Finite Group Databasebeta

← #46823495

Subgroups — #46823495

Order 34432.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771326C211yesmaximal, characteristic
2#31771313C211yesmaximal, characteristic
3#31771002C211yesmaximal, characteristic, complemented
4#31770945C211yesmaximal, characteristic, complemented
5#31770943C211yesmaximal, characteristic, complemented
6#31770934C211yesmaximal, characteristic, complemented
7#31770933C211yesmaximal, characteristic, complemented
8#31770898C211yesmaximal, characteristic, complemented
9#31770888C211yesmaximal, characteristic, complemented
10#31770860C211yesmaximal, characteristic, complemented
11Q16×Dic269C211yesmaximal, characteristic, complemented
12#31770681C211yesmaximal, characteristic
13#31770647C211yesmaximal, characteristic, complemented
14#31770644C211yesmaximal, characteristic, complemented
15#470×C269C211yesmaximal, characteristic, complemented
16C2×#16864385C2211yescharacteristic
17#22575452C2211yescharacteristic
18#22575452C2211yescharacteristic
19#22575448—21no—
20#22575446—21no—
21#22575443—21no—
22#22575439—21no—
23#22575433—21no—
24#22575430—21no—
25#22575429C2211yescharacteristic
26#22575429C2211yescharacteristic
27Q8×Dic269C2211yescharacteristic
28Q8×Dic269C2211yescharacteristic
29#22575418—21no—
30C4×C22⋊D269C2211yescharacteristic
31#22575413—21no—
32#22575412C2211yescharacteristic
33#22575411C2211yescharacteristic
34C2×C4⋊Dic269C2211yescharacteristic
35C4⋊C4×D269C2211yescharacteristic
36#22575387C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575385C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575384C2211yescharacteristic
43#22575384C2211yescharacteristic
44#22575382C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575377C2211yescharacteristic
47#22575377C2211yescharacteristic
48#22575377C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×Q16×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic
54C22⋊Q8×C269C2211yescharacteristic
55C22⋊Q8×C269C2211yescharacteristic
56#106×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#103×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538—21no—
61C2×C22⋊D269D411yescharacteristic
62C22×Dic269—21no—
63C2×C4×D269—21no—
64C2×C4×D269C2311yescharacteristic
65#16864385—22no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C4⋊Dic269—21no—
69C4⋊Dic269—21no—
70C4⋊Dic269—21no—
71C4⋊Dic269C2311yescharacteristic
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74C4⋊Dic269C2311yescharacteristic
75#16864381—21no—
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381D411yescharacteristic
81C22⋊Dic269—21no—
82C22⋊Dic269D411yescharacteristic
83#16864379—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379D411yescharacteristic
87C4×Dic269—21no—
88C4×Dic269—21no—
89C4×Dic269—21no—
90C4×Dic269—21no—
91C4×Dic269C2311yescharacteristic
92C2×Q8×C269C2311yescharacteristic
93C2×Q8×C269C2311yescharacteristic
94C22×C4×C269C2311yescharacteristic
95Q16×C269—22no—
96M16×C269—22no—
97C2×C8×C269C2311yescharacteristic
98C2×C8×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
100C4⋊C4×C269—21no—
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22×D269C2×D411yescharacteristic
106Dic538—42no—
107Dic538—81no—
108C22⋊D269—22no—
109C2×Dic269—21no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—81no—
114C2×Dic269C2×D411yescharacteristic
115C2×Dic269C2×D411yescharacteristic
116C2×Dic269C2×D411yescharacteristic
117C4×D269—21no—
118C4×D269—21no—
119C4×D269—81no—
120#14524130—21no—
121#14524130—21no—
122C23×C269C2×D411yescharacteristic
123Q8×C269—21no—
124Q8×C269—21no—
125Q8×C269—21no—
126Q8×C269—21no—
127C2×C4×C269—21no—
128C2×C4×C269—21no—
129C2×C4×C269—21no—
130C2×C4×C269—21no—
131C2×C4×C269—21no—
132C2×C4×C269—21no—
133C2×C4×C269C2×D411yescharacteristic
134C2×C4×C269C2411yescharacteristic
135C8×C269—21no—
136C8×C269#4112yes—
137C22×C269—21no—
138C22×C269—21no—
139C22×C269—21no—
140C22×C269C22×D411yescharacteristic
141C2×D269—21no—
142C2×D269—21no—
143C2×D269—21no—
144C4×C269—21no—
145C4×C269—41no—
146C4×C269—41no—
147C4×C269—41no—
148C4×C269—41no—
149C4×C269C2×#4111yescharacteristic
150C4×C2692−1+411yescharacteristic
151Dic269—21no—
152Dic269—21no—
153Dic269—41no—
154Dic269—41no—
155Dic269—41no—
156Dic269—41no—
157Dic269#13712yes—
158C2×C269—41no—
159C2×C269#54711yescharacteristic
160C2×C269C2×#13711yescharacteristic
161C2×C269#57711yescharacteristic
162D269—41no—
163C269#335611yescharacteristic, complemented, Sylow
164#3356—2691nomaximal, Sylow
165#556—2691no—
166#554—2691no—
167#498—2691no—
168#485—2691no—
169#485—2691no—
170#483—2691no—
171#483—2691no—
172#477—2691no—
173#475—2691no—
174#470—2691no—
175#470D26911yescharacteristic, complemented
176C4×Q16—2691no—
177#436—2691no—
178#428—2691no—
179#428—2691no—
180C2×Q16C2×D26911yescharacteristic
181C2×M16—2691no—
182C2×M16C2×D26911yescharacteristic
183C4⋊Q8—2691no—
184C4⋊Q8—2691no—
185#126—5381no—
186#126—5381no—
187#125—5381no—
188#124—5381no—
189C22⋊Q8—2691no—
190C22⋊Q8—2691no—
191C22⋊Q8—5381no—
192C22⋊Q8—5381no—
193C22⋊Q8C2×D26911yescharacteristic
194C22⋊Q8C2×D26911yescharacteristic
195C4×Q8—2691no—
196C4×Q8—2691no—
197C4×Q8—5381no—
198C4×D4—2691no—
199C4×D4—5381no—
200#117—2691no—
201#117—2691no—
202C2×C4⋊C4—2691no—
203C2×C4⋊C4—2691no—
204#107—2691no—
205#107—2691no—
206#107—2691no—
207#106—2691no—
208#106—2691no—
209#106—2691no—
210#106—2691no—
211#106—2691no—
212#106C2×D26911yescharacteristic
213C4⋊C8—2691no—
214#103—2691no—
215#103—2691no—
216#103—2691no—
217#103—2691no—
218#103C2×D26911yescharacteristic
219#103C2×D26911yescharacteristic
220C22⋊C8—2691no—
221C22⋊C8—2691no—
222C4×C8—2691no—
223C2×Q8—5381no—
224C2×Q8—5381no—
225C2×Q8C22×D26911yescharacteristic
226C2×Q8C22×D26911yescharacteristic
227C2×D4—2691no—
228C22×C4—2691no—
229C22×C4—5381no—
230C22×C4—5381no—
231C22×C4C22×D26911yescharacteristic
232Q16—22no—
233M16—22no—
234M16—5382no—
235C2×C8—2691no—
236C2×C8—2691no—
237C2×C8C22×D26911yescharacteristic
238C2×C8C22×D26911yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—2691no—
242C4⋊C4—2691no—
243C4⋊C4—2691no—
244C4⋊C4—2691no—
245C4⋊C4—2691no—
246C4⋊C4—5381no—
247C4⋊C4—5381no—
248C4⋊C4—5381no—
249C4⋊C4—5381no—
250C4⋊C4—5381no—
251C4⋊C4—5381no—
252C4⋊C4—5381no—
253C4⋊C4—5381no—
254C4⋊C4C22×D26911yescharacteristic
255C4⋊C4C22×D26911yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—2691no—
259C22⋊C4—2691no—
260C22⋊C4—5381no—
261C22⋊C4—5381no—
262C22⋊C4—5381no—
263C22⋊C4—5381no—
264C42—2691no—
265C42—5381no—
266C42—5381no—
267C42—5381no—
268C42—5381no—
269C23—2691no—
270C23D4×D26911yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—10762no—
276Q8—21521no—
277D4—5382no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—2691no—
285C2×C4—2691no—
286C2×C4—2691no—
287C2×C4—5381no—
288C2×C4—5381no—
289C2×C4—5381no—
290C2×C4—5381no—
291C2×C4—5381no—
292C2×C4—5381no—
293C2×C4—21521no—
294C2×C4—21521no—
295C2×C4D4×D26911yescharacteristic
296C2×C4C23×D26911yescharacteristic
297C8—21no—
298C8—5381no—
299C8—5381no—
300C8#1686440112yes—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—5381no—
305C22—5381no—
306C22—5381no—
307C22C2×D4×D26911yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—2692no—
314C4—5381no—
315C4—5381no—
316C4—10761no—
317C4—10761no—
318C4—10761no—
319C4—10761no—
320C4C2×#1686440111yescharacteristic
321C4#2257550011yescharacteristic
322C2—41no—
323C2—10761no—
324C2#3177125511yescharacteristic
325C2#137×D26911yescharacteristic
326C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#470×C269C2
socleC22×C269C22×D4
prime radicalC2×C22⋊D269D4
hypercenter#470D269
hypocenterC269#3356
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #31771002C2semidirect
4. #31770945C2semidirect
5. #31770943C2semidirect
6. #31770934C2semidirect
7. #31770933C2semidirect
8. #31770898C2semidirect
9. #31770888C2semidirect
10. #31770860C2semidirect
11. Q16×Dic269C2semidirect
13. #31770647C2semidirect
14. #31770644C2semidirect
15. #470×C269C2semidirect
163. C269#3356semidirect
175. #470D269semidirect

Maximal subgroups

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

subgroupindex
#317713262
#317713132
#317710022
#317709452
#317709432
#317709342
#317709332
#317708982
#317708882
#317708602
Q16×Dic2692
#317706812
#317706472
#317706442
#470×C2692
#3356269