Finite Group Databasebeta

← #46823611

Subgroups — #46823611

Order 34432.

382 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771332C211yesmaximal, characteristic, complemented
2#31771275C211yesmaximal, characteristic
3#31770968C211yesmaximal, characteristic, complemented
4#31770950C211yesmaximal, characteristic, complemented
5#31770903C211yesmaximal, characteristic, complemented
6#31770902C211yesmaximal, characteristic, complemented
7#31770883C211yesmaximal, characteristic, complemented
8#31770804C211yesmaximal, characteristic, complemented
9#31770804C211yesmaximal, characteristic, complemented
10#31770802C211yesmaximal, characteristic, complemented
11#31770733C211yesmaximal, characteristic, complemented
12#31770707C211yesmaximal, characteristic, complemented
13#31770702C211yesmaximal, characteristic, complemented
14#31770696C211yesmaximal, characteristic, complemented
15#490×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×#16864391C2211yescharacteristic
18C2×#16864390C2211yescharacteristic
19C2×#16864390C2211yescharacteristic
20#22575452—21no—
21#22575452C2211yescharacteristic
22#22575449C2211yescharacteristic
23#22575449C2211yescharacteristic
24#22575444C2211yescharacteristic
25#22575443—21no—
26#22575443—21no—
27#22575425—21no—
28#22575425—21no—
29#22575424—21no—
30C4×Dic538C2211yescharacteristic
31C4×Dic538C2211yescharacteristic
32#22575418—21no—
33#22575418C2211yescharacteristic
34#22575418C2211yescharacteristic
35C4×D1076C2211yescharacteristic
36D4×Dic269C2211yescharacteristic
37D4×Dic269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575386C2211yescharacteristic
40#22575384C2211yescharacteristic
41C4⋊#14524130C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575382C2211yescharacteristic
44#22575378C2211yescharacteristic
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575363C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52#127×C269C2211yescharacteristic
53#125×C269C2211yescharacteristic
54#102×C269C2211yescharacteristic
55#102×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#97×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×Dic538D411yescharacteristic
62C2×Dic538C2311yescharacteristic
63C2×D1076C2311yescharacteristic
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68C22×Dic269—21no—
69C22×Dic269—21no—
70C2×C4×D269—21no—
71#16864391—22no—
72#16864390—22no—
73#16864390—22no—
74C2×#14524130C2311yescharacteristic
75C2×#14524130C2311yescharacteristic
76C4⋊Dic269—21no—
77C4⋊Dic269D411yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85#16864381D411yescharacteristic
86#16864381D411yescharacteristic
87C22⋊Dic269—21no—
88C22⋊Dic269—21no—
89#16864379—21no—
90C4×Dic269—21no—
91C4×Dic269—21no—
92C4×Dic269—21no—
93C4×Dic269C2311yescharacteristic
94C4×Dic269C2311yescharacteristic
95C2×D4×C269—21no—
96C2×D4×C269—21no—
97C2×D4×C269C2311yescharacteristic
98C2×D4×C269C2311yescharacteristic
99C2×C8×C269C2311yescharacteristic
100C2×C8×C269C2311yescharacteristic
101C4⋊C4×C269—21no—
102C4⋊C4×C269—21no—
103C4⋊C4×C269C2311yescharacteristic
104C4⋊C4×C269C2311yescharacteristic
105C42×C269C2311yescharacteristic
106C22×D269—21no—
107Dic538—21no—
108Dic538—22no—
109Dic538—42no—
110Dic538#4112yescomplemented
111D1076—21no—
112D1076—21no—
113C22⋊D269—42no—
114C22⋊D269—42no—
115C22⋊D269—81no—
116C22⋊D269—81no—
117C2×Dic269—21no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—42no—
121C2×Dic269—42no—
122C2×Dic269C2×D411yescharacteristic
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C4×D269—42no—
127#14524130—21no—
128#14524130—21no—
129C23×C269—21no—
130C23×C269—21no—
131D4×C269—21no—
132D4×C269—21no—
133D4×C269—21no—
134D4×C269—21no—
135D4×C269—42no—
136D4×C269—81no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269C2×D411yescharacteristic
140C2×C4×C269C2×D411yescharacteristic
141C2×C4×C269C2411yescharacteristic
142C8×C269—21no—
143C8×C269—21no—
144C22×C269—41no—
145C22×C269—41no—
146C22×C269—41no—
147C22×C269—41no—
148C22×C269—41no—
149C22×C269—41no—
150C22×C269C22×D411yescharacteristic
151C2×D269—41no—
152C2×D269—41no—
153C2×D269—41no—
154C4×C269—21no—
155C4×C269—21no—
156C4×C269—41no—
157C4×C269—41no—
158C4×C269C2×#4111yescharacteristic
159C4×C2692−1+411yescharacteristic
160Dic269—22no—
161Dic269—21no—
162Dic269—21no—
163Dic269—21no—
164Dic269—41no—
165Dic269—41no—
166Dic269AGL(1,ℤ/8)12yes—
167C2×C269—81no—
168C2×C269—81no—
169C2×C269#54811yescharacteristic
170C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
171C2×C269#57711yescharacteristic
172D269—81no—
173C269#336811yescharacteristic, complemented, Sylow
174#3368—2691nomaximal, Sylow
175#557—2691no—
176#550—2691no—
177#490—2691no—
178#490D26911yescharacteristic, complemented
179#486—2691no—
180#478—2691no—
181#478—2691no—
182#474—2691no—
183#460—2691no—
184#460—2691no—
185#460—2691no—
186#446—2691no—
187#440—2691no—
188#440—2691no—
189C4×SD16—2691no—
190C2×#41—5381no—
191C2×SD16—2691no—
192C2×SD16—2691no—
193C2×SD16—2691no—
194C4⋊Q8—2691no—
195C4⋊Q8—5381no—
196#127—2691no—
197#127—2691no—
198#127C2×D26911yescharacteristic
199#125—2691no—
200#125—5381no—
201#125—5381no—
202#125C2×D26911yescharacteristic
203#121—5381no—
204#121—5381no—
205#121—5381no—
206C4×Q8—2691no—
207C4×Q8—2691no—
208C4×Q8—2691no—
209C4×Q8—2691no—
210C4×Q8—5381no—
211C4×D4—2691no—
212C4×D4—2691no—
213C4×D4—2691no—
214#107—2691no—
215#107—2691no—
216#106—2691no—
217C4⋊C8—2691no—
218C4⋊C8—2691no—
219C4⋊C8—2691no—
220#103—2691no—
221#102—2691no—
222#102—2691no—
223#102—2691no—
224#102—2691no—
225#102—2691no—
226#102C2×D26911yescharacteristic
227#102C2×D26911yescharacteristic
228#102C2×D26911yescharacteristic
229#102C2×D26911yescharacteristic
230#97—2691no—
231#97C2×D26911yescharacteristic
232C4×C8—2691no—
233#41—10764no—
234C2×Q8—2691no—
235C2×Q8—2691no—
236C2×Q8—5381no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—2691no—
240C2×D4—5381no—
241C2×D4—5381no—
242C2×D4—5381no—
243C2×D4—5381no—
244C2×D4C22×D26911yescharacteristic
245C2×D4C22×D26911yescharacteristic
246C22×C4—5381no—
247C22×C4—5381no—
248C22×C4—5381no—
249SD16—5382no—
250SD16—5382no—
251SD16—5382no—
252C2×C8—2691no—
253C2×C8—2691no—
254C2×C8C22×D26911yescharacteristic
255C2×C8C22×D26911yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—2691no—
259C4⋊C4—2691no—
260C4⋊C4—2691no—
261C4⋊C4—2691no—
262C4⋊C4—2691no—
263C4⋊C4—5381no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4—5381no—
269C4⋊C4C22×D26911yescharacteristic
270C4⋊C4C22×D26911yescharacteristic
271C22⋊C4—5381no—
272C22⋊C4—5381no—
273C22⋊C4—5381no—
274C42—2691no—
275C42—2691no—
276C42—5381no—
277C42—5381no—
278C42—5381no—
279C42C22×D26911yescharacteristic
280C23—21no—
281C23—21no—
282C23—5381no—
283Q8—2692no—
284Q8—5382no—
285Q8—5381no—
286Q8—10762no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—81no—
293D4—5381no—
294D4—5381no—
295D4—10762no—
296D4—10762no—
297D4—21521no—
298D4—21521no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2691no—
302C2×C4—2691no—
303C2×C4—2691no—
304C2×C4—2691no—
305C2×C4—5381no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—10762no—
309C2×C4—10762no—
310C2×C4—10762no—
311C2×C4D4×D26911yescharacteristic
312C2×C4D4×D26911yescharacteristic
313C2×C4C23×D26911yescharacteristic
314C8—21no—
315C8—21no—
316C8—5381no—
317C8—5381no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—10761no—
325C22—10761no—
326C22—10761no—
327C22C2×D4×D26911yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—2692no—
333C4—5381no—
334C4—5381no—
335C4—5382no—
336C4—5381no—
337C4—10761no—
338C4—10761no—
339C4#41×D26911yescharacteristic
340C4#2257550011yescharacteristic
341C2—81no—
342C2—81no—
343C2—21521no—
344C2#3177126311yescharacteristic
345C2AGL(1,ℤ/8)×D26911yescharacteristic
346C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#490×C269C2
socleC22×C269C22×D4
prime radical#31771275C2
hypercenter#490D269
hypocenterC269#3368
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #31771332C2semidirect
3. #31770968C2semidirect
4. #31770950C2semidirect
5. #31770903C2semidirect
6. #31770902C2semidirect
7. #31770883C2semidirect
8. #31770804C2semidirect
9. #31770804C2semidirect
10. #31770802C2semidirect
11. #31770733C2semidirect
12. #31770707C2semidirect
13. #31770702C2semidirect
14. #31770696C2semidirect
15. #490×C269C2semidirect
110. Dic538#41semidirect
173. C269#3368semidirect
178. #490D269semidirect

Maximal subgroups

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

subgroupindex
#317713322
#317712752
#317709682
#317709502
#317709032
#317709022
#317708832
#317708042
#317708022
#317707332
#317707072
#317707022
#317706962
#490×C2692
#3368269