Finite Group Databasebeta

← #49132745

Subgroups — #49132745

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×#22583637C211yesmaximal, characteristic, complemented
2C22×#17985377C211yesmaximal, characteristic, complemented
3#31827106C211yesmaximal, characteristic, complemented
4D4×C22⋊D283C211yesmaximal, characteristic, complemented
5#31826851C211yesmaximal, characteristic, complemented
6#31826717C211yesmaximal, characteristic, complemented
7#31826696C211yesmaximal, characteristic, complemented
8#31826682C211yesmaximal, characteristic, complemented
9#31826659C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826601C211yesmaximal, characteristic, complemented
12#31826600C211yesmaximal, characteristic, complemented
13#31826574C211yesmaximal, characteristic, complemented
14#31826533C211yesmaximal, characteristic, complemented
15#450×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic, complemented
17C22×D1132C2211yescharacteristic, complemented
18C22×C22⋊D283—21no—
19C2×D4×D283C2211yescharacteristic, complemented
20#22583637—24no—
21C2×#17985379C2211yescharacteristic, complemented
22C2×D2264C2211yescharacteristic, complemented
23C2×D2264C2211yescharacteristic, complemented
24C2×#17985377—24no—
25C2×#17985377C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic, complemented
28C2×#17985377C2211yescharacteristic, complemented
29C2×#17985375C2211yescharacteristic, complemented
30C22×C283⋊C8C2211yescharacteristic, complemented
31#22583606C2211yescharacteristic, complemented
32#22583599C2211yescharacteristic, complemented
33#22583595—21no—
34#22583585—21no—
35#22583585C2211yescharacteristic, complemented
36#22583584C2211yescharacteristic
37#22583582—21no—
38#22583581C2211yescharacteristic, complemented
39#22583580—21no—
40#22583579—21no—
41#22583578—21no—
42D4×Dic283C2211yescharacteristic, complemented
43#22583572C2211yescharacteristic, complemented
44C4×C22⋊D283C2211yescharacteristic
45C22⋊C4×D283—21no—
46#22583541C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583534C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×SD16×C283C2211yescharacteristic, complemented
57C22⋊Q8×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C23×D283—21no—
63#17985393—22no—
64#17985393—41no—
65C2×D1132—22no—
66C2×D1132—21no—
67C2×D1132—21no—
68C2×D1132—21no—
69C2×D1132—41no—
70C2×D1132C2311yescharacteristic
71C2×D1132C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—44no—
79C2×C22⋊D283D411yescharacteristic
80D4×D283—22no—
81D4×D283—41no—
82C22×Dic283—21no—
83C2×C4×D283—21no—
84C2×C4×D283—21no—
85C2×C4×D283C2311yescharacteristic
86#17985379—22no—
87D2264—22no—
88D2264—41no—
89#17985377—22no—
90#17985377—22no—
91#17985377—22no—
92#17985377D414yescomplemented
93#17985375—22no—
94C2×C283⋊C8—22no—
95C2×C283⋊C8C2311yescharacteristic
96C2×C283⋊C8C2311yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98Dic283⋊C4D411yescharacteristic
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283D411yescharacteristic
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108C4×Dic283C2311yescharacteristic
109C24×C283—21no—
110C2×Q8×C283C2311yescharacteristic
111C2×D4×C283—22no—
112C2×D4×C283—21no—
113C2×D4×C283—41no—
114C2×D4×C283C2311yescharacteristic
115C2×D4×C283C2311yescharacteristic
116C22×C4×C283C2311yescharacteristic
117SD16×C283—22no—
118SD16×C283—41no—
119C2×C8×C283C2311yescharacteristic
120C2×C8×C283C2311yescharacteristic
121C4⋊C4×C283—21no—
122C4⋊C4×C283C2311yescharacteristic
123C22⋊C4×C283—21no—
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
134D1132—21no—
135D1132—21no—
136D1132—21no—
137D1132—21no—
138D1132—21no—
139D1132—42no—
140D1132—42no—
141D1132—81no—
142D1132C2×D412yescomplemented
143C22⋊D283—22no—
144C22⋊D283—42no—
145C22⋊D283—42no—
146C22⋊D283—42no—
147C22⋊D283—42no—
148C22⋊D283—81no—
149C2×Dic283—21no—
150C2×Dic283—42no—
151C2×Dic283C2×D411yescharacteristic
152C2×Dic283C2×D411yescharacteristic
153C2×Dic283C2×D411yescharacteristic
154C4×D283—21no—
155C4×D283—21no—
156C4×D283—42no—
157C4×D283—81no—
158C283⋊C8—21no—
159C283⋊C8C2×D412yescomplemented
160C23×C283—21no—
161C23×C283—21no—
162C23×C283—42no—
163C23×C283—42no—
164C23×C283—81no—
165C23×C283C2×D411yescharacteristic
166Q8×C283—21no—
167Q8×C283—21no—
168D4×C283—21no—
169D4×C283—21no—
170D4×C283—21no—
171D4×C283—42no—
172D4×C283C2×D412yes—
173C2×C4×C283—21no—
174C2×C4×C283—21no—
175C2×C4×C283—21no—
176C2×C4×C283—21no—
177C2×C4×C283C2×D411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C8×C283—21no—
180C8×C283—21no—
181C22×C283—21no—
182C22×C283—21no—
183C22×C283—22no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283—41no—
188C22×C283—41no—
189C22×C283—84no—
190C22×C283AGL(1,ℤ/8)12yescomplemented
191C22×C283C22×D411yescharacteristic
192C2×D283—21no—
193C2×D283—22no—
194C2×D283—21no—
195C2×D283—21no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—41no—
203C2×D283—41no—
204C2×D283—84no—
205C2×D283—84no—
206C4×C283—21no—
207C4×C283—41no—
208C4×C283—41no—
209C4×C283C22×D411yescharacteristic
210C4×C283H(5,𝔽2)11yescharacteristic
211Dic283—21no—
212Dic283—21no—
213Dic283—21no—
214Dic283—41no—
215C2×C283—22no—
216C2×C283—42no—
217C2×C283—81no—
218C2×C283D4211yescharacteristic
219C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C283#57611yescharacteristic
221D283—41no—
222D283—42no—
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#450D28311yescharacteristic, complemented
237#449—2831no—
238#447—2831no—
239#447—2831no—
240#442—2831no—
241#435—2831no—
242C2×#41—2831no—
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—5661no—
246C22×D4C2×D28311yescharacteristic
247AGL(1,ℤ/8)—5664no—
248C2×SD16—2831no—
249C2×SD16C2×D28311yescharacteristic, complemented
250C2×SD16C2×D28311yescharacteristic, complemented
251C2×D8—2831no—
252C2×D8—2831no—
253C2×D8—2831no—
254C2×D8—2831no—
255C2×D8—2831no—
256C2×D8—2831no—
257C2×D8—5664no—
258C2×M16—2831no—
259C22×C8—2831no—
260#127—2831no—
261#124—2831no—
262#123—5661no—
263C22⋊Q8C2×D28311yescharacteristic, complemented
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×D4—2831no—
274C4×D4—2831no—
275C2×C22⋊C4—5661no—
276#106—2831no—
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⋊C8C2×D28311yescharacteristic, complemented
286#97—2831no—
287C24—21no—
288C24—5661no—
289C24—5661no—
290#41—5662no—
291#41—11321no—
292C2×Q8C22×D28311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—2831no—
297C2×D4—2831no—
298C2×D4—2831no—
299C2×D4—2831no—
300C2×D4—5662no—
301C2×D4—5661no—
302C2×D4—5661no—
303C2×D4—5661no—
304C2×D4—5661no—
305C2×D4—5661no—
306C2×D4—5662no—
307C2×D4—5661no—
308C2×D4—5661no—
309C2×D4—5661no—
310C2×D4—11321no—
311C2×D4—11321no—
312C2×D4—11324no—
313C2×D4C22×D28311yescharacteristic
314C2×D4C22×D28311yescharacteristic
315C22×C4—2831no—
316C22×C4—5661no—
317C22×C4—5661no—
318C22×C4—5661no—
319C22×C4C22×D28311yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—5662no—
323D8—2834no—
324D8—5662no—
325D8—5662no—
326D8—5662no—
327D8—5662no—
328D8—11321no—
329M16—5662no—
330C2×C8—2831no—
331C2×C8—2831no—
332C2×C8—5662no—
333C2×C8C22×D28311yescharacteristic
334C2×C8C22×D28311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—2831no—
337C4⋊C4—2831no—
338C4⋊C4C22×D28311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2831no—
341C22⋊C4—2831no—
342C22⋊C4—5661no—
343C22⋊C4—5661no—
344C22⋊C4—5661no—
345C22⋊C4—5661no—
346C22⋊C4—5661no—
347C22⋊C4—5661no—
348C22⋊C4—5661no—
349C42—2831no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—2831no—
356C23—5661no—
357C23—5661no—
358C23—5661no—
359C23—11322no—
360C23—11322no—
361C23—11322no—
362C23—11322no—
363C23—22641no—
364C23—22641no—
365C23D4×D28311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—2832no—
373D4—5662no—
374D4—5661no—
375D4—5661no—
376D4—5661no—
377D4—5661no—
378D4—5661no—
379D4—11322no—
380D4—11322no—
381D4—11322no—
382D4—11322no—
383D4—11322no—
384D4—11322no—
385D4—22641no—
386D4—22641no—
387D4D4×D28312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2831no—
393C2×C4—2831no—
394C2×C4—2831no—
395C2×C4—5661no—
396C2×C4—5661no—
397C2×C4—5661no—
398C2×C4—11322no—
399C2×C4—11322no—
400C2×C4—22641no—
401C2×C4D4×D28311yescharacteristic
402C2×C4C23×D28311yescharacteristic
403C8—21no—
404C8—21no—
405C8—2832no—
406C8—5661no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—5661no—
417C22—5661no—
418C22—5661no—
419C22—5662no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22—11321no—
425C22—11321no—
426C22—11321no—
427C22—11321no—
428C22—22644no—
429C22—22644no—
430C22#2258363712yescomplemented
431C22C2×D4×D28311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5661no—
436C4—5661no—
437C4—5661no—
438C4—11321no—
439C4C2×D4×D28311yescharacteristic
440C4#2258365511yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—11321no—
445C2—11322no—
446C2—22641no—
447C2—22641no—
448C2#3182709711yescharacteristic
449C2C2×#2258363711yescharacteristic
450C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
hypercenter#450D283
hypocenterC283#3264
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C22×#17985377C2semidirect
3. #31827106C2semidirect
4. D4×C22⋊D283C2semidirect
5. #31826851C2semidirect
6. #31826717C2semidirect
7. #31826696C2semidirect
8. #31826682C2semidirect
9. #31826659C2semidirect
10. #31826609C2semidirect
11. #31826601C2semidirect
12. #31826600C2semidirect
13. #31826574C2semidirect
14. #31826533C2semidirect
15. #450×C283C2semidirect
16. C2×#17985393C22semidirect
17. C22×D1132C22semidirect
19. C2×D4×D283C22semidirect
21. C2×#17985379C22semidirect
22. C2×D2264C22semidirect
23. C2×D2264C22semidirect
27. C2×#17985377C22semidirect
28. C2×#17985377C22semidirect
29. C2×#17985375C22semidirect
30. C22×C283⋊C8C22semidirect
31. #22583606C22semidirect
32. #22583599C22semidirect
35. #22583585C22semidirect
38. #22583581C22semidirect
42. D4×Dic283C22semidirect
43. #22583572C22semidirect
46. #22583541C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583534C22semidirect
49. #22583532C22semidirect
50. #22583532C22semidirect
51. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
55. C2×SD16×C283C22semidirect
56. C2×SD16×C283C22semidirect
57. C22⋊Q8×C283C22semidirect
58. #102×C283C22semidirect
59. #102×C283C22semidirect
60. C22⋊C8×C283C22semidirect
92. #17985377D4semidirect
142. D1132C2×D4semidirect
159. C283⋊C8C2×D4semidirect
190. C22×C283AGL(1,ℤ/8)semidirect
225. C283#3264semidirect
236. #450D283semidirect
249. C2×SD16C2×D283semidirect
250. C2×SD16C2×D283semidirect
263. C22⋊Q8C2×D283semidirect
282. #102C2×D283semidirect
283. #102C2×D283semidirect
285. C22⋊C8C2×D283semidirect
430. C22#22583637semidirect

Maximal subgroups

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

subgroupindex
C2×#225836372
C22×#179853772
#318271062
D4×C22⋊D2832
#318268512
#318267172
#318266962
#318266822
#318266592
#318266092
#318266012
#318266002
#318265742
#318265332
#450×C2832
#3264283