Finite Group Databasebeta

← #46825665

Subgroups — #46825665

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
2#31771416C218yesmaximal, complemented
3#31771408C218yesmaximal, complemented
4#31771403C216yesmaximal, complemented
5C2×#22575479C216yesmaximal, complemented
6#31771372C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C269C211yesmaximal, characteristic, complemented
8#22575501—21no—
9#22575500C2214yescomplemented
10#22575499C2214yescomplemented
11#22575498—23no—
12C2×#16864402C2216yescomplemented
13#41×D269—24no—
14#22575484C2214yescomplemented
15#22575483C22112yescomplemented
16#22575479C22112yescomplemented
17#22575479C22124yescomplemented
18#22575478C22112yescomplemented
19#22575473C22124yescomplemented
20C2×#16864390C22112yescomplemented
21C2×#16864387C22112yescomplemented
22#22575460C2218yescomplemented
23C2×#16864386C2216yescomplemented
24H(5,𝔽2)×C269—24no—
25H(5,𝔽2)×C269C2218yescomplemented
26C2×#41×C269—21no—
27C2×#41×C269C2211yescharacteristic
28C2×#41×C269C2213yescomplemented
29C22×D4×C269—23no—
30C22×D4×C269C2213yescomplemented
31#16864403—24no—
32#16864402—26no—
33#16864402—26no—
34#16864402C2314yes—
35#16864402C23112yescomplemented
36#16864401—212no—
37C2×Dic538C2316yescomplemented
38Q8×D269—24no—
39C2×D1076C2316yescomplemented
40C2×C22⋊D269—26no—
41D4×D269—212no—
42C2×C4×D269—26no—
43#16864392C2318yescomplemented
44#16864390C23124yescomplemented
45#16864389C2318yescomplemented
46#16864387C23124yescomplemented
47#16864386C2314yes—
48#16864386C23112yescomplemented
49C2×#14524130C23112yescomplemented
50C24×C269—23no—
51#41×C269—28no—
52#41×C269—26no—
53#41×C269D414yes—
54#41×C269C23112yes—
55#41×C269C2314yes—
56C2×Q8×C269D411yescharacteristic
57C2×Q8×C269C2311yescharacteristic
58C2×D4×C269—23no—
59C2×D4×C269—224no—
60C2×D4×C269—23no—
61C2×D4×C269—26no—
62C2×D4×C269D413yes—
63C2×D4×C269C23112yes—
64C2×D4×C269C2313yes—
65C22×C4×C269—23no—
66C22×C4×C269C2313yes—
67C22×D269—26no—
68Dic538—26no—
69Dic538C2414yes—
70D1076—26no—
71D1076C2414yes—
72C22⋊D269—212no—
73C22⋊D269—24no—
74C2×Dic269—26no—
75C4×D269—24no—
76C4×D269—212no—
77#14524130C2418yes—
78C23×C269—23no—
79C23×C269—212no—
80C23×C269—86no—
81C23×C269C2×D413yes—
82Q8×C269—21no—
83Q8×C269C2×D414yes—
84Q8×C269C2412yes—
85D4×C269—212no—
86D4×C269—212no—
87D4×C269—23no—
88D4×C269C2×D4112yes—
89D4×C269C2416yes—
90C2×C4×C269—21no—
91C2×C4×C269—212no—
92C2×C4×C269C2×D413yes—
93C2×C4×C269C2×D416yes—
94C2×C4×C269C2411yescharacteristic
95C2×C4×C269C2416yes—
96C22×C269—26no—
97C22×C269—43no—
98C22×C269—46no—
99C22×C269—812no—
100C22×C269C22×D411yescharacteristic
101C22×C269C22×D416yes—
102C2×D269—24no—
103C2×D269—86no—
104C4×C269—22no—
105C4×C269C22×D416yes—
106C4×C269C22×D411yescharacteristic
107C4×C269C2511yescharacteristic
108Dic269—24no—
109C2×C269—26no—
110C2×C269—21no—
111C2×C269—46no—
112C2×C269C23×D411yescharacteristic
113D269—44no—
114C269#358511yescharacteristic, complemented, Sylow
115#3585—2691nomaximal, Sylow
116#585—2691no—
117C2×H(5,𝔽2)D26911yescharacteristic, complemented
118#577—2698no—
119#576—2698no—
120#575—2696no—
121C2×AGL(1,ℤ/8)—2696no—
122#568—2691no—
1232−1+4—2694no—
1242−1+4—5381no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—2694no—
127H(5,𝔽2)—5383no—
128H(5,𝔽2)C2×D26918yescomplemented
129C2×#41—21no—
130C2×#41—2696no—
131C2×#41—5384no—
132C2×#41C2×D26911yescharacteristic
133C2×#41C2×D26913yescomplemented
134C22×D4—23no—
135C22×D4C2×D26913yescomplemented
136#137—26912no—
137#137—2694no—
138AGL(1,ℤ/8)—26912no—
139AGL(1,ℤ/8)—26912no—
140AGL(1,ℤ/8)—26924no—
141#135—26924no—
142C2×SD16—26912no—
143C2×D8—26912no—
144#131—2698no—
145C2×M16—2696no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—26912no—
150#41—2694no—
151#41—5384no—
152#41—53812no—
153#41—5386no—
154#41—5386no—
155#41C22⋊D26914yes—
156#41C22×D269112yes—
157#41C22×D26914yes—
158C2×Q8—2696no—
159C2×Q8—5384no—
160C2×Q8C22⋊D26911yescharacteristic
161C2×Q8C22×D26911yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—2696no—
167C2×D4—5386no—
168C2×D4—53812no—
169C2×D4C22⋊D26913yes—
170C2×D4C22×D269112yes—
171C2×D4C22×D26913yes—
172C22×C4—23no—
173C22×C4—5386no—
174C22×C4C22×D26913yes—
175Q16—2698no—
176SD16—2698no—
177SD16—26924no—
178D8—26924no—
179M16—2694no—
180M16—26912no—
181C2×C8—26912no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—5386no—
186C23C2×C22⋊D26913yes—
187Q8—21no—
188Q8—2694no—
189Q8—5386no—
190Q8C2×C22⋊D26914yes—
191Q8C23×D26912yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—2694no—
196D4—5384no—
197D4—53812no—
198D4—5386no—
199D4C2×C22⋊D269112yes—
200D4C23×D26916yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—5386no—
204C2×C4—5384no—
205C2×C4—53812no—
206C2×C4C2×C22⋊D26913yes—
207C2×C4C2×C22⋊D26916yes—
208C2×C4C23×D26916yes—
209C2×C4C23×D26911yescharacteristic
210C8—2698no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—5384no—
216C22—21526no—
217C22C22×C22⋊D26911yescharacteristic
218C22C22×C22⋊D26916yes—
219C4—22no—
220C4—5384no—
221C4C22×C22⋊D26916yes—
222C4C22×C22⋊D26911yescharacteristic
223C4C24×D26911yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—10764no—
228C2C23×C22⋊D26911yescharacteristic
subgroupG/H
centerC2C23×C22⋊D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
FittingC2×H(5,𝔽2)×C269C2
socleC2×C269C23×D4
hypercenterC2×H(5,𝔽2)D269
hypocenterC269#3585
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771416C2semidirect
3. #31771408C2semidirect
4. #31771403C2semidirect
5. C2×#22575479C2semidirect
6. #31771372C2semidirect
7. C2×H(5,𝔽2)×C269C2semidirect
9. #22575500C22semidirect
10. #22575499C22semidirect
12. C2×#16864402C22semidirect
14. #22575484C22semidirect
15. #22575483C22semidirect
16. #22575479C22semidirect
17. #22575479C22semidirect
18. #22575478C22semidirect
19. #22575473C22semidirect
20. C2×#16864390C22semidirect
21. C2×#16864387C22semidirect
22. #22575460C22semidirect
23. C2×#16864386C22semidirect
25. H(5,𝔽2)×C269C22semidirect
28. C2×#41×C269C22semidirect
30. C22×D4×C269C22semidirect
35. #16864402C23semidirect
37. C2×Dic538C23semidirect
39. C2×D1076C23semidirect
43. #16864392C23semidirect
44. #16864390C23semidirect
45. #16864389C23semidirect
46. #16864387C23semidirect
48. #16864386C23semidirect
49. C2×#14524130C23semidirect
114. C269#3585semidirect
117. C2×H(5,𝔽2)D269semidirect
128. H(5,𝔽2)C2×D269semidirect
133. C2×#41C2×D269semidirect
135. C22×D4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317714162
#317714082
#317714032
C2×#225754792
#317713722
C2×H(5,𝔽2)×C2692
#3585269