Finite Group Databasebeta

← #144583

Subgroups — #144583

Order 512.

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

normal subgroups only

This group has 327 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, complemented
2#11744C212yesmaximal
3#9852C211yesmaximal, characteristic
4#7302C212yesmaximal, complemented
5#7143C211yesmaximal, characteristic, complemented
6#3169C2211yescharacteristic
7C4×C4⋊Q8—21no—
8#2163C2212yes—
9#2084—22no—
10#2058—22no—
11#1962—21no—
12C4×#106—21no—
13C4×#106C2212yes—
14#1744C2211yescharacteristic
15#1566C2211yescharacteristic
16#1492—21no—
17#1347—22no—
18#1347—22no—
19#532—21no—
20C2×C4⋊Q8—21no—
21C2×C4×Q8—41no—
22#500Q811yescharacteristic, complemented
23#493Q811yescharacteristic, complemented
24#486—21no—
25#466—22no—
26C4×D8—22no—
27C2×#106—22no—
28C2×#106—22no—
29#423—21no—
30C2×#104—41no—
31C2×#104—41no—
32C2×C4×C8D411yescharacteristic
33C2×C4×C8C2311yescharacteristic
34#391—41no—
35#389—22no—
36#389—41no—
37#384—21no—
38C4×C4⋊C4—22no—
39C4×C4⋊C4—41no—
40C43D411yescharacteristic
41C4⋊C16—22no—
42C22⋊C16—22no—
43#346—22no—
44#340—21no—
45#340—21no—
46#340D412yes—
47#340D412yes—
48#336—22no—
49#334—22no—
50D4⋊C8—42no—
51C2×#41—41no—
52C22×Q8—41no—
53C2×D8—22no—
54C2×M16—41no—
55C22×C8—21no—
56C22×C8C2×D412yes—
57C4⋊Q8—21no—
58C4⋊Q8—44no—
59#127—21no—
60#124—41no—
61#121—41no—
62#121—41no—
63C4×Q8—84no—
64C4×D4—41no—
65C4×D4—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#104—82no—
85#104—82no—
86#102—42no—
87C4×C8—21no—
88C4×C8SD1612yes—
89C4×C8C2×Q811yescharacteristic
90C4×C8#4111yescharacteristic
91C4×C8#4112yes—
92#95—42no—
93#95—42no—
94#41—81no—
95#41—81no—
96#41—81no—
97#41—81no—
98C2×Q8—44no—
99C2×Q8—41no—
100C2×Q8—82no—
101C2×Q8—82no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—81no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—81no—
111C22×C4#12011yescharacteristic
112C22×C4C2×D811yescharacteristic
113C22×C4AGL(1,ℤ/8)11yescharacteristic
114D8—82no—
115M16—81no—
116M16—81no—
117C2×C8—22no—
118C2×C8—42no—
119C2×C8—42no—
120C2×C8—81no—
121C2×C8C22⋊Q812yes—
122C2×C8#12412yes—
123C4⋊C4—24no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—44no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—84no—
131C4⋊C4—82no—
132C4⋊C4—82no—
133C22⋊C4—81no—
134C22⋊C4—81no—
135C42—24no—
136C42—41no—
137C42—41no—
138C42—42no—
139C42—84no—
140C42C22⋊Q811yescharacteristic
141C42#12411yescharacteristic
142C42C2×SD1611yescharacteristic
143C42#13711yescharacteristic
144C16—82no—
145C23—81no—
146C23#44711yescharacteristic
147Q8—81no—
148Q8—81no—
149Q8—88no—
150D4—81no—
151D4—81no—
152D4—82no—
153D4—82no—
154D4—161no—
155D4—161no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—81no—
173C2×C4—161no—
174C2×C4—161no—
175C2×C4#39311yescharacteristic
176C2×C4#44811yescharacteristic
177C2×C4#47411yescharacteristic
178C2×C4#47611yescharacteristic
179C2×C4#48611yescharacteristic
180C2×C4#48911yescharacteristic
181C8—22no—
182C8—22no—
183C8—42no—
184C8—161no—
185C22—21no—
186C22—21no—
187C22—161no—
188C22—161no—
189C22—161no—
190C22#202311yescharacteristic
191C22#219011yescharacteristic
192C22#219011yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—84no—
200C4—162no—
201C4—161no—
202C4#203112yes—
203C4#223012yes—
204C2—21no—
205C2—21no—
206C2—321no—
207C2#1037011yescharacteristic
208C2#1336711yescharacteristic
209C2#1337111yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
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
1. #23837C2semidirect
4. #7302C2semidirect
5. #7143C2semidirect
22. #500Q8semidirect
23. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#238372
#117442
#98522
#73022
#71432