Finite Group Databasebeta

← #136365

Subgroups — #136365

Order 512.

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

normal subgroups only

This group has 329 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×#1336C211yesmaximal, characteristic
5#8825C211yesmaximal, characteristic, complemented
6#7270C211yesmaximal, characteristic, complemented
7#7182C211yesmaximal, 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×#322C2211yescharacteristic
23#1388—21no—
24#1385—21no—
25#1336C414yescomplemented
26C2×#135D411yescharacteristic
27#568—22no—
28C2×#131—21no—
29C2×M32—21no—
30C2×M32—21no—
31#444—42no—
32#443—21no—
33#443—21no—
34#443—42no—
35#443C2×C412yes—
36#443Q812yes—
37#430—22no—
38#430—42no—
39C2×#108—21no—
40C2×#108—21no—
41C2×#108—21no—
42C2×#108—21no—
43C2×#108—42no—
44C2×#108D411yescharacteristic
45C2×#108D411yescharacteristic
46#427D411yescharacteristic
47#423—21no—
48#423D412yes—
49#421—22no—
50C2×#104—21no—
51#405—21no—
52C2×C4×C8—21no—
53C2×C4×C8C2311yescharacteristic
54#365—42no—
55#364—21no—
56#364—21no—
57#364—21no—
58#364C2×C412yes—
59#362—41no—
60#361—41no—
61#344—22no—
62#339—21no—
63#334—44no—
64#322—21no—
65#322C2×C412yes—
66C2×#41—21no—
67#135—21no—
68#135—21no—
69#135C4⋊C412yes—
70#135SD1612yes—
71C2×Q16SD1611yescharacteristic
72C2×SD16—21no—
73C2×D8SD1611yescharacteristic
74#131—22no—
75#131—22no—
76#131—41no—
77#131—41no—
78#131—44no—
79#131—82no—
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—44no—
85C2×M16—41no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90#117—21no—
91C2×C42C2×D411yescharacteristic
92M32—22no—
93M32—41no—
94M32—41no—
95M32—41no—
96C2×C16—41no—
97C2×C16—41no—
98#108—21no—
99#108—21no—
100#108—44no—
101#108—41no—
102#108—41no—
103#108—41no—
104#108—41no—
105#108—41no—
106#108—41no—
107#108—41no—
108#108—41no—
109#108C4⋊C412yes—
110#108SD1612yes—
111#107—21no—
112#106—21no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#104—22no—
116#104—22no—
117#104—44no—
118#97—41no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8—41no—
122C4×C8C22×C411yescharacteristic
123C4×C8C2×Q811yescharacteristic
124C4×C8#4111yescharacteristic
125C4×C8#4111yescharacteristic
126#41—22no—
127#41—22no—
128#41—41no—
129#41—41no—
130C2×Q8—21no—
131C2×D4—21no—
132C2×D4—41no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4#12011yescharacteristic
136Q16—21no—
137Q16#10612yes—
138SD16—22no—
139SD16—22no—
140D8—21no—
141D8#10612yes—
142M16—22no—
143M16—22no—
144M16—41no—
145M16—41no—
146M16—42no—
147M16—42no—
148M16—42no—
149M16—42no—
150M16—84no—
151M16—81no—
152M16—81no—
153C2×C8—24no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—82no—
160C2×C8—84no—
161C2×C8—81no—
162C2×C8C2×C4⋊C411yescharacteristic
163C2×C8C4×D411yescharacteristic
164C2×C8C22⋊Q811yescharacteristic
165C2×C8#12311yescharacteristic
166C2×C8C2×SD1611yescharacteristic
167C2×C8C2×SD1611yescharacteristic
168C2×C8C2×SD1611yescharacteristic
169C2×C8#13511yescharacteristic
170C2×C8AGL(1,ℤ/8)11yescharacteristic
171C2×C8AGL(1,ℤ/8)11yescharacteristic
172C2×C8AGL(1,ℤ/8)11yescharacteristic
173C2×C8#13711yescharacteristic
174C4⋊C4—41no—
175C22⋊C4—41no—
176C42—21no—
177C42—41no—
178C42C4×D411yescharacteristic
179C42C22⋊Q811yescharacteristic
180C16—42no—
181C16—81no—
182C23—41no—
183C23#45611yescharacteristic
184Q8—22no—
185Q8—41no—
186D4—22no—
187D4—42no—
188D4—42no—
189D4—41no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—81no—
197C2×C4#38511yescharacteristic
198C2×C4#44811yescharacteristic
199C2×C4#44911yescharacteristic
200C2×C4#45011yescharacteristic
201C2×C4#45011yescharacteristic
202C2×C4#45311yescharacteristic
203C8—44no—
204C8—42no—
205C8—42no—
206C8—82no—
207C8—161no—
208C8C2×#10611yescharacteristic
209C8#42811yescharacteristic
210C8C4×SD1611yescharacteristic
211C8#44211yescharacteristic
212C8#47611yescharacteristic
213C8#47811yescharacteristic
214C8#48111yescharacteristic
215C8#48311yescharacteristic
216C22—41no—
217C22—42no—
218C22—161no—
219C22#165111yescharacteristic
220C22#165411yescharacteristic
221C22#185911yescharacteristic
222C4—22no—
223C4—41no—
224C4—42no—
225C4—161no—
226C4#162211yescharacteristic
227C4#162911yescharacteristic
228C4#185311yescharacteristic
229C4#186411yescharacteristic
230C2—21no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#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. #8825C2semidirect
6. #7270C2semidirect
7. #7182C2semidirect
25. #1336C4semidirect

Maximal subgroups

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

subgroupindex
#226942
#131882
#128252
C2×#13362
#88252
#72702
#71822