Finite Group Databasebeta

← #46821349

Subgroups — #46821349

Order 34432.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22575484C212yesmaximal, complemented
2C2×#22575479C212yesmaximal, complemented
3#31771155C211yesmaximal, characteristic, complemented
4#31771000C212yesmaximal, complemented
5#31770995C212yesmaximal, complemented
6#31770987C214yesmaximal, complemented
7C4×#16864386C211yesmaximal, characteristic, complemented
8C2×#124×C269C211yesmaximal, characteristic, complemented
9C2×#16864402C2212yes—
10#22575484—28no—
11#22575479—28no—
12C2×#16864392C2214yescomplemented
13C2×#16864390C2214yes—
14C2×#16864389C2214yescomplemented
15C2×#16864387C2214yes—
16C2×#16864386C2212yes—
17#22575453C2211yescharacteristic, complemented
18#22575450C2211yescharacteristic, complemented
19#22575441C2212yescomplemented
20#22575427—22no—
21C4×D1076—22no—
22#22575364C2212yescomplemented
23C4×#14524130C2212yescomplemented
24C22×Q8×C269C2211yescharacteristic
25C22×D4×C269C2211yescharacteristic
26#124×C269—22no—
27#124×C269C2212yes—
28#124×C269C2212yes—
29C2×C22⋊C4×C269—22no—
30C2×C42×C269C2211yescharacteristic
31#16864402—24no—
32#16864402—24no—
33C2×Dic538C2312yes—
34C2×D1076—22no—
35C2×D1076C2312yes—
36C2×C22⋊D269—22no—
37C2×C4×D269—22no—
38#16864392—28no—
39#16864390—28no—
40#16864389—28no—
41#16864387—28no—
42#16864386D418yescomplemented
43C2×#14524130C2314yes—
44C4⋊Dic269—22no—
45#16864379—24no—
46C24×C269—21no—
47C2×Q8×C269—21no—
48C2×Q8×C269—24no—
49C2×Q8×C269C2312yes—
50C2×D4×C269—24no—
51C2×D4×C269—21no—
52C2×D4×C269C2312yes—
53C22×C4×C269—21no—
54C22×C4×C269D412yes—
55C22×C4×C269C2311yescharacteristic
56C22⋊C4×C269—24no—
57C22⋊C4×C269—24no—
58C42×C269D412yes—
59C42×C269C2311yescharacteristic
60C42×C269C2311yescharacteristic
61C22×D269—22no—
62Dic538—22no—
63Dic538—22no—
64D1076—22no—
65D1076—22no—
66D1076—44no—
67C22⋊D269—44no—
68C2×Dic269—22no—
69C4×D269—44no—
70#14524130C2×D418yes—
71C23×C269—22no—
72C23×C269—44no—
73C23×C269—42no—
74C23×C269C2×D411yescharacteristic
75Q8×C269—24no—
76Q8×C269—42no—
77D4×C269—24no—
78D4×C269—42no—
79C2×C4×C269—22no—
80C2×C4×C269—42no—
81C2×C4×C269—42no—
82C2×C4×C269C2×D414yes—
83C2×C4×C269C2×D412yes—
84C2×C4×C269C2×D412yes—
85C2×C4×C269C2×D411yescharacteristic
86C2×C4×C269C2411yescharacteristic
87C22×C269—22no—
88C22×C269—42no—
89C22×C269—44no—
90C22×C269—84no—
91C22×C269#12712yes—
92C22×C269C22×D411yescharacteristic
93C2×D269—42no—
94C2×D269—44no—
95C4×C269—22no—
96C4×C269—22no—
97C4×C269—42no—
98C4×C269#12712yes—
99C4×C269C22×D412yes—
100Dic269—42no—
101C2×C269—22no—
102C2×C269—82no—
103C2×C269C2×#12711yescharacteristic
104C2×C269#57512yes—
105D269—82no—
106C269#315311yescharacteristic, complemented, Sylow
107#3153—2691nomaximal, Sylow
108C2×#137—2692no—
109C2×AGL(1,ℤ/8)—2692no—
110#532—2691no—
111C2×#124D26911yescharacteristic, complemented
112#497—2692no—
113#496—2692no—
114#495—2694no—
115C4×M16—2691no—
116C2×#41—2692no—
117C22×Q8C2×D26911yescharacteristic
118C22×D4C2×D26911yescharacteristic
119#137—5388no—
120AGL(1,ℤ/8)—5388no—
121C2×Q16—2694no—
122C2×SD16—2694no—
123C2×SD16—2694no—
124C2×D8—2694no—
125C2×M16—2692no—
126C4⋊Q8—2691no—
127#127—2691no—
128#124—22no—
129#124—2692no—
130#124C2×D26912yes—
131#124C2×D26912yes—
132#121—5382no—
133C4×D4—5382no—
134C2×C22⋊C4—22no—
135C2×C42C2×D26911yescharacteristic
136#97—2692no—
137C4×C8—2692no—
138C24—21no—
139#41—5384no—
140#41—5384no—
141C2×Q8—21no—
142C2×Q8—24no—
143C2×Q8—2692no—
144C2×Q8C22×D26912yes—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—2692no—
148C2×D4—5382no—
149C2×D4—5382no—
150C2×D4C22×D26912yes—
151C22×C4—21no—
152C22×C4—5382no—
153C22×C4C22⋊D26912yes—
154C22×C4C22×D26911yescharacteristic
155Q16—5388no—
156SD16—5388no—
157SD16—5388no—
158D8—5388no—
159M16—2698no—
160C2×C8—2694no—
161C4⋊C4—5382no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—5384no—
165C42C22⋊D26912yes—
166C42C22×D26911yescharacteristic
167C42C22×D26911yescharacteristic
168C23—22no—
169C23—44no—
170C23—42no—
171C23—5382no—
172C23C2×C22⋊D26911yescharacteristic
173Q8—24no—
174Q8—42no—
175Q8—5382no—
176Q8—5382no—
177D4—24no—
178D4—42no—
179D4—5382no—
180D4—5382no—
181D4—10764no—
182D4—10764no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—5382no—
187C2×C4—10764no—
188C2×C4D4×D26914yes—
189C2×C4C2×C22⋊D26912yes—
190C2×C4C2×C22⋊D26912yes—
191C2×C4C2×C22⋊D26911yescharacteristic
192C2×C4C23×D26911yescharacteristic
193C8—2698no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—84no—
198C22—10762no—
199C22—10764no—
200C22#2257544912yes—
201C22C22×C22⋊D26911yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—10762no—
206C4#2257544912yes—
207C4C2×D4×D26912yes—
208C2—22no—
209C2—82no—
210C2—21522no—
211C2C2×#2257544911yescharacteristic
212C2#3177140312yes—
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#124×C269C2
socleC22×C269C22×D4
hypercenterC2×#124D269
hypocenterC269#3153
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#22575484C2semidirect
2. C2×#22575479C2semidirect
3. #31771155C2semidirect
4. #31771000C2semidirect
5. #31770995C2semidirect
6. #31770987C2semidirect
7. C4×#16864386C2semidirect
8. C2×#124×C269C2semidirect
12. C2×#16864392C22semidirect
14. C2×#16864389C22semidirect
17. #22575453C22semidirect
18. #22575450C22semidirect
19. #22575441C22semidirect
22. #22575364C22semidirect
23. C4×#14524130C22semidirect
42. #16864386D4semidirect
106. C269#3153semidirect
111. C2×#124D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754842
C2×#225754792
#317711552
#317710002
#317709952
#317709872
C4×#168643862
C2×#124×C2692
#3153269