Finite Group Databasebeta

← #145340

Subgroups — #145340

Order 512.

269 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 60 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#19286C211yesmaximal, characteristic
2#12029C211yesmaximal, characteristic, complemented
3#11948C211yesmaximal, characteristic
4#11864C211yesmaximal, characteristic
5#11495C211yesmaximal, characteristic, complemented
6#8665C211yesmaximal, characteristic, complemented
7#7219C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3037—21no—
10C2×#446C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1837C2211yescharacteristic
14#1702—22no—
15#1694—21no—
16#1688—21no—
17#1566C2211yescharacteristic
18#1564C2211yescharacteristic
19#1547—21no—
20#1487—21no—
21#1368—22no—
22#1365—21no—
23#1361—21no—
24#1349—22no—
25#1322—22no—
26#1301—21no—
27C2×#137—41no—
28#532—21no—
29C2×C4×Q8D411yescharacteristic
30#500D411yescharacteristic
31#493D411yescharacteristic
32#486—21no—
33#466—21no—
34#466—21no—
35#464—42no—
36#462—22no—
37#460—22no—
38#446—21no—
39#446—21no—
40#446—21no—
41#446—21no—
42C4×D8—21no—
43C4×D8—21no—
44#431—21no—
45#423—21no—
46#423—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49#417—41no—
50C2×#97—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54C22⋊C16—21no—
55C22⋊C16D412yes—
56#346—21no—
57C4×C16—21no—
58#341—21no—
59#341—21no—
60#335—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#332—22no—
65#331—22no—
66Q8⋊C8—42no—
67C2×#41—41no—
68C22×Q8SD1611yescharacteristic
69#137—84no—
70C2×Q16—42no—
71C2×SD16—42no—
72C2×D8—21no—
73C2×D8—21no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8—21no—
77C22×C8C2×D411yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—21no—
80#127—21no—
81#124—41no—
82#121—41no—
83#121—41no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4SD1611yescharacteristic
92C2×C42—21no—
93C2×C42C2×D411yescharacteristic
94C2×C16—22no—
95C2×C16—41no—
96#107—21no—
97#107—21no—
98C4⋊C8—21no—
99C4⋊C8—22no—
100C4⋊C8—21no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107#103—42no—
108#102—42no—
109#102—41no—
110#102—41no—
111C22⋊C8—41no—
112#97—41no—
113#97—41no—
114C4×C8—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8#4111yescharacteristic
118C4×C8#4111yescharacteristic
119#41—82no—
120#41—82no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—81no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—81no—
132C22×C4#12111yescharacteristic
133C22×C4C2×SD1611yescharacteristic
134C22×C4#13511yescharacteristic
135Q16—84no—
136SD16—84no—
137D8—81no—
138D8—81no—
139M16—82no—
140M16—81no—
141M16—81no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—42no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#12111yescharacteristic
150C2×C8#12111yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8#12411yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C22⋊C4—81no—
160C22⋊C4—81no—
161C42—41no—
162C42—41no—
163C42#12011yescharacteristic
164C42#12311yescharacteristic
165C42AGL(1,ℤ/8)11yescharacteristic
166C42AGL(1,ℤ/8)11yescharacteristic
167C16—42no—
168C16—81no—
169C23—81no—
170C23#46511yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—81no—
174Q8—81no—
175D4—81no—
176D4—81no—
177D4—81no—
178D4—81no—
179D4—161no—
180D4—162no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—162no—
191C2×C4#39411yescharacteristic
192C2×C4#44911yescharacteristic
193C2×C4#45011yescharacteristic
194C2×C4#46911yescharacteristic
195C2×C4#48111yescharacteristic
196C2×C4#48211yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C8—82no—
202C8—81no—
203C8—161no—
204C8—161no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—161no—
209C22—161no—
210C22—161no—
211C22#203411yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—81no—
217C4—81no—
218C4—161no—
219C4#204911yescharacteristic
220C4#205311yescharacteristic
221C4#223812yes—
222C2—41no—
223C2—321no—
224C2#1046711yescharacteristic
225C2#1339811yescharacteristic
226C2#1340611yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #12029C2semidirect
5. #11495C2semidirect
6. #8665C2semidirect
7. #7219C2semidirect

Maximal subgroups

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

subgroupindex
#192862
#120292
#119482
#118642
#114952
#86652
#72192