Finite Group Databasebeta

← #145445

Subgroups — #145445

Order 512.

254 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 74 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#33637C211yesmaximal, characteristic
2#13285C211yesmaximal, characteristic, complemented
3#13195C211yesmaximal, characteristic
4#12822C211yesmaximal, characteristic
5C2×#1373C211yesmaximal, characteristic, complemented
6#11956C211yesmaximal, characteristic, complemented
7#7270C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2231—22no—
11#2230—22no—
12#2155—21no—
13C2×#364C2211yescharacteristic
14C2×#364C2211yescharacteristic
15C2×#364C2211yescharacteristic
16#2145—21no—
17#2143—21no—
18C2×#350—21no—
19#2113—21no—
20C2×C4×C16C2211yescharacteristic
21#1848C2211yescharacteristic
22#1388—21no—
23#1382—21no—
24#1382—21no—
25#1373—22no—
26#1366—22no—
27C2×#135D411yescharacteristic
28C2×#131—21no—
29#533—21no—
30C2×M32—21no—
31C2×M32—21no—
32C2×M32—21no—
33C22×C16D411yescharacteristic
34C22×C16D411yescharacteristic
35#500—21no—
36#499—21no—
37#499D412yes—
38#498—21no—
39#443—21no—
40#443—21no—
41#443—21no—
42#443—21no—
43C2×#108—21no—
44C2×#108—21no—
45C2×#108D411yescharacteristic
46#427—21no—
47#427—21no—
48#423—21no—
49C2×#104—21no—
50C2×C4×C8C2311yescharacteristic
51#365—24no—
52#364—21no—
53#364—21no—
54#364—21no—
55#364—21no—
56#364—21no—
57#364D412yes—
58#360—24no—
59#359—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—
78C2×M16—41no—
79C22×C8—41no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C4⋊Q8—41no—
83#125—22no—
84C22⋊Q8—81no—
85C4×Q8—81no—
86#117—41no—
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#108—41no—
106#108—41no—
107#108—41no—
108#108—41no—
109#107—22no—
110#107—41no—
111#106—22no—
112#106—41no—
113C4⋊C8—81no—
114#104—41no—
115#104—41no—
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119C4×C8#4111yescharacteristic
120#41—41no—
121#41—41no—
122#41—41no—
123#41—41no—
124C2×Q8—21no—
125C2×Q8—81no—
126C2×D4—21no—
127C2×D4—41no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4#12111yescharacteristic
131Q16—21no—
132Q16—21no—
133SD16—41no—
134SD16—41no—
135D8—21no—
136D8—21no—
137M16—41no—
138M16—41no—
139M16—81no—
140M16—81no—
141M16—81no—
142M16—81no—
143C2×C8—41no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8#12011yescharacteristic
148C2×C8#12111yescharacteristic
149C2×C8#12111yescharacteristic
150C2×C8#12311yescharacteristic
151C2×C8C2×SD1611yescharacteristic
152C2×C8C2×Q1611yescharacteristic
153C2×C8#13511yescharacteristic
154C2×C8#13511yescharacteristic
155C2×C8AGL(1,ℤ/8)11yescharacteristic
156C2×C8AGL(1,ℤ/8)11yescharacteristic
157C2×C8#13711yescharacteristic
158C2×C8#13711yescharacteristic
159C4⋊C4—42no—
160C4⋊C4—82no—
161C4⋊C4—81no—
162C22⋊C4—81no—
163C42—21no—
164C42—81no—
165C42#12311yescharacteristic
166C42#12411yescharacteristic
167C16—22no—
168C16—42no—
169C16—41no—
170C16—81no—
171C16—81no—
172C23—41no—
173C23#47111yescharacteristic
174Q8—41no—
175Q8—41no—
176Q8—162no—
177D4—41no—
178D4—41no—
179D4—81no—
180D4—81no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4#39411yescharacteristic
189C2×C4#46111yescharacteristic
190C2×C4#46311yescharacteristic
191C2×C4#46711yescharacteristic
192C2×C4#46811yescharacteristic
193C2×C4#47211yescharacteristic
194C8—81no—
195C8—161no—
196C8#44911yescharacteristic
197C8#45111yescharacteristic
198C8#46011yescharacteristic
199C8#46411yescharacteristic
200C8#46511yescharacteristic
201C8#46811yescharacteristic
202C8#48411yescharacteristic
203C8#48511yescharacteristic
204C22—41no—
205C22—82no—
206C22—81no—
207C22#168311yescharacteristic
208C22#169111yescharacteristic
209C22#205111yescharacteristic
210C4—22no—
211C4—41no—
212C4—81no—
213C4—162no—
214C4#167511yescharacteristic
215C4#169711yescharacteristic
216C4#203611yescharacteristic
217C4#205511yescharacteristic
218C2—21no—
219C2—21no—
220C2—21no—
221C2—161no—
222C2#1048111yescharacteristic
subgroupG/H
centerC4#1697
commutator, Frattini, cosocleC2×C4×C8C23
socleC2#10481
prime radicalC2×#135D4
LCSG ▷ C2×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. #13285C2semidirect
5. C2×#1373C2semidirect
6. #11956C2semidirect
7. #7270C2semidirect

Maximal subgroups

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

subgroupindex
#336372
#132852
#131952
#128222
C2×#13732
#119562
#72702