Finite Group Databasebeta

← #136363

Subgroups — #136363

Order 512.

333 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 94 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#22694C211yesmaximal, characteristic
2#13188C211yesmaximal, characteristic, complemented
3#12825C211yesmaximal, characteristic
4C2×#1335C211yesmaximal, characteristic
5#8822C211yesmaximal, characteristic, complemented
6#7270C211yesmaximal, characteristic, complemented
7#7179C211yesmaximal, characteristic, complemented
8#2957—21no—
9C2×#443C2211yescharacteristic
10#2916—21no—
11#2155—21no—
12C2×#364C2211yescharacteristic
13C2×#364C2211yescharacteristic
14#2148—21no—
15#1953—21no—
16#1861—22no—
17#1848C2211yescharacteristic
18#1848C2211yescharacteristic
19#1814—21no—
20#1772C2211yescharacteristic
21#1566—22no—
22C2×C82C2211yescharacteristic
23#1388—21no—
24#1385—21no—
25#1335C412yescomplemented
26#1335C412yescomplemented
27C2×#135D411yescharacteristic
28#568—22no—
29C2×#131—21no—
30C2×M32—21no—
31C2×M32—21no—
32#444—42no—
33#443—21no—
34#443—21no—
35#443—42no—
36#443C2×C412yes—
37#443Q812yes—
38#430—22no—
39#430—42no—
40C2×#108—21no—
41C2×#108—21no—
42C2×#108—21no—
43C2×#108—21no—
44C2×#108—42no—
45C2×#108D411yescharacteristic
46C2×#108D411yescharacteristic
47#427D411yescharacteristic
48#423—21no—
49#423D412yes—
50#421—22no—
51C2×#104—21no—
52#405—21no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55#365—42no—
56#364—21no—
57#364—21no—
58#364—21no—
59#364C2×C412yes—
60#362—41no—
61#361—41no—
62#344—22no—
63#339—21no—
64#334—44no—
65C82—21no—
66C82C2×C411yescharacteristic
67C82C2×C411yescharacteristic
68C2×#41—21no—
69#135—21no—
70#135—21no—
71#135C4⋊C412yes—
72#135SD1612yes—
73C2×Q16SD1611yescharacteristic
74C2×SD16—21no—
75C2×D8SD1611yescharacteristic
76#131—22no—
77#131—22no—
78#131—41no—
79#131—41no—
80#131—44no—
81#131—82no—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—22no—
85C2×M16—41no—
86C2×M16—44no—
87C2×M16—41no—
88C22×C8—22no—
89C22×C8—41no—
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
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—44no—
103#108—41no—
104#108—41no—
105#108—41no—
106#108—41no—
107#108—41no—
108#108—41no—
109#108—41no—
110#108—41no—
111#108C4⋊C412yes—
112#108SD1612yes—
113#107—21no—
114#106—21no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#104—22no—
118#104—22no—
119#104—44no—
120#97—41no—
121C4×C8—22no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—41no—
125C4×C8C22×C411yescharacteristic
126C4×C8C2×Q811yescharacteristic
127C4×C8#4111yescharacteristic
128C4×C8#4111yescharacteristic
129#41—22no—
130#41—22no—
131#41—41no—
132#41—41no—
133C2×Q8—21no—
134C2×D4—21no—
135C2×D4—41no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4#12011yescharacteristic
139Q16—21no—
140Q16#10612yes—
141SD16—22no—
142SD16—22no—
143D8—21no—
144D8#10612yes—
145M16—22no—
146M16—22no—
147M16—41no—
148M16—41no—
149M16—42no—
150M16—42no—
151M16—42no—
152M16—42no—
153M16—84no—
154M16—81no—
155M16—81no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—44no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—84no—
165C2×C8—81no—
166C2×C8C2×C4⋊C411yescharacteristic
167C2×C8C4×D411yescharacteristic
168C2×C8C22⋊Q811yescharacteristic
169C2×C8#12311yescharacteristic
170C2×C8C2×SD1611yescharacteristic
171C2×C8C2×SD1611yescharacteristic
172C2×C8C2×SD1611yescharacteristic
173C2×C8#13511yescharacteristic
174C2×C8AGL(1,ℤ/8)11yescharacteristic
175C2×C8AGL(1,ℤ/8)11yescharacteristic
176C2×C8AGL(1,ℤ/8)11yescharacteristic
177C2×C8#13711yescharacteristic
178C4⋊C4—41no—
179C22⋊C4—41no—
180C42—21no—
181C42—41no—
182C42C4×D411yescharacteristic
183C42C22⋊Q811yescharacteristic
184C16—42no—
185C16—81no—
186C23—41no—
187C23#45611yescharacteristic
188Q8—22no—
189Q8—41no—
190D4—22no—
191D4—42no—
192D4—42no—
193D4—41no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4#38511yescharacteristic
202C2×C4#44811yescharacteristic
203C2×C4#44911yescharacteristic
204C2×C4#45011yescharacteristic
205C2×C4#45011yescharacteristic
206C2×C4#45311yescharacteristic
207C8—24no—
208C8—42no—
209C8—42no—
210C8—42no—
211C8—82no—
212C8—161no—
213C8C2×#10611yescharacteristic
214C8#42811yescharacteristic
215C8C4×SD1611yescharacteristic
216C8#44211yescharacteristic
217C8#47611yescharacteristic
218C8#47811yescharacteristic
219C8#48111yescharacteristic
220C8#48311yescharacteristic
221C22—41no—
222C22—42no—
223C22—161no—
224C22#165111yescharacteristic
225C22#165411yescharacteristic
226C22#185911yescharacteristic
227C4—22no—
228C4—41no—
229C4—42no—
230C4—161no—
231C4#162211yescharacteristic
232C4#162911yescharacteristic
233C4#185311yescharacteristic
234C4#186411yescharacteristic
235C2—21no—
236C2—21no—
237C2—21no—
238C2—82no—
239C2#888611yescharacteristic
subgroupG/H
centerC4#1629
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC2#8886
prime radicalC2×#135D4
LCSG ▷ 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
5. #8822C2semidirect
6. #7270C2semidirect
7. #7179C2semidirect
25. #1335C4semidirect
26. #1335C4semidirect

Maximal subgroups

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

subgroupindex
#226942
#131882
#128252
C2×#13352
#88222
#72702
#71792