#44911yescharacteristic
215C2×C4#45011yescharacteristic
216C2×C4#45011yescharacteristic
217C2×C4#45611yescharacteristic
218C8—28no—
219C8—42no—
220C8—81no—
221C8C2×#10611yescharacteristic
222C8#42811yescharacteristic
223C8C4×SD1611yescharacteristic
224C8#44211yescharacteristic
225C8#47611yescharacteristic
226C8#47811yescharacteristic
227C8#48111yescharacteristic
228C8#48311yescharacteristic
229C22—22no—
230C22—42no—
231C22—82no—
232C22#162211yescharacteristic
233C22#162911yescharacteristic
234C22#185911yescharacteristic
235C4—22no—
236C4—22no—
237C4—42no—
238C4—82no—
239C4—162no—
240C4—321no—
241C4#165111yescharacteristic
242C4#165411yescharacteristic
243C4#185311yescharacteristic
244C4#186411yescharacteristic
245C2—21no—
246C2—21no—
247C2—82no—
248C2#888611yescharacteristic
249C2#1236412yes—
subgroupG/H
center, socleC22#1629
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #13186C2semidirect
5. #8813C2semidirect
6. #7268C2semidirect
7. #7179C2semidirect
25. #1335C4semidirect
61. #364C2×C4semidirect
66. C82C2×C4semidirect
146. Q16#106semidirect
150. D8#106semidirect

Maximal subgroups

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

subgroupindex
#226892
#131862
#128232
C2×#13352
#88132
#72682
#71792