Finite Group Databasebeta

← #46820429

Subgroups — #46820429

Order 34432.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22575479C211yesmaximal, characteristic, complemented
2C2×#22575475C211yesmaximal, characteristic, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31771086C211yesmaximal, characteristic, complemented
5#31770826C212yesmaximal, complemented
6#31770819C212yesmaximal, complemented
7#31770804C212yesmaximal, complemented
8#31770799C212yesmaximal, complemented
9#31770631C211yesmaximal, characteristic, complemented
10#31770604C211yesmaximal, characteristic, complemented
11C2×#102×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14#22575479—24no—
15#22575475—24no—
16C2×Dic1076C2211yescharacteristic, complemented
17C2×#16864391C2211yescharacteristic, complemented
18C2×#16864391C2211yescharacteristic, complemented
19C2×#16864390C2212yes—
20C2×D2152C2211yescharacteristic, complemented
21C2×#16864387C2212yes—
22C2×#16864386C2211yescharacteristic
23#22575449C2212yes—
24#22575440C2212yes—
25#22575424—21no—
26#22575424—21no—
27#22575422—22no—
28#22575418C2212yescomplemented
29#22575415C2212yescomplemented
30C4×C22⋊D269—22no—
31#22575412C2211yescharacteristic
32#22575411—21no—
33C4⋊C4×D269—21no—
34#22575382C2212yescomplemented
35#22575382C2212yescomplemented
36#22575376C2212yescomplemented
37#22575373C2212yescomplemented
38C22×D4×C269C2211yescharacteristic
39C22×C8×C269C2211yescharacteristic
40C2×C4⋊C4×C269C2211yescharacteristic
41#102×C269C2212yes—
42#102×C269C2212yes—
43#16864402—22no—
44#16864402—22no—
45#16864402—22no—
46#16864402D414yescomplemented
47C2×Dic538C2311yescharacteristic
48C2×Dic538C2311yescharacteristic
49C2×D1076C2311yescharacteristic
50C2×D1076C2311yescharacteristic
51C2×C22⋊D269—22no—
52C2×C22⋊D269—22no—
53C2×C22⋊D269—21no—
54C2×C22⋊D269—21no—
55C2×C4×D269—22no—
56C2×C4×D269—21no—
57C2×C4×D269—21no—
58Dic1076—22no—
59#16864391—22no—
60#16864391—22no—
61#16864390—24no—
62D2152—22no—
63#16864387—24no—
64#16864386—22no—
65C2×#14524130C2312yes—
66C4⋊Dic269D411yescharacteristic
67C4⋊Dic269D411yescharacteristic
68#16864381—22no—
69C22⋊Dic269—22no—
70C22⋊Dic269—22no—
71C22⋊Dic269D412yes—
72#16864379—22no—
73C4×Dic269—22no—
74C4×Dic269C2312yes—
75C24×C269—21no—
76C2×D4×C269—22no—
77C2×D4×C269—21no—
78C2×D4×C269—22no—
79C2×D4×C269C2312yes—
80C22×C4×C269—21no—
81C22×C4×C269C2311yescharacteristic
82C2×C8×C269—22no—
83C2×C8×C269C2311yescharacteristic
84C2×C8×C269C2311yescharacteristic
85C4⋊C4×C269—21no—
86C4⋊C4×C269C2312yes—
87C22×D269—21no—
88C22×D269—21no—
89Dic538—21no—
90Dic538—21no—
91Dic538—21no—
92Dic538C2×D412yes—
93D1076—21no—
94D1076—21no—
95D1076—21no—
96D1076C2×D412yes—
97C22⋊D269—24no—
98C22⋊D269—42no—
99C22⋊D269—82no—
100C22⋊D269—82no—
101C2×Dic269—21no—
102C2×Dic269—21no—
103C2×Dic269C2×D412yes—
104C2×Dic269C2×D412yes—
105C4×D269—24no—
106C4×D269—42no—
107C4×D269—82no—
108#14524130—22no—
109C23×C269—22no—
110C23×C269—42no—
111C23×C269—81no—
112C23×C269—81no—
113C23×C269C2×D411yescharacteristic
114D4×C269—22no—
115D4×C269—22no—
116D4×C269—42no—
117C2×C4×C269—22no—
118C2×C4×C269—81no—
119C2×C4×C269C2×D411yescharacteristic
120C2×C4×C269C2×D412yes—
121C2×C4×C269#4112yes—
122C2×C4×C269C2411yescharacteristic
123C8×C269—21no—
124C8×C269—21no—
125C22×C269—21no—
126C22×C269—21no—
127C22×C269—42no—
128C22×C269—42no—
129C22×C269—42no—
130C22×C269—84no—
131C22×C269#12112yes—
132C22×C269C22×D411yescharacteristic
133C2×D269—22no—
134C2×D269—41no—
135C2×D269—41no—
136C2×D269—41no—
137C2×D269—41no—
138C2×D269—41no—
139C4×C269—42no—
140C4×C269#12112yes—
141C4×C269C22×D411yescharacteristic
142C4×C269C2×#4111yescharacteristic
143Dic269—22no—
144Dic269—22no—
145Dic269—22no—
146Dic269—41no—
147C2×C269—22no—
148C2×C269—82no—
149C2×C269C2×#12111yescharacteristic
150C2×C269#57611yescharacteristic
151C2×C269#57711yescharacteristic
152D269—42no—
153D269—81no—
154C269#304511yescharacteristic, complemented, Sylow
155#3045—2691nomaximal, Sylow
156C2×AGL(1,ℤ/8)—2691no—
157C2×#135—2691no—
158#535—2691no—
159#520—2691no—
160#464—2692no—
161#463—2692no—
162#460—2692no—
163#459—2692no—
164#424—2691no—
165#417—2691no—
166C2×#102D26911yescharacteristic, complemented
167C2×#41—2691no—
168C2×#41—2691no—
169C22×D4C2×D26911yescharacteristic
170AGL(1,ℤ/8)—5384no—
171#135—5384no—
172C2×Q16—2691no—
173C2×SD16—2692no—
174C2×SD16—2691no—
175C2×SD16—2691no—
176C2×D8—2692no—
177C2×D8—2691no—
178C2×M16—2691no—
179C22×C8C2×D26911yescharacteristic
180#127—2692no—
181#124—2692no—
182#121—5381no—
183#121—5381no—
184#120—5382no—
185C4×Q8—2692no—
186C4×D4—2692no—
187C4×D4—5382no—
188#117—2691no—
189#117—5381no—
190C2×C4⋊C4—5381no—
191C2×C4⋊C4C2×D26911yescharacteristic
192C4⋊C8—2692no—
193C4⋊C8—2692no—
194#103—2692no—
195#102—2692no—
196#102C2×D26912yes—
197#102C2×D26912yes—
198C24—21no—
199#41—2694no—
200#41—5382no—
201#41—5382no—
202#41—5382no—
203C2×Q8—2691no—
204C2×Q8—2691no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—2691no—
209C2×D4—2691no—
210C2×D4—5382no—
211C2×D4—5382no—
212C2×D4—5381no—
213C2×D4—5381no—
214C2×D4C22×D26912yes—
215C22×C4—21no—
216C22×C4—5382no—
217C22×C4—5381no—
218C22×C4—5381no—
219C22×C4C22×D26911yescharacteristic
220Q16—5382no—
221SD16—5382no—
222SD16—5382no—
223SD16—5384no—
224D8—5382no—
225D8—5384no—
226M16—5382no—
227C2×C8—22no—
228C2×C8—2692no—
229C2×C8C22×D26911yescharacteristic
230C2×C8C22×D26911yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—2691no—
233C4⋊C4—2691no—
234C4⋊C4—5382no—
235C4⋊C4C22×D26912yes—
236C22⋊C4—2692no—
237C22⋊C4—5382no—
238C22⋊C4—5382no—
239C22⋊C4—5382no—
240C42—2692no—
241C42—5382no—
242C23—22no—
243C23—42no—
244C23—81no—
245C23—81no—
246C23—5381no—
247C23—5381no—
248C23D4×D26911yescharacteristic
249Q8—2692no—
250Q8—5381no—
251Q8—5381no—
252Q8—5381no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2692no—
257D4—5384no—
258D4—5381no—
259D4—5381no—
260D4—5381no—
261D4—10762no—
262D4—21522no—
263D4—21522no—
264C2×C4—22no—
265C2×C4—81no—
266C2×C4—2692no—
267C2×C4—2692no—
268C2×C4—5381no—
269C2×C4—5384no—
270C2×C4—5381no—
271C2×C4—10762no—
272C2×C4—21522no—
273C2×C4D4×D26911yescharacteristic
274C2×C4D4×D26912yes—
275C2×C4#1686440212yes—
276C2×C4C23×D26911yescharacteristic
277C8—21no—
278C8—21no—
279C8—5382no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—5382no—
287C22—10761no—
288C22—10761no—
289C22—10761no—
290C22—10761no—
291C22—10761no—
292C22#2257542612yes—
293C22C2×D4×D26911yescharacteristic
294C4—42no—
295C4—5382no—
296C4—5382no—
297C4—5382no—
298C4—10761no—
299C4#2257542612yes—
300C4C2×D4×D26911yescharacteristic
301C4C2×#1686440211yescharacteristic
302C2—22no—
303C2—82no—
304C2—10762no—
305C2—21521no—
306C2C2×#2257542611yescharacteristic
307C2#3177140611yescharacteristic
308C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#102×C269C2
socleC22×C269C22×D4
hypercenterC2×#102D269
hypocenterC269#3045
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#22575479C2semidirect
2. C2×#22575475C2semidirect
3. #31771172C2semidirect
4. #31771086C2semidirect
5. #31770826C2semidirect
6. #31770819C2semidirect
7. #31770804C2semidirect
8. #31770799C2semidirect
9. #31770631C2semidirect
10. #31770604C2semidirect
11. C2×#102×C269C2semidirect
16. C2×Dic1076C22semidirect
17. C2×#16864391C22semidirect
18. C2×#16864391C22semidirect
20. C2×D2152C22semidirect
28. #22575418C22semidirect
29. #22575415C22semidirect
34. #22575382C22semidirect
35. #22575382C22semidirect
36. #22575376C22semidirect
37. #22575373C22semidirect
46. #16864402D4semidirect
154. C269#3045semidirect
166. C2×#102D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754792
C2×#225754752
#317711722
#317710862
#317708262
#317708192
#317708042
#317707992
#317706312
#317706042
C2×#102×C2692
#3045269