Finite Group Databasebeta

← #145443

Subgroups — #145443

Order 512.

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

normal subgroups only

This group has 252 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#13286C211yesmaximal, characteristic, complemented
3#13196C211yesmaximal, characteristic
4#12825C211yesmaximal, characteristic
5C2×#1369C211yesmaximal, characteristic, complemented
6#11956C211yesmaximal, characteristic, complemented
7#7272C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9C2×#443C2211yescharacteristic
10#2231—22no—
11#2230—22no—
12#2155—21no—
13C2×#364C2211yescharacteristic
14C2×#364C2211yescharacteristic
15C2×#364C2211yescharacteristic
16#2148—21no—
17#2145—21no—
18C2×#350—21no—
19#2113—21no—
20C2×#346C2211yescharacteristic
21#1848C2211yescharacteristic
22#1385—21no—
23#1382—21no—
24#1382—21no—
25#1369—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#362—41no—
59#361—41no—
60#360—24no—
61#350—42no—
62C22⋊C16—42no—
63#346—21no—
64#346—21no—
65C2×#41—21no—
66#135—21no—
67#135—21no—
68#135—21no—
69#135—21no—
70C2×Q16SD1611yescharacteristic
71C2×SD16—21no—
72C2×D8SD1611yescharacteristic
73#131—41no—
74#131—41no—
75#131—41no—
76#131—41no—
77C2×M16—21no—
78C2×M16—41no—
79C2×M16—41no—
80C22×C8—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—41no—
84#125—22no—
85C22⋊Q8—81no—
86C4×Q8—81no—
87#117—41no—
88C2×C42#4111yescharacteristic
89M32—22no—
90M32—41no—
91M32—41no—
92M32—41no—
93M32—41no—
94M32—41no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—21no—
100C2×C16—41no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16SD1612yes—
104#108—21no—
105#108—21no—
106#108—41no—
107#108—41no—
108#108—41no—
109#108—41no—
110#107—22no—
111#107—41no—
112#106—22no—
113#106—41no—
114C4⋊C8—81no—
115#104—41no—
116#104—41no—
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121#41—41no—
122#41—41no—
123#41—41no—
124#41—41no—
125C2×Q8—21no—
126C2×Q8—81no—
127C2×D4—21no—
128C2×D4—41no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4#12111yescharacteristic
132Q16—21no—
133Q16—21no—
134SD16—41no—
135SD16—41no—
136D8—21no—
137D8—21no—
138M16—41no—
139M16—41no—
140M16—81no—
141M16—81no—
142M16—81no—
143M16—81no—
144C2×C8—41no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8#12011yescharacteristic
149C2×C8#12111yescharacteristic
150C2×C8#12111yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8C2×SD1611yescharacteristic
153C2×C8C2×SD1611yescharacteristic
154C2×C8#13511yescharacteristic
155C2×C8#13511yescharacteristic
156C2×C8AGL(1,ℤ/8)11yescharacteristic
157C2×C8#13711yescharacteristic
158C2×C8#13711yescharacteristic
159C2×C8#13711yescharacteristic
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—81no—
163C22⋊C4—81no—
164C42—21no—
165C42—81no—
166C42#12311yescharacteristic
167C42#12411yescharacteristic
168C16—22no—
169C16—42no—
170C16—41no—
171C16—81no—
172C16—81no—
173C23—41no—
174C23#47111yescharacteristic
175Q8—41no—
176Q8—41no—
177Q8—162no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4#39411yescharacteristic
190C2×C4#46111yescharacteristic
191C2×C4#46311yescharacteristic
192C2×C4#46511yescharacteristic
193C2×C4#47011yescharacteristic
194C2×C4#47211yescharacteristic
195C8—81no—
196C8—161no—
197C8#44811yescharacteristic
198C8#45211yescharacteristic
199C8#46011yescharacteristic
200C8#46411yescharacteristic
201C8#46511yescharacteristic
202C8#46811yescharacteristic
203C8#48311yescharacteristic
204C8#48511yescharacteristic
205C22—41no—
206C22—82no—
207C22—81no—
208C22#167111yescharacteristic
209C22#170311yescharacteristic
210C22#205111yescharacteristic
211C4—41no—
212C4—41no—
213C4—81no—
214C4—162no—
215C4#167711yescharacteristic
216C4#169511yescharacteristic
217C4#203511yescharacteristic
218C4#205511yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—161no—
223C2#1048011yescharacteristic
subgroupG/H
centerC4#1695
commutator, Frattini, cosocleC2×C4×C8C23
socleC2#10480
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. #13286C2semidirect
5. C2×#1369C2semidirect
6. #11956C2semidirect
7. #7272C2semidirect

Maximal subgroups

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

subgroupindex
#336372
#132862
#131962
#128252
C2×#13692
#119562
#72722