Finite Group Databasebeta

← #49132743

Subgroups — #49132743

Order 36224.

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×#22583634C211yesmaximal, characteristic, complemented
2C2×D8×D283C211yesmaximal, characteristic, complemented
3#31827110C211yesmaximal, characteristic, complemented
4D4×D1132C211yesmaximal, characteristic, complemented
5#31826854C211yesmaximal, characteristic, complemented
6#31826719C211yesmaximal, characteristic, complemented
7#31826695C211yesmaximal, characteristic, complemented
8#31826680C211yesmaximal, characteristic, complemented
9#31826660C211yesmaximal, characteristic, complemented
10#31826617C211yesmaximal, characteristic, complemented
11#31826611C211yesmaximal, characteristic, complemented
12#31826600C211yesmaximal, characteristic, complemented
13#31826600C211yesmaximal, characteristic, complemented
14#31826538C211yesmaximal, characteristic, complemented
15#442×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic, complemented
17C22×D1132—21no—
18C2×D4×D283C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic, complemented
20C2×D4×D283C2211yescharacteristic, complemented
21#22583634—24no—
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic, complemented
24C2×#17985380C2211yescharacteristic, complemented
25C2×D2264C2211yescharacteristic, complemented
26C2×D2264C2211yescharacteristic, complemented
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985377C2211yescharacteristic, complemented
30D8×D283—24no—
31C2×#17985375C2211yescharacteristic, complemented
32C2×C8×D283C2211yescharacteristic, complemented
33#22583607C2211yescharacteristic, complemented
34#22583598C2211yescharacteristic, complemented
35#22583594—21no—
36#22583590C2211yescharacteristic, complemented
37#22583585C2211yescharacteristic
38#22583585C2211yescharacteristic, complemented
39#22583585C2211yescharacteristic, complemented
40#22583584—21no—
41#22583584—21no—
42#22583580—21no—
43#22583580—21no—
44#22583580—21no—
45C4×D1132C2211yescharacteristic
46C2×(C2×D283)⋊C4—21no—
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×D8×C283C2211yescharacteristic
55C4×D4×C283C2211yescharacteristic, complemented
56C4×D4×C283C2211yescharacteristic, complemented
57#106×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60#97×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63C23×D283—21no—
64#17985391—22no—
65#17985391—41no—
66C2×Dic566C2311yescharacteristic
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132—44no—
73C2×D1132D411yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×D1132C2311yescharacteristic
76C2×D1132C2311yescharacteristic
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79C2×C22⋊D283—21no—
80C2×C22⋊D283—21no—
81D4×D283—22no—
82D4×D283—22no—
83D4×D283—22no—
84D4×D283—41no—
85D4×D283—41no—
86D4×D283—41no—
87C22×Dic283—21no—
88C2×C4×D283C2311yescharacteristic
89C2×C4×D283C2311yescharacteristic
90#17985381—22no—
91#17985381—41no—
92#17985380—22no—
93D2264—22no—
94D2264—41no—
95#17985377—22no—
96#17985377—22no—
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×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283—21no—
116D8×C283D414yescomplemented
117C2×C8×C283C2311yescharacteristic
118C2×C8×C283C2311yescharacteristic
119C4⋊C4×C283C2311yescharacteristic
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C22⋊C4×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—42no—
132C22×D283—42no—
133C22×D283—81no—
134C22×D283—81no—
135C22×D283—81no—
136C22×D283C2×D411yescharacteristic
137C22×D283C2×D411yescharacteristic
138Dic566—21no—
139Dic566—21no—
140D1132—22no—
141D1132—21no—
142D1132—21no—
143D1132—21no—
144D1132—21no—
145D1132—21no—
146D1132—21no—
147D1132—42no—
148D1132—42no—
149D1132—42no—
150D1132—81no—
151D1132—81no—
152C22⋊D283—42no—
153C22⋊D283—42no—
154C22⋊D283—42no—
155C22⋊D283—42no—
156C2×Dic283—21no—
157C2×Dic283—42no—
158C2×Dic283C2×D411yescharacteristic
159C2×Dic283C2×D411yescharacteristic
160C4×D283—21no—
161C4×D283—21no—
162C4×D283—21no—
163C4×D283—21no—
164C283⋊C8—21no—
165C283⋊C8—21no—
166C23×C283—21no—
167C23×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283C2×D412yes—
171D4×C283C2×D412yescomplemented
172C2×C4×C283—21no—
173C2×C4×C283—21no—
174C2×C4×C283—42no—
175C2×C4×C283—81no—
176C2×C4×C283C2×D411yescharacteristic
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283C2×D412yescomplemented
181C22×C283—22no—
182C22×C283—22no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283C22×D411yescharacteristic
188C2×D283—21no—
189C2×D283—21no—
190C2×D283—21no—
191C2×D283—21no—
192C2×D283—21no—
193C2×D283—41no—
194C2×D283—41no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—84no—
203C2×D283—84no—
204C2×D283—84no—
205C2×D283AGL(1,ℤ/8)12yescomplemented
206C4×C283—21no—
207C4×C283—21no—
208C4×C283—41no—
209C4×C283—41no—
210C4×C283C22×D411yescharacteristic
211C4×C283H(5,𝔽2)11yescharacteristic
212Dic283—21no—
213Dic283—21no—
214Dic283—41no—
215C2×C283—42no—
216C2×C283—42no—
217C2×C283D4211yescharacteristic
218C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C283#57611yescharacteristic
220D283—22no—
221D283—41no—
222D283—81no—
223D283—81no—
224D283—81no—
225C283#326411yescharacteristic, complemented, Sylow
226#3264—2831nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2831no—
228C22×D8—2831no—
229#546—2831no—
230D42—2831no—
231#496—2831no—
232#469—2831no—
233#465—2831no—
234#463—2831no—
235#459—2831no—
236#450—2831no—
237#449—2831no—
238#447—2831no—
239#447—2831no—
240#442D28311yescharacteristic, complemented
241#435—2831no—
242C2×#41—2831no—
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—2831no—
246C22×D4—5661no—
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16—2831no—
250C2×SD16—2831no—
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—2831no—
256C2×D8—5664no—
257C2×D8C2×D28311yescharacteristic
258C2×M16—2831no—
259C22×C8—2831no—
260#127—2831no—
261#124—2831no—
262#123—5661no—
263C22⋊Q8—2831no—
264#121—2831no—
265#121—2831no—
266#121—2831no—
267#121—5661no—
268#121—5661no—
269#120—5661no—
270#120—5661no—
271#120—5661no—
272C4×D4—2831no—
273C4×D4C2×D28311yescharacteristic, complemented
274C4×D4C2×D28311yescharacteristic, complemented
275C2×C22⋊C4—5661no—
276#106C2×D28311yescharacteristic, complemented
277C4⋊C8—2831no—
278#103—2831no—
279#102—2831no—
280#102—2831no—
281#102—2831no—
282#102C2×D28311yescharacteristic, complemented
283#102C2×D28311yescharacteristic, complemented
284C22⋊C8—2831no—
285C22⋊C8—2831no—
286#97C2×D28311yescharacteristic, complemented
287C24—5661no—
288C24—5661no—
289C24—5661no—
290#41—5662no—
291#41—11321no—
292C2×Q8—2831no—
293C2×D4—2831no—
294C2×D4—2831no—
295C2×D4—2831no—
296C2×D4—2831no—
297C2×D4—5661no—
298C2×D4—5661no—
299C2×D4—5661no—
300C2×D4—5661no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—5662no—
304C2×D4—5662no—
305C2×D4—5662no—
306C2×D4—5661no—
307C2×D4—5661no—
308C2×D4—5661no—
309C2×D4—11324no—
310C2×D4—11321no—
311C2×D4—11321no—
312C2×D4—11321no—
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2831no—
318C22×C4—2831no—
319C22×C4—5661no—
320SD16—5662no—
321SD16—5662no—
322SD16—11321no—
323D8—5662no—
324D8—5662no—
325D8—5662no—
326D8—5662no—
327D8—11321no—
328D8D113214yescomplemented
329M16—5662no—
330C2×C8—2831no—
331C2×C8—2831no—
332C2×C8—5662no—
333C2×C8C22×D28311yescharacteristic
334C2×C8C22×D28311yescharacteristic
335C4⋊C4—2831no—
336C4⋊C4—5661no—
337C4⋊C4C22×D28311yescharacteristic
338C4⋊C4C22×D28311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2831no—
342C22⋊C4—2831no—
343C22⋊C4—5661no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C22⋊C4—5661no—
348C22⋊C4—5661no—
349C42C22×D28311yescharacteristic
350C23—21no—
351C23—21no—
352C23—2831no—
353C23—2831no—
354C23—5661no—
355C23—5661no—
356C23—5661no—
357C23—11322no—
358C23—11322no—
359C23—11322no—
360C23—11322no—
361C23—11322no—
362C23—11322no—
363C23—22641no—
364C23—22641no—
365C23—22641no—
366Q8—5661no—
367Q8—5661no—
368D4—21no—
369D4—21no—
370D4—5662no—
371D4—5661no—
372D4—5661no—
373D4—5661no—
374D4—5661no—
375D4—5661no—
376D4—5661no—
377D4—11322no—
378D4—11322no—
379D4—11322no—
380D4—11322no—
381D4—11322no—
382D4—11322no—
383D4—11322no—
384D4—22641no—
385D4—22641no—
386D4C2×D113212yes—
387D4C2×D113212yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—2831no—
393C2×C4—2831no—
394C2×C4—5661no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—5661no—
398C2×C4—5661no—
399C2×C4—11322no—
400C2×C4D4×D28311yescharacteristic
401C2×C4D4×D28311yescharacteristic
402C2×C4C23×D28311yescharacteristic
403C8—21no—
404C8—5661no—
405C8—5661no—
406C8C2×D113212yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2832no—
414C22—5661no—
415C22—5661no—
416C22—5661no—
417C22—5661no—
418C22—5661no—
419C22—11321no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22—22644no—
430C22—22644no—
431C22C2×D4×D28311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5661no—
437C4—5661no—
438C4—11321no—
439C4C22×D113211yescharacteristic
440C4#2258365611yescharacteristic
441C2—42no—
442C2—42no—
443C2—5662no—
444C2—11321no—
445C2—22641no—
446C2—22641no—
447C2—22641no—
448C2D4×D113211yescharacteristic
449C2AGL(1,ℤ/8)×D28311yescharacteristic
450C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#442×C283C2
socleC22×C283C22×D4
hypercenter#442D283
hypocenterC283#3264
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×D8×D283C2semidirect
3. #31827110C2semidirect
4. D4×D1132C2semidirect
5. #31826854C2semidirect
6. #31826719C2semidirect
7. #31826695C2semidirect
8. #31826680C2semidirect
9. #31826660C2semidirect
10. #31826617C2semidirect
11. #31826611C2semidirect
12. #31826600C2semidirect
13. #31826600C2semidirect
14. #31826538C2semidirect
15. #442×C283C2semidirect
16. C2×#17985391C22semidirect
19. C2×D4×D283C22semidirect
20. C2×D4×D283C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985381C22semidirect
24. C2×#17985380C22semidirect
25. C2×D2264C22semidirect
26. C2×D2264C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985377C22semidirect
31. C2×#17985375C22semidirect
32. C2×C8×D283C22semidirect
33. #22583607C22semidirect
34. #22583598C22semidirect
36. #22583590C22semidirect
38. #22583585C22semidirect
39. #22583585C22semidirect
47. C4⋊(C283⋊C8)C22semidirect
48. #22583533C22semidirect
49. #22583530C22semidirect
50. #22583530C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. (C2×D283)⋊C8C22semidirect
55. C4×D4×C283C22semidirect
56. C4×D4×C283C22semidirect
57. #106×C283C22semidirect
58. #102×C283C22semidirect
59. #102×C283C22semidirect
60. #97×C283C22semidirect
116. D8×C283D4semidirect
171. D4×C283C2×D4semidirect
180. C8×C283C2×D4semidirect
205. C2×D283AGL(1,ℤ/8)semidirect
225. C283#3264semidirect
240. #442D283semidirect
273. C4×D4C2×D283semidirect
274. C4×D4C2×D283semidirect
276. #106C2×D283semidirect
282. #102C2×D283semidirect
283. #102C2×D283semidirect
286. #97C2×D283semidirect
328. D8D1132semidirect
387. D4C2×D1132semidirect
406. C8C2×D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
C2×D8×D2832
#318271102
D4×D11322
#318268542
#318267192
#318266952
#318266802
#318266602
#318266172
#318266112
#318266002
#318265382
#442×C2832
#3264283