Finite Group Databasebeta

← #46820154

Subgroups — #46820154

Order 34432.

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

normal subgroups only

This group has 784 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×#22575498C211yesmaximal, characteristic, complemented
2C2×#22575481C211yesmaximal, characteristic, complemented
3C22×#16864387C211yesmaximal, characteristic, complemented
4#31771174C211yesmaximal, characteristic, complemented
5#31770751C214yesmaximal, complemented
6#31770740C214yesmaximal, complemented
7#31770599C211yesmaximal, characteristic, complemented
8#31770573C211yesmaximal, characteristic, complemented
9#418×C269C211yesmaximal, characteristic, complemented
10#22575498—24no—
11#22575498—24no—
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14C2×#16864401—21no—
15C2×#16864401C2212yescomplemented
16C22×D1076C2211yescharacteristic, complemented
17C22×C22⋊D269—22no—
18C2×D4×D269—21no—
19C2×D4×D269C2212yescomplemented
20#22575481—24no—
21C2×#16864391C2212yescomplemented
22C2×D2152C2212yescomplemented
23C2×#16864387—24no—
24C2×#16864387C2214yescomplemented
25C22×#14524130C2211yescharacteristic, complemented
26#22575450—21no—
27#22575441—21no—
28#22575428C2214yescomplemented
29#22575423—22no—
30#22575376C2212yescomplemented
31#22575373C2212yescomplemented
32#22575365C2214yescomplemented
33C22×D4×C269C2211yescharacteristic
34C2×M16×C269C2211yescharacteristic, complemented
35#117×C269C2211yescharacteristic, complemented
36#102×C269C2214yescomplemented
37C23×D269—22no—
38C23×D269—21no—
39#16864402—22no—
40#16864402—22no—
41#16864402—22no—
42#16864402D414yescomplemented
43#16864401—28no—
44#16864401—24no—
45#16864401—24no—
46C2×Dic538D411yescharacteristic
47C2×Dic538C2311yescharacteristic
48C2×D1076—24no—
49C2×D1076—22no—
50C2×D1076—22no—
51C2×D1076—21no—
52C2×D1076D411yescharacteristic
53C2×D1076C2312yes—
54C2×D1076C2311yescharacteristic
55C2×C22⋊D269—216no—
56C2×C22⋊D269—22no—
57C2×C22⋊D269—22no—
58C2×C22⋊D269—21no—
59C2×C22⋊D269D412yes—
60D4×D269—28no—
61D4×D269—24no—
62D4×D269—24no—
63C22×Dic269—22no—
64C2×C4×D269—21no—
65C2×C4×D269C2312yes—
66#16864391—24no—
67D2152—24no—
68#16864387—28no—
69C2×#14524130—22no—
70C2×#14524130C2312yes—
71#16864379—24no—
72C24×C269—21no—
73C2×D4×C269—21no—
74C2×D4×C269—22no—
75C2×D4×C269D414yescomplemented
76C2×D4×C269C2312yes—
77C22×C4×C269C2311yescharacteristic
78M16×C269—22no—
79C2×C8×C269C2312yes—
80C4⋊C4×C269C2312yes—
81C22⋊C4×C269—21no—
82C42×C269—21no—
83C22×D269—21no—
84C22×D269—28no—
85C22×D269—22no—
86C22×D269—42no—
87C22×D269—84no—
88C22×D269—82no—
89C22×D269C2×D412yes—
90Dic538—22no—
91Dic538—21no—
92Dic538C2×D412yes—
93D1076—22no—
94D1076—21no—
95D1076—22no—
96D1076—22no—
97D1076—42no—
98D1076—84no—
99D1076C2×D412yes—
100C22⋊D269—28no—
101C22⋊D269—28no—
102C22⋊D269—24no—
103C22⋊D269—24no—
104C2×Dic269—28no—
105C2×Dic269—21no—
106C2×Dic269C2×D412yes—
107C4×D269—24no—
108C4×D269—22no—
109C4×D269—22no—
110#14524130—22no—
111C23×C269—22no—
112C23×C269—24no—
113C23×C269—82no—
114C23×C269C2×D411yescharacteristic
115D4×C269—24no—
116D4×C269—22no—
117D4×C269C2×D414yes—
118C2×C4×C269—22no—
119C2×C4×C269C2×D411yescharacteristic
120C2×C4×C269C2×D412yes—
121C2×C4×C269C2×D412yes—
122C2×C4×C269C2411yescharacteristic
123C8×C269—22no—
124C22×C269—22no—
125C22×C269—24no—
126C22×C269—44no—
127C22×C269—84no—
128C22×C269#12012yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—22no—
131C2×D269—22no—
132C2×D269—24no—
133C2×D269—42no—
134C2×D269—42no—
135C2×D269—44no—
136C2×D269—88no—
137C2×D269—84no—
138C4×C269—42no—
139C4×C269#12012yes—
140C4×C269C22×D411yescharacteristic
141C4×C269C22×D411yescharacteristic
142Dic269—22no—
143Dic269—22no—
144C2×C269—22no—
145C2×C269—44no—
146C2×C269C2×#12011yescharacteristic
147C2×C269#57612yes—
148D269—42no—
149D269—42no—
150D269—82no—
151C269#300811yescharacteristic, complemented, Sylow
152#3008—2691nomaximal, Sylow
153C2×H(5,𝔽2)—2691no—
154C2×AGL(1,ℤ/8)—2691no—
155C22×D8—2691no—
156#535—2691no—
157#449—2694no—
158#447—2694no—
159#418D26911yescharacteristic, complemented
160#416—2691no—
161#408—2691no—
162H(5,𝔽2)—5384no—
163H(5,𝔽2)—5384no—
164C2×#41—2691no—
165C2×#41—2692no—
166C2×#41—2691no—
167C2×#41—5381no—
168C22×D4—2692no—
169C22×D4—2691no—
170C22×D4—5381no—
171C22×D4—5382no—
172C22×D4C2×D26911yescharacteristic
173AGL(1,ℤ/8)—5384no—
174C2×SD16—2692no—
175C2×D8—2694no—
176C2×D8—2692no—
177C2×D8—5384no—
178C2×M16C2×D26911yescharacteristic, complemented
179C22×C8—2691no—
180#127—5381no—
181#124—5381no—
182#121—2694no—
183#120—5382no—
184#117C2×D26911yescharacteristic, complemented
185#103—2692no—
186#102—2692no—
187#102C2×D26914yescomplemented
188C22⋊C8—2694no—
189C24—21no—
190C24—5382no—
191C24—5381no—
192#41—2694no—
193#41—5384no—
194#41—5388no—
195#41—5384no—
196#41—5382no—
197#41—5382no—
198#41—5382no—
199C2×Q8—2691no—
200C2×Q8—2691no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—2692no—
204C2×D4—2691no—
205C2×D4—2691no—
206C2×D4—2692no—
207C2×D4—5384no—
208C2×D4—5382no—
209C2×D4—5384no—
210C2×D4—5388no—
211C2×D4—5382no—
212C2×D4—53816no—
213C2×D4—5381no—
214C2×D4—5384no—
215C2×D4—5382no—
216C2×D4—5381no—
217C2×D4—5382no—
218C2×D4D107614yescomplemented
219C2×D4C22×D26912yes—
220C22×C4—2692no—
221C22×C4—5382no—
222C22×C4—5381no—
223C22×C4C22×D26911yescharacteristic
224SD16—5384no—
225D8—5384no—
226D8—5388no—
227M16—22no—
228C2×C8—2692no—
229C2×C8—5382no—
230C2×C8C22×D26912yes—
231C4⋊C4C22×D26912yes—
232C22⋊C4—21no—
233C22⋊C4—5384no—
234C42—21no—
235C23—22no—
236C23—24no—
237C23—82no—
238C23—2692no—
239C23—5388no—
240C23—5381no—
241C23—5382no—
242C23—10762no—
243C23—21522no—
244C23—21524no—
245C23D4×D26911yescharacteristic
246Q8—2692no—
247Q8—5382no—
248Q8—5381no—
249D4—24no—
250D4—22no—
251D4—2692no—
252D4—5388no—
253D4—5388no—
254D4—5384no—
255D4—5384no—
256D4—5382no—
257D4—5382no—
258D4—5381no—
259D4—5382no—
260D4—10762no—
261D4—21524no—
262D4C2×D107614yes—
263C2×C4—22no—
264C2×C4—2692no—
265C2×C4—5388no—
266C2×C4—5381no—
267C2×C4—5384no—
268C2×C4—5382no—
269C2×C4—5382no—
270C2×C4D4×D26911yescharacteristic
271C2×C4D4×D26912yes—
272C2×C4C2×D107612yes—
273C2×C4C23×D26911yescharacteristic
274C8—22no—
275C8—5382no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—84no—
280C22—5384no—
281C22—5382no—
282C22—5382no—
283C22—10762no—
284C22—10764no—
285C22—10762no—
286C22—21524no—
287C22—21528no—
288C22#2257542312yes—
289C22C2×D4×D26911yescharacteristic
290C4—42no—
291C4—5382no—
292C4—5382no—
293C4#2257542312yes—
294C4C2×D4×D26911yescharacteristic
295C4C22×D107611yescharacteristic
296C2—22no—
297C2—44no—
298C2—10762no—
299C2—10762no—
300C2—21522no—
301C2C2×#2257542311yescharacteristic
302C2#3177140912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#418×C269C2
socleC22×C269C22×D4
hypercenter#418D269
hypocenterC269#3008
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. C2×#22575481C2semidirect
3. C22×#16864387C2semidirect
4. #31771174C2semidirect
5. #31770751C2semidirect
6. #31770740C2semidirect
7. #31770599C2semidirect
8. #31770573C2semidirect
9. #418×C269C2semidirect
15. C2×#16864401C22semidirect
16. C22×D1076C22semidirect
19. C2×D4×D269C22semidirect
21. C2×#16864391C22semidirect
22. C2×D2152C22semidirect
24. C2×#16864387C22semidirect
25. C22×#14524130C22semidirect
28. #22575428C22semidirect
30. #22575376C22semidirect
31. #22575373C22semidirect
32. #22575365C22semidirect
34. C2×M16×C269C22semidirect
35. #117×C269C22semidirect
36. #102×C269C22semidirect
42. #16864402D4semidirect
75. C2×D4×C269D4semidirect
151. C269#3008semidirect
159. #418D269semidirect
178. C2×M16C2×D269semidirect
184. #117C2×D269semidirect
187. #102C2×D269semidirect
218. C2×D4D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
C2×#225754812
C22×#168643872
#317711742
#317707512
#317707402
#317705992
#317705732
#418×C2692
#3008269