Finite Group Databasebeta

← #27206628

Subgroups — #27206628

Order 13184.

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×#15693048C211yesmaximal, characteristic, complemented
2C22×#13110090C211yesmaximal, characteristic, complemented
3#20268432C211yesmaximal, characteristic, complemented
4D4×C22⋊D103C211yesmaximal, characteristic, complemented
5#20268177C211yesmaximal, characteristic, complemented
6#20268043C211yesmaximal, characteristic, complemented
7#20268022C211yesmaximal, characteristic, complemented
8#20268008C211yesmaximal, characteristic, complemented
9#20267985C211yesmaximal, characteristic, complemented
10#20267935C211yesmaximal, characteristic, complemented
11#20267927C211yesmaximal, characteristic, complemented
12#20267926C211yesmaximal, characteristic, complemented
13#20267900C211yesmaximal, characteristic, complemented
14#20267859C211yesmaximal, characteristic, complemented
15#450×C103C211yesmaximal, characteristic, complemented
16C2×#13110106C2211yescharacteristic, complemented
17C22×D412C2211yescharacteristic, complemented
18C22×C22⋊D103—21no—
19C2×D4×D103C2211yescharacteristic, complemented
20#15693048—24no—
21C2×#13110092C2211yescharacteristic, complemented
22C2×D824C2211yescharacteristic, complemented
23C2×D824C2211yescharacteristic, complemented
24C2×#13110090—24no—
25C2×#13110090C2211yescharacteristic
26C2×#13110090C2211yescharacteristic
27C2×#13110090C2211yescharacteristic, complemented
28C2×#13110090C2211yescharacteristic, complemented
29C2×#13110088C2211yescharacteristic, complemented
30C22×C103⋊C8C2211yescharacteristic, complemented
31#15693017C2211yescharacteristic, complemented
32#15693010C2211yescharacteristic, complemented
33#15693006—21no—
34#15692996—21no—
35#15692996C2211yescharacteristic, complemented
36#15692995C2211yescharacteristic
37#15692993—21no—
38#15692992C2211yescharacteristic, complemented
39#15692991—21no—
40#15692990—21no—
41#15692989—21no—
42D4×Dic103C2211yescharacteristic, complemented
43#15692983C2211yescharacteristic, complemented
44C4×C22⋊D103C2211yescharacteristic
45C22⋊C4×D103—21no—
46#15692952C2211yescharacteristic, complemented
47Dic103⋊C8C2211yescharacteristic, complemented
48#15692945C2211yescharacteristic, complemented
49#15692943C2211yescharacteristic, complemented
50#15692943C2211yescharacteristic, complemented
51#15692941C2211yescharacteristic, complemented
52(C2×D103)⋊C8C2211yescharacteristic, complemented
53#15692931C2211yescharacteristic, complemented
54C22×D4×C103C2211yescharacteristic
55C2×SD16×C103C2211yescharacteristic, complemented
56C2×SD16×C103C2211yescharacteristic, complemented
57C22⋊Q8×C103C2211yescharacteristic, complemented
58#102×C103C2211yescharacteristic, complemented
59#102×C103C2211yescharacteristic, complemented
60C22⋊C8×C103C2211yescharacteristic, complemented
61C23×D103—21no—
62C23×D103—21no—
63#13110106—22no—
64#13110106—41no—
65C2×D412—22no—
66C2×D412—21no—
67C2×D412—21no—
68C2×D412—21no—
69C2×D412—41no—
70C2×D412C2311yescharacteristic
71C2×D412C2311yescharacteristic
72C2×D412C2311yescharacteristic
73C2×C22⋊D103—21no—
74C2×C22⋊D103—21no—
75C2×C22⋊D103—21no—
76C2×C22⋊D103—21no—
77C2×C22⋊D103—21no—
78C2×C22⋊D103—44no—
79C2×C22⋊D103D411yescharacteristic
80D4×D103—22no—
81D4×D103—41no—
82C22×Dic103—21no—
83C2×C4×D103—21no—
84C2×C4×D103—21no—
85C2×C4×D103C2311yescharacteristic
86#13110092—22no—
87D824—22no—
88D824—41no—
89#13110090—22no—
90#13110090—22no—
91#13110090—22no—
92#13110090D414yescomplemented
93#13110088—22no—
94C2×C103⋊C8—22no—
95C2×C103⋊C8C2311yescharacteristic
96C2×C103⋊C8C2311yescharacteristic
97C4⋊Dic103C2311yescharacteristic
98Dic103⋊C4D411yescharacteristic
99C22⋊Dic103—21no—
100C22⋊Dic103—21no—
101C22⋊Dic103D411yescharacteristic
102(C2×D103)⋊C4—21no—
103(C2×D103)⋊C4—21no—
104(C2×D103)⋊C4—21no—
105(C2×D103)⋊C4—21no—
106(C2×D103)⋊C4—21no—
107(C2×D103)⋊C4D411yescharacteristic
108C4×Dic103C2311yescharacteristic
109C24×C103—21no—
110C2×Q8×C103C2311yescharacteristic
111C2×D4×C103—22no—
112C2×D4×C103—21no—
113C2×D4×C103—41no—
114C2×D4×C103C2311yescharacteristic
115C2×D4×C103C2311yescharacteristic
116C22×C4×C103C2311yescharacteristic
117SD16×C103—22no—
118SD16×C103—41no—
119C2×C8×C103C2311yescharacteristic
120C2×C8×C103C2311yescharacteristic
121C4⋊C4×C103—21no—
122C4⋊C4×C103C2311yescharacteristic
123C22⋊C4×C103—21no—
124C23×C103—21no—
125C23×C103—21no—
126C23×C103—42no—
127C23×C103—42no—
128C23×C103—81no—
129C23×C103C2×D411yescharacteristic
130C22×D103—21no—
131C22×D103—21no—
132C22×D103—21no—
133C22×D103—42no—
134C22×D103—42no—
135C22×D103—42no—
136C22×D103—42no—
137C22×D103—81no—
138C22×D103—81no—
139C22×D103C2×D411yescharacteristic
140Q8×C103—21no—
141Q8×C103—21no—
142D4×C103—21no—
143D4×C103—21no—
144D4×C103—21no—
145D4×C103—42no—
146D4×C103C2×D412yes—
147C2×C4×C103—21no—
148C2×C4×C103—21no—
149C2×C4×C103—21no—
150C2×C4×C103—21no—
151C2×C4×C103C2×D411yescharacteristic
152C2×C4×C103C2411yescharacteristic
153C22⋊D103—22no—
154C22⋊D103—42no—
155C22⋊D103—42no—
156C22⋊D103—42no—
157C22⋊D103—42no—
158C22⋊D103—81no—
159C2×Dic103—21no—
160C2×Dic103—42no—
161C2×Dic103C2×D411yescharacteristic
162C2×Dic103C2×D411yescharacteristic
163C2×Dic103C2×D411yescharacteristic
164D412—21no—
165D412—21no—
166D412—21no—
167D412—21no—
168D412—21no—
169D412—42no—
170D412—42no—
171D412—81no—
172D412C2×D412yescomplemented
173C4×D103—21no—
174C4×D103—21no—
175C4×D103—42no—
176C4×D103—81no—
177C8×C103—21no—
178C8×C103—21no—
179C103⋊C8—21no—
180C103⋊C8C2×D412yescomplemented
181C22×C103—21no—
182C22×C103—21no—
183C22×C103—22no—
184C22×C103—41no—
185C22×C103—41no—
186C22×C103—41no—
187C22×C103—41no—
188C22×C103—41no—
189C22×C103—84no—
190C22×C103AGL(1,ℤ/8)12yescomplemented
191C22×C103C22×D411yescharacteristic
192C2×D103—21no—
193C2×D103—22no—
194C2×D103—21no—
195C2×D103—21no—
196C2×D103—41no—
197C2×D103—41no—
198C2×D103—41no—
199C2×D103—41no—
200C2×D103—41no—
201C2×D103—41no—
202C2×D103—41no—
203C2×D103—41no—
204C2×D103—84no—
205C2×D103—84no—
206C4×C103—21no—
207C4×C103—41no—
208C4×C103—41no—
209C4×C103C22×D411yescharacteristic
210C4×C103H(5,𝔽2)11yescharacteristic
211Dic103—21no—
212Dic103—21no—
213Dic103—21no—
214Dic103—41no—
215C2×C103—22no—
216C2×C103—42no—
217C2×C103—81no—
218C2×C103D4211yescharacteristic
219C2×C103C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C103#57611yescharacteristic
221D103—41no—
222D103—42no—
223D103—81no—
224D103—81no—
225#3264—1031nomaximal, Sylow
226C103#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1031no—
228C22×D8—1031no—
229#546—1031no—
230D42—1031no—
231#496—1031no—
232#469—1031no—
233#465—1031no—
234#463—1031no—
235#459—1031no—
236#450D10311yescharacteristic, complemented
237#449—1031no—
238#447—1031no—
239#447—1031no—
240#442—1031no—
241#435—1031no—
242C2×#41—1031no—
243C22×D4—1031no—
244C22×D4—1031no—
245C22×D4—2061no—
246C22×D4C2×D10311yescharacteristic
247AGL(1,ℤ/8)—2064no—
248C2×SD16—1031no—
249C2×SD16C2×D10311yescharacteristic, complemented
250C2×SD16C2×D10311yescharacteristic, complemented
251C2×D8—1031no—
252C2×D8—1031no—
253C2×D8—1031no—
254C2×D8—1031no—
255C2×D8—1031no—
256C2×D8—1031no—
257C2×D8—2064no—
258C2×M16—1031no—
259C22×C8—1031no—
260#127—1031no—
261#124—1031no—
262#123—2061no—
263C22⋊Q8C2×D10311yescharacteristic, complemented
264#121—1031no—
265#121—1031no—
266#121—1031no—
267#121—2061no—
268#121—2061no—
269#120—2061no—
270#120—2061no—
271#120—2061no—
272C4×D4—1031no—
273C4×D4—1031no—
274C4×D4—1031no—
275C2×C22⋊C4—2061no—
276#106—1031no—
277C4⋊C8—1031no—
278#103—1031no—
279#102—1031no—
280#102—1031no—
281#102—1031no—
282#102C2×D10311yescharacteristic, complemented
283#102C2×D10311yescharacteristic, complemented
284C22⋊C8—1031no—
285C22⋊C8C2×D10311yescharacteristic, complemented
286#97—1031no—
287C24—21no—
288C24—2061no—
289C24—2061no—
290#41—4121no—
291#41—2062no—
292C2×Q8C22×D10311yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1031no—
297C2×D4—1031no—
298C2×D4—1031no—
299C2×D4—1031no—
300C2×D4—4121no—
301C2×D4—4124no—
302C2×D4—4121no—
303C2×D4—2061no—
304C2×D4—2062no—
305C2×D4—2061no—
306C2×D4—2061no—
307C2×D4—2061no—
308C2×D4—2061no—
309C2×D4—2061no—
310C2×D4—2061no—
311C2×D4—2062no—
312C2×D4—2061no—
313C2×D4C22×D10311yescharacteristic
314C2×D4C22×D10311yescharacteristic
315C22×C4—1031no—
316C22×C4—2061no—
317C22×C4—2061no—
318C22×C4—2061no—
319C22×C4C22×D10311yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—2062no—
323D8—1034no—
324D8—4121no—
325D8—2062no—
326D8—2062no—
327D8—2062no—
328D8—2062no—
329M16—2062no—
330C2×C8—1031no—
331C2×C8—1031no—
332C2×C8—2062no—
333C2×C8C22×D10311yescharacteristic
334C2×C8C22×D10311yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1031no—
337C4⋊C4—1031no—
338C4⋊C4C22×D10311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1031no—
341C22⋊C4—1031no—
342C22⋊C4—2061no—
343C22⋊C4—2061no—
344C22⋊C4—2061no—
345C22⋊C4—2061no—
346C22⋊C4—2061no—
347C22⋊C4—2061no—
348C22⋊C4—2061no—
349C42—1031no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—8241no—
356C23—8241no—
357C23—1031no—
358C23—4122no—
359C23—4122no—
360C23—4122no—
361C23—4122no—
362C23—2061no—
363C23—2061no—
364C23—2061no—
365C23D4×D10311yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—8241no—
373D4—8241no—
374D4—1032no—
375D4—4122no—
376D4—4122no—
377D4—4122no—
378D4—4122no—
379D4—4122no—
380D4—4122no—
381D4—2062no—
382D4—2061no—
383D4—2061no—
384D4—2061no—
385D4—2061no—
386D4—2061no—
387D4D4×D10312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—8241no—
393C2×C4—1031no—
394C2×C4—1031no—
395C2×C4—1031no—
396C2×C4—4122no—
397C2×C4—4122no—
398C2×C4—2061no—
399C2×C4—2061no—
400C2×C4—2061no—
401C2×C4D4×D10311yescharacteristic
402C2×C4C23×D10311yescharacteristic
403C8—21no—
404C8—21no—
405C8—1032no—
406C8—2061no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—8244no—
417C22—8244no—
418C22—4121no—
419C22—4121no—
420C22—4121no—
421C22—4121no—
422C22—4121no—
423C22—4121no—
424C22—4121no—
425C22—4121no—
426C22—2061no—
427C22—2061no—
428C22—2062no—
429C22—2061no—
430C22#1569304812yescomplemented
431C22C2×D4×D10311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4121no—
436C4—2061no—
437C4—2061no—
438C4—2061no—
439C4C2×D4×D10311yescharacteristic
440C4#1569306611yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—8241no—
445C2—8241no—
446C2—4121no—
447C2—4122no—
448C2#2026842311yescharacteristic
449C2C2×#1569304811yescharacteristic
450C2#2026859511yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#450×C103C2
socleC22×C103C22×D4
hypercenter#450D103
hypocenterC103#3264
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#15693048C2semidirect
2. C22×#13110090C2semidirect
3. #20268432C2semidirect
4. D4×C22⋊D103C2semidirect
5. #20268177C2semidirect
6. #20268043C2semidirect
7. #20268022C2semidirect
8. #20268008C2semidirect
9. #20267985C2semidirect
10. #20267935C2semidirect
11. #20267927C2semidirect
12. #20267926C2semidirect
13. #20267900C2semidirect
14. #20267859C2semidirect
15. #450×C103C2semidirect
16. C2×#13110106C22semidirect
17. C22×D412C22semidirect
19. C2×D4×D103C22semidirect
21. C2×#13110092C22semidirect
22. C2×D824C22semidirect
23. C2×D824C22semidirect
27. C2×#13110090C22semidirect
28. C2×#13110090C22semidirect
29. C2×#13110088C22semidirect
30. C22×C103⋊C8C22semidirect
31. #15693017C22semidirect
32. #15693010C22semidirect
35. #15692996C22semidirect
38. #15692992C22semidirect
42. D4×Dic103C22semidirect
43. #15692983C22semidirect
46. #15692952C22semidirect
47. Dic103⋊C8C22semidirect
48. #15692945C22semidirect
49. #15692943C22semidirect
50. #15692943C22semidirect
51. #15692941C22semidirect
52. (C2×D103)⋊C8C22semidirect
53. #15692931C22semidirect
55. C2×SD16×C103C22semidirect
56. C2×SD16×C103C22semidirect
57. C22⋊Q8×C103C22semidirect
58. #102×C103C22semidirect
59. #102×C103C22semidirect
60. C22⋊C8×C103C22semidirect
92. #13110090D4semidirect
172. D412C2×D4semidirect
180. C103⋊C8C2×D4semidirect
190. C22×C103AGL(1,ℤ/8)semidirect
226. C103#3264semidirect
236. #450D103semidirect
249. C2×SD16C2×D103semidirect
250. C2×SD16C2×D103semidirect
263. C22⋊Q8C2×D103semidirect
282. #102C2×D103semidirect
283. #102C2×D103semidirect
285. C22⋊C8C2×D103semidirect
430. C22#15693048semidirect

Maximal subgroups

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

subgroupindex
C2×#156930482
C22×#131100902
#202684322
D4×C22⋊D1032
#202681772
#202680432
#202680222
#202680082
#202679852
#202679352
#202679272
#202679262
#202679002
#202678592
#450×C1032
#3264103