Finite Group Databasebeta

← #144525

Subgroups — #144525

Order 512.

309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 80 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#28824C211yesmaximal, characteristic
2#13272C212yesmaximal
3#9819C211yesmaximal, characteristic
4#7268C211yesmaximal, characteristic
5#7179C212yesmaximal
6#3163—22no—
7C2×#498C2211yescharacteristic
8C2×#364C2212yes—
9#2060—21no—
10#2060—21no—
11#1990—21no—
12#1956—21no—
13#1951—22no—
14#1844C2211yescharacteristic
15#1772C2212yes—
16C2×C82C2211yescharacteristic
17#1386—22no—
18#1383—22no—
19#544—21no—
20#544—21no—
21C2×C4⋊Q8—21no—
22C2×M32—22no—
23#501—42no—
24#501—42no—
25#499—42no—
26#498—22no—
27#498—21no—
28#498—42no—
29#498Q812yescomplemented
30C2×#108D412yes—
31#428—22no—
32#428—22no—
33#427D412yes—
34C2×#106—22no—
35C2×#106—22no—
36C2×#103—42no—
37#405—22no—
38C2×C4×C8—21no—
39C2×C4×C8C2311yescharacteristic
40#384D411yescharacteristic
41#384D411yescharacteristic
42#364—22no—
43#364—22no—
44#343—21no—
45#343—21no—
46#340—42no—
47#339—22no—
48C82—22no—
49C22×Q8—41no—
50C2×M16—22no—
51C22×C8—22no—
52C22×C8C2×D412yes—
53C4⋊Q8—22no—
54C4⋊Q8—41no—
55C4⋊Q8—41no—
56C4⋊Q8—42no—
57C4⋊Q8—42no—
58C4⋊Q8—41no—
59#125—42no—
60#125—42no—
61C22⋊Q8—81no—
62C22⋊Q8—81no—
63#117—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C42C2×D411yescharacteristic
71M32—42no—
72M32—42no—
73C2×C16—42no—
74#108—22no—
75#108Q1614yes—
76#107—42no—
77#107—42no—
78#107SD1612yes—
79#107SD1612yes—
80#106—22no—
81#106—24no—
82#106—42no—
83#106—42no—
84#106—42no—
85#106—42no—
86#106—42no—
87#103—84no—
88#97—42no—
89C4×C8—41no—
90C4×C8—42no—
91C4×C8—41no—
92C4×C8C2×Q811yescharacteristic
93C4×C8#4111yescharacteristic
94C4×C8#4112yes—
95C2×Q8—42no—
96C2×Q8—81no—
97C2×Q8—82no—
98C2×Q8—82no—
99C2×Q8—81no—
100C2×Q8—81no—
101C22×C4—21no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—81no—
105C22×C4#12011yescharacteristic
106M16—24no—
107M16—42no—
108C2×C8—24no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—42no—
112C2×C8C22⋊Q812yes—
113C2×C8#12412yes—
114C2×C8C2×SD1612yes—
115C2×C8C2×Q1612yes—
116C2×C8AGL(1,ℤ/8)12yes—
117C2×C8#13712yes—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—81no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—81no—
128C4⋊C4—81no—
129C4⋊C4—82no—
130C4⋊C4—82no—
131C4⋊C4—81no—
132C22⋊C4—42no—
133C42—21no—
134C42—42no—
135C42C22⋊Q811yescharacteristic
136C42#12411yescharacteristic
137C16—82no—
138C23#45311yescharacteristic
139Q8—84no—
140Q8—162no—
141Q8—321no—
142Q8—321no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—82no—
152C2×C4—161no—
153C2×C4—161no—
154C2×C4—161no—
155C2×C4—161no—
156C2×C4#39311yescharacteristic
157C2×C4#45012yes—
158C2×C4#45112yes—
159C2×C4#45411yescharacteristic
160C8—44no—
161C8—44no—
162C8#47512yes—
163C8#47712yes—
164C8#48712yes—
165C8#48912yes—
166C22—21no—
167C22—21no—
168C22#162911yescharacteristic
169C22#163311yescharacteristic
170C22#202711yescharacteristic
171C4—22no—
172C4—22no—
173C4—82no—
174C4—82no—
175C4—162no—
176C4—161no—
177C4—161no—
178C4#163712yes—
179C4#202412yes—
180C2—21no—
181C2—21no—
182C2#1034911yescharacteristic
183C2#1236411yescharacteristic
184C2#1236511yescharacteristic
subgroupG/H
center, socleC22#1629
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#453
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
29. #498Q8semidirect

Maximal subgroups

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

subgroupindex
#288242
#132722
#98192
#72682
#71792