Finite Group Databasebeta

← #46821351

Subgroups — #46821351

Order 34432.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#22575486C212yesmaximal, complemented
2C2×#22575481C212yesmaximal, complemented
3#31771155C211yesmaximal, characteristic, complemented
4C2×#22575441C211yesmaximal, characteristic, complemented
5#31770999C212yesmaximal, complemented
6#31770994C212yesmaximal, complemented
7#31770989C214yesmaximal, complemented
8C4×M16×C269C211yesmaximal, characteristic, complemented
9C2×#16864402C2212yes—
10C22×Dic538C2211yescharacteristic
11C22×D1076C2211yescharacteristic
12#22575486—28no—
13#22575481—28no—
14C2×Dic1076C2214yescomplemented
15C2×#16864391C2214yes—
16C2×#16864391C2214yescomplemented
17C2×D2152C2214yes—
18#22575453C2211yescharacteristic, complemented
19#22575450C2211yescharacteristic, complemented
20#22575441—22no—
21#22575441C2212yes—
22#22575441C2212yes—
23#22575441C2212yescomplemented
24#22575427—22no—
25C4×D1076—22no—
26C2×#16864379—22no—
27C2×M16×C269C2212yes—
28C2×C42×C269C2211yescharacteristic
29#97×C269C2212yescomplemented
30C4×C8×C269C2212yescomplemented
31C23×D269—21no—
32#16864402—24no—
33#16864402—24no—
34C2×Dic538—21no—
35C2×Dic538—24no—
36C2×Dic538C2312yes—
37C2×Dic538C2312yes—
38C2×D1076—24no—
39C2×D1076—22no—
40C2×D1076—21no—
41C2×D1076C2312yes—
42C2×D1076C2312yes—
43C2×C22⋊D269—22no—
44C22×Dic269—21no—
45C2×C4×D269—22no—
46Dic1076—28no—
47#16864391—28no—
48#16864391—28no—
49D2152—28no—
50C4⋊Dic269—22no—
51#16864379—24no—
52#16864379—24no—
53#16864379—24no—
54C22×C4×C269D412yes—
55C22×C4×C269C2311yescharacteristic
56M16×C269D418yescomplemented
57C2×C8×C269C2314yes—
58C42×C269D412yes—
59C42×C269C2311yescharacteristic
60C42×C269C2311yescharacteristic
61C22×D269—22no—
62C22×D269—22no—
63C22×D269—44no—
64C22×D269—42no—
65Dic538—22no—
66Dic538—24no—
67Dic538—22no—
68Dic538—42no—
69D1076—22no—
70D1076—24no—
71D1076—22no—
72D1076—44no—
73D1076—42no—
74C22⋊D269—44no—
75C2×Dic269—22no—
76C2×Dic269—22no—
77C2×Dic269—42no—
78C4×D269—44no—
79C23×C269C2×D411yescharacteristic
80C2×C4×C269—42no—
81C2×C4×C269C2×D412yes—
82C2×C4×C269C2×D412yes—
83C2×C4×C269C2×D411yescharacteristic
84C2×C4×C269C2×D414yes—
85C2×C4×C269C2411yescharacteristic
86C8×C269C2×D418yes—
87C22×C269—22no—
88C22×C269#12712yes—
89C22×C269C22×D411yescharacteristic
90C2×D269—42no—
91C2×D269—42no—
92C2×D269—44no—
93C2×D269—44no—
94C2×D269—84no—
95C4×C269—22no—
96C4×C269—22no—
97C4×C269#12712yes—
98C4×C269C22×D412yes—
99Dic269—42no—
100Dic269—42no—
101C2×C269—22no—
102C2×C269C2×#12711yescharacteristic
103C2×C269#57512yes—
104D269—82no—
105D269—82no—
106C269#315311yescharacteristic, complemented, Sylow
107#3153—2691nomaximal, Sylow
108C2×#137—2692no—
109C2×AGL(1,ℤ/8)—2692no—
110#532—2691no—
111C2×#124—2691no—
112#497—2692no—
113#496—2692no—
114#495—2694no—
115C4×M16D26911yescharacteristic, complemented
116C2×#41—2692no—
117C22×Q8—2691no—
118C22×D4—2691no—
119#137—5388no—
120AGL(1,ℤ/8)—5388no—
121C2×Q16—2694no—
122C2×SD16—2694no—
123C2×SD16—2694no—
124C2×D8—2694no—
125C2×M16C2×D26912yes—
126C4⋊Q8—2691no—
127#127—2691no—
128#124—2692no—
129#124—2692no—
130#124—2692no—
131#124—5382no—
132#121—5382no—
133C4×D4—5382no—
134C2×C22⋊C4—5382no—
135C2×C42C2×D26911yescharacteristic
136#97C2×D26912yescomplemented
137C4×C8C2×D26912yescomplemented
138C24—5381no—
139#41—5384no—
140#41—5384no—
141C2×Q8—2692no—
142C2×Q8—2692no—
143C2×Q8—5381no—
144C2×Q8—5384no—
145C2×D4—2692no—
146C2×D4—2692no—
147C2×D4—5382no—
148C2×D4—5384no—
149C2×D4—5382no—
150C2×D4—5381no—
151C22×C4—5381no—
152C22×C4—5382no—
153C22×C4D107612yes—
154C22×C4C22×D26911yescharacteristic
155Q16—5388no—
156SD16—5388no—
157SD16—5388no—
158D8—5388no—
159M16D107618yescomplemented
160C2×C8C22×D26914yes—
161C4⋊C4—5382no—
162C22⋊C4—5384no—
163C22⋊C4—5384no—
164C22⋊C4—5384no—
165C42D107612yes—
166C42C22×D26911yescharacteristic
167C42C22×D26911yescharacteristic
168C23—5382no—
169C23—5382no—
170C23—10764no—
171C23—10762no—
172C23C2×D107611yescharacteristic
173Q8—5382no—
174Q8—5384no—
175Q8—5382no—
176Q8—10762no—
177D4—5382no—
178D4—5384no—
179D4—5382no—
180D4—10764no—
181D4—10764no—
182D4—10762no—
183C2×C4—42no—
184C2×C4—5382no—
185C2×C4—5382no—
186C2×C4—10762no—
187C2×C4—10764no—
188C2×C4C2×D107612yes—
189C2×C4C2×D107612yes—
190C2×C4C2×D107611yescharacteristic
191C2×C4C2×D107614yes—
192C2×C4C23×D26911yescharacteristic
193C8C2×D107618yes—
194C22—22no—
195C22—10762no—
196C22—10762no—
197C22—10764no—
198C22—10764no—
199C22—21524no—
200C22#2257545012yes—
201C22C22×D107611yescharacteristic
202C4—22no—
203C4—22no—
204C4—10762no—
205C4—10762no—
206C4#2257545012yes—
207C4C22×D107612yes—
208C2—22no—
209C2—21522no—
210C2—21522no—
211C2C2×#2257545011yescharacteristic
212C2#3177140512yes—
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC4×M16×C269C2
socleC22×C269C22×D4
hypercenterC4×M16D269
hypocenterC269#3153
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#22575486C2semidirect
2. C2×#22575481C2semidirect
3. #31771155C2semidirect
4. C2×#22575441C2semidirect
5. #31770999C2semidirect
6. #31770994C2semidirect
7. #31770989C2semidirect
8. C4×M16×C269C2semidirect
14. C2×Dic1076C22semidirect
16. C2×#16864391C22semidirect
18. #22575453C22semidirect
19. #22575450C22semidirect
23. #22575441C22semidirect
29. #97×C269C22semidirect
30. C4×C8×C269C22semidirect
56. M16×C269D4semidirect
106. C269#3153semidirect
115. C4×M16D269semidirect
136. #97C2×D269semidirect
137. C4×C8C2×D269semidirect
159. M16D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754862
C2×#225754812
#317711552
C2×#225754412
#317709992
#317709942
#317709892
C4×M16×C2692
#3153269