Finite Group Databasebeta

← #34056178

Subgroups — #34056178

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×#18011939C211yesmaximal, characteristic, complemented
2C2×D8×D149C211yesmaximal, characteristic, complemented
3#23750272C211yesmaximal, characteristic, complemented
4#23750259C211yesmaximal, characteristic, complemented
5#23750013C211yesmaximal, characteristic, complemented
6#23749855C211yesmaximal, characteristic, complemented
7#23749839C211yesmaximal, characteristic, complemented
8#23749819C211yesmaximal, characteristic, complemented
9#23749777C211yesmaximal, characteristic, complemented
10#23749768C211yesmaximal, characteristic, complemented
11#23749760C211yesmaximal, characteristic, complemented
12#23749758C211yesmaximal, characteristic, complemented
13#23749736C211yesmaximal, characteristic, complemented
14#23749694C211yesmaximal, characteristic, complemented
15#469×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic, complemented
17C22×D596C2211yescharacteristic, complemented
18C2×D4×D149—21no—
19C2×D4×D149C2211yescharacteristic
20C2×D4×D149C2211yescharacteristic, complemented
21#18011939—24no—
22C2×#14558725C2211yescharacteristic, complemented
23C2×#14558725C2211yescharacteristic, complemented
24C2×#14558724C2211yescharacteristic, complemented
25C2×D1192C2211yescharacteristic
26C2×D1192C2211yescharacteristic, complemented
27C2×#14558721C2211yescharacteristic
28C2×#14558721C2211yescharacteristic, complemented
29C2×#14558721C2211yescharacteristic, complemented
30D8×D149—24no—
31C2×C8×D149C2211yescharacteristic, complemented
32#18011907C2211yescharacteristic, complemented
33#18011898C2211yescharacteristic, complemented
34#18011893—21no—
35#18011891C2211yescharacteristic, complemented
36#18011886C2211yescharacteristic
37#18011884—21no—
38#18011882—21no—
39#18011881—21no—
40#18011879—21no—
41#18011879—21no—
42#18011873C2211yescharacteristic, complemented
43#18011873C2211yescharacteristic, complemented
44C4×C22⋊D149C2211yescharacteristic
45C22⋊C4×D149—21no—
46#18011840C2211yescharacteristic, complemented
47#18011834C2211yescharacteristic, complemented
48#18011831C2211yescharacteristic, complemented
49#18011831C2211yescharacteristic, complemented
50#18011831C2211yescharacteristic, complemented
51C22⋊#11924746C2211yescharacteristic, complemented
52#18011823C2211yescharacteristic, complemented
53#18011821C2211yescharacteristic, complemented
54C2×D8×C149C2211yescharacteristic, complemented
55C2×M16×C149C2211yescharacteristic, complemented
56#121×C149C2211yescharacteristic, complemented
57#121×C149C2211yescharacteristic, complemented
58#106×C149C2211yescharacteristic, complemented
59#102×C149C2211yescharacteristic, complemented
60#102×C149C2211yescharacteristic, complemented
61C23×D149—21no—
62C23×D149—21no—
63C23×D149—21no—
64#14558736—22no—
65#14558736—41no—
66C2×Dic298C2311yescharacteristic
67C2×D596—22no—
68C2×D596—21no—
69C2×D596—21no—
70C2×D596—41no—
71C2×D596C2311yescharacteristic
72C2×D596C2311yescharacteristic
73C2×D596C2311yescharacteristic
74C2×C22⋊D149—21no—
75C2×C22⋊D149—21no—
76C2×C22⋊D149—21no—
77C2×C22⋊D149—21no—
78C2×C22⋊D149—21no—
79C2×C22⋊D149D411yescharacteristic
80D4×D149—22no—
81D4×D149—22no—
82D4×D149—41no—
83D4×D149—41no—
84D4×D149—44no—
85C2×C4×D149—21no—
86C2×C4×D149—21no—
87C2×C4×D149—21no—
88C2×C4×D149C2311yescharacteristic
89#14558725—22no—
90#14558725—22no—
91#14558724—41no—
92D1192—22no—
93D1192D414yescomplemented
94#14558721—22no—
95#14558721—22no—
96#14558721—41no—
97C2×#11924746C2311yescharacteristic
98C2×#11924746C2311yescharacteristic
99C8×D149—22no—
100C4⋊Dic149—21no—
101#14558715D411yescharacteristic
102C22⋊Dic149—21no—
103C22⋊Dic149—21no—
104C22⋊Dic149D411yescharacteristic
105#14558713—21no—
106#14558713—21no—
107#14558713—21no—
108#14558713—21no—
109#14558713D411yescharacteristic
110C4×Dic149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149—21no—
113C2×D4×C149C2311yescharacteristic
114C2×D4×C149C2311yescharacteristic
115C22×C4×C149C2311yescharacteristic
116D8×C149—22no—
117M16×C149—22no—
118C2×C8×C149C2311yescharacteristic
119C2×C8×C149C2311yescharacteristic
120C4⋊C4×C149C2311yescharacteristic
121C4⋊C4×C149C2311yescharacteristic
122C22⋊C4×C149—21no—
123C22⋊C4×C149—21no—
124C22×D149—21no—
125C22×D149—21no—
126C22×D149—21no—
127C22×D149—42no—
128C22×D149—42no—
129C22×D149—42no—
130C22×D149—42no—
131C22×D149—42no—
132C22×D149—42no—
133C22×D149—81no—
134C22×D149—81no—
135C22×D149—81no—
136C22×D149C2×D411yescharacteristic
137Dic298—21no—
138Dic298—21no—
139D596—21no—
140D596—21no—
141D596—21no—
142D596—21no—
143D596—42no—
144D596—42no—
145D596C2×D412yes—
146D596C2×D412yescomplemented
147C22⋊D149—22no—
148C22⋊D149—42no—
149C22⋊D149—42no—
150C22⋊D149—42no—
151C22⋊D149—42no—
152C22⋊D149—81no—
153C2×Dic149—21no—
154C2×Dic149C2×D411yescharacteristic
155C2×Dic149C2×D411yescharacteristic
156C2×Dic149C2×D411yescharacteristic
157C4×D149—21no—
158C4×D149—21no—
159C4×D149—42no—
160C4×D149—42no—
161C4×D149—81no—
162#11924746—21no—
163#11924746—21no—
164C23×C149—21no—
165C23×C149—21no—
166C23×C149C2×D411yescharacteristic
167D4×C149—21no—
168D4×C149—21no—
169D4×C149—21no—
170D4×C149—21no—
171D4×C149—42no—
172D4×C149—81no—
173C2×C4×C149—21no—
174C2×C4×C149—21no—
175C2×C4×C149—21no—
176C2×C4×C149—21no—
177C2×C4×C149C2×D411yescharacteristic
178C2×C4×C149C2411yescharacteristic
179C8×C149—21no—
180C8×C149C2×D412yescomplemented
181C22×C149—21no—
182C22×C149—21no—
183C22×C149—21no—
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—22no—
194C2×D149—22no—
195C2×D149—41no—
196C2×D149—41no—
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—41no—
208C4×C149—41no—
209C4×C149C22×D411yescharacteristic
210C4×C149H(5,𝔽2)11yescharacteristic
211Dic149—21no—
212Dic149—21no—
213Dic149—21no—
214Dic149—41no—
215C2×C149—41no—
216C2×C149—81no—
217C2×C149—81no—
218C2×C149D4211yescharacteristic
219C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C149#57611yescharacteristic
221D149—22no—
222D149—42no—
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#469D14911yescharacteristic, complemented
233#465—1491no—
234#463—1491no—
235#459—1491no—
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×D8—1491no—
257C2×D8C2×D14911yescharacteristic, complemented
258C2×M16C2×D14911yescharacteristic, complemented
259C22×C8—1491no—
260#127—1491no—
261#124—1491no—
262#123—2981no—
263C22⋊Q8—1491no—
264#121—2981no—
265#121—2981no—
266#121—1491no—
267#121C2×D14911yescharacteristic, complemented
268#121C2×D14911yescharacteristic, complemented
269#120—2981no—
270#120—2981no—
271#120—2981no—
272C4×D4—1491no—
273C4×D4—1491no—
274C4×D4—1491no—
275C2×C22⋊C4—2981no—
276#106C2×D14911yescharacteristic, complemented
277C4⋊C8—1491no—
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—2982no—
296C2×D4—2981no—
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×D4—1491no—
313C2×D4C22×D14911yescharacteristic
314C2×D4C22×D14911yescharacteristic
315C22×C4—2981no—
316C22×C4—2981no—
317C22×C4—2981no—
318C22×C4—1491no—
319C22×C4C22×D14911yescharacteristic
320SD16—2982no—
321SD16—2982no—
322SD16—5961no—
323D8—22no—
324D8—2982no—
325D8—2982no—
326D8—2982no—
327D8—5961no—
328D8—1494no—
329M16—22no—
330C2×C8—2982no—
331C2×C8—1491no—
332C2×C8—1491no—
333C2×C8C22×D14911yescharacteristic
334C2×C8C22×D14911yescharacteristic
335C4⋊C4—2981no—
336C4⋊C4—1491no—
337C4⋊C4C22×D14911yescharacteristic
338C4⋊C4C22×D14911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2981no—
342C22⋊C4—2981no—
343C22⋊C4—2981no—
344C22⋊C4—2981no—
345C22⋊C4—2981no—
346C22⋊C4—2981no—
347C22⋊C4—1491no—
348C22⋊C4—1491no—
349C42—1491no—
350C23—21no—
351C23—21no—
352C23—2981no—
353C23—2981no—
354C23—2981no—
355C23—5962no—
356C23—5962no—
357C23—5962no—
358C23—5962no—
359C23—5962no—
360C23—5962no—
361C23—1491no—
362C23—11921no—
363C23—11921no—
364C23—11921no—
365C23D4×D14911yescharacteristic
366Q8—2981no—
367Q8—2981no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—2982no—
375D4—2981no—
376D4—2981no—
377D4—2981no—
378D4—2981no—
379D4—5962no—
380D4—5962no—
381D4—5962no—
382D4—5962no—
383D4—5962no—
384D4—5962no—
385D4—1492no—
386D4—1492no—
387D4—11921no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2981no—
393C2×C4—2981no—
394C2×C4—2981no—
395C2×C4—5962no—
396C2×C4—5962no—
397C2×C4—1491no—
398C2×C4—1491no—
399C2×C4—1491no—
400C2×C4—11921no—
401C2×C4D4×D14911yescharacteristic
402C2×C4C23×D14911yescharacteristic
403C8—21no—
404C8—2981no—
405C8—2981no—
406C8D4×D14912yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2981no—
417C22—2981no—
418C22—2982no—
419C22—2982no—
420C22—5961no—
421C22—5961no—
422C22—5961no—
423C22—5961no—
424C22—5961no—
425C22—5961no—
426C22—5961no—
427C22—1492no—
428C22—11924no—
429C22—11924no—
430C22—11924no—
431C22C2×D4×D14911yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—2981no—
436C4—2981no—
437C4—2981no—
438C4—5961no—
439C4C2×D4×D14911yescharacteristic
440C4#1801195711yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—2982no—
445C2—5962no—
446C2—5962no—
447C2—11921no—
448C2#2375025911yescharacteristic
449C2AGL(1,ℤ/8)×D14911yescharacteristic
450C2#2375042911yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#469×C149C2
socleC22×C149C22×D4
hypercenter#469D149
hypocenterC149#3264
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#18011939C2semidirect
2. C2×D8×D149C2semidirect
3. #23750272C2semidirect
4. #23750259C2semidirect
5. #23750013C2semidirect
6. #23749855C2semidirect
7. #23749839C2semidirect
8. #23749819C2semidirect
9. #23749777C2semidirect
10. #23749768C2semidirect
11. #23749760C2semidirect
12. #23749758C2semidirect
13. #23749736C2semidirect
14. #23749694C2semidirect
15. #469×C149C2semidirect
16. C2×#14558736C22semidirect
17. C22×D596C22semidirect
20. C2×D4×D149C22semidirect
22. C2×#14558725C22semidirect
23. C2×#14558725C22semidirect
24. C2×#14558724C22semidirect
26. C2×D1192C22semidirect
28. C2×#14558721C22semidirect
29. C2×#14558721C22semidirect
31. C2×C8×D149C22semidirect
32. #18011907C22semidirect
33. #18011898C22semidirect
35. #18011891C22semidirect
42. #18011873C22semidirect
43. #18011873C22semidirect
46. #18011840C22semidirect
47. #18011834C22semidirect
48. #18011831C22semidirect
49. #18011831C22semidirect
50. #18011831C22semidirect
51. C22⋊#11924746C22semidirect
52. #18011823C22semidirect
53. #18011821C22semidirect
54. C2×D8×C149C22semidirect
55. C2×M16×C149C22semidirect
56. #121×C149C22semidirect
57. #121×C149C22semidirect
58. #106×C149C22semidirect
59. #102×C149C22semidirect
60. #102×C149C22semidirect
93. D1192D4semidirect
146. D596C2×D4semidirect
180. C8×C149C2×D4semidirect
205. C2×D149AGL(1,ℤ/8)semidirect
225. C149#3264semidirect
232. #469D149semidirect
257. C2×D8C2×D149semidirect
258. C2×M16C2×D149semidirect
267. #121C2×D149semidirect
268. #121C2×D149semidirect
276. #106C2×D149semidirect
282. #102C2×D149semidirect
283. #102C2×D149semidirect
406. C8D4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119392
C2×D8×D1492
#237502722
#237502592
#237500132
#237498552
#237498392
#237498192
#237497772
#237497682
#237497602
#237497582
#237497362
#237496942
#469×C1492
#3264149