Finite Group Databasebeta

← #145157

Subgroups — #145157

Order 512.

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

normal subgroups only

This group has 361 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
1#28246C211yesmaximal, characteristic
2#12823C212yesmaximal, complemented
3C2×#1335C212yesmaximal, complemented
4#8815C211yesmaximal, characteristic, complemented
5#7268C211yesmaximal, characteristic, complemented
6#3060—22no—
7C2×#443C2212yes—
8C2×#364C2212yes—
9C2×#360—22no—
10#2039—22no—
11#1844C2211yescharacteristic
12#1844C2211yescharacteristic
13#1834—22no—
14#1815—22no—
15C2×C82C2211yescharacteristic
16#1383—22no—
17#1335—24no—
18C2×#135D412yes—
19C2×#131—22no—
20#537—22no—
21C2×M32—22no—
22#470—42no—
23#469—42no—
24#465—44no—
25#443—22no—
26#443—22no—
27#443—22no—
28#443—22no—
29C2×#108D412yes—
30C2×#106—22no—
31C2×#106—22no—
32C2×#104—22no—
33#417—24no—
34C2×C4×C8—21no—
35C2×C4×C8C2311yescharacteristic
36#384—21no—
37#384D412yes—
38#364—22no—
39#364—22no—
40#360—44no—
41#343—22no—
42#340—44no—
43C82—21no—
44C82D412yescomplemented
45C2×#41—22no—
46#135—22no—
47#135—22no—
48#135—22no—
49#135—22no—
50C2×Q16SD1612yes—
51C2×SD16—22no—
52C2×D8SD1612yes—
53#131—42no—
54#131—42no—
55#131—42no—
56#131—42no—
57C2×M16—22no—
58C2×M16—42no—
59C22×C8—22no—
60C22×C8—42no—
61C22×C8C2×D412yes—
62C22⋊Q8—44no—
63#121—44no—
64#121—82no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C42C2×D411yescharacteristic
70M32—42no—
71M32—42no—
72C2×C16—42no—
73#108—22no—
74#108—22no—
75#106—42no—
76#106—42no—
77#106—42no—
78#106—42no—
79#104—42no—
80#104—42no—
81#103—44no—
82#102—44no—
83C4×C8—22no—
84C4×C8—22no—
85C4×C8C2×D411yescharacteristic
86C4×C8#4111yescharacteristic
87C4×C8#4112yes—
88#41—42no—
89#41—42no—
90#41—42no—
91#41—42no—
92C2×Q8—22no—
93C2×D4—22no—
94C2×D4—42no—
95C2×D4—82no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—81no—
100C22×C4#12011yescharacteristic
101Q16—22no—
102Q16—22no—
103SD16—42no—
104SD16—42no—
105D8—22no—
106D8—22no—
107M16—42no—
108M16—42no—
109M16—82no—
110M16—82no—
111C2×C8—24no—
112C2×C8—44no—
113C2×C8—42no—
114C2×C8—82no—
115C2×C8—82no—
116C2×C8#12112yes—
117C2×C8#12312yes—
118C2×C8C2×SD1612yes—
119C2×C8#13512yes—
120C2×C8AGL(1,ℤ/8)12yes—
121C2×C8AGL(1,ℤ/8)12yes—
122C4⋊C4—44no—
123C4⋊C4—84no—
124C4⋊C4—84no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C22⋊C4—84no—
128C42—21no—
129C42#12111yescharacteristic
130C42#12411yescharacteristic
131C16—82no—
132C23—42no—
133C23#45311yescharacteristic
134Q8—42no—
135Q8—42no—
136D4—42no—
137D4—42no—
138D4—82no—
139D4—82no—
140D4—322no—
141C2×C4—24no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—162no—
149C2×C4—162no—
150C2×C4—161no—
151C2×C4#39411yescharacteristic
152C2×C4#44912yes—
153C2×C4#45012yes—
154C2×C4#45611yescharacteristic
155C8—44no—
156C8—82no—
157C8#46512yes—
158C8#46912yes—
159C8#48112yes—
160C8#48212yes—
161C22—22no—
162C22—84no—
163C22—82no—
164C22#162411yescharacteristic
165C22#162911yescharacteristic
166C22#203911yescharacteristic
167C4—22no—
168C4—22no—
169C4—82no—
170C4—162no—
171C4—321no—
172C4#165412yes—
173C4#203412yes—
174C2—21no—
175C2—21no—
176C2—162no—
177C2#1043211yescharacteristic
178C2#1236512yes—
subgroupG/H
center, socleC22#1629
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#28246C2
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. #12823C2semidirect
3. C2×#1335C2semidirect
4. #8815C2semidirect
5. #7268C2semidirect
44. C82D4semidirect

Maximal subgroups

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

subgroupindex
#282462
#128232
C2×#13352
#88152
#72682