Finite Group Databasebeta

← #145444

Subgroups — #145444

Order 512.

258 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×#2231C211yesmaximal, characteristic
2#13281C211yesmaximal, characteristic, complemented
3#13195C211yesmaximal, characteristic
4#12824C211yesmaximal, characteristic
5C2×#1373C211yesmaximal, characteristic, complemented
6C2×#1366C211yesmaximal, characteristic, complemented
7#7271C211yesmaximal, characteristic, complemented
8C2×#499C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2231—22no—
11#2231—22no—
12C2×#365—21no—
13C2×#364C2211yescharacteristic
14C2×#364C2211yescharacteristic
15C2×#364C2211yescharacteristic
16C2×#360—21no—
17C2×#360—21no—
18C2×#350—21no—
19#2113—21no—
20C2×C4×C16C2211yescharacteristic
21#1846C2211yescharacteristic
22#1383—21no—
23#1380—21no—
24#1380—21no—
25#1373—22no—
26#1366—22no—
27C2×#135D411yescharacteristic
28C2×#131—21no—
29C2×#125—21no—
30C2×M32—21no—
31C2×M32—21no—
32C2×M32—21no—
33C22×C16D411yescharacteristic
34C22×C16D411yescharacteristic
35#499—21no—
36#499—21no—
37#499—21no—
38#499D412yes—
39#443—21no—
40#443—21no—
41#443—21no—
42#443—21no—
43C2×#108D411yescharacteristic
44C2×#107—21no—
45C2×#107—21no—
46C2×#106—21no—
47C2×#106—21no—
48C2×#104—21no—
49C2×C4×C8C2311yescharacteristic
50#382—21no—
51#365—24no—
52#364—21no—
53#364—21no—
54#364—21no—
55#364—21no—
56#364—21no—
57#364D412yes—
58#360—24no—
59#360—42no—
60#350—42no—
61C22⋊C16—42no—
62C4×C16—21no—
63C4×C16—21no—
64C2×#41—21no—
65#135—21no—
66#135—21no—
67#135—21no—
68#135—21no—
69C2×Q16Q1611yescharacteristic
70C2×SD16—21no—
71C2×D8Q1611yescharacteristic
72#131—41no—
73#131—41no—
74#131—41no—
75#131—41no—
76C2×M16—21no—
77C2×M16—41no—
78C22×C8—41no—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#125—22no—
82#125—41no—
83#125—81no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—81no—
87C2×C42#4111yescharacteristic
88M32—22no—
89M32—41no—
90M32—41no—
91M32—41no—
92M32—41no—
93M32—41no—
94C2×C16—21no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—41no—
100C2×C16—41no—
101C2×C16—41no—
102C2×C16SD1612yes—
103#108—21no—
104#108—21no—
105#107—22no—
106#107—41no—
107#107—41no—
108#107—41no—
109#106—22no—
110#106—41no—
111#106—41no—
112#106—41no—
113#104—41no—
114#104—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8#4111yescharacteristic
118C4×C8#4111yescharacteristic
119#95—81no—
120#41—41no—
121#41—41no—
122#41—41no—
123#41—41no—
124C2×Q8—21no—
125C2×D4—21no—
126C2×D4—41no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4#12111yescharacteristic
132Q16—21no—
133Q16—21no—
134SD16—41no—
135SD16—41no—
136D8—21no—
137D8—21no—
138M16—41no—
139M16—41no—
140M16—81no—
141M16—81no—
142C2×C8—41no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#12011yescharacteristic
146C2×C8#12111yescharacteristic
147C2×C8#12111yescharacteristic
148C2×C8#12311yescharacteristic
149C2×C8C2×SD1611yescharacteristic
150C2×C8C2×Q1611yescharacteristic
151C2×C8#13511yescharacteristic
152C2×C8#13511yescharacteristic
153C2×C8AGL(1,ℤ/8)11yescharacteristic
154C2×C8AGL(1,ℤ/8)11yescharacteristic
155C2×C8#13711yescharacteristic
156C2×C8#13711yescharacteristic
157C4⋊C4—42no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C42—21no—
164C42#12311yescharacteristic
165C42#12411yescharacteristic
166C16—22no—
167C16—42no—
168C16—41no—
169C16—81no—
170C16—81no—
171C23—41no—
172C23#47211yescharacteristic
173Q8—41no—
174Q8—41no—
175D4—41no—
176D4—41no—
177D4—81no—
178D4—81no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—161no—
188C2×C4—161no—
189C2×C4—161no—
190C2×C4—161no—
191C2×C4—161no—
192C2×C4#39411yescharacteristic
193C2×C4#46111yescharacteristic
194C2×C4#46311yescharacteristic
195C2×C4#46711yescharacteristic
196C2×C4#46811yescharacteristic
197C2×C4#47111yescharacteristic
198C8—81no—
199C8#44911yescharacteristic
200C8#45111yescharacteristic
201C8#46011yescharacteristic
202C8#46411yescharacteristic
203C8#46511yescharacteristic
204C8#46811yescharacteristic
205C8#48411yescharacteristic
206C8#48511yescharacteristic
207C22—21no—
208C22—21no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22#167511yescharacteristic
213C22#169711yescharacteristic
214C22#205111yescharacteristic
215C4—41no—
216C4—41no—
217C4—81no—
218C4—162no—
219C4—321no—
220C4#168311yescharacteristic
221C4#169111yescharacteristic
222C4#203611yescharacteristic
223C4#205511yescharacteristic
224C2—21no—
225C2—21no—
226C2—161no—
227C2#1048111yescharacteristic
228C2#1246311yescharacteristic
229C2#1246411yescharacteristic
subgroupG/H
center, socleC22#1675
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C2×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. #13281C2semidirect
5. C2×#1373C2semidirect
6. C2×#1366C2semidirect
7. #7271C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22312
#132812
#131952
#128242
C2×#13732
C2×#13662
#72712