Finite Group Databasebeta

← #46822362

Subgroups — #46822362

Order 34432.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575480C211yesmaximal, characteristic, complemented
2C22×D2152C211yesmaximal, characteristic, complemented
3#31771246C211yesmaximal, characteristic, complemented
4#31771239C211yesmaximal, characteristic, complemented
5#31770991C211yesmaximal, characteristic, complemented
6#31770859C211yesmaximal, characteristic, complemented
7#31770823C211yesmaximal, characteristic, complemented
8#31770799C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770753C211yesmaximal, characteristic, complemented
11#31770740C211yesmaximal, characteristic, complemented
12#31770738C211yesmaximal, characteristic, complemented
13#31770716C211yesmaximal, characteristic, complemented
14#31770676C211yesmaximal, characteristic, complemented
15#465×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic, complemented
17C22×D1076C2211yescharacteristic
18C22×D1076C2211yescharacteristic, complemented
19C2×D4×D269—21no—
20C2×D4×D269C2211yescharacteristic, complemented
21#22575480—24no—
22C2×#16864389C2211yescharacteristic, complemented
23C2×#16864389C2211yescharacteristic, complemented
24C2×D2152—24no—
25C2×D2152C2211yescharacteristic
26C2×D2152C2211yescharacteristic
27C2×D2152C2211yescharacteristic, complemented
28C2×D2152C2211yescharacteristic, complemented
29C2×#16864387C2211yescharacteristic, complemented
30C2×#16864387C2211yescharacteristic, complemented
31C2×#16864385C2211yescharacteristic, complemented
32#22575449C2211yescharacteristic, complemented
33#22575442C2211yescharacteristic, complemented
34#22575438—21no—
35#22575428—21no—
36#22575428C2211yescharacteristic, complemented
37#22575428C2211yescharacteristic, complemented
38#22575426—21no—
39#22575423—21no—
40#22575423—21no—
41#22575421—21no—
42#22575415C2211yescharacteristic, complemented
43#22575415C2211yescharacteristic, complemented
44C4×C22⋊D269C2211yescharacteristic
45C22⋊C4×D269—21no—
46#22575382C2211yescharacteristic, complemented
47#22575373C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic, complemented
49#22575373C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic, complemented
51C22⋊#14524130C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic, complemented
55C22×C8×C269C2211yescharacteristic, complemented
56C22⋊Q8×C269C2211yescharacteristic, complemented
57#121×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63C23×D269—21no—
64#16864403—22no—
65#16864403—41no—
66C2×D1076—22no—
67C2×D1076—22no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076—21no—
72C2×D1076—21no—
73C2×D1076—41no—
74C2×D1076—41no—
75C2×D1076C2311yescharacteristic
76C2×D1076C2311yescharacteristic
77C2×D1076C2311yescharacteristic
78C2×D1076C2311yescharacteristic
79C2×C22⋊D269—21no—
80C2×C22⋊D269—21no—
81C2×C22⋊D269—21no—
82C2×C22⋊D269D411yescharacteristic
83D4×D269—22no—
84D4×D269—41no—
85D4×D269—44no—
86C2×C4×D269—21no—
87C2×C4×D269—21no—
88C2×C4×D269—21no—
89C2×C4×D269C2311yescharacteristic
90#16864389—22no—
91#16864389—41no—
92D2152—22no—
93D2152—22no—
94D2152—22no—
95D2152D414yescomplemented
96#16864387—22no—
97#16864387—41no—
98#16864385—22no—
99C2×#14524130C2311yescharacteristic
100C2×#14524130C2311yescharacteristic
101#16864381D411yescharacteristic
102C22⋊Dic269D411yescharacteristic
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379—21no—
109#16864379D411yescharacteristic
110C4×Dic269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C2×D4×C269—21no—
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269C2311yescharacteristic
115SD16×C269—22no—
116C2×C8×C269—22no—
117C2×C8×C269C2311yescharacteristic
118C2×C8×C269C2311yescharacteristic
119C4⋊C4×C269—21no—
120C4⋊C4×C269C2311yescharacteristic
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22⋊C4×C269—21no—
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—21no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—42no—
131C22×D269—42no—
132C22×D269—42no—
133C22×D269—42no—
134C22×D269—81no—
135C22×D269—81no—
136C22×D269—81no—
137C22×D269C2×D411yescharacteristic
138D1076—21no—
139D1076—21no—
140D1076—21no—
141D1076—21no—
142D1076—21no—
143D1076—21no—
144D1076—42no—
145D1076—42no—
146D1076—42no—
147D1076—42no—
148D1076—81no—
149D1076C2×D412yes—
150D1076C2×D412yescomplemented
151C22⋊D269—22no—
152C22⋊D269—42no—
153C22⋊D269—42no—
154C22⋊D269—81no—
155C2×Dic269C2×D411yescharacteristic
156C2×Dic269C2×D411yescharacteristic
157C2×Dic269C2×D411yescharacteristic
158C4×D269—21no—
159C4×D269—21no—
160C4×D269—42no—
161C4×D269—42no—
162C4×D269—81no—
163#14524130—21no—
164#14524130—21no—
165C23×C269—21no—
166C23×C269C2×D411yescharacteristic
167Q8×C269—21no—
168Q8×C269—21no—
169D4×C269—21no—
170D4×C269—21no—
171D4×C269—42no—
172C2×C4×C269—21no—
173C2×C4×C269—21no—
174C2×C4×C269—21no—
175C2×C4×C269—21no—
176C2×C4×C269—21no—
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269C2×D412yescomplemented
181C22×C269—21no—
182C22×C269—21no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269AGL(1,ℤ/8)12yescomplemented
187C22×C269C22×D411yescharacteristic
188C2×D269—21no—
189C2×D269—22no—
190C2×D269—22no—
191C2×D269—21no—
192C2×D269—21no—
193C2×D269—41no—
194C2×D269—41no—
195C2×D269—41no—
196C2×D269—41no—
197C2×D269—41no—
198C2×D269—41no—
199C2×D269—41no—
200C2×D269—41no—
201C2×D269—41no—
202C2×D269—41no—
203C2×D269—84no—
204C2×D269—84no—
205C2×D269—84no—
206C4×C269—21no—
207C4×C269—41no—
208C4×C269—41no—
209C4×C269—41no—
210C4×C269C22×D411yescharacteristic
211C4×C269H(5,𝔽2)11yescharacteristic
212Dic269—21no—
213Dic269—21no—
214Dic269—21no—
215C2×C269—22no—
216C2×C269—81no—
217C2×C269D4211yescharacteristic
218C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C269#57611yescharacteristic
220D269—41no—
221D269—42no—
222D269—42no—
223D269—81no—
224D269—81no—
225C269#326411yescharacteristic, complemented, Sylow
226#3264—2691nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2691no—
228C22×D8—2691no—
229#546—2691no—
230D42—2691no—
231#496—2691no—
232#469—2691no—
233#465D26911yescharacteristic, complemented
234#463—2691no—
235#459—2691no—
236#450—2691no—
237#449—2691no—
238#447—2691no—
239#447—2691no—
240#442—2691no—
241#435—2691no—
242C2×#41—2691no—
243C22×D4—2691no—
244C22×D4—2691no—
245C22×D4—2691no—
246C22×D4—5381no—
247AGL(1,ℤ/8)—5384no—
248C2×SD16—2691no—
249C2×SD16—2691no—
250C2×SD16C2×D26911yescharacteristic, complemented
251C2×D8—2691no—
252C2×D8—2691no—
253C2×D8—2691no—
254C2×D8—2691no—
255C2×D8—2691no—
256C2×D8—2691no—
257C2×D8—5384no—
258C2×M16—2691no—
259C22×C8C2×D26911yescharacteristic, complemented
260#127—2691no—
261#124—2691no—
262#123—5381no—
263C22⋊Q8C2×D26911yescharacteristic, complemented
264#121—2691no—
265#121—2691no—
266#121—5381no—
267#121—5381no—
268#121C2×D26911yescharacteristic
269#120—5381no—
270#120—5381no—
271#120—5381no—
272C4×D4—2691no—
273C4×D4—2691no—
274C4×D4—2691no—
275C2×C22⋊C4—5381no—
276#106C2×D26911yescharacteristic, complemented
277C4⋊C8—2691no—
278#103C2×D26911yescharacteristic, complemented
279#102—2691no—
280#102—2691no—
281#102—2691no—
282#102—2691no—
283#102C2×D26911yescharacteristic, complemented
284C22⋊C8—2691no—
285C22⋊C8—2691no—
286#97—2691no—
287C24—5381no—
288C24—5381no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8C22×D26911yescharacteristic
293C2×D4—21no—
294C2×D4—2691no—
295C2×D4—2691no—
296C2×D4—2691no—
297C2×D4—2691no—
298C2×D4—2691no—
299C2×D4—5382no—
300C2×D4—5382no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—5381no—
304C2×D4—5381no—
305C2×D4—5381no—
306C2×D4—5381no—
307C2×D4—5382no—
308C2×D4—5381no—
309C2×D4—5381no—
310C2×D4—10761no—
311C2×D4—10761no—
312C2×D4—10761no—
313C2×D4—10764no—
314C2×D4C22×D26911yescharacteristic
315C22×C4—2691no—
316C22×C4—5381no—
317C22×C4—5381no—
318C22×C4—5381no—
319C22×C4C22×D26911yescharacteristic
320SD16—22no—
321SD16—5382no—
322SD16—10761no—
323D8—2694no—
324D8—5382no—
325D8—5382no—
326D8—5382no—
327D8—5382no—
328D8—10761no—
329M16—5382no—
330C2×C8—22no—
331C2×C8—2691no—
332C2×C8—2691no—
333C2×C8C22×D26911yescharacteristic
334C2×C8C22×D26911yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—2691no—
337C4⋊C4C22×D26911yescharacteristic
338C4⋊C4C22×D26911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2691no—
342C22⋊C4—2691no—
343C22⋊C4—5381no—
344C22⋊C4—5381no—
345C22⋊C4—5381no—
346C22⋊C4—5381no—
347C22⋊C4—5381no—
348C22⋊C4—5381no—
349C42—2691no—
350C23—21no—
351C23—2691no—
352C23—5381no—
353C23—5381no—
354C23—5381no—
355C23—5381no—
356C23—10762no—
357C23—10762no—
358C23—10762no—
359C23—10762no—
360C23—10762no—
361C23—10762no—
362C23—21521no—
363C23—21521no—
364C23—21521no—
365C23D4×D26911yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—42no—
371D4—2692no—
372D4—2692no—
373D4—5382no—
374D4—5381no—
375D4—5381no—
376D4—5381no—
377D4—5381no—
378D4—5381no—
379D4—5381no—
380D4—10762no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—10762no—
385D4—10762no—
386D4—21521no—
387D4—21521no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—2691no—
394C2×C4—2691no—
395C2×C4—2691no—
396C2×C4—5381no—
397C2×C4—5381no—
398C2×C4—10762no—
399C2×C4—10762no—
400C2×C4—21521no—
401C2×C4D4×D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—21no—
404C8—5381no—
405C8—5381no—
406C8D4×D26912yescomplemented
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—5381no—
413C22—5381no—
414C22—5381no—
415C22—5382no—
416C22—5382no—
417C22—10761no—
418C22—10761no—
419C22—10761no—
420C22—10761no—
421C22—10761no—
422C22—10761no—
423C22—10761no—
424C22—10761no—
425C22—10761no—
426C22—10761no—
427C22—21524no—
428C22—21524no—
429C22—21524no—
430C22#2257548012yescomplemented
431C22C2×D4×D26911yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—5381no—
437C4—5381no—
438C4—5381no—
439C4C2×D4×D26911yescharacteristic
440C4#2257549911yescharacteristic
441C2—22no—
442C2—81no—
443C2—10761no—
444C2—10762no—
445C2—10762no—
446C2—21521no—
447C2—21521no—
448C2#3177123911yescharacteristic
449C2C2×#2257548011yescharacteristic
450C2#3177140711yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#465×C269C2
socleC22×C269C22×D4
hypercenter#465D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22575480C2semidirect
2. C22×D2152C2semidirect
3. #31771246C2semidirect
4. #31771239C2semidirect
5. #31770991C2semidirect
6. #31770859C2semidirect
7. #31770823C2semidirect
8. #31770799C2semidirect
9. #31770755C2semidirect
10. #31770753C2semidirect
11. #31770740C2semidirect
12. #31770738C2semidirect
13. #31770716C2semidirect
14. #31770676C2semidirect
15. #465×C269C2semidirect
16. C2×#16864403C22semidirect
18. C22×D1076C22semidirect
20. C2×D4×D269C22semidirect
22. C2×#16864389C22semidirect
23. C2×#16864389C22semidirect
27. C2×D2152C22semidirect
28. C2×D2152C22semidirect
29. C2×#16864387C22semidirect
30. C2×#16864387C22semidirect
31. C2×#16864385C22semidirect
32. #22575449C22semidirect
33. #22575442C22semidirect
36. #22575428C22semidirect
37. #22575428C22semidirect
42. #22575415C22semidirect
43. #22575415C22semidirect
46. #22575382C22semidirect
47. #22575373C22semidirect
48. #22575373C22semidirect
49. #22575373C22semidirect
50. #22575373C22semidirect
51. C22⋊#14524130C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
54. C2×SD16×C269C22semidirect
55. C22×C8×C269C22semidirect
56. C22⋊Q8×C269C22semidirect
58. #106×C269C22semidirect
59. #103×C269C22semidirect
60. #102×C269C22semidirect
95. D2152D4semidirect
150. D1076C2×D4semidirect
180. C8×C269C2×D4semidirect
186. C22×C269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
233. #465D269semidirect
250. C2×SD16C2×D269semidirect
259. C22×C8C2×D269semidirect
263. C22⋊Q8C2×D269semidirect
276. #106C2×D269semidirect
278. #103C2×D269semidirect
283. #102C2×D269semidirect
406. C8D4×D269semidirect
430. C22#22575480semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C22×D21522
#317712462
#317712392
#317709912
#317708592
#317708232
#317707992
#317707552
#317707532
#317707402
#317707382
#317707162
#317706762
#465×C2692
#3264269