Finite Group Databasebeta

← #34056166

Subgroups — #34056166

Order 19072.

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×#18011935C211yesmaximal, characteristic, complemented
2C2×D8×D149C211yesmaximal, characteristic, complemented
3#23750270C211yesmaximal, characteristic, complemented
4#23750260C211yesmaximal, characteristic, complemented
5#23750012C211yesmaximal, characteristic, complemented
6#23749877C211yesmaximal, characteristic, complemented
7#23749853C211yesmaximal, characteristic, complemented
8#23749841C211yesmaximal, characteristic, complemented
9#23749777C211yesmaximal, characteristic, complemented
10#23749767C211yesmaximal, characteristic, complemented
11#23749760C211yesmaximal, characteristic, complemented
12#23749757C211yesmaximal, characteristic, complemented
13#23749735C211yesmaximal, characteristic, complemented
14#23749695C211yesmaximal, characteristic, complemented
15#459×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic, complemented
17C2×D4×D149—21no—
18C2×D4×D149C2211yescharacteristic
19C2×D4×D149C2211yescharacteristic, complemented
20C2×D4×D149C2211yescharacteristic, complemented
21#18011935—24no—
22C2×#14558725C2211yescharacteristic, complemented
23C2×#14558724C2211yescharacteristic, complemented
24C2×#14558724C2211yescharacteristic, complemented
25C2×D1192C2211yescharacteristic
26C2×#14558721C2211yescharacteristic
27C2×#14558721C2211yescharacteristic, complemented
28C2×#14558721C2211yescharacteristic, complemented
29D8×D149—24no—
30C2×#14558719C2211yescharacteristic, complemented
31C2×C8×D149C2211yescharacteristic, complemented
32#18011899C2211yescharacteristic, complemented
33#18011895—21no—
34#18011891C2211yescharacteristic, complemented
35#18011886C2211yescharacteristic, complemented
36#18011885—21no—
37#18011882—21no—
38#18011882C2211yescharacteristic
39#18011882C2211yescharacteristic, complemented
40#18011881—21no—
41#18011879—21no—
42#18011879—21no—
43C4×D596C2211yescharacteristic
44C4×D596C2211yescharacteristic, complemented
45C2×#14558713—21no—
46#18011842C2211yescharacteristic, complemented
47#18011836C2211yescharacteristic, complemented
48#18011833C2211yescharacteristic, complemented
49#18011832C2211yescharacteristic, complemented
50#18011831C2211yescharacteristic, complemented
51#18011823C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53#18011822C2211yescharacteristic, complemented
54C2×D8×C149C2211yescharacteristic, complemented
55C2×D8×C149C2211yescharacteristic, complemented
56#127×C149C2211yescharacteristic, complemented
57C4×D4×C149C2211yescharacteristic, complemented
58C4⋊C8×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic, complemented
60#102×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C23×D149—21no—
63C23×D149—21no—
64#14558735—22no—
65#14558735—41no—
66C2×Dic298C2311yescharacteristic
67C2×D596—21no—
68C2×D596D411yescharacteristic
69C2×D596C2311yescharacteristic
70C2×D596C2311yescharacteristic
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149—21no—
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149—21no—
77D4×D149—22no—
78D4×D149—22no—
79D4×D149—22no—
80D4×D149—41no—
81D4×D149—41no—
82D4×D149—41no—
83D4×D149—44no—
84C22×Dic149—21no—
85C2×C4×D149—21no—
86C2×C4×D149C2311yescharacteristic
87C2×C4×D149C2311yescharacteristic
88#14558725—22no—
89#14558724—22no—
90#14558724—41no—
91D1192—22no—
92#14558721—22no—
93#14558721—41no—
94#14558721D414yescomplemented
95#14558719—22no—
96C2×#11924746C2311yescharacteristic
97C2×#11924746C2311yescharacteristic
98C8×D149—22no—
99C4⋊Dic149—21no—
100C4⋊Dic149D411yescharacteristic
101C4⋊Dic149C2311yescharacteristic
102C22⋊Dic149—21no—
103C22⋊Dic149—21no—
104#14558713—21no—
105#14558713—21no—
106#14558713—21no—
107#14558713—21no—
108#14558713—21no—
109#14558713D411yescharacteristic
110#14558713D411yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149—21no—
113C2×D4×C149C2311yescharacteristic
114C2×D4×C149C2311yescharacteristic
115C2×D4×C149C2311yescharacteristic
116C22×C4×C149—21no—
117D8×C149—22no—
118D8×C149—22no—
119C2×C8×C149C2311yescharacteristic
120C2×C8×C149C2311yescharacteristic
121C4⋊C4×C149C2311yescharacteristic
122C22⋊C4×C149—21no—
123C42×C149C2311yescharacteristic
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—42no—
127C22×D149—42no—
128C22×D149—42no—
129C22×D149—42no—
130C22×D149—42no—
131C22×D149—42no—
132C22×D149—81no—
133C22×D149—81no—
134C22×D149—81no—
135C22×D149C2×D411yescharacteristic
136C22×D149C2×D411yescharacteristic
137Dic298—21no—
138Dic298—21no—
139D596—22no—
140D596—21no—
141D596—21no—
142D596—21no—
143D596—81no—
144D596C2×D412yes—
145C22⋊D149—42no—
146C22⋊D149—42no—
147C22⋊D149—42no—
148C22⋊D149—42no—
149C22⋊D149—42no—
150C22⋊D149—42no—
151C2×Dic149—21no—
152C2×Dic149—21no—
153C2×Dic149—42no—
154C2×Dic149C2×D411yescharacteristic
155C2×Dic149C2×D411yescharacteristic
156C4×D149—21no—
157C4×D149—21no—
158C4×D149—21no—
159C4×D149—21no—
160C4×D149—42no—
161#11924746—21no—
162#11924746C2×D412yescomplemented
163C23×C149—21no—
164C23×C149—21no—
165C23×C149—21no—
166D4×C149—21no—
167D4×C149—21no—
168D4×C149—21no—
169D4×C149—21no—
170D4×C149—21no—
171D4×C149—42no—
172D4×C149—81no—
173D4×C149C2×D412yescomplemented
174C2×C4×C149—21no—
175C2×C4×C149—81no—
176C2×C4×C149C2×D411yescharacteristic
177C2×C4×C149C2×D411yescharacteristic
178C2×C4×C149C2411yescharacteristic
179C8×C149—21no—
180C8×C149—21no—
181C22×C149—22no—
182C22×C149—41no—
183C22×C149—41no—
184C22×C149—41no—
185C22×C149—41no—
186C22×C149—41no—
187C22×C149—41no—
188C22×C149—41no—
189C22×C149—41no—
190C22×C149C22×D411yescharacteristic
191C2×D149—21no—
192C2×D149—21no—
193C2×D149—21no—
194C2×D149—22no—
195C2×D149—21no—
196C2×D149—21no—
197C2×D149—41no—
198C2×D149—41no—
199C2×D149—41no—
200C2×D149—41no—
201C2×D149—41no—
202C2×D149—84no—
203C2×D149—84no—
204C2×D149—84no—
205C2×D149AGL(1,ℤ/8)12yescomplemented
206C4×C149—21no—
207C4×C149—21no—
208C4×C149—41no—
209C4×C149C22×D411yescharacteristic
210C4×C149H(5,𝔽2)11yescharacteristic
211Dic149—21no—
212Dic149—21no—
213Dic149—41no—
214Dic149—41no—
215C2×C149—42no—
216C2×C149—81no—
217C2×C149—81no—
218C2×C149D4211yescharacteristic
219C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C149#57611yescharacteristic
221D149—22no—
222D149—41no—
223D149—42no—
224D149—81no—
225C149#326411yescharacteristic, complemented, Sylow
226#3264—1491nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1491no—
228C22×D8—1491no—
229#546—1491no—
230D42—1491no—
231#496—1491no—
232#469—1491no—
233#465—1491no—
234#463—1491no—
235#459D14911yescharacteristic, complemented
236#450—1491no—
237#449—1491no—
238#447—1491no—
239#447—1491no—
240#442—1491no—
241#435—1491no—
242C2×#41—1491no—
243C22×D4—2981no—
244C22×D4—1491no—
245C22×D4—1491no—
246C22×D4—1491no—
247AGL(1,ℤ/8)—2984no—
248C2×SD16—1491no—
249C2×SD16—1491no—
250C2×SD16—1491no—
251C2×D8—2984no—
252C2×D8—1491no—
253C2×D8—1491no—
254C2×D8—1491no—
255C2×D8—1491no—
256C2×D8C2×D14911yescharacteristic, complemented
257C2×D8C2×D14911yescharacteristic, complemented
258C2×M16—1491no—
259C22×C8—1491no—
260#127C2×D14911yescharacteristic, complemented
261#124—1491no—
262#123—2981no—
263C22⋊Q8—1491no—
264#121—2981no—
265#121—2981no—
266#121—1491no—
267#121—1491no—
268#121—1491no—
269#120—2981no—
270#120—2981no—
271#120—2981no—
272C4×D4—1491no—
273C4×D4—1491no—
274C4×D4C2×D14911yescharacteristic, complemented
275C2×C22⋊C4—2981no—
276#106—1491no—
277C4⋊C8C2×D14911yescharacteristic, complemented
278#103—1491no—
279#102—1491no—
280#102—1491no—
281#102—1491no—
282#102C2×D14911yescharacteristic, complemented
283#102C2×D14911yescharacteristic, complemented
284C22⋊C8—1491no—
285C22⋊C8—1491no—
286#97—1491no—
287C24—2981no—
288C24—2981no—
289C24—2981no—
290#41—2982no—
291#41—5961no—
292C2×Q8—1491no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2981no—
296C2×D4—2982no—
297C2×D4—2982no—
298C2×D4—2982no—
299C2×D4—2981no—
300C2×D4—2981no—
301C2×D4—2981no—
302C2×D4—2981no—
303C2×D4—2981no—
304C2×D4—2981no—
305C2×D4—5961no—
306C2×D4—5961no—
307C2×D4—5961no—
308C2×D4—5964no—
309C2×D4—1491no—
310C2×D4—1491no—
311C2×D4—1491no—
312C2×D4C22×D14911yescharacteristic
313C2×D4C22×D14911yescharacteristic
314C2×D4C22×D14911yescharacteristic
315C22×C4—21no—
316C22×C4—2981no—
317C22×C4—2981no—
318C22×C4—1491no—
319C22×C4—1491no—
320SD16—2982no—
321SD16—2982no—
322SD16—5961no—
323D8—22no—
324D8—22no—
325D8—2982no—
326D8—2982no—
327D8—5961no—
328D8—1494no—
329M16—2982no—
330C2×C8—2982no—
331C2×C8—1491no—
332C2×C8—1491no—
333C2×C8C22×D14911yescharacteristic
334C2×C8C22×D14911yescharacteristic
335C4⋊C4—2981no—
336C4⋊C4—1491no—
337C4⋊C4—1491no—
338C4⋊C4C22×D14911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2981no—
341C22⋊C4—2981no—
342C22⋊C4—2981no—
343C22⋊C4—2981no—
344C22⋊C4—2981no—
345C22⋊C4—2981no—
346C22⋊C4—2981no—
347C22⋊C4—1491no—
348C22⋊C4—1491no—
349C42C22×D14911yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—2981no—
354C23—2981no—
355C23—5962no—
356C23—5962no—
357C23—5962no—
358C23—5962no—
359C23—5962no—
360C23—5962no—
361C23—1491no—
362C23—1491no—
363C23—11921no—
364C23—11921no—
365C23—11921no—
366Q8—2981no—
367Q8—2981no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—2982no—
376D4—2981no—
377D4—2981no—
378D4—2981no—
379D4—5962no—
380D4—5962no—
381D4—5962no—
382D4—5962no—
383D4—5962no—
384D4—5962no—
385D4—1492no—
386D4—11921no—
387D4D4×D14912yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—2981no—
391C2×C4—2981no—
392C2×C4—2981no—
393C2×C4—2981no—
394C2×C4—2981no—
395C2×C4—2981no—
396C2×C4—5962no—
397C2×C4—5962no—
398C2×C4—1491no—
399C2×C4—1491no—
400C2×C4D4×D14911yescharacteristic
401C2×C4D4×D14911yescharacteristic
402C2×C4C23×D14911yescharacteristic
403C8—21no—
404C8—21no—
405C8—2981no—
406C8—1492no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2981no—
417C22—2981no—
418C22—2981no—
419C22—2981no—
420C22—2981no—
421C22—2982no—
422C22—5961no—
423C22—5961no—
424C22—5961no—
425C22—5961no—
426C22—5961no—
427C22—1492no—
428C22—11924no—
429C22—11924no—
430C22—11924no—
431C22C2×D4×D14911yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—2981no—
436C4—2981no—
437C4—5961no—
438C4—5961no—
439C4C2×D4×D14911yescharacteristic
440C4#1801195711yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—2982no—
445C2—5961no—
446C2—5962no—
447C2—11921no—
448C2#2375025911yescharacteristic
449C2AGL(1,ℤ/8)×D14911yescharacteristic
450C2#2375042611yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#459×C149C2
socleC22×C149C22×D4
hypercenter#459D149
hypocenterC149#3264
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#18011935C2semidirect
2. C2×D8×D149C2semidirect
3. #23750270C2semidirect
4. #23750260C2semidirect
5. #23750012C2semidirect
6. #23749877C2semidirect
7. #23749853C2semidirect
8. #23749841C2semidirect
9. #23749777C2semidirect
10. #23749767C2semidirect
11. #23749760C2semidirect
12. #23749757C2semidirect
13. #23749735C2semidirect
14. #23749695C2semidirect
15. #459×C149C2semidirect
16. C2×#14558735C22semidirect
19. C2×D4×D149C22semidirect
20. C2×D4×D149C22semidirect
22. C2×#14558725C22semidirect
23. C2×#14558724C22semidirect
24. C2×#14558724C22semidirect
27. C2×#14558721C22semidirect
28. C2×#14558721C22semidirect
30. C2×#14558719C22semidirect
31. C2×C8×D149C22semidirect
32. #18011899C22semidirect
34. #18011891C22semidirect
35. #18011886C22semidirect
39. #18011882C22semidirect
44. C4×D596C22semidirect
46. #18011842C22semidirect
47. #18011836C22semidirect
48. #18011833C22semidirect
49. #18011832C22semidirect
50. #18011831C22semidirect
51. #18011823C22semidirect
52. #18011823C22semidirect
53. #18011822C22semidirect
54. C2×D8×C149C22semidirect
55. C2×D8×C149C22semidirect
56. #127×C149C22semidirect
57. C4×D4×C149C22semidirect
58. C4⋊C8×C149C22semidirect
59. #102×C149C22semidirect
60. #102×C149C22semidirect
94. #14558721D4semidirect
162. #11924746C2×D4semidirect
173. D4×C149C2×D4semidirect
205. C2×D149AGL(1,ℤ/8)semidirect
225. C149#3264semidirect
235. #459D149semidirect
256. C2×D8C2×D149semidirect
257. C2×D8C2×D149semidirect
260. #127C2×D149semidirect
274. C4×D4C2×D149semidirect
277. C4⋊C8C2×D149semidirect
282. #102C2×D149semidirect
283. #102C2×D149semidirect
387. D4D4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119352
C2×D8×D1492
#237502702
#237502602
#237500122
#237498772
#237498532
#237498412
#237497772
#237497672
#237497602
#237497572
#237497352
#237496952
#459×C1492
#3264149