Finite Group Databasebeta

← #46822366

Subgroups — #46822366

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×#22575478C211yesmaximal, characteristic, complemented
2C22×#16864387C211yesmaximal, characteristic, complemented
3D4×D1076C211yesmaximal, characteristic, complemented
4#31770995C211yesmaximal, characteristic, complemented
5#31770856C211yesmaximal, characteristic, complemented
6#31770836C211yesmaximal, characteristic, complemented
7#31770817C211yesmaximal, characteristic, complemented
8#31770800C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770750C211yesmaximal, characteristic, complemented
11#31770739C211yesmaximal, characteristic, complemented
12#31770738C211yesmaximal, characteristic, complemented
13#31770715C211yesmaximal, characteristic, complemented
14#31770677C211yesmaximal, characteristic, complemented
15#546×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C22×D1076C2211yescharacteristic, complemented
18C2×D4×D269—21no—
19#22575478—24no—
20C2×#16864389C2211yescharacteristic, complemented
21C2×#16864389C2211yescharacteristic, complemented
22C2×#16864389C2211yescharacteristic, complemented
23C2×#16864387—24no—
24C2×#16864387C2211yescharacteristic
25C2×#16864387C2211yescharacteristic
26C2×#16864387C2211yescharacteristic, complemented
27C2×#16864387C2211yescharacteristic, complemented
28C2×#16864387C2211yescharacteristic, complemented
29C2×#16864387C2211yescharacteristic, complemented
30C2×#16864386C2211yescharacteristic, complemented
31C22×#14524130C2211yescharacteristic, complemented
32#22575450C2211yescharacteristic, complemented
33#22575428—21no—
34#22575427C2211yescharacteristic
35#22575427C2211yescharacteristic, complemented
36#22575423—21no—
37#22575423—21no—
38C4×D1076C2211yescharacteristic, complemented
39#22575384C2211yescharacteristic, complemented
40C4⋊#14524130C2211yescharacteristic, complemented
41#22575377C2211yescharacteristic, complemented
42#22575374C2211yescharacteristic, complemented
43#22575374C2211yescharacteristic, complemented
44#22575373C2211yescharacteristic, complemented
45#22575373C2211yescharacteristic, complemented
46#22575373C2211yescharacteristic, complemented
47C22⋊#14524130C2211yescharacteristic, complemented
48C22⋊#14524130C2211yescharacteristic, complemented
49#22575364C2211yescharacteristic, complemented
50C2×#41×C269C2211yescharacteristic, complemented
51C22×D4×C269C2211yescharacteristic, complemented
52#124×C269C2211yescharacteristic, complemented
53#123×C269—21no—
54C22⋊Q8×C269C2211yescharacteristic, complemented
55#121×C269—21no—
56#121×C269C2211yescharacteristic, complemented
57#120×C269—21no—
58C4×D4×C269C2211yescharacteristic
59C4×D4×C269C2211yescharacteristic, complemented
60C2×C22⋊C4×C269—21no—
61C23×D269—21no—
62C23×D269—21no—
63C2×D1076—22no—
64C2×D1076—22no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—41no—
70C2×D1076—41no—
71C2×D1076C2311yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×D1076C2311yescharacteristic
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76D4×D269—44no—
77C2×C4×D269—21no—
78#16864389—22no—
79#16864389—22no—
80#16864389—41no—
81#16864387—22no—
82#16864387—22no—
83#16864387—22no—
84#16864387—22no—
85#16864387—41no—
86#16864387D414yescomplemented
87#16864386—22no—
88C2×#14524130—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C2×#14524130C2311yescharacteristic
92C2×#14524130C2311yescharacteristic
93C4⋊Dic269C2311yescharacteristic
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97C24×C269—21no—
98#41×C269—22no—
99#41×C269—41no—
100C2×Q8×C269C2311yescharacteristic
101C2×D4×C269—22no—
102C2×D4×C269—21no—
103C2×D4×C269—21no—
104C2×D4×C269—21no—
105C2×D4×C269—41no—
106C2×D4×C269D411yescharacteristic
107C2×D4×C269C2311yescharacteristic
108C2×D4×C269C2311yescharacteristic
109C22×C4×C269—21no—
110C22×C4×C269—21no—
111C22×C4×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113C4⋊C4×C269—21no—
114C4⋊C4×C269D411yescharacteristic
115C4⋊C4×C269C2311yescharacteristic
116C22⋊C4×C269—21no—
117C22⋊C4×C269—21no—
118C22⋊C4×C269—21no—
119C22⋊C4×C269—21no—
120C22⋊C4×C269—21no—
121C22⋊C4×C269D411yescharacteristic
122C22⋊C4×C269D411yescharacteristic
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—81no—
132C22×D269—81no—
133D1076—21no—
134D1076—21no—
135D1076—21no—
136D1076—21no—
137D1076—21no—
138D1076—42no—
139D1076—42no—
140D1076—42no—
141D1076—81no—
142D1076C2×D412yes—
143C22⋊D269—42no—
144C22⋊D269—42no—
145C2×Dic269—21no—
146C4×D269—42no—
147#14524130—21no—
148#14524130—21no—
149#14524130—21no—
150#14524130C2×D412yescomplemented
151C23×C269—21no—
152C23×C269—21no—
153C23×C269—42no—
154C23×C269—42no—
155C23×C269—81no—
156C23×C269C2×D411yescharacteristic
157C23×C269C2×D411yescharacteristic
158Q8×C269—21no—
159Q8×C269—21no—
160D4×C269—22no—
161D4×C269—21no—
162D4×C269—21no—
163D4×C269—21no—
164D4×C269—42no—
165D4×C269—42no—
166D4×C269—81no—
167D4×C269C2×D412yescomplemented
168C2×C4×C269—21no—
169C2×C4×C269—21no—
170C2×C4×C269—21no—
171C2×C4×C269—21no—
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×C269C2×D411yescharacteristic
179C2×C4×C269C2×D411yescharacteristic
180C2×C4×C269C2411yescharacteristic
181C22×C269—21no—
182C22×C269—21no—
183C22×C269—21no—
184C22×C269—22no—
185C22×C269—21no—
186C22×C269—21no—
187C22×C269—41no—
188C22×C269—41no—
189C22×C269—41no—
190C22×C269—41no—
191C22×C269—41no—
192C22×C269—84no—
193C22×C269AGL(1,ℤ/8)12yescomplemented
194C22×C269C22×D411yescharacteristic
195C2×D269—22no—
196C2×D269—41no—
197C2×D269—41no—
198C2×D269—41no—
199C2×D269—41no—
200C2×D269—41no—
201C2×D269—41no—
202C2×D269—41no—
203C2×D269—41no—
204C2×D269—84no—
205C2×D269—84no—
206C4×C269—21no—
207C4×C269—21no—
208C4×C269—21no—
209C4×C269—21no—
210C4×C269—41no—
211C4×C269—41no—
212C4×C269C22×D411yescharacteristic
213C4×C269H(5,𝔽2)11yescharacteristic
214Dic269—41no—
215C2×C269—22no—
216C2×C269—41no—
217C2×C269—42no—
218C2×C269—81no—
219C2×C269D4211yescharacteristic
220C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C269#57611yescharacteristic
222D269—42no—
223D269—81no—
224D269—81no—
225C269#326411yescharacteristic, complemented, Sylow
226#3264—2691nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2691no—
228C22×D8—2691no—
229#546D26911yescharacteristic, complemented
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#442—2691no—
241#435—2691no—
242C2×#41C2×D26911yescharacteristic, complemented
243C22×D4—2691no—
244C22×D4—2691no—
245C22×D4—5381no—
246C22×D4C2×D26911yescharacteristic, complemented
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—2691no—
257C2×D8—5384no—
258C2×M16—2691no—
259C22×C8—2691no—
260#127—2691no—
261#124C2×D26911yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D26911yescharacteristic, complemented
264#121—21no—
265#121—2691no—
266#121—2691no—
267#121—5381no—
268#121C2×D26911yescharacteristic, complemented
269#120—21no—
270#120—5381no—
271#120—5381no—
272C4×D4—2691no—
273C4×D4C2×D26911yescharacteristic
274C4×D4C2×D26911yescharacteristic, complemented
275C2×C22⋊C4—21no—
276#106—2691no—
277C4⋊C8—2691no—
278#103—2691no—
279#102—2691no—
280#102—2691no—
281#102—2691no—
282#102—2691no—
283#102—2691no—
284C22⋊C8—2691no—
285C22⋊C8—2691no—
286#97—2691no—
287C24—21no—
288C24—5381no—
289C24—5381no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D26911yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—2691no—
299C2×D4—2691no—
300C2×D4—2691no—
301C2×D4—5382no—
302C2×D4—5382no—
303C2×D4—5381no—
304C2×D4—5381no—
305C2×D4—5381no—
306C2×D4—5381no—
307C2×D4—5381no—
308C2×D4—5381no—
309C2×D4—10761no—
310C2×D4—10761no—
311C2×D4—10764no—
312C2×D4C22⋊D26911yescharacteristic
313C2×D4C22×D26911yescharacteristic
314C2×D4C22×D26911yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—5381no—
318C22×C4C22×D26911yescharacteristic
319C22×C4C22×D26911yescharacteristic
320SD16—5382no—
321SD16—5382no—
322SD16—10761no—
323D8—2694no—
324D8—5382no—
325D8—5382no—
326D8—5382no—
327D8—5382no—
328D8—10761no—
329M16—5382no—
330C2×C8—2691no—
331C2×C8—2691no—
332C2×C8—2691no—
333C2×C8—2691no—
334C2×C8—5382no—
335C4⋊C4—21no—
336C4⋊C4—2691no—
337C4⋊C4C22⋊D26911yescharacteristic
338C4⋊C4C22×D26911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—5381no—
345C22⋊C4—5381no—
346C22⋊C4—5381no—
347C22⋊C4C22⋊D26911yescharacteristic
348C22⋊C4C22⋊D26911yescharacteristic
349C42C22×D26911yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—5381no—
356C23—5381no—
357C23—5381no—
358C23—10762no—
359C23—10762no—
360C23—10762no—
361C23—10762no—
362C23—21521no—
363C23—21521no—
364C23C2×C22⋊D26911yescharacteristic
365C23C2×C22⋊D26911yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—2692no—
376D4—5381no—
377D4—5381no—
378D4—5381no—
379D4—5381no—
380D4—5381no—
381D4—10762no—
382D4—10762no—
383D4—10762no—
384D4—10762no—
385D4—10762no—
386D4—21521no—
387D4D4×D26912yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—5381no—
397C2×C4—10762no—
398C2×C4C2×C22⋊D26911yescharacteristic
399C2×C4C2×C22⋊D26911yescharacteristic
400C2×C4C2×C22⋊D26911yescharacteristic
401C2×C4C2×C22⋊D26911yescharacteristic
402C2×C4C23×D26911yescharacteristic
403C8—2692no—
404C8—5381no—
405C8—5381no—
406C8—5381no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—5382no—
420C22—10761no—
421C22—10761no—
422C22—10761no—
423C22—10761no—
424C22—10761no—
425C22—10761no—
426C22—10761no—
427C22—10761no—
428C22—21524no—
429C22—21524no—
430C22#2257547812yescomplemented
431C22C22×C22⋊D26911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—10761no—
439C4C2×D4×D26911yescharacteristic
440C4#2257549811yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—10762no—
446C2—21521no—
447C2—21521no—
448C2D4×C22⋊D26911yescharacteristic
449C2C2×#2257547811yescharacteristic
450C2#3177140811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#546×C269C2
socleC22×C269C22×D4
hypercenter#546D269
hypocenterC269#3264
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. C22×#16864387C2semidirect
3. D4×D1076C2semidirect
4. #31770995C2semidirect
5. #31770856C2semidirect
6. #31770836C2semidirect
7. #31770817C2semidirect
8. #31770800C2semidirect
9. #31770755C2semidirect
10. #31770750C2semidirect
11. #31770739C2semidirect
12. #31770738C2semidirect
13. #31770715C2semidirect
14. #31770677C2semidirect
15. #546×C269C2semidirect
17. C22×D1076C22semidirect
20. C2×#16864389C22semidirect
21. C2×#16864389C22semidirect
22. C2×#16864389C22semidirect
26. C2×#16864387C22semidirect
27. C2×#16864387C22semidirect
28. C2×#16864387C22semidirect
29. C2×#16864387C22semidirect
30. C2×#16864386C22semidirect
31. C22×#14524130C22semidirect
32. #22575450C22semidirect
35. #22575427C22semidirect
38. C4×D1076C22semidirect
39. #22575384C22semidirect
40. C4⋊#14524130C22semidirect
41. #22575377C22semidirect
42. #22575374C22semidirect
43. #22575374C22semidirect
44. #22575373C22semidirect
45. #22575373C22semidirect
46. #22575373C22semidirect
47. C22⋊#14524130C22semidirect
48. C22⋊#14524130C22semidirect
49. #22575364C22semidirect
50. C2×#41×C269C22semidirect
51. C22×D4×C269C22semidirect
52. #124×C269C22semidirect
54. C22⋊Q8×C269C22semidirect
56. #121×C269C22semidirect
59. C4×D4×C269C22semidirect
86. #16864387D4semidirect
150. #14524130C2×D4semidirect
167. D4×C269C2×D4semidirect
193. C22×C269AGL(1,ℤ/8)semidirect
225. C269#3264semidirect
229. #546D269semidirect
242. C2×#41C2×D269semidirect
246. C22×D4C2×D269semidirect
261. #124C2×D269semidirect
263. C22⋊Q8C2×D269semidirect
268. #121C2×D269semidirect
274. C4×D4C2×D269semidirect
387. D4D4×D269semidirect
430. C22#22575478semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
C22×#168643872
D4×D10762
#317709952
#317708562
#317708362
#317708172
#317708002
#317707552
#317707502
#317707392
#317707382
#317707152
#317706772
#546×C2692
#3264269