Finite Group Databasebeta

← #16828797

Subgroups — #16828797

Order 3712.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745359C211yesmaximal, characteristic, complemented
2C2×#11745276C211yesmaximal, characteristic, complemented
3C2×#11745278C211yesmaximal, characteristic, complemented
4#14271743C211yesmaximal, characteristic, complemented
5#14271323C211yesmaximal, characteristic, complemented
6#14271322C211yesmaximal, characteristic, complemented
7#14271312C211yesmaximal, characteristic, complemented
8#14271310C211yesmaximal, characteristic, complemented
9#14271309C211yesmaximal, characteristic, complemented
10#14271298C211yesmaximal, characteristic, complemented
11#14271297C211yesmaximal, characteristic, complemented
12#14271161C211yesmaximal, characteristic, complemented
13#102×D29C211yesmaximal, characteristic, complemented
14#14271128C211yesmaximal, characteristic, complemented
15C29×#450C211yesmaximal, characteristic, complemented
16C22×D4×C29C2211yescharacteristic
17C22×C22⋊D29—21no—
18C22×C22⋊D29—21no—
19C2×#90809C2211yescharacteristic, complemented
20#11745359—24no—
21#11745359—24no—
22C2×#90807—21no—
23C2×#90807C2211yescharacteristic
24C2×#90807C2211yescharacteristic
25C2×D4×D29—21no—
26C2×D4×D29C2211yescharacteristic
27C2×D4×D29C2211yescharacteristic
28C2×#90805C2211yescharacteristic
29C2×#90805C2211yescharacteristic
30C2×SD16×C29C2211yescharacteristic, complemented
31C2×SD16×C29C2211yescharacteristic, complemented
32C29×C22⋊Q8C2211yescharacteristic, complemented
33#11745308C2211yescharacteristic, complemented
34C2×#90785C2211yescharacteristic, complemented
35C2×#90784C2211yescharacteristic, complemented
36C2×#90782C2211yescharacteristic, complemented
37C2×#90782C2211yescharacteristic, complemented
38#11745292C2211yescharacteristic, complemented
39#11745278—24no—
40#11745276—24no—
41C2×D232C2211yescharacteristic, complemented
42C2×#90773C2211yescharacteristic, complemented
43C2×#90772C2211yescharacteristic, complemented
44C2×C8×D29C2211yescharacteristic, complemented
45#11745258—21no—
46#11745257—21no—
47#11745257C2211yescharacteristic, complemented
48C4⋊C4×D29C2211yescharacteristic, complemented
49#11745246—21no—
50#11745246—21no—
51C29×#102C2211yescharacteristic, complemented
52C29×#102C2211yescharacteristic, complemented
53C29×C22⋊C8C2211yescharacteristic, complemented
54#11745180C2211yescharacteristic, complemented
55#11745180C2211yescharacteristic, complemented
56C22⋊#6597C2211yescharacteristic, complemented
57#11745169C2211yescharacteristic, complemented
58#11745168C2211yescharacteristic, complemented
59#11745168C2211yescharacteristic, complemented
60#11745164C2211yescharacteristic, complemented
61#11745158C2211yescharacteristic, complemented
62#11745157C2211yescharacteristic, complemented
63C24×C29—21no—
64C23×D29—21no—
65C23×D29—21no—
66C2×Q8×C29C2311yescharacteristic
67C2×D4×C29—22no—
68C2×D4×C29—22no—
69C2×D4×C29—21no—
70C2×D4×C29C2311yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C22×C4×C29C2311yescharacteristic
73C2×C22⋊D29—28no—
74C2×C22⋊D29—28no—
75C2×C22⋊D29—21no—
76C2×C22⋊D29—21no—
77C2×C22⋊D29—21no—
78C2×C22⋊D29—21no—
79C2×C22⋊D29—21no—
80C2×C22⋊D29—21no—
81C2×C22⋊D29D411yescharacteristic
82C2×C22⋊D29D411yescharacteristic
83C22×Dic29—21no—
84C22×Dic29—21no—
85#90809—22no—
86#90809—22no—
87#90807—28no—
88#90807—22no—
89#90807—22no—
90#90807—22no—
91#90807D414yescomplemented
92D4×D29—28no—
93D4×D29—22no—
94D4×D29—22no—
95D4×D29—22no—
96D4×D29D414yescomplemented
97#90805—22no—
98#90805—22no—
99#90805—22no—
100#90805—22no—
101C2×D116—21no—
102C2×D116—21no—
103C2×D116D411yescharacteristic
104C2×D116C2311yescharacteristic
105C2×D116C2311yescharacteristic
106C2×C4×D29—21no—
107C2×C4×D29—21no—
108C2×C4×D29—21no—
109C2×C4×D29C2311yescharacteristic
110C2×C4×D29C2311yescharacteristic
111C2×Dic58D411yescharacteristic
112C2×Dic58C2311yescharacteristic
113SD16×C29—22no—
114SD16×C29—22no—
115C2×C8×C29C2311yescharacteristic
116C2×C8×C29C2311yescharacteristic
117C4⋊C4×C29—21no—
118C4⋊C4×C29C2311yescharacteristic
119C22⋊C4×C29—21no—
120#90785—22no—
121#90784—22no—
122#90782—22no—
123#90782—22no—
124#90781—21no—
125#90781—21no—
126#90781—21no—
127C4⋊Dic29C2311yescharacteristic
128#90779—21no—
129#90779—21no—
130C2×#6597C2311yescharacteristic
131C2×#6597C2311yescharacteristic
132D232—22no—
133#90773—22no—
134#90772—22no—
135C8×D29—22no—
136C23×C29—24no—
137C23×C29—21no—
138C23×C29—21no—
139C23×C29—81no—
140C23×C29—81no—
141C23×C29C2×D411yescharacteristic
142C22×D29—21no—
143C22×D29—24no—
144C22×D29—24no—
145C22×D29—21no—
146C22×D29—81no—
147C22×D29—81no—
148C22×D29—81no—
149C22×D29—81no—
150C22×D29C2×D411yescharacteristic
151C22×D29C2×D411yescharacteristic
152Q8×C29—21no—
153Q8×C29—21no—
154D4×C29—24no—
155D4×C29—21no—
156D4×C29—21no—
157D4×C29C2×D412yes—
158D4×C29C2×D412yes—
159C2×C4×C29—21no—
160C2×C4×C29—21no—
161C2×C4×C29—21no—
162C2×C4×C29—21no—
163C2×C4×C29C2×D411yescharacteristic
164C2×C4×C29C2411yescharacteristic
165C22⋊D29—24no—
166C22⋊D29—24no—
167C22⋊D29—24no—
168C22⋊D29—22no—
169C22⋊D29—24no—
170C22⋊D29—22no—
171C22⋊D29—24no—
172C22⋊D29—81no—
173C2×Dic29—24no—
174C2×Dic29—21no—
175C2×Dic29—24no—
176C2×Dic29—21no—
177C2×Dic29C2×D411yescharacteristic
178C2×Dic29C2×D411yescharacteristic
179D116—22no—
180D116—21no—
181D116—21no—
182D116—21no—
183D116—42no—
184D116—81no—
185D116C2×D412yes—
186C4×D29—24no—
187C4×D29—21no—
188C4×D29—21no—
189C4×D29—42no—
190C4×D29—81no—
191C4×D29C2×D412yes—
192C4×D29C2×D412yes—
193Dic58—21no—
194Dic58—22no—
195Dic58C2×D412yes—
196C8×C29—21no—
197C8×C29—21no—
198#6597—21no—
199#6597—21no—
200#3012—291nomaximal, Sylow
201C22×C29—22no—
202C22×C29—22no—
203C22×C29—21no—
204C22×C29—21no—
205C22×C29—21no—
206C22×C29—41no—
207C22×C29—41no—
208C22×C29—41no—
209C22×C29—41no—
210C22×C29—84no—
211C22×C29C22×D411yescharacteristic
212C2×D29—21no—
213C2×D29—21no—
214C2×D29—21no—
215C2×D29—22no—
216C2×D29—21no—
217C2×D29—21no—
218C2×D29—41no—
219C2×D29—41no—
220C2×D29—41no—
221C2×D29—41no—
222C2×D29—41no—
223C2×D29—84no—
224C2×D29—84no—
225C2×D29#12012yes—
226C4×C29—21no—
227C4×C29—41no—
228C4×C29—41no—
229C4×C29C22×D411yescharacteristic
230C4×C29C22×D411yescharacteristic
231Dic29—21no—
232Dic29—22no—
233Dic29—41no—
234Dic29#12012yes—
235C2×H(5,𝔽2)—291no—
236C2×AGL(1,ℤ/8)—291no—
237C2×#135—291no—
238#537—291no—
239#452—291no—
240#452—291no—
241#450—291no—
242#450D2911yescharacteristic, complemented
243#449—291no—
244#449—291no—
245#447—291no—
246#447—291no—
247#417—291no—
248C2×#102—291no—
249#408—291no—
250C2×C29—42no—
251C2×C29—42no—
252C2×C29—41no—
253C2×C29C2×#12011yescharacteristic
254C2×C29#57611yescharacteristic
255C2×C29#57711yescharacteristic
256D29—22no—
257D29—41no—
258D29—42no—
259D29—81no—
260H(5,𝔽2)—584no—
261H(5,𝔽2)—584no—
262C2×#41—291no—
263C2×#41—291no—
264C2×#41—291no—
265C2×#41—291no—
266C2×#41—291no—
267C2×#41—581no—
268C22×D4—291no—
269C22×D4—291no—
270C22×D4—581no—
271C22×D4—581no—
272C22×D4—581no—
273C22×D4C2×D2911yescharacteristic
274AGL(1,ℤ/8)—584no—
275#135—584no—
276C2×Q16—291no—
277C2×SD16—291no—
278C2×SD16—291no—
279C2×SD16C2×D2911yescharacteristic, complemented
280C2×SD16C2×D2911yescharacteristic, complemented
281C2×D8—291no—
282C2×D8—291no—
283C2×D8—291no—
284C2×M16—291no—
285C22×C8—291no—
286C22⋊Q8—291no—
287C22⋊Q8—581no—
288C22⋊Q8C2×D2911yescharacteristic, complemented
289#121—291no—
290#121—291no—
291#121—581no—
292#121—581no—
293#121—581no—
294C2×C4⋊C4—291no—
295#103—291no—
296#103—291no—
297#102—291no—
298#102—291no—
299#102—291no—
300#102—291no—
301#102C2×D2911yescharacteristic, complemented
302#102C2×D2911yescharacteristic, complemented
303C22⋊C8—291no—
304C22⋊C8—291no—
305C22⋊C8—291no—
306C22⋊C8C2×D2911yescharacteristic, complemented
307C29#301211yescharacteristic, complemented, Sylow
308C24—21no—
309C24—581no—
310C24—581no—
311#41—294no—
312#41—582no—
313#41—582no—
314#41—582no—
315#41—582no—
316#41—582no—
317#41—582no—
318#41—582no—
319#41—582no—
320#41—588no—
321#41—582no—
322C2×Q8—291no—
323C2×Q8—291no—
324C2×Q8C22×D2911yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—294no—
329C2×D4—291no—
330C2×D4—291no—
331C2×D4—291no—
332C2×D4—291no—
333C2×D4—291no—
334C2×D4—581no—
335C2×D4—581no—
336C2×D4—582no—
337C2×D4—581no—
338C2×D4—581no—
339C2×D4—582no—
340C2×D4—588no—
341C2×D4—581no—
342C2×D4—581no—
343C2×D4—588no—
344C2×D4—588no—
345C2×D4—581no—
346C2×D4—582no—
347C2×D4—581no—
348C2×D4C22×D2911yescharacteristic
349C2×D4C22×D2911yescharacteristic
350C22×C4—291no—
351C22×C4—291no—
352C22×C4—581no—
353C22×C4—581no—
354C22×C4—581no—
355C22×C4—581no—
356C22×C4—581no—
357C22×C4C22×D2911yescharacteristic
358Q16—582no—
359SD16—22no—
360SD16—22no—
361SD16—582no—
362SD16—582no—
363D8—582no—
364D8—582no—
365D8—582no—
366M16—582no—
367C2×C8—291no—
368C2×C8—291no—
369C2×C8—582no—
370C2×C8C22×D2911yescharacteristic
371C2×C8C22×D2911yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—291no—
374C4⋊C4—581no—
375C4⋊C4—581no—
376C4⋊C4C22×D2911yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—581no—
379C22⋊C4—581no—
380C22⋊C4—581no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—81no—
385C23—81no—
386C23—291no—
387C23—291no—
388C23—584no—
389C23—584no—
390C23—581no—
391C23—581no—
392C23—2321no—
393C23—2321no—
394C23—2321no—
395C23—2321no—
396C23D4×D2911yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—292no—
400Q8—582no—
401Q8—581no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—292no—
406D4—582no—
407D4—582no—
408D4—584no—
409D4—582no—
410D4—584no—
411D4—584no—
412D4—584no—
413D4—584no—
414D4—581no—
415D4—581no—
416D4—581no—
417D4—1162no—
418D4—2321no—
419D4—2321no—
420D4D4×D2912yes—
421D4D4×D2912yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—291no—
427C2×C4—292no—
428C2×C4—292no—
429C2×C4—291no—
430C2×C4—584no—
431C2×C4—584no—
432C2×C4—584no—
433C2×C4—581no—
434C2×C4—581no—
435C2×C4—581no—
436C2×C4—581no—
437C2×C4—1162no—
438C2×C4—2321no—
439C2×C4D4×D2911yescharacteristic
440C2×C4C23×D2911yescharacteristic
441C8—21no—
442C8—21no—
443C8—581no—
444C8—581no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—84no—
455C22—292no—
456C22—581no—
457C22—581no—
458C22—582no—
459C22—581no—
460C22—581no—
461C22—581no—
462C22—1161no—
463C22—1161no—
464C22—1161no—
465C22—1161no—
466C22—1161no—
467C22—2324no—
468C22—2324no—
469C22C2×D4×D2911yescharacteristic
470C4—21no—
471C4—41no—
472C4—41no—
473C4—292no—
474C4—581no—
475C4—582no—
476C4—1161no—
477C4C2×D4×D2911yescharacteristic
478C4C2×D4×D2911yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—582no—
483C2—1161no—
484C2—1162no—
485C2—2321no—
486C2#120×D2911yescharacteristic
487C2#1427196711yescharacteristic
488C2#1427196911yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#450C2
socleC22×C29C22×D4
hypercenter#450D29
hypocenterC29#3012
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#11745359C2semidirect
2. C2×#11745276C2semidirect
3. C2×#11745278C2semidirect
4. #14271743C2semidirect
5. #14271323C2semidirect
6. #14271322C2semidirect
7. #14271312C2semidirect
8. #14271310C2semidirect
9. #14271309C2semidirect
10. #14271298C2semidirect
11. #14271297C2semidirect
12. #14271161C2semidirect
13. #102×D29C2semidirect
14. #14271128C2semidirect
15. C29×#450C2semidirect
19. C2×#90809C22semidirect
30. C2×SD16×C29C22semidirect
31. C2×SD16×C29C22semidirect
32. C29×C22⋊Q8C22semidirect
33. #11745308C22semidirect
34. C2×#90785C22semidirect
35. C2×#90784C22semidirect
36. C2×#90782C22semidirect
37. C2×#90782C22semidirect
38. #11745292C22semidirect
41. C2×D232C22semidirect
42. C2×#90773C22semidirect
43. C2×#90772C22semidirect
44. C2×C8×D29C22semidirect
47. #11745257C22semidirect
48. C4⋊C4×D29C22semidirect
51. C29×#102C22semidirect
52. C29×#102C22semidirect
53. C29×C22⋊C8C22semidirect
54. #11745180C22semidirect
55. #11745180C22semidirect
56. C22⋊#6597C22semidirect
57. #11745169C22semidirect
58. #11745168C22semidirect
59. #11745168C22semidirect
60. #11745164C22semidirect
61. #11745158C22semidirect
62. #11745157C22semidirect
91. #90807D4semidirect
96. D4×D29D4semidirect
242. #450D29semidirect
279. C2×SD16C2×D29semidirect
280. C2×SD16C2×D29semidirect
288. C22⋊Q8C2×D29semidirect
301. #102C2×D29semidirect
302. #102C2×D29semidirect
306. C22⋊C8C2×D29semidirect
307. C29#3012semidirect

Maximal subgroups

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

subgroupindex
C2×#117453592
C2×#117452762
C2×#117452782
#142717432
#142713232
#142713222
#142713122
#142713102
#142713092
#142712982
#142712972
#142711612
#102×D292
#142711282
C29×#4502
#301229