Finite Group Databasebeta

← #46820245

Subgroups — #46820245

Order 34432.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#22575498C211yesmaximal, characteristic, complemented
2#31771405C212yesmaximal, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31770785C214yesmaximal, complemented
5#31770775C214yesmaximal, complemented
6#31770621C212yesmaximal, complemented
7C2×#100×C269C211yesmaximal, characteristic, complemented
8#22575498—24no—
9#22575498—22no—
10#22575498C2214yes—
11C2×#16864402C2212yes—
12C2×#16864402C2211yescharacteristic, complemented
13C2×#16864401—22no—
14C22×C22⋊D269—22no—
15C2×D4×D269—22no—
16#22575486—22no—
17#22575486C2214yescomplemented
18#22575481—22no—
19#22575481C2214yescomplemented
20#22575475—24no—
21#22575449C2212yescomplemented
22#22575440C2212yescomplemented
23#22575424—22no—
24#22575422—22no—
25#22575412C2211yescharacteristic, complemented
26#22575379C2218yescomplemented
27C22×D4×C269C2211yescharacteristic
28C2×M16×C269C2212yescomplemented
29#100×C269C2214yescomplemented
30C23×D269—22no—
31#16864402—22no—
32#16864402—22no—
33#16864402—22no—
34#16864402—22no—
35#16864402D414yescomplemented
36#16864402C2314yes—
37#16864401—28no—
38#16864401—28no—
39C2×Dic538D412yescomplemented
40C2×Dic538C2311yescharacteristic, complemented
41C2×D1076D412yescomplemented
42C2×D1076C2311yescharacteristic, complemented
43C2×C22⋊D269—24no—
44C2×C22⋊D269—28no—
45C2×C22⋊D269—22no—
46C2×C22⋊D269—28no—
47C2×C22⋊D269—21no—
48C2×C22⋊D269—24no—
49D4×D269—28no—
50D4×D269—28no—
51C22×Dic269—22no—
52C2×C4×D269—22no—
53C2×C4×D269—21no—
54Dic1076—24no—
55#16864391—24no—
56#16864391—24no—
57D2152—24no—
58C4⋊Dic269—21no—
59C22⋊Dic269—21no—
60C22⋊Dic269—24no—
61C4×Dic269C2312yescomplemented
62C24×C269D412yes—
63C2×D4×C269—22no—
64C2×D4×C269—22no—
65C2×D4×C269D412yes—
66C2×D4×C269C2312yes—
67C22×C4×C269C2311yescharacteristic
68M16×C269—22no—
69M16×C269C2314yescomplemented
70C2×C8×C269—22no—
71C22×D269—22no—
72C22×D269—28no—
73C22×D269—21no—
74C22×D269—84no—
75Dic538—22no—
76Dic538—22no—
77Dic538C2×D414yes—
78D1076—22no—
79D1076—22no—
80D1076C2×D414yes—
81C22⋊D269—216no—
82C22⋊D269—24no—
83C22⋊D269—24no—
84C22⋊D269—41no—
85C22⋊D269—41no—
86C22⋊D269—84no—
87C2×Dic269—28no—
88C2×Dic269—22no—
89C2×Dic269—22no—
90C2×Dic269—21no—
91C4×D269—24no—
92C4×D269—24no—
93C4×D269—42no—
94C23×C269—22no—
95C23×C269—22no—
96C23×C269—82no—
97C23×C269C2×D414yes—
98C23×C269C2×D411yescharacteristic
99D4×C269—24no—
100D4×C269—24no—
101C2×C4×C269C2×D412yes—
102C2×C4×C269C2×D412yes—
103C2×C4×C269C2×D411yescharacteristic
104C2×C4×C269C2411yescharacteristic
105C8×C269—24no—
106C22×C269—24no—
107C22×C269—41no—
108C22×C269—44no—
109C22×C269—44no—
110C22×C269—82no—
111C22×C269#12011yescharacteristic
112C22×C269#12011yescharacteristic
113C22×C269C22×D411yescharacteristic
114C2×D269—24no—
115C2×D269—44no—
116C2×D269—42no—
117C2×D269—41no—
118C2×D269—88no—
119C4×C269#12012yes—
120C4×C269C22×D412yes—
121Dic269—24no—
122Dic269—42no—
123Dic269—41no—
124C2×C269—21no—
125C2×C269—21no—
126C2×C269—21no—
127C2×C269—44no—
128C2×C269C2×#12011yescharacteristic
129D269—44no—
130D269—81no—
131C269#301911yescharacteristic, complemented, Sylow
132#3019—2691nomaximal, Sylow
133C2×H(5,𝔽2)—2691no—
134#575—2692no—
135#535—2691no—
136#455—2694no—
137#453—2694no—
138#421—2692no—
139C2×#100D26911yescharacteristic, complemented
140H(5,𝔽2)—2694no—
141H(5,𝔽2)—5384no—
142H(5,𝔽2)—5382no—
143C2×#41—2692no—
144C2×#41—2691no—
145C2×#41—5382no—
146C22×D4—5382no—
147C22×D4—5382no—
148C22×D4C2×D26911yescharacteristic
149#137—2694no—
150#137—5382no—
151AGL(1,ℤ/8)—2694no—
152AGL(1,ℤ/8)—5382no—
153#135—5384no—
154C2×M16C2×D26912yescomplemented
155#127—2692no—
156#124—2692no—
157#121—5382no—
158#120—5382no—
159#117—2691no—
160#104—2698no—
161#100C2×D26914yescomplemented
162C24—5382no—
163C24D107612yes—
164#41—2694no—
165#41—2694no—
166#41—5388no—
167#41—5388no—
168#41—5382no—
169#41—5382no—
170#41—5382no—
171#41—5382no—
172C2×Q8—2692no—
173C2×Q8—2691no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—2692no—
177C2×D4—2691no—
178C2×D4—5388no—
179C2×D4—5388no—
180C2×D4—5384no—
181C2×D4—5388no—
182C2×D4—5382no—
183C2×D4—5388no—
184C2×D4—5381no—
185C2×D4—5384no—
186C2×D4D107612yes—
187C2×D4C22×D26912yes—
188C22×C4—5382no—
189C22×C4—5382no—
190C22×C4—5381no—
191C22×C4C22×D26911yescharacteristic
192Q16—5384no—
193SD16—5384no—
194SD16—5384no—
195D8—5384no—
196M16—22no—
197M16C22×D26914yescomplemented
198C2×C8—22no—
199C4⋊C4—5381no—
200C22⋊C4—5381no—
201C22⋊C4—5384no—
202C42—2692no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—5388no—
207C23—5382no—
208C23—5381no—
209C23—21524no—
210C23C2×D107614yes—
211C23C2×D107611yescharacteristic
212Q8—2694no—
213Q8—5382no—
214Q8—5382no—
215D4—24no—
216D4—24no—
217D4—2694no—
218D4—53816no—
219D4—5384no—
220D4—5384no—
221D4—5382no—
222D4—5382no—
223D4—10761no—
224D4—10761no—
225D4—21524no—
226C2×C4—5388no—
227C2×C4—5382no—
228C2×C4—5384no—
229C2×C4—5384no—
230C2×C4—5382no—
231C2×C4—5381no—
232C2×C4—10762no—
233C2×C4D4×D26912yes—
234C2×C4D4×D26912yes—
235C2×C4C2×D107611yescharacteristic
236C2×C4C23×D26911yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—5384no—
244C22—10764no—
245C22—10762no—
246C22—10761no—
247C22—21528no—
248C22#2257542311yescharacteristic
249C22#2257542311yescharacteristic
250C22C22×D107611yescharacteristic
251C4—5384no—
252C4—10762no—
253C4—10761no—
254C4#2257542312yes—
255C4C2×D4×D26912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—10764no—
261C2—21521no—
262C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#100×C269C2
socleC2×C269C2×#120
hypercenterC2×#100D269
hypocenterC269#3019
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. #31771405C2semidirect
3. #31771172C2semidirect
4. #31770785C2semidirect
5. #31770775C2semidirect
6. #31770621C2semidirect
7. C2×#100×C269C2semidirect
12. C2×#16864402C22semidirect
17. #22575486C22semidirect
19. #22575481C22semidirect
21. #22575449C22semidirect
22. #22575440C22semidirect
25. #22575412C22semidirect
26. #22575379C22semidirect
28. C2×M16×C269C22semidirect
29. #100×C269C22semidirect
35. #16864402D4semidirect
39. C2×Dic538D4semidirect
40. C2×Dic538C23semidirect
41. C2×D1076D4semidirect
42. C2×D1076C23semidirect
61. C4×Dic269C23semidirect
69. M16×C269C23semidirect
131. C269#3019semidirect
139. C2×#100D269semidirect
154. C2×M16C2×D269semidirect
161. #100C2×D269semidirect
197. M16C22×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
#317714052
#317711722
#317707852
#317707752
#317706212
C2×#100×C2692
#3019269