Finite Group Databasebeta

← #46822904

Subgroups — #46822904

Order 34432.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771290C211yesmaximal, characteristic, complemented
2#31771239C211yesmaximal, characteristic
3#31770968C211yesmaximal, characteristic, complemented
4#31770932C211yesmaximal, characteristic, complemented
5#31770908C211yesmaximal, characteristic, complemented
6#31770852C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770804C211yesmaximal, characteristic, complemented
9#31770758C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770704C211yesmaximal, characteristic, complemented
12#31770673C211yesmaximal, characteristic, complemented
13#31770608C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#480×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C2×D4×D269—21no—
18C2×D4×D269C2211yescharacteristic
19C2×#16864391C2211yescharacteristic
20C2×#16864391C2211yescharacteristic, complemented
21C2×#16864390C2211yescharacteristic, complemented
22C2×#16864385C2211yescharacteristic, complemented
23C22×#14524130C2211yescharacteristic, complemented
24#22575449C2211yescharacteristic
25#22575448—21no—
26#22575444C2211yescharacteristic, complemented
27#22575443—21no—
28#22575436—21no—
29#22575433C2211yescharacteristic, complemented
30#22575432C2211yescharacteristic, complemented
31#22575428C2211yescharacteristic
32#22575426—21no—
33#22575423—21no—
34#22575421—21no—
35#22575418C2211yescharacteristic, complemented
36#22575415C2211yescharacteristic
37C4×C22⋊D269C2211yescharacteristic
38#22575412—21no—
39#22575411C2211yescharacteristic, complemented
40#22575384C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575376C2211yescharacteristic, complemented
43#22575375C2211yescharacteristic
44#22575375C2211yescharacteristic, complemented
45#22575374C2211yescharacteristic, complemented
46#22575373C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic, complemented
49#22575373C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51#22575363C2211yescharacteristic, complemented
52#121×C269C2211yescharacteristic
53C2×C4⋊C4×C269C2211yescharacteristic, complemented
54#107×C269C2211yescharacteristic, complemented
55#107×C269C2211yescharacteristic, complemented
56#102×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C23×D269—21no—
61C2×Dic538C2311yescharacteristic
62C2×D1076—22no—
63C2×D1076—21no—
64C2×D1076—21no—
65C2×D1076—41no—
66C2×D1076C2311yescharacteristic
67C2×D1076C2311yescharacteristic
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269D411yescharacteristic
72D4×D269—22no—
73D4×D269—41no—
74D4×D269—44no—
75C2×C4×D269—21no—
76C2×C4×D269C2311yescharacteristic
77#16864391—22no—
78#16864391—41no—
79#16864390—41no—
80#16864385—41no—
81C2×#14524130—41no—
82C2×#14524130C2311yescharacteristic
83C2×#14524130C2311yescharacteristic
84C4⋊Dic269—21no—
85C4⋊Dic269—21no—
86C4⋊Dic269C2311yescharacteristic
87#16864381—21no—
88#16864381—21no—
89#16864381—21no—
90#16864381D411yescharacteristic
91C22⋊Dic269—21no—
92C22⋊Dic269D411yescharacteristic
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379D411yescharacteristic
98C4×Dic269—21no—
99C4×Dic269—21no—
100C4×Dic269C2311yescharacteristic
101C2×D4×C269—21no—
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269—21no—
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22⋊C4×C269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—42no—
118C22×D269—81no—
119C22×D269—81no—
120C22×D269C2×D411yescharacteristic
121Dic538—21no—
122Dic538—21no—
123D1076—21no—
124D1076—21no—
125D1076—21no—
126D1076—42no—
127D1076—42no—
128D1076#4112yes—
129C22⋊D269—22no—
130C22⋊D269—42no—
131C22⋊D269—42no—
132C22⋊D269—81no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269C2×D411yescharacteristic
136C2×Dic269C2×D411yescharacteristic
137C2×Dic269C2×D411yescharacteristic
138C4×D269—21no—
139C4×D269—21no—
140C4×D269—42no—
141#14524130—21no—
142#14524130—21no—
143C23×C269—21no—
144C23×C269C2×D411yescharacteristic
145D4×C269—21no—
146D4×C269—21no—
147D4×C269—42no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—21no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—42no—
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2411yescharacteristic
156C8×C269—21no—
157C8×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269AGL(1,ℤ/8)12yes—
164C22×C269C22×D411yescharacteristic
165C2×D269—21no—
166C2×D269—22no—
167C2×D269—21no—
168C2×D269—21no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—84no—
175C2×D269—84no—
176C4×C269—21no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269—41no—
180C4×C269C2×#4111yescharacteristic
181C4×C269H(5,𝔽2)11yescharacteristic
182Dic269—21no—
183Dic269—21no—
184Dic269—21no—
185Dic269—41no—
186Dic269—41no—
187C2×C269—22no—
188C2×C269—81no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—41no—
193D269—42no—
194D269—81no—
195C269#331011yescharacteristic, complemented, Sylow
196#3310—2691nomaximal, Sylow
197#552—2691no—
198D42—2691no—
199#490—2691no—
200#482—2691no—
201#480D26911yescharacteristic, complemented
202#478—2691no—
203#468—2691no—
204#465—2691no—
205#460—2691no—
206#450—2691no—
207#450—2691no—
208#440—2691no—
209#435—2691no—
210#418—2691no—
211C2×#102—2691no—
212C22×D4—2691no—
213C22×D4—2691no—
214C22×D4—5381no—
215C2×SD16—2691no—
216C2×SD16—2691no—
217C2×SD16—2691no—
218C2×M16—2691no—
219C22×C8—2691no—
220#127—2691no—
221#126—5381no—
222#125—2691no—
223#125—5381no—
224#123—5381no—
225C22⋊Q8—2691no—
226C22⋊Q8—2691no—
227#121—2691no—
228#121—5381no—
229#121C2×D26911yescharacteristic
230#120—5381no—
231#120—5381no—
232C4×Q8—2691no—
233C4×D4—2691no—
234C4×D4—2691no—
235#117—2691no—
236#117—5381no—
237C2×C4⋊C4C2×D26911yescharacteristic, complemented
238#107C2×D26911yescharacteristic, complemented
239#107C2×D26911yescharacteristic, complemented
240#106—2691no—
241C4⋊C8—2691no—
242#103—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102—2691no—
249#102—2691no—
250#102C2×D26911yescharacteristic, complemented
251#102C2×D26911yescharacteristic, complemented
252C22⋊C8—2691no—
253C22⋊C8C2×D26911yescharacteristic, complemented
254#97—2691no—
255C24—5381no—
256C24—5381no—
257C2×Q8—2691no—
258C2×D4—21no—
259C2×D4—2691no—
260C2×D4—2691no—
261C2×D4—2691no—
262C2×D4—5382no—
263C2×D4—5381no—
264C2×D4—5381no—
265C2×D4—5381no—
266C2×D4—5381no—
267C2×D4—5382no—
268C2×D4—5381no—
269C2×D4—10761no—
270C2×D4—10761no—
271C2×D4—10764no—
272C2×D4C22×D26911yescharacteristic
273C22×C4—21no—
274C22×C4—2691no—
275C22×C4—5381no—
276C22×C4C22×D26911yescharacteristic
277SD16—5382no—
278SD16—10761no—
279SD16—10761no—
280M16—10761no—
281C2×C8—2691no—
282C2×C8—2691no—
283C2×C8—10761no—
284C2×C8C22×D26911yescharacteristic
285C2×C8C22×D26911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2691no—
288C4⋊C4—2691no—
289C4⋊C4—5381no—
290C4⋊C4—5381no—
291C4⋊C4—5381no—
292C4⋊C4—5381no—
293C4⋊C4—5381no—
294C4⋊C4C22×D26911yescharacteristic
295C4⋊C4C22×D26911yescharacteristic
296C4⋊C4C22×D26911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—2691no—
299C22⋊C4—2691no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C42—2691no—
306C42—5381no—
307C42—5381no—
308C23—21no—
309C23—2691no—
310C23—5381no—
311C23—5381no—
312C23—10762no—
313C23—10762no—
314C23—10762no—
315C23—10762no—
316C23—21521no—
317C23—21521no—
318C23D4×D26911yescharacteristic
319Q8—5381no—
320Q8—5381no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2692no—
325D4—5382no—
326D4—5381no—
327D4—5381no—
328D4—5381no—
329D4—10762no—
330D4—10762no—
331D4—10762no—
332D4—10762no—
333D4—21521no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—2691no—
341C2×C4—2691no—
342C2×C4—2691no—
343C2×C4—5381no—
344C2×C4—5381no—
345C2×C4—5381no—
346C2×C4—5381no—
347C2×C4—10762no—
348C2×C4D4×D26911yescharacteristic
349C2×C4C23×D26911yescharacteristic
350C8—21no—
351C8—21no—
352C8—5381no—
353C8—5381no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—5381no—
360C22—5381no—
361C22—5382no—
362C22—5381no—
363C22—10761no—
364C22—10761no—
365C22—10761no—
366C22—10761no—
367C22—10761no—
368C22—21524no—
369C22—21524no—
370C22#2257547712yes—
371C22C2×D4×D26911yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—5381no—
377C4—5381no—
378C4—5381no—
379C4—10761no—
380C4—10761no—
381C4#41×D26911yescharacteristic
382C4#2257549911yescharacteristic
383C2—22no—
384C2—81no—
385C2—10761no—
386C2—10762no—
387C2—21521no—
388C2#3177124711yescharacteristic
389C2C2×#2257547711yescharacteristic
390C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#480×C269C2
socleC22×C269C22×D4
prime radical#31771239C2
hypercenter#480D269
hypocenterC269#3310
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #31771290C2semidirect
3. #31770968C2semidirect
4. #31770932C2semidirect
5. #31770908C2semidirect
6. #31770852C2semidirect
7. #31770832C2semidirect
8. #31770804C2semidirect
9. #31770758C2semidirect
10. #31770757C2semidirect
11. #31770704C2semidirect
12. #31770673C2semidirect
13. #31770608C2semidirect
14. C2×#22575373C2semidirect
15. #480×C269C2semidirect
20. C2×#16864391C22semidirect
21. C2×#16864390C22semidirect
22. C2×#16864385C22semidirect
23. C22×#14524130C22semidirect
26. #22575444C22semidirect
29. #22575433C22semidirect
30. #22575432C22semidirect
35. #22575418C22semidirect
39. #22575411C22semidirect
40. #22575384C22semidirect
41. #22575382C22semidirect
42. #22575376C22semidirect
44. #22575375C22semidirect
45. #22575374C22semidirect
48. #22575373C22semidirect
49. #22575373C22semidirect
50. #22575365C22semidirect
51. #22575363C22semidirect
53. C2×C4⋊C4×C269C22semidirect
54. #107×C269C22semidirect
55. #107×C269C22semidirect
56. #102×C269C22semidirect
57. #102×C269C22semidirect
58. C22⋊C8×C269C22semidirect
195. C269#3310semidirect
201. #480D269semidirect
237. C2×C4⋊C4C2×D269semidirect
238. #107C2×D269semidirect
239. #107C2×D269semidirect
250. #102C2×D269semidirect
251. #102C2×D269semidirect
253. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712902
#317712392
#317709682
#317709322
#317709082
#317708522
#317708322
#317708042
#317707582
#317707572
#317707042
#317706732
#317706082
C2×#225753732
#480×C2692
#3310269