Finite Group Databasebeta

← #16829218

Subgroups — #16829218

Order 3712.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14271959C211yesmaximal, characteristic, complemented
2C2×#11745273C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D29C211yesmaximal, characteristic, complemented
4#14271928C211yesmaximal, characteristic, complemented
5#14271432C211yesmaximal, characteristic, complemented
6#14271431C211yesmaximal, characteristic, complemented
7#14271430C211yesmaximal, characteristic, complemented
8#14271427C211yesmaximal, characteristic, complemented
9#14271426C211yesmaximal, characteristic, complemented
10#14271425C211yesmaximal, characteristic, complemented
11#14271424C211yesmaximal, characteristic, complemented
12#14271210C211yesmaximal, characteristic, complemented
13#14271144C211yesmaximal, characteristic, complemented
14#100×D29C211yesmaximal, characteristic, complemented
15C29×#472C211yesmaximal, characteristic, complemented
16#41×D29C2211yescharacteristic
17C2×#90807C2211yescharacteristic
18C2×D4×D29C2211yescharacteristic
19C29×AGL(1,ℤ/8)C2211yescharacteristic
20C29×AGL(1,ℤ/8)C2211yescharacteristic, complemented
21C2×D8×C29C2211yescharacteristic, complemented
22C29×#131C2211yescharacteristic, complemented
23#11745313C2211yescharacteristic, complemented
24#11745311C2211yescharacteristic
25#11745310C2211yescharacteristic, complemented
26C2×#90783C2211yescharacteristic, complemented
27#11745294C2211yescharacteristic
28#11745294C2211yescharacteristic, complemented
29C2×#90782C2211yescharacteristic, complemented
30#11745278—21no—
31#11745277—21no—
32#11745276—21no—
33SD16×D29—21no—
34#11745274—21no—
35#11745273—21no—
36#11745273—21no—
37#11745273—21no—
38#11745273—21no—
39#11745273—21no—
40#11745273—21no—
41D8×D29—21no—
42#11745271C2211yescharacteristic, complemented
43#11745270C2211yescharacteristic
44#11745269—21no—
45#11745269C2211yescharacteristic, complemented
46M16×D29—21no—
47M16×D29C2211yescharacteristic
48M16×D29C2211yescharacteristic
49M16×D29C2211yescharacteristic
50C2×#90773C2211yescharacteristic, complemented
51#11745263—21no—
52#11745263C2211yescharacteristic, complemented
53C2×#90772—21no—
54C2×#90772C2211yescharacteristic
55C29×#108C2211yescharacteristic, complemented
56C29×#100C2211yescharacteristic, complemented
57C29×#100C2211yescharacteristic, complemented
58#11745181C2211yescharacteristic, complemented
59#11745181C2211yescharacteristic, complemented
60#11745172C2211yescharacteristic, complemented
61#11745172C2211yescharacteristic, complemented
62#11745171C2211yescharacteristic, complemented
63#11745171C2211yescharacteristic, complemented
64#11745170C2211yescharacteristic, complemented
65#11745170C2211yescharacteristic, complemented
66#11745167C2211yescharacteristic, complemented
67C23×D29—21no—
68#41×C29C2311yescharacteristic
69C2×D4×C29C2311yescharacteristic
70C2×D4×C29C2311yescharacteristic
71C2×C22⋊D29—21no—
72C2×C22⋊D29—21no—
73C22×Dic29—21no—
74#90809D411yescharacteristic
75Q8×D29D411yescharacteristic
76#90807—21no—
77#90807—21no—
78#90807—21no—
79#90807—21no—
80#90807—21no—
81#90807D411yescharacteristic
82D4×D29—21no—
83D4×D29—21no—
84D4×D29—21no—
85D4×D29—21no—
86D4×D29—21no—
87D4×D29D411yescharacteristic
88#90805—21no—
89#90805C2311yescharacteristic
90C2×D116C2311yescharacteristic
91C2×C4×D29—21no—
92C2×C4×D29C2311yescharacteristic
93C2×Dic58C2311yescharacteristic
94SD16×C29—21no—
95SD16×C29—21no—
96D8×C29—21no—
97D8×C29—21no—
98D8×C29—21no—
99D8×C29—21no—
100M16×C29—21no—
101M16×C29C2311yescharacteristic
102M16×C29C2311yescharacteristic
103M16×C29C2311yescharacteristic
104C2×C8×C29—21no—
105C2×C8×C29C2311yescharacteristic
106#90785—21no—
107#90784—21no—
108#90783—21no—
109#90783—21no—
110#90783—21no—
111#90783—21no—
112#90783—21no—
113#90782—21no—
114#90782—21no—
115#90782—21no—
116#90782—21no—
117#90782—21no—
118#90777—21no—
119#90777C2311yescharacteristic
120#90777C2311yescharacteristic
121#90777C2311yescharacteristic
122C2×#6597—21no—
123C2×#6597C2311yescharacteristic
124Dic116—21no—
125D232—21no—
126#90773—21no—
127#90773—21no—
128#90773—21no—
129#90773—21no—
130#90772—21no—
131#90772—21no—
132#90772—21no—
133#90772—21no—
134#90772—21no—
135#90772D411yescharacteristic
136#90772D411yescharacteristic
137#90772D411yescharacteristic
138#90772D411yescharacteristic
139C8×D29—21no—
140C8×D29—21no—
141C8×D29—21no—
142C8×D29—21no—
143C8×D29—21no—
144C23×C29—21no—
145C23×C29—21no—
146C22×D29—21no—
147C22×D29—21no—
148C22×D29—41no—
149C22×D29—41no—
150C22×D29—82no—
151C22×D29#4111yescharacteristic
152Q8×C29C2×D411yescharacteristic
153D4×C29—21no—
154D4×C29—21no—
155D4×C29—21no—
156D4×C29—21no—
157D4×C29—21no—
158D4×C29C2×D411yescharacteristic
159C2×C4×C29—21no—
160C2×C4×C29C2411yescharacteristic
161C22⋊D29—21no—
162C22⋊D29—21no—
163C22⋊D29—21no—
164C22⋊D29—41no—
165C22⋊D29—41no—
166C22⋊D29—41no—
167C22⋊D29—41no—
168C2×Dic29—21no—
169C2×Dic29—21no—
170C2×Dic29—41no—
171C2×Dic29—41no—
172C2×Dic29#4111yescharacteristic
173D116—21no—
174D116—21no—
175D116—21no—
176D116C2×D411yescharacteristic
177C4×D29—21no—
178C4×D29—21no—
179C4×D29—21no—
180C4×D29C2×D411yescharacteristic
181C4×D29C2×D411yescharacteristic
182C4×D29C2×D411yescharacteristic
183C4×D29C2×D411yescharacteristic
184Dic58—21no—
185Dic58—21no—
186Dic58—21no—
187Dic58C2×D411yescharacteristic
188C8×C29—21no—
189C8×C29—21no—
190C8×C29—21no—
191C8×C29C2×D411yescharacteristic
192C8×C29C2×D411yescharacteristic
193#6597—21no—
194#6597—21no—
195#6597—21no—
196#6597C2×D411yescharacteristic
197#6597C2×D411yescharacteristic
198#3068—291nomaximal, Sylow
199C22×C29—21no—
200C22×C29—41no—
201C22×C29—41no—
202C22×C29—81no—
203C22×C29—81no—
204C22×C29C2×#4111yescharacteristic
205C2×D29—21no—
206C2×D29—41no—
207C2×D29—41no—
208C2×D29—81no—
209C2×D29—82no—
210C2×D29—82no—
211C2×D29—81no—
212C2×D29#12111yescharacteristic
213C2×D29#12111yescharacteristic
214C4×C29—21no—
215C4×C29C22×D411yescharacteristic
216C4×C29C22×D411yescharacteristic
217Dic29—21no—
218Dic29—41no—
219Dic29#12111yescharacteristic
220Dic29#12111yescharacteristic
221#575—291no—
222C2×AGL(1,ℤ/8)—291no—
223C2×AGL(1,ℤ/8)—291no—
224#568—291no—
225#472—291no—
226#472—291no—
227#472—291no—
228#472D2911yescharacteristic, complemented
229#471—291no—
230#471—291no—
231#471—291no—
232#471—291no—
233#430—291no—
234#413—291no—
235C2×#100—291no—
236C2×C29—21no—
237C2×C29—41no—
238C2×C29—81no—
239C2×C29—81no—
240C2×C29C2×#12111yescharacteristic
241D29—21no—
242D29—21no—
243D29—41no—
244D29—81no—
245C2×#41—291no—
246C2×#41—291no—
247C22×D4—291no—
248#137—291no—
249#137—291no—
250#137—581no—
251AGL(1,ℤ/8)—291no—
252AGL(1,ℤ/8)—291no—
253AGL(1,ℤ/8)—291no—
254AGL(1,ℤ/8)—291no—
255AGL(1,ℤ/8)—581no—
256AGL(1,ℤ/8)—581no—
257AGL(1,ℤ/8)—581no—
258AGL(1,ℤ/8)—581no—
259AGL(1,ℤ/8)—581no—
260AGL(1,ℤ/8)—581no—
261AGL(1,ℤ/8)—581no—
262AGL(1,ℤ/8)C2×D2911yescharacteristic
263AGL(1,ℤ/8)C2×D2911yescharacteristic, complemented
264#135—581no—
265#135—581no—
266C2×SD16—291no—
267C2×SD16—291no—
268C2×SD16—581no—
269C2×D8—291no—
270C2×D8—581no—
271C2×D8C2×D2911yescharacteristic, complemented
272#131—291no—
273#131—291no—
274#131—291no—
275#131—581no—
276#131—581no—
277#131C2×D2911yescharacteristic, complemented
278C2×M16—291no—
279C2×M16—291no—
280C2×M16—291no—
281C2×M16—291no—
282C2×M16—581no—
283C2×M16—581no—
284#108—291no—
285#108—291no—
286#108—291no—
287#108C2×D2911yescharacteristic, complemented
288#101—291no—
289#101—291no—
290#100—291no—
291#100—291no—
292#100—291no—
293#100—291no—
294#100C2×D2911yescharacteristic, complemented
295#100C2×D2911yescharacteristic, complemented
296C29#306811yescharacteristic, complemented, Sylow
297C24—581no—
298#41—291no—
299#41—291no—
300#41—291no—
301#41—581no—
302#41—581no—
303#41—581no—
304#41—581no—
305#41—581no—
306#41—581no—
307#41C22×D2911yescharacteristic
308C2×Q8—291no—
309C2×Q8—291no—
310C2×D4—291no—
311C2×D4—291no—
312C2×D4—581no—
313C2×D4—581no—
314C2×D4—581no—
315C2×D4—581no—
316C2×D4—581no—
317C2×D4—581no—
318C2×D4—581no—
319C2×D4C22×D2911yescharacteristic
320C2×D4C22×D2911yescharacteristic
321C22×C4—291no—
322C22×C4—581no—
323C22×C4—581no—
324Q16—581no—
325Q16—581no—
326SD16—21no—
327SD16—21no—
328SD16—581no—
329SD16—581no—
330SD16—581no—
331SD16—581no—
332SD16—581no—
333SD16—581no—
334SD16—581no—
335SD16—581no—
336SD16—581no—
337SD16—581no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—581no—
343D8—581no—
344D8—581no—
345D8—581no—
346D8—581no—
347D8—581no—
348M16—21no—
349M16—291no—
350M16—291no—
351M16—291no—
352M16—291no—
353M16—291no—
354M16—291no—
355M16—291no—
356M16—581no—
357M16—581no—
358M16—581no—
359M16—581no—
360M16—581no—
361M16—581no—
362M16C22×D2911yescharacteristic
363M16C22×D2911yescharacteristic
364M16C22×D2911yescharacteristic
365C2×C8—21no—
366C2×C8—291no—
367C2×C8—581no—
368C2×C8—581no—
369C2×C8—581no—
370C2×C8—581no—
371C2×C8—581no—
372C2×C8—581no—
373C2×C8C22×D2911yescharacteristic
374C23—21no—
375C23—21no—
376C23—291no—
377C23—581no—
378C23—581no—
379C23—1161no—
380C23—1161no—
381C23—2322no—
382Q8—291no—
383Q8—581no—
384Q8—581no—
385Q8—581no—
386Q8D4×D2911yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—291no—
393D4—581no—
394D4—581no—
395D4—581no—
396D4—581no—
397D4—581no—
398D4—581no—
399D4—1161no—
400D4—1161no—
401D4—1161no—
402D4—1161no—
403D4D4×D2911yescharacteristic
404C2×C4—21no—
405C2×C4—291no—
406C2×C4—291no—
407C2×C4—291no—
408C2×C4—291no—
409C2×C4—291no—
410C2×C4—581no—
411C2×C4—581no—
412C2×C4—581no—
413C2×C4—581no—
414C2×C4—581no—
415C2×C4—1161no—
416C2×C4—1161no—
417C2×C4C23×D2911yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—291no—
422C8—291no—
423C8—581no—
424C8—581no—
425C8—581no—
426C8D4×D2911yescharacteristic
427C8D4×D2911yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—291no—
434C22—291no—
435C22—581no—
436C22—1161no—
437C22—1161no—
438C22—2321no—
439C22—2321no—
440C22—2322no—
441C22—2322no—
442C22#41×D2911yescharacteristic
443C4—21no—
444C4—291no—
445C4—291no—
446C4—581no—
447C4—1161no—
448C4C2×D4×D2911yescharacteristic
449C4C2×D4×D2911yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—581no—
455C2—581no—
456C2—1161no—
457C2—2321no—
458C2#121×D2911yescharacteristic
subgroupG/H
centerC2#121×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#472C2
socleC2×C29C2×#121
hypercenter#472D29
hypocenterC29#3068
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #14271959C2semidirect
2. C2×#11745273C2semidirect
3. AGL(1,ℤ/8)×D29C2semidirect
4. #14271928C2semidirect
5. #14271432C2semidirect
6. #14271431C2semidirect
7. #14271430C2semidirect
8. #14271427C2semidirect
9. #14271426C2semidirect
10. #14271425C2semidirect
11. #14271424C2semidirect
12. #14271210C2semidirect
13. #14271144C2semidirect
14. #100×D29C2semidirect
15. C29×#472C2semidirect
20. C29×AGL(1,ℤ/8)C22semidirect
21. C2×D8×C29C22semidirect
22. C29×#131C22semidirect
23. #11745313C22semidirect
25. #11745310C22semidirect
26. C2×#90783C22semidirect
28. #11745294C22semidirect
29. C2×#90782C22semidirect
42. #11745271C22semidirect
45. #11745269C22semidirect
50. C2×#90773C22semidirect
52. #11745263C22semidirect
55. C29×#108C22semidirect
56. C29×#100C22semidirect
57. C29×#100C22semidirect
58. #11745181C22semidirect
59. #11745181C22semidirect
60. #11745172C22semidirect
61. #11745172C22semidirect
62. #11745171C22semidirect
63. #11745171C22semidirect
64. #11745170C22semidirect
65. #11745170C22semidirect
66. #11745167C22semidirect
228. #472D29semidirect
263. AGL(1,ℤ/8)C2×D29semidirect
271. C2×D8C2×D29semidirect
277. #131C2×D29semidirect
287. #108C2×D29semidirect
294. #100C2×D29semidirect
295. #100C2×D29semidirect
296. C29#3068semidirect

Maximal subgroups

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

subgroupindex
#142719592
C2×#117452732
AGL(1,ℤ/8)×D292
#142719282
#142714322
#142714312
#142714302
#142714272
#142714262
#142714252
#142714242
#142712102
#142711442
#100×D292
C29×#4722
#306829