Finite Group Databasebeta

← #49132758

Subgroups — #49132758

Order 36224.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583637C211yesmaximal, characteristic, complemented
2C2×SD16×D283C211yesmaximal, characteristic, complemented
3#31827127C211yesmaximal, characteristic, complemented
4D4×D1132C211yesmaximal, characteristic, complemented
5#31826854C211yesmaximal, characteristic, complemented
6#31826712C211yesmaximal, characteristic, complemented
7#31826702C211yesmaximal, characteristic, complemented
8#31826680C211yesmaximal, characteristic, complemented
9#31826665C211yesmaximal, characteristic, complemented
10#31826617C211yesmaximal, characteristic, complemented
11#31826613C211yesmaximal, characteristic, complemented
12#31826605C211yesmaximal, characteristic, complemented
13#31826600C211yesmaximal, characteristic, complemented
14#31826538C211yesmaximal, characteristic, complemented
15#440×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C2×Q8×D283C2211yescharacteristic, complemented
18C22×D1132—21no—
19C2×D4×D283C2211yescharacteristic
20C2×D4×D283C2211yescharacteristic
21#22583637—24no—
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic, complemented
24C2×#17985380C2211yescharacteristic
25C2×#17985379C2211yescharacteristic, complemented
26C2×#17985379C2211yescharacteristic, complemented
27SD16×D283—24no—
28C2×D2264C2211yescharacteristic, complemented
29C2×D2264C2211yescharacteristic, complemented
30C2×#17985377C2211yescharacteristic
31C2×#17985375C2211yescharacteristic, complemented
32C2×C8×D283C2211yescharacteristic, complemented
33#22583607C2211yescharacteristic
34#22583598—21no—
35#22583598C2211yescharacteristic, complemented
36#22583590—21no—
37#22583590C2211yescharacteristic, complemented
38#22583585—21no—
39#22583585C2211yescharacteristic
40#22583585C2211yescharacteristic
41#22583585C2211yescharacteristic, complemented
42#22583584—21no—
43#22583580—21no—
44#22583580—21no—
45C4×D1132—21no—
46C4×D1132C2211yescharacteristic
47C4⋊(C283⋊C8)C2211yescharacteristic, complemented
48#22583533C2211yescharacteristic, complemented
49#22583530C2211yescharacteristic, complemented
50#22583530C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53(C2×D283)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic
55C4×Q8×C283C2211yescharacteristic, complemented
56C4×D4×C283C2211yescharacteristic
57#107×C283C2211yescharacteristic, complemented
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic
60#97×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63#17985393—22no—
64#17985393—41no—
65C2×Dic566—21no—
66C2×Dic566C2311yescharacteristic
67Q8×D283—22no—
68Q8×D283—41no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132—21no—
73C2×D1132—21no—
74C2×D1132—21no—
75C2×D1132—44no—
76C2×D1132D411yescharacteristic
77C2×D1132C2311yescharacteristic
78C2×D1132C2311yescharacteristic
79C2×D1132C2311yescharacteristic
80C2×C22⋊D283—21no—
81C2×C22⋊D283—21no—
82D4×D283—22no—
83D4×D283—22no—
84D4×D283—41no—
85D4×D283—41no—
86C2×C4×D283—21no—
87C2×C4×D283—21no—
88C2×C4×D283C2311yescharacteristic
89C2×C4×D283C2311yescharacteristic
90#17985381—22no—
91#17985381—41no—
92#17985380—22no—
93#17985379—22no—
94#17985379—22no—
95D2264—22no—
96D2264—41no—
97#17985377—22no—
98#17985375—22no—
99C2×C283⋊C8C2311yescharacteristic
100C2×C283⋊C8C2311yescharacteristic
101C8×D283—22no—
102C4⋊Dic283—21no—
103C4⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4—21no—
109(C2×D283)⋊C4—21no—
110(C2×D283)⋊C4D411yescharacteristic
111(C2×D283)⋊C4D411yescharacteristic
112C2×Q8×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115SD16×C283D414yescomplemented
116C2×C8×C283C2311yescharacteristic
117C2×C8×C283C2311yescharacteristic
118C4⋊C4×C283—21no—
119C4⋊C4×C283C2311yescharacteristic
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C42×C283—21no—
123C42×C283C2311yescharacteristic
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—21no—
127C22×D283—42no—
128C22×D283—42no—
129C22×D283—42no—
130C22×D283—42no—
131C22×D283—81no—
132C22×D283—81no—
133C22×D283C2×D411yescharacteristic
134C22×D283C2×D411yescharacteristic
135Dic566—21no—
136Dic566—21no—
137Dic566—42no—
138D1132—22no—
139D1132—21no—
140D1132—21no—
141D1132—21no—
142D1132—21no—
143D1132—21no—
144D1132—21no—
145D1132—42no—
146D1132—42no—
147D1132—42no—
148D1132—42no—
149D1132—42no—
150D1132—81no—
151C22⋊D283—42no—
152C22⋊D283—42no—
153C2×Dic283—21no—
154C2×Dic283C2×D411yescharacteristic
155C2×Dic283C2×D411yescharacteristic
156C4×D283—21no—
157C4×D283—21no—
158C4×D283—21no—
159C4×D283—21no—
160C4×D283—42no—
161C4×D283—42no—
162C283⋊C8—21no—
163C283⋊C8—21no—
164C23×C283—21no—
165Q8×C283—21no—
166Q8×C283C2×D412yescomplemented
167D4×C283—21no—
168D4×C283C2×D412yes—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283—42no—
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2×D411yescharacteristic
175C2×C4×C283C2411yescharacteristic
176C8×C283—21no—
177C8×C283C2×D412yescomplemented
178C22×C283—22no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283C22×D411yescharacteristic
182C2×D283—21no—
183C2×D283—21no—
184C2×D283—21no—
185C2×D283—21no—
186C2×D283—21no—
187C2×D283—41no—
188C2×D283—41no—
189C2×D283—41no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—84no—
197C2×D283—84no—
198C2×D283AGL(1,ℤ/8)12yescomplemented
199C4×C283—22no—
200C4×C283—21no—
201C4×C283—21no—
202C4×C283—41no—
203C4×C283—41no—
204C4×C283C22×D411yescharacteristic
205C4×C283H(5,𝔽2)11yescharacteristic
206Dic283—21no—
207Dic283—21no—
208Dic283—41no—
209C2×C283—42no—
210C2×C283D4211yescharacteristic
211C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
212C2×C283#57711yescharacteristic
213D283—22no—
214D283—41no—
215D283—81no—
216D283—81no—
217D283—81no—
218C283#326511yescharacteristic, complemented, Sylow
219#3265—2831nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2831no—
221C22×SD16—2831no—
222#548—2831no—
223D42—2831no—
224#496—2831no—
225#468—2831no—
226#466—2831no—
227#463—2831no—
228#460—2831no—
229#450—2831no—
230#449—2831no—
231#448—2831no—
232#447—2831no—
233#440D28311yescharacteristic, complemented
234#435—2831no—
235C2×#41—2831no—
236C22×Q8—2831no—
237C22×D4—2831no—
238C22×D4—2831no—
239C22×D4—5661no—
240AGL(1,ℤ/8)—5664no—
241C2×SD16—2831no—
242C2×SD16—2831no—
243C2×SD16—2831no—
244C2×SD16—2831no—
245C2×SD16—2831no—
246C2×SD16—5664no—
247C2×SD16C2×D28311yescharacteristic
248C2×D8—2831no—
249C2×D8—2831no—
250C2×D8—2831no—
251C2×M16—2831no—
252C22×C8—2831no—
253#127—2831no—
254#124—2831no—
255#124—5661no—
256C22⋊Q8—2831no—
257C22⋊Q8—5661no—
258#121—2831no—
259#121—2831no—
260#121—2831no—
261#121—5661no—
262#121—5661no—
263#120—5661no—
264#120—5661no—
265C4×Q8C2×D28311yescharacteristic, complemented
266C4×D4—2831no—
267C4×D4—5661no—
268C4×D4C2×D28311yescharacteristic
269#107C2×D28311yescharacteristic, complemented
270C4⋊C8—2831no—
271#103—2831no—
272#103C2×D28311yescharacteristic, complemented
273#102—2831no—
274#102—2831no—
275#102—2831no—
276#102C2×D28311yescharacteristic
277C22⋊C8—2831no—
278C22⋊C8—2831no—
279#97C2×D28311yescharacteristic, complemented
280C24—5661no—
281C24—5661no—
282#41—5662no—
283#41—11321no—
284C2×Q8—2831no—
285C2×Q8—5661no—
286C2×Q8—5662no—
287C2×Q8—11321no—
288C2×Q8C22×D28311yescharacteristic
289C2×D4—2831no—
290C2×D4—2831no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5662no—
300C2×D4—5662no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—11324no—
304C2×D4—11321no—
305C2×D4—11321no—
306C2×D4C22×D28311yescharacteristic
307C22×C4—21no—
308C22×C4—2831no—
309C22×C4—2831no—
310C22×C4—5661no—
311C22×C4—5661no—
312SD16—5662no—
313SD16—5662no—
314SD16—5662no—
315SD16—5662no—
316SD16—11321no—
317SD16D113214yescomplemented
318D8—5662no—
319D8—5662no—
320D8—11321no—
321M16—5662no—
322C2×C8—2831no—
323C2×C8—2831no—
324C2×C8—5662no—
325C2×C8C22×D28311yescharacteristic
326C2×C8C22×D28311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2831no—
329C4⋊C4—5661no—
330C4⋊C4C22×D28311yescharacteristic
331C4⋊C4C22×D28311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2831no—
334C22⋊C4—2831no—
335C22⋊C4—5661no—
336C22⋊C4—5661no—
337C22⋊C4—5661no—
338C22⋊C4—5661no—
339C22⋊C4—5661no—
340C22⋊C4—5661no—
341C42—21no—
342C42C22×D28311yescharacteristic
343C23—21no—
344C23—2831no—
345C23—2831no—
346C23—5661no—
347C23—5661no—
348C23—5661no—
349C23—11322no—
350C23—11322no—
351C23—11322no—
352C23—11322no—
353C23—22641no—
354C23—22641no—
355Q8—21no—
356Q8—5661no—
357Q8—5661no—
358Q8—11322no—
359Q8C2×D113212yescomplemented
360D4—21no—
361D4—5662no—
362D4—5661no—
363D4—5661no—
364D4—5661no—
365D4—5661no—
366D4—5661no—
367D4—5661no—
368D4—11322no—
369D4—11322no—
370D4—11322no—
371D4—11322no—
372D4—11322no—
373D4—11322no—
374D4—11322no—
375D4—22641no—
376D4C2×D113212yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—2831no—
382C2×C4—2831no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—5661no—
388C2×C4—11322no—
389C2×C4—11322no—
390C2×C4D4×D28311yescharacteristic
391C2×C4D4×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C8—21no—
394C8—5661no—
395C8—5661no—
396C8C2×D113212yescomplemented
397C22—22no—
398C22—41no—
399C22—41no—
400C22—2832no—
401C22—5661no—
402C22—5661no—
403C22—5661no—
404C22—5661no—
405C22—5661no—
406C22—11321no—
407C22—11321no—
408C22—11321no—
409C22—11321no—
410C22—11321no—
411C22—11321no—
412C22—11321no—
413C22—11321no—
414C22—11321no—
415C22—22644no—
416C22—22644no—
417C22C2×D4×D28311yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—5661no—
424C4—5661no—
425C4—11321no—
426C4C22×D113211yescharacteristic
427C4#2258365611yescharacteristic
428C2—42no—
429C2—5662no—
430C2—11321no—
431C2—22641no—
432C2—22641no—
433C2—22641no—
434C2D4×D113211yescharacteristic
435C2AGL(1,ℤ/8)×D28311yescharacteristic
436C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#440×C283C2
socleC22×C283C22×D4
hypercenter#440D283
hypocenterC283#3265
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827127C2semidirect
4. D4×D1132C2semidirect
5. #31826854C2semidirect
6. #31826712C2semidirect
7. #31826702C2semidirect
8. #31826680C2semidirect
9. #31826665C2semidirect
10. #31826617C2semidirect
11. #31826613C2semidirect
12. #31826605C2semidirect
13. #31826600C2semidirect
14. #31826538C2semidirect
15. #440×C283C2semidirect
16. C2×#17985393C22semidirect
17. C2×Q8×D283C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985381C22semidirect
25. C2×#17985379C22semidirect
26. C2×#17985379C22semidirect
28. C2×D2264C22semidirect
29. C2×D2264C22semidirect
31. C2×#17985375C22semidirect
32. C2×C8×D283C22semidirect
35. #22583598C22semidirect
37. #22583590C22semidirect
41. #22583585C22semidirect
47. C4⋊(C283⋊C8)C22semidirect
48. #22583533C22semidirect
49. #22583530C22semidirect
50. #22583530C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. (C2×D283)⋊C8C22semidirect
55. C4×Q8×C283C22semidirect
57. #107×C283C22semidirect
58. #103×C283C22semidirect
60. #97×C283C22semidirect
115. SD16×C283D4semidirect
166. Q8×C283C2×D4semidirect
177. C8×C283C2×D4semidirect
198. C2×D283AGL(1,ℤ/8)semidirect
218. C283#3265semidirect
233. #440D283semidirect
265. C4×Q8C2×D283semidirect
269. #107C2×D283semidirect
272. #103C2×D283semidirect
279. #97C2×D283semidirect
317. SD16D1132semidirect
359. Q8C2×D1132semidirect
396. C8C2×D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C2×SD16×D2832
#318271272
D4×D11322
#318268542
#318267122
#318267022
#318266802
#318266652
#318266172
#318266132
#318266052
#318266002
#318265382
#440×C2832
#3265283