Finite Group Databasebeta

← #140322

Subgroups — #140322

Order 512.

232 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 68 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#12029C211yesmaximal, characteristic, complemented
2#11951C211yesmaximal, characteristic
3#11652C211yesmaximal, characteristic
4#9849C211yesmaximal, characteristic
5#8269C211yesmaximal, characteristic, complemented
6#7295C211yesmaximal, characteristic, complemented
7#7149C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#2163C2211yescharacteristic
10#2163C2211yescharacteristic
11#1962—21no—
12C4×#97C2211yescharacteristic
13#1566C2211yescharacteristic
14#1564C2211yescharacteristic
15#1564C2211yescharacteristic
16#1492—21no—
17#1452—21no—
18#1368—22no—
19#1364C412yescomplemented
20#1361C412yescomplemented
21#1333—22no—
22#1276—21no—
23#532—21no—
24#500C2×C411yescharacteristic
25#493C2×C411yescharacteristic
26#486—21no—
27#466—21no—
28#466—21no—
29C4×D8—21no—
30C4×D8—21no—
31#431—21no—
32#431—21no—
33#423—21no—
34C2×#104—42no—
35C2×#97—21no—
36C2×#97—21no—
37C2×C4×C8D411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39C43D411yescharacteristic
40C4⋊C16—21no—
41C4⋊C16C2×C412yes—
42C22⋊C16—21no—
43C22⋊C16—21no—
44#346—21no—
45#346—21no—
46#335—21no—
47#335C2×C412yes—
48#334—21no—
49#334—21no—
50#334—21no—
51#334D412yes—
52D4⋊C8—42no—
53#322—42no—
54C2×#41—41no—
55C2×D8—21no—
56C2×D8—21no—
57C2×M16—41no—
58C22×C8—21no—
59C22×C8#4111yescharacteristic
60C22×C8#4111yescharacteristic
61C4⋊Q8—21no—
62#127—21no—
63#124—41no—
64#121—41no—
65#121—41no—
66C4×D4—41no—
67C4×D4—41no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C2×D411yescharacteristic
72C2×C16—22no—
73C2×C16—41no—
74#107—21no—
75#107—21no—
76C4⋊C8—22no—
77C4⋊C8—22no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80#104—84no—
81#102—41no—
82#102—41no—
83C22⋊C8—41no—
84C22⋊C8—41no—
85#97—42no—
86#97—42no—
87C4×C8—21no—
88C4×C8—41no—
89C4×C8—41no—
90C4×C8SD1612yes—
91C4×C8C22×C411yescharacteristic
92C4×C8C2×D411yescharacteristic
93C4×C8#4111yescharacteristic
94C4×C8#4111yescharacteristic
95#41—82no—
96#41—82no—
97C2×Q8—41no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—81no—
102C22×C4—41no—
103C22×C4—81no—
104C22×C4#12311yescharacteristic
105C22×C4#13511yescharacteristic
106C22×C4AGL(1,ℤ/8)11yescharacteristic
107D8—42no—
108D8—81no—
109M16—82no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—42no—
116C2×C8—42no—
117C2×C8—81no—
118C2×C8#11711yescharacteristic
119C2×C8C4×D411yescharacteristic
120C2×C8#12411yescharacteristic
121C2×C8#12611yescharacteristic
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—81no—
125C22⋊C4—81no—
126C42—22no—
127C42—41no—
128C42—42no—
129C42—81no—
130C42C4×D411yescharacteristic
131C42#12111yescharacteristic
132C42C2×SD1611yescharacteristic
133C42AGL(1,ℤ/8)11yescharacteristic
134C16—42no—
135C16—81no—
136C23—81no—
137C23#48211yescharacteristic
138Q8—82no—
139D4—82no—
140D4—82no—
141D4—81no—
142D4—81no—
143D4—162no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—161no—
153C2×C4—161no—
154C2×C4#38811yescharacteristic
155C2×C4C4×SD1611yescharacteristic
156C2×C4#44211yescharacteristic
157C2×C4#46011yescharacteristic
158C2×C4#46311yescharacteristic
159C2×C4#48111yescharacteristic
160C8—21no—
161C8—21no—
162C8—21no—
163C8—42no—
164C8—82no—
165C8—82no—
166C8—161no—
167C8#44412yes—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—161no—
172C22—161no—
173C22—161no—
174C22#192611yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—82no—
180C4—161no—
181C4#193211yescharacteristic
182C4#193811yescharacteristic
183C4AGL(1,ℤ/16)12yes—
184C2—41no—
185C2—321no—
186C2#958211yescharacteristic
187C2#1314011yescharacteristic
188C2#1314211yescharacteristic
subgroupG/H
center, socleC22#1926
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #12029C2semidirect
5. #8269C2semidirect
6. #7295C2semidirect
7. #7149C2semidirect
19. #1364C4semidirect
20. #1361C4semidirect

Maximal subgroups

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

subgroupindex
#120292
#119512
#116522
#98492
#82692
#72952
#71492