Finite Group Databasebeta

← #46825666

Subgroups — #46825666

Order 34432.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#31771441C211yesmaximal, characteristic, complemented
2C2×#22575499C211yesmaximal, characteristic, complemented
3#31771417C218yesmaximal, complemented
4#31771410C218yesmaximal, complemented
5#31771405C216yesmaximal, complemented
6C2×#22575481C216yesmaximal, complemented
7#568×C269C211yesmaximal, characteristic, complemented
8#22575501—21no—
9#22575500C2214yescomplemented
10#22575499—24no—
11#22575499C2218yescomplemented
12#22575499C2214yescomplemented
13#22575498—23no—
14C2×#16864403—21no—
15C2×#16864402C2213yescomplemented
16C2×#16864402C2216yescomplemented
17#41×D269—24no—
18C22×D1076C2213yescomplemented
19C2×D4×D269—23no—
20#22575486C2214yescomplemented
21#22575486C22112yescomplemented
22#22575481C22112yescomplemented
23#22575481C22124yescomplemented
24#22575481C22112yescomplemented
25#22575475C22124yescomplemented
26C2×#16864391C22112yescomplemented
27C2×D2152C22112yescomplemented
28C2×#41×C269C2211yescharacteristic
29#131×C269C2218yescomplemented
30C2×M16×C269C2216yescomplemented
31C23×D269—23no—
32#16864403—24no—
33#16864403—28no—
34#16864402—26no—
35#16864402—26no—
36#16864402—26no—
37#16864402C23112yes—
38#16864402C2314yes—
39#16864402C23112yescomplemented
40#16864401—212no—
41C2×Dic538C2311yescharacteristic
42C2×Dic538C2316yescomplemented
43Q8×D269—24no—
44C2×D1076—23no—
45C2×D1076—26no—
46C2×D1076C23112yes—
47C2×D1076C2313yes—
48C2×D1076C2316yescomplemented
49C2×C22⋊D269—23no—
50C2×C22⋊D269—26no—
51D4×D269—224no—
52D4×D269—212no—
53C2×C4×D269—23no—
54C2×C4×D269—26no—
55Dic1076C2318yescomplemented
56#16864391C23124yescomplemented
57#16864391C2318yescomplemented
58D2152C23124yescomplemented
59#41×C269D414yes—
60#41×C269C2314yes—
61C2×Q8×C269D411yescharacteristic
62C2×D4×C269D413yes—
63C22×C4×C269C2313yes—
64M16×C269C2314yes—
65M16×C269C23112yescomplemented
66C2×C8×C269C23112yescomplemented
67C22×D269—23no—
68C22×D269—26no—
69C22×D269—212no—
70C22×D269—86no—
71Dic538—26no—
72Dic538—21no—
73Dic538C2414yes—
74Dic538C2412yes—
75D1076—212no—
76D1076—26no—
77D1076—23no—
78D1076C2414yes—
79D1076C2416yes—
80C22⋊D269—212no—
81C22⋊D269—212no—
82C22⋊D269—24no—
83C2×Dic269—21no—
84C2×Dic269—26no—
85C4×D269—24no—
86C4×D269—212no—
87C4×D269—212no—
88C23×C269C2×D413yes—
89Q8×C269C2×D414yes—
90D4×C269C2×D4112yes—
91C2×C4×C269C2×D413yes—
92C2×C4×C269C2×D416yes—
93C2×C4×C269C2416yes—
94C2×C4×C269C2411yescharacteristic
95C8×C269C2418yes—
96C22×C269—43no—
97C22×C269C22×D411yescharacteristic
98C22×C269C22×D416yes—
99C2×D269—24no—
100C2×D269—26no—
101C2×D269—46no—
102C2×D269—86no—
103C2×D269—812no—
104C4×C269C22×D411yescharacteristic
105C4×C269C22×D416yes—
106C4×C269C2511yescharacteristic
107Dic269—22no—
108Dic269—24no—
109C2×C269—21no—
110C2×C269—26no—
111C2×C269C23×D411yescharacteristic
112D269—46no—
113D269—44no—
114C269#358511yescharacteristic, complemented, Sylow
115#3585—2691nomaximal, Sylow
116#585—2691no—
117C2×H(5,𝔽2)—2691no—
118#577—2698no—
119#576—2698no—
120#575—2696no—
121C2×AGL(1,ℤ/8)—2696no—
122#568D26911yescharacteristic, complemented
1232−1+4—2694no—
1242−1+4—5381no—
125H(5,𝔽2)—2694no—
126H(5,𝔽2)—2698no—
127H(5,𝔽2)—5384no—
128H(5,𝔽2)—5383no—
129C2×#41—2696no—
130C2×#41—2693no—
131C2×#41—5384no—
132C2×#41—5381no—
133C2×#41C2×D26911yescharacteristic
134C22×D4—2693no—
135C22×D4—5383no—
136#137—2694no—
137#137—26912no—
138AGL(1,ℤ/8)—26912no—
139AGL(1,ℤ/8)—26924no—
140AGL(1,ℤ/8)—26912no—
141#135—26924no—
142C2×SD16—26912no—
143C2×D8—26912no—
144#131C2×D26918yescomplemented
145C2×M16C2×D26916yescomplemented
146C24—5383no—
147#41—26912no—
148#41—26912no—
149#41—2694no—
150#41—5388no—
151#41—5384no—
152#41—53812no—
153#41—5386no—
154#41—5386no—
155#41—5386no—
156#41D107614yes—
157#41C22×D26914yes—
158C2×Q8—2696no—
159C2×Q8—2691no—
160C2×Q8—5384no—
161C2×Q8D107611yescharacteristic
162C2×D4—2696no—
163C2×D4—26912no—
164C2×D4—2693no—
165C2×D4—5383no—
166C2×D4—5386no—
167C2×D4—53824no—
168C2×D4—53812no—
169C2×D4—5383no—
170C2×D4—5386no—
171C2×D4D107613yes—
172C22×C4—5383no—
173C22×C4—5386no—
174C22×C4C22×D26913yes—
175Q16—2698no—
176SD16—26924no—
177SD16—2698no—
178D8—26924no—
179M16C22×D26914yes—
180M16C22×D269112yescomplemented
181C2×C8C22×D269112yescomplemented
182C23—5383no—
183C23—5386no—
184C23—53812no—
185C23—21526no—
186C23C2×D107613yes—
187Q8—2694no—
188Q8—2692no—
189Q8—5386no—
190Q8—5381no—
191Q8C2×D107614yes—
192D4—2694no—
193D4—2696no—
194D4—53812no—
195D4—5384no—
196D4—53812no—
197D4—5383no—
198D4—53812no—
199D4—5386no—
200D4C2×D1076112yes—
201C2×C4—5381no—
202C2×C4—5386no—
203C2×C4—5384no—
204C2×C4—53812no—
205C2×C4—53812no—
206C2×C4C2×D107613yes—
207C2×C4C2×D107616yes—
208C2×C4C23×D26916yes—
209C2×C4C23×D26911yescharacteristic
210C8C23×D26918yes—
211C22—43no—
212C22—5384no—
213C22—5386no—
214C22—10766no—
215C22—215212no—
216C22—21526no—
217C22C22×D107611yescharacteristic
218C22C22×D107616yes—
219C4—5382no—
220C4—5384no—
221C4C22×D107611yescharacteristic
222C4C22×D107616yes—
223C4C24×D26911yescharacteristic
224C2—21no—
225C2—26no—
226C2—10766no—
227C2—10764no—
228C2C23×D107611yescharacteristic
subgroupG/H
centerC2C23×D1076
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
Fitting#568×C269C2
socleC2×C269C23×D4
hypercenter#568D269
hypocenterC269#3585
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. C2×#22575499C2semidirect
3. #31771417C2semidirect
4. #31771410C2semidirect
5. #31771405C2semidirect
6. C2×#22575481C2semidirect
7. #568×C269C2semidirect
9. #22575500C22semidirect
11. #22575499C22semidirect
12. #22575499C22semidirect
15. C2×#16864402C22semidirect
16. C2×#16864402C22semidirect
18. C22×D1076C22semidirect
20. #22575486C22semidirect
21. #22575486C22semidirect
22. #22575481C22semidirect
23. #22575481C22semidirect
24. #22575481C22semidirect
25. #22575475C22semidirect
26. C2×#16864391C22semidirect
27. C2×D2152C22semidirect
29. #131×C269C22semidirect
30. C2×M16×C269C22semidirect
39. #16864402C23semidirect
42. C2×Dic538C23semidirect
48. C2×D1076C23semidirect
55. Dic1076C23semidirect
56. #16864391C23semidirect
57. #16864391C23semidirect
58. D2152C23semidirect
65. M16×C269C23semidirect
66. C2×C8×C269C23semidirect
114. C269#3585semidirect
122. #568D269semidirect
144. #131C2×D269semidirect
145. C2×M16C2×D269semidirect
180. M16C22×D269semidirect
181. C2×C8C22×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754992
#317714172
#317714102
#317714052
C2×#225754812
#568×C2692
#3585269