Finite Group Databasebeta

← #144584

Subgroups — #144584

Order 512.

308 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 64 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#23837C211yesmaximal, characteristic
2#11745C212yesmaximal
3#9853C211yesmaximal, characteristic
4#7302C212yesmaximal
5#7143C211yesmaximal, characteristic
6#3170C2211yescharacteristic
7C4×C4⋊Q8—21no—
8#2163C2212yes—
9#2084—22no—
10#2058—22no—
11#1994—21no—
12C4×#106—21no—
13C4×#106C2212yes—
14#1744C2211yescharacteristic
15#1566C2211yescharacteristic
16#1493—21no—
17#1348—22no—
18#1348—22no—
19#533—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8—41no—
22#500Q811yescharacteristic, complemented
23#494Q811yescharacteristic, complemented
24#487—21no—
25#467—22no—
26C4×Q16—22no—
27C2×#106—22no—
28C2×#106—22no—
29#423—21no—
30C2×C4×C8D411yescharacteristic
31C2×C4×C8C2311yescharacteristic
32#391—41no—
33#389—22no—
34#389—41no—
35#384—21no—
36C4×C4⋊C4—22no—
37C4×C4⋊C4—41no—
38C43D411yescharacteristic
39C4⋊C16—22no—
40C22⋊C16—22no—
41#346—22no—
42#340—21no—
43#340—21no—
44#340D412yes—
45#340D412yes—
46#339—41no—
47#339—41no—
48#336—22no—
49#334—22no—
50Q8⋊C8—42no—
51C22×Q8—41no—
52C2×Q16—22no—
53C2×M16—41no—
54C22×C8—21no—
55C22×C8C2×D412yes—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—44no—
59#125—41no—
60C22⋊Q8—41no—
61C22⋊Q8—41no—
62C4×Q8—41no—
63C4×Q8—41no—
64C4×Q8—84no—
65#117—41no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—42no—
75C2×C42D811yescharacteristic
76C2×C42D811yescharacteristic
77C2×C42C2×D411yescharacteristic
78C2×C16—42no—
79#107—22no—
80#106—44no—
81#106—44no—
82C4⋊C8—41no—
83C4⋊C8—41no—
84#103—42no—
85C4×C8—21no—
86C4×C8SD1612yes—
87C4×C8C2×Q811yescharacteristic
88C4×C8#4111yescharacteristic
89C4×C8#4112yes—
90#95—42no—
91#95—42no—
92C2×Q8—44no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8—82no—
96C2×Q8—82no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#12011yescharacteristic
102C22×C4C2×D811yescharacteristic
103C22×C4AGL(1,ℤ/8)11yescharacteristic
104Q16—82no—
105M16—81no—
106M16—81no—
107C2×C8—22no—
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—81no—
111C2×C8C22⋊Q812yes—
112C2×C8#12412yes—
113C4⋊C4—24no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—84no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C4⋊C4—81no—
126C4⋊C4—81no—
127C22⋊C4—81no—
128C42—24no—
129C42—41no—
130C42—41no—
131C42—42no—
132C42—84no—
133C42—81no—
134C42C22⋊Q811yescharacteristic
135C42#12411yescharacteristic
136C42C2×SD1611yescharacteristic
137C42#13711yescharacteristic
138C16—82no—
139C23#44711yescharacteristic
140Q8—88no—
141Q8—82no—
142Q8—82no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—44no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4#39311yescharacteristic
162C2×C4#44811yescharacteristic
163C2×C4#47411yescharacteristic
164C2×C4#47611yescharacteristic
165C2×C4#48611yescharacteristic
166C2×C4#48911yescharacteristic
167C8—22no—
168C8—22no—
169C8—42no—
170C8—161no—
171C22—21no—
172C22—21no—
173C22#202311yescharacteristic
174C22#219111yescharacteristic
175C22#219111yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—42no—
181C4—42no—
182C4—84no—
183C4—162no—
184C4—161no—
185C4—161no—
186C4#203112yes—
187C4#223012yes—
188C2—21no—
189C2—21no—
190C2#1037011yescharacteristic
191C2#1336811yescharacteristic
192C2#1337111yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#447
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #500Q8semidirect
23. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#238372
#117452
#98532
#73022
#71432