Finite Group Databasebeta

← #152255

Subgroups — #152255

Order 512.

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

normal subgroups only

This group has 397 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#22276C211yesmaximal, characteristic
2C2×#1408C212yesmaximal
3#12799C212yesmaximal, complemented
4#7504C212yesmaximal, complemented
5C2×#493C2211yescharacteristic
6#2328C411yescharacteristic, complemented
7#2328C2211yescharacteristic
8C2×#353C2214yes—
9#1961C411yescharacteristic, complemented
10#1961C2211yescharacteristic
11#1879—22no—
12#1879—22no—
13#1835C411yescharacteristic, complemented
14#1799—22no—
15#1746C411yescharacteristic, complemented
16#1408—24no—
17#1393—22no—
18C22×D8—22no—
19C2×#127C2×C411yescharacteristic
20C2×#127C2311yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4D412yes—
23#493—22no—
24#493D412yes—
25#493Q812yes—
26C2×#102—22no—
27C2×#102—22no—
28C2×#99—24no—
29C2×#97C2×C411yescharacteristic
30C2×C4×C8C2×C411yescharacteristic
31#390—21no—
32#390D412yes—
33#386—42no—
34#386—42no—
35#382C2×C411yescharacteristic
36#382C2×C411yescharacteristic
37#380—42no—
38#376C2×C411yescharacteristic
39#353—24no—
40#353—24no—
41#342—42no—
42#340—22no—
43#336—21no—
44#336—21no—
45C22×D4—21no—
46C22×D4—41no—
47C22×D4—41no—
48C22×D4C2×D412yes—
49C23×C4—22no—
50C23×C4—41no—
51C2×D8—24no—
52C2×D8—24no—
53C2×D8—82no—
54C2×D8—82no—
55C22×C8—21no—
56C22×C8—21no—
57#127—21no—
58#127—41no—
59#127—41no—
60#127C4⋊C412yescomplemented
61#127C2×D411yescharacteristic
62#127C2×Q811yescharacteristic
63#127#4112yes—
64C4×D4—82no—
65C4×D4—82no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—82no—
68C2×C4⋊C4C22⋊C412yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—41no—
71C2×C22⋊C4—44no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22×C411yescharacteristic
77#102—82no—
78#102—82no—
79#99—44no—
80#99—44no—
81#97—41no—
82C4×C8—21no—
83C4×C8C4⋊C412yescomplemented
84#95—22no—
85#95—41no—
86#95—41no—
87#95—82no—
88C24—22no—
89C24—41no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—42no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—41no—
102C2×D4—162no—
103C2×D4—162no—
104C2×D4#12112yes—
105C2×D4C22⋊Q812yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—41no—
112C22×C4—82no—
113C22×C4—84no—
114C22×C4—162no—
115C22×C4—162no—
116C22×C4#9912yes—
117C22×C4C2×C22⋊C411yescharacteristic
118D8—84no—
119D8—84no—
120C2×C8—22no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—164no—
129C22⋊C4—82no—
130C22⋊C4—82no—
131C22⋊C4—84no—
132C22⋊C4—84no—
133C22⋊C4—162no—
134C22⋊C4—162no—
135C42—21no—
136C42—41no—
137C42—41no—
138C42C2×C4⋊C411yescharacteristic
139C42#11711yescharacteristic
140C23—22no—
141C23—22no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—82no—
146C23—82no—
147C23—82no—
148C23—82no—
149C23—81no—
150C23—81no—
151C23C2×#9911yescharacteristic
152D4—42no—
153D4—42no—
154D4—44no—
155D4—82no—
156D4—82no—
157D4—161no—
158D4—161no—
159D4—162no—
160D4—162no—
161D4—162no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—84no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—81no—
176C2×C4—162no—
177C2×C4—164no—
178C2×C4—164no—
179C2×C4—164no—
180C2×C4—164no—
181C2×C4#38011yescharacteristic
182C2×C4#41011yescharacteristic
183C8—42no—
184C8—81no—
185C22—22no—
186C22—42no—
187C22—42no—
188C22—82no—
189C22—82no—
190C22—84no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22—162no—
195C22—161no—
196C22—161no—
197C22—162no—
198C22#179911yescharacteristic
199C22#212711yescharacteristic
200C22#212811yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—42no—
206C4—81no—
207C4—82no—
208C4—161no—
209C4—324no—
210C2—21no—
211C2—21no—
212C2—82no—
213C2—82no—
214C2—162no—
215C2#1279911yescharacteristic
216C2#1360912yes—
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocleC2×#127C23
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2328 ◁ G
LECSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12799C2semidirect
4. #7504C2semidirect
6. #2328C4semidirect
9. #1961C4semidirect
13. #1835C4semidirect
15. #1746C4semidirect
60. #127C4⋊C4semidirect
83. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#222762
C2×#14082
#127992
#75042