Finite Group Databasebeta

← #46822367

Subgroups — #46822367

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×#22575477C211yesmaximal, characteristic, complemented
2C2×D8×D269C211yesmaximal, characteristic, complemented
3#31771250C211yesmaximal, characteristic, complemented
4D4×D1076C211yesmaximal, characteristic, complemented
5#31770994C211yesmaximal, characteristic, complemented
6#31770859C211yesmaximal, characteristic, complemented
7#31770835C211yesmaximal, characteristic, complemented
8#31770820C211yesmaximal, characteristic, complemented
9#31770800C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770751C211yesmaximal, characteristic, complemented
12#31770740C211yesmaximal, characteristic, complemented
13#31770740C211yesmaximal, characteristic, complemented
14#31770678C211yesmaximal, characteristic, complemented
15#442×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic, complemented
17C22×D1076—21no—
18C2×D4×D269C2211yescharacteristic
19C2×D4×D269C2211yescharacteristic, complemented
20C2×D4×D269C2211yescharacteristic, complemented
21#22575477—24no—
22C2×#16864391C2211yescharacteristic, complemented
23C2×#16864391C2211yescharacteristic, complemented
24C2×#16864390C2211yescharacteristic, complemented
25C2×D2152C2211yescharacteristic, complemented
26C2×D2152C2211yescharacteristic, complemented
27C2×#16864387C2211yescharacteristic
28C2×#16864387C2211yescharacteristic, complemented
29C2×#16864387C2211yescharacteristic, complemented
30D8×D269—24no—
31C2×#16864385C2211yescharacteristic, complemented
32C2×C8×D269C2211yescharacteristic, complemented
33#22575450C2211yescharacteristic, complemented
34#22575441C2211yescharacteristic, complemented
35#22575437—21no—
36#22575433C2211yescharacteristic, complemented
37#22575428C2211yescharacteristic
38#22575428C2211yescharacteristic, complemented
39#22575428C2211yescharacteristic, complemented
40#22575427—21no—
41#22575427—21no—
42#22575423—21no—
43#22575423—21no—
44#22575423—21no—
45C4×D1076C2211yescharacteristic
46C2×#16864379—21no—
47C4⋊#14524130C2211yescharacteristic, complemented
48#22575376C2211yescharacteristic, complemented
49#22575373C2211yescharacteristic, complemented
50#22575373C2211yescharacteristic, complemented
51#22575373C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575365C2211yescharacteristic, complemented
54C2×D8×C269C2211yescharacteristic
55C4×D4×C269C2211yescharacteristic, complemented
56C4×D4×C269C2211yescharacteristic, complemented
57#106×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic, complemented
59#102×C269C2211yescharacteristic, complemented
60#97×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62C23×D269—21no—
63C23×D269—21no—
64#16864401—22no—
65#16864401—41no—
66C2×Dic538C2311yescharacteristic
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—21no—
70C2×D1076—21no—
71C2×D1076—21no—
72C2×D1076—44no—
73C2×D1076D411yescharacteristic
74C2×D1076C2311yescharacteristic
75C2×D1076C2311yescharacteristic
76C2×D1076C2311yescharacteristic
77C2×C22⋊D269—21no—
78C2×C22⋊D269—21no—
79C2×C22⋊D269—21no—
80C2×C22⋊D269—21no—
81D4×D269—22no—
82D4×D269—22no—
83D4×D269—22no—
84D4×D269—41no—
85D4×D269—41no—
86D4×D269—41no—
87C22×Dic269—21no—
88C2×C4×D269C2311yescharacteristic
89C2×C4×D269C2311yescharacteristic
90#16864391—22no—
91#16864391—41no—
92#16864390—22no—
93D2152—22no—
94D2152—41no—
95#16864387—22no—
96#16864387—22no—
97#16864387—22no—
98#16864385—22no—
99C2×#14524130C2311yescharacteristic
100C2×#14524130C2311yescharacteristic
101C8×D269—22no—
102C4⋊Dic269—21no—
103C4⋊Dic269D411yescharacteristic
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379—21no—
109#16864379—21no—
110#16864379D411yescharacteristic
111#16864379D411yescharacteristic
112C2×D4×C269C2311yescharacteristic
113C2×D4×C269C2311yescharacteristic
114C22×C4×C269—21no—
115C22×C4×C269—21no—
116D8×C269D414yescomplemented
117C2×C8×C269C2311yescharacteristic
118C2×C8×C269C2311yescharacteristic
119C4⋊C4×C269C2311yescharacteristic
120C4⋊C4×C269C2311yescharacteristic
121C22⋊C4×C269—21no—
122C22⋊C4×C269—21no—
123C42×C269C2311yescharacteristic
124C22×D269—21no—
125C22×D269—21no—
126C22×D269—21no—
127C22×D269—42no—
128C22×D269—42no—
129C22×D269—42no—
130C22×D269—42no—
131C22×D269—42no—
132C22×D269—42no—
133C22×D269—81no—
134C22×D269—81no—
135C22×D269—81no—
136C22×D269C2×D411yescharacteristic
137C22×D269C2×D411yescharacteristic
138Dic538—21no—
139Dic538—21no—
140D1076—22no—
141D1076—21no—
142D1076—21no—
143D1076—21no—
144D1076—21no—
145D1076—21no—
146D1076—21no—
147D1076—42no—
148D1076—42no—
149D1076—42no—
150D1076—81no—
151D1076—81no—
152C22⋊D269—42no—
153C22⋊D269—42no—
154C22⋊D269—42no—
155C22⋊D269—42no—
156C2×Dic269—21no—
157C2×Dic269—42no—
158C2×Dic269C2×D411yescharacteristic
159C2×Dic269C2×D411yescharacteristic
160C4×D269—21no—
161C4×D269—21no—
162C4×D269—21no—
163C4×D269—21no—
164#14524130—21no—
165#14524130—21no—
166C23×C269—21no—
167C23×C269—21no—
168D4×C269—21no—
169D4×C269—21no—
170D4×C269C2×D412yes—
171D4×C269C2×D412yescomplemented
172C2×C4×C269—21no—
173C2×C4×C269—21no—
174C2×C4×C269—42no—
175C2×C4×C269—81no—
176C2×C4×C269C2×D411yescharacteristic
177C2×C4×C269C2×D411yescharacteristic
178C2×C4×C269C2411yescharacteristic
179C8×C269—21no—
180C8×C269C2×D412yescomplemented
181C22×C269—22no—
182C22×C269—22no—
183C22×C269—41no—
184C22×C269—41no—
185C22×C269—41no—
186C22×C269—41no—
187C22×C269C22×D411yescharacteristic
188C2×D269—21no—
189C2×D269—21no—
190C2×D269—21no—
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—84no—
203C2×D269—84no—
204C2×D269—84no—
205C2×D269AGL(1,ℤ/8)12yescomplemented
206C4×C269—21no—
207C4×C269—21no—
208C4×C269—41no—
209C4×C269—41no—
210C4×C269C22×D411yescharacteristic
211C4×C269H(5,𝔽2)11yescharacteristic
212Dic269—21no—
213Dic269—21no—
214Dic269—41no—
215C2×C269—42no—
216C2×C269—42no—
217C2×C269D4211yescharacteristic
218C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C269#57611yescharacteristic
220D269—22no—
221D269—41no—
222D269—81no—
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#465—2691no—
234#463—2691no—
235#459—2691no—
236#450—2691no—
237#449—2691no—
238#447—2691no—
239#447—2691no—
240#442D26911yescharacteristic, complemented
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×SD16—2691no—
251C2×D8—2691no—
252C2×D8—2691no—
253C2×D8—2691no—
254C2×D8—2691no—
255C2×D8—2691no—
256C2×D8—5384no—
257C2×D8C2×D26911yescharacteristic
258C2×M16—2691no—
259C22×C8—2691no—
260#127—2691no—
261#124—2691no—
262#123—5381no—
263C22⋊Q8—2691no—
264#121—2691no—
265#121—2691no—
266#121—2691no—
267#121—5381no—
268#121—5381no—
269#120—5381no—
270#120—5381no—
271#120—5381no—
272C4×D4—2691no—
273C4×D4C2×D26911yescharacteristic, complemented
274C4×D4C2×D26911yescharacteristic, complemented
275C2×C22⋊C4—5381no—
276#106C2×D26911yescharacteristic, complemented
277C4⋊C8—2691no—
278#103—2691no—
279#102—2691no—
280#102—2691no—
281#102—2691no—
282#102C2×D26911yescharacteristic, complemented
283#102C2×D26911yescharacteristic, complemented
284C22⋊C8—2691no—
285C22⋊C8—2691no—
286#97C2×D26911yescharacteristic, complemented
287C24—5381no—
288C24—5381no—
289C24—5381no—
290#41—5382no—
291#41—10761no—
292C2×Q8—2691no—
293C2×D4—2691no—
294C2×D4—2691no—
295C2×D4—2691no—
296C2×D4—2691no—
297C2×D4—5381no—
298C2×D4—5381no—
299C2×D4—5381no—
300C2×D4—5381no—
301C2×D4—5381no—
302C2×D4—5381no—
303C2×D4—5382no—
304C2×D4—5382no—
305C2×D4—5382no—
306C2×D4—5381no—
307C2×D4—5381no—
308C2×D4—5381no—
309C2×D4—10764no—
310C2×D4—10761no—
311C2×D4—10761no—
312C2×D4—10761no—
313C2×D4C22×D26911yescharacteristic
314C2×D4C22×D26911yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2691no—
318C22×C4—2691no—
319C22×C4—5381no—
320SD16—5382no—
321SD16—5382no—
322SD16—10761no—
323D8—5382no—
324D8—5382no—
325D8—5382no—
326D8—5382no—
327D8—10761no—
328D8D107614yescomplemented
329M16—5382no—
330C2×C8—2691no—
331C2×C8—2691no—
332C2×C8—5382no—
333C2×C8C22×D26911yescharacteristic
334C2×C8C22×D26911yescharacteristic
335C4⋊C4—2691no—
336C4⋊C4—5381no—
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—
349C42C22×D26911yescharacteristic
350C23—21no—
351C23—21no—
352C23—2691no—
353C23—2691no—
354C23—5381no—
355C23—5381no—
356C23—5381no—
357C23—10762no—
358C23—10762no—
359C23—10762no—
360C23—10762no—
361C23—10762no—
362C23—10762no—
363C23—21521no—
364C23—21521no—
365C23—21521no—
366Q8—5381no—
367Q8—5381no—
368D4—21no—
369D4—21no—
370D4—5382no—
371D4—5381no—
372D4—5381no—
373D4—5381no—
374D4—5381no—
375D4—5381no—
376D4—5381no—
377D4—10762no—
378D4—10762no—
379D4—10762no—
380D4—10762no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—21521no—
385D4—21521no—
386D4C2×D107612yes—
387D4C2×D107612yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—2691no—
393C2×C4—2691no—
394C2×C4—5381no—
395C2×C4—5381no—
396C2×C4—5381no—
397C2×C4—5381no—
398C2×C4—5381no—
399C2×C4—10762no—
400C2×C4D4×D26911yescharacteristic
401C2×C4D4×D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—21no—
404C8—5381no—
405C8—5381no—
406C8C2×D107612yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2692no—
414C22—5381no—
415C22—5381no—
416C22—5381no—
417C22—5381no—
418C22—5381no—
419C22—10761no—
420C22—10761no—
421C22—10761no—
422C22—10761no—
423C22—10761no—
424C22—10761no—
425C22—10761no—
426C22—10761no—
427C22—10761no—
428C22—21524no—
429C22—21524no—
430C22—21524no—
431C22C2×D4×D26911yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5381no—
437C4—5381no—
438C4—10761no—
439C4C22×D107611yescharacteristic
440C4#2257549911yescharacteristic
441C2—42no—
442C2—42no—
443C2—5382no—
444C2—10761no—
445C2—21521no—
446C2—21521no—
447C2—21521no—
448C2D4×D107611yescharacteristic
449C2AGL(1,ℤ/8)×D26911yescharacteristic
450C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#442×C269C2
socleC22×C269C22×D4
hypercenter#442D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#22575477C2semidirect
2. C2×D8×D269C2semidirect
3. #31771250C2semidirect
4. D4×D1076C2semidirect
5. #31770994C2semidirect
6. #31770859C2semidirect
7. #31770835C2semidirect
8. #31770820C2semidirect
9. #31770800C2semidirect
10. #31770757C2semidirect
11. #31770751C2semidirect
12. #31770740C2semidirect
13. #31770740C2semidirect
14. #31770678C2semidirect
15. #442×C269C2semidirect
16. C2×#16864401C22semidirect
19. C2×D4×D269C22semidirect
20. C2×D4×D269C22semidirect
22. C2×#16864391C22semidirect
23. C2×#16864391C22semidirect
24. C2×#16864390C22semidirect
25. C2×D2152C22semidirect
26. C2×D2152C22semidirect
28. C2×#16864387C22semidirect
29. C2×#16864387C22semidirect
31. C2×#16864385C22semidirect
32. C2×C8×D269C22semidirect
33. #22575450C22semidirect
34. #22575441C22semidirect
36. #22575433C22semidirect
38. #22575428C22semidirect
39. #22575428C22semidirect
47. C4⋊#14524130C22semidirect
48. #22575376C22semidirect
49. #22575373C22semidirect
50. #22575373C22semidirect
51. #22575373C22semidirect
52. #22575365C22semidirect
53. #22575365C22semidirect
55. C4×D4×C269C22semidirect
56. C4×D4×C269C22semidirect
57. #106×C269C22semidirect
58. #102×C269C22semidirect
59. #102×C269C22semidirect
60. #97×C269C22semidirect
116. D8×C269D4semidirect
171. D4×C269C2×D4semidirect
180. C8×C269C2×D4semidirect
205. C2×D269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
240. #442D269semidirect
273. C4×D4C2×D269semidirect
274. C4×D4C2×D269semidirect
276. #106C2×D269semidirect
282. #102C2×D269semidirect
283. #102C2×D269semidirect
286. #97C2×D269semidirect
328. D8D1076semidirect
387. D4C2×D1076semidirect
406. C8C2×D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754772
C2×D8×D2692
#317712502
D4×D10762
#317709942
#317708592
#317708352
#317708202
#317708002
#317707572
#317707512
#317707402
#317706782
#442×C2692
#3264269