Finite Group Databasebeta

← #152256

Subgroups — #152256

Order 512.

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

normal subgroups only

This group has 369 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#22275C211yesmaximal, characteristic
2C2×#1409C212yesmaximal
3#12799C212yesmaximal
4#7504C212yesmaximal
5C2×#492C2211yescharacteristic
6#2328C2211yescharacteristic
7#2327C411yescharacteristic, complemented
8C2×#353C2214yes—
9#1961C2211yescharacteristic
10#1960C411yescharacteristic, complemented
11#1880—22no—
12#1877—22no—
13#1835C411yescharacteristic, complemented
14#1799—22no—
15#1746C411yescharacteristic, complemented
16#1409—24no—
17#1393—22no—
18C22×SD16—22no—
19C2×C4⋊Q8C2×C411yescharacteristic
20C2×#127C2311yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×D4D412yes—
23#492—22no—
24#492D412yes—
25#492Q812yes—
26C2×#103—22no—
27C2×#102—22no—
28C2×#99—24no—
29C2×#97C2×C411yescharacteristic
30C2×C4×C8C2×C411yescharacteristic
31#391—21no—
32#390D412yes—
33#387—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×Q8—21no—
46C22×D4—41no—
47C22×D4C2×D412yes—
48C23×C4—22no—
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—82no—
52C2×SD16—82no—
53C22×C8—21no—
54C22×C8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—41no—
57C4⋊Q8C4⋊C412yescomplemented
58#127—41no—
59#127C2×D411yescharacteristic
60#127C2×Q811yescharacteristic
61#127#4112yes—
62C4×Q8—82no—
63C4×D4—82no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—82no—
68C2×C4⋊C4C22⋊C412yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—44no—
71C2×C22⋊C4—42no—
72C2×C42—41no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22×C411yescharacteristic
76#103—82no—
77#102—82no—
78#99—44no—
79#99—44no—
80#97—41no—
81C4×C8—21no—
82C4×C8C4⋊C412yescomplemented
83#95—22no—
84#95—41no—
85#95—41no—
86#95—42no—
87#95—82no—
88C24—22no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—41no—
93C2×D4—22no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—42no—
99C2×D4—162no—
100C2×D4#12112yes—
101C2×D4C22⋊Q812yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—84no—
110C22×C4—162no—
111C22×C4#9912yes—
112C22×C4C2×C22⋊C411yescharacteristic
113SD16—84no—
114SD16—84no—
115C2×C8—22no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C4⋊C4—164no—
127C22⋊C4—82no—
128C22⋊C4—84no—
129C22⋊C4—84no—
130C22⋊C4—162no—
131C42—21no—
132C42—41no—
133C42—41no—
134C42—82no—
135C42C2×C4⋊C411yescharacteristic
136C42#11711yescharacteristic
137C23—22no—
138C23—22no—
139C23—42no—
140C23—42no—
141C23—82no—
142C23—82no—
143C23C2×#9911yescharacteristic
144Q8—44no—
145Q8—82no—
146D4—42no—
147D4—42no—
148D4—82no—
149D4—161no—
150D4—161no—
151D4—162no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—84no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—162no—
170C2×C4—164no—
171C2×C4—164no—
172C2×C4—164no—
173C2×C4#38011yescharacteristic
174C2×C4#41011yescharacteristic
175C8—42no—
176C8—81no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22—82no—
184C22—162no—
185C22#179911yescharacteristic
186C22#212711yescharacteristic
187C22#212811yescharacteristic
188C4—21no—
189C4—21no—
190C4—41no—
191C4—41no—
192C4—42no—
193C4—81no—
194C4—82no—
195C4—82no—
196C4—161no—
197C4—324no—
198C2—21no—
199C2—21no—
200C2—82no—
201C2—82no—
202C2#1279911yescharacteristic
203C2#1361012yes—
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, prime radical, cosocleC2×#127C23
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
7. #2327C4semidirect
10. #1960C4semidirect
13. #1835C4semidirect
15. #1746C4semidirect
57. C4⋊Q8C4⋊C4semidirect
82. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#222752
C2×#14092
#127992
#75042