Finite Group Databasebeta

← #16830979

Subgroups — #16830979

Order 3712.

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×#11745270C211yesmaximal, characteristic, complemented
2C2×D8×D29C211yesmaximal, characteristic, complemented
3#14271810C211yesmaximal, characteristic, complemented
4#14271797C211yesmaximal, characteristic, complemented
5#14271551C211yesmaximal, characteristic, complemented
6#14271393C211yesmaximal, characteristic, complemented
7#14271377C211yesmaximal, characteristic, complemented
8#14271357C211yesmaximal, characteristic, complemented
9#14271315C211yesmaximal, characteristic, complemented
10#14271306C211yesmaximal, characteristic, complemented
11#14271298C211yesmaximal, characteristic, complemented
12#14271296C211yesmaximal, characteristic, complemented
13#14271274C211yesmaximal, characteristic, complemented
14#14271232C211yesmaximal, characteristic, complemented
15C29×#469C211yesmaximal, characteristic, complemented
16C2×D4×D29—21no—
17C2×D4×D29C2211yescharacteristic
18C2×D4×D29C2211yescharacteristic, complemented
19C2×#90805C2211yescharacteristic, complemented
20C22×D116C2211yescharacteristic, complemented
21C2×D8×C29C2211yescharacteristic, complemented
22C2×M16×C29C2211yescharacteristic, complemented
23C29×#121C2211yescharacteristic, complemented
24C29×#121C2211yescharacteristic, complemented
25#11745302C2211yescharacteristic, complemented
26#11745300—21no—
27#11745299—21no—
28#11745299—21no—
29#11745298C2211yescharacteristic, complemented
30C2×#90783C2211yescharacteristic, complemented
31C2×#90782C2211yescharacteristic
32C2×#90782C2211yescharacteristic, complemented
33C2×#90782C2211yescharacteristic, complemented
34C4×C22⋊D29C2211yescharacteristic
35D8×D29—24no—
36#11745270—24no—
37C2×D232C2211yescharacteristic
38C2×D232C2211yescharacteristic, complemented
39C2×#90773C2211yescharacteristic, complemented
40C2×#90773C2211yescharacteristic, complemented
41C2×C8×D29C2211yescharacteristic, complemented
42#11745259C2211yescharacteristic, complemented
43#11745257C2211yescharacteristic
44#11745255C2211yescharacteristic, complemented
45#11745255C2211yescharacteristic, complemented
46#11745246—21no—
47#11745245—21no—
48#11745244—21no—
49C22⋊C4×D29—21no—
50C29×#106C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic, complemented
52C29×#102C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic, complemented
54#11745168C2211yescharacteristic, complemented
55#11745163C2211yescharacteristic, complemented
56#11745162C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745157C2211yescharacteristic, complemented
59#11745157C2211yescharacteristic, complemented
60#11745157C2211yescharacteristic, complemented
61C23×D29—21no—
62C23×D29—21no—
63C23×D29—21no—
64C2×D4×C29—21no—
65C2×D4×C29—21no—
66C2×D4×C29C2311yescharacteristic
67C2×D4×C29C2311yescharacteristic
68C22×C4×C29C2311yescharacteristic
69C2×C22⋊D29—21no—
70C2×C22⋊D29—21no—
71C2×C22⋊D29—21no—
72C2×C22⋊D29—21no—
73C2×C22⋊D29—21no—
74C2×C22⋊D29D411yescharacteristic
75D4×D29—22no—
76D4×D29—22no—
77D4×D29—41no—
78D4×D29—41no—
79D4×D29—44no—
80#90805—22no—
81#90805—41no—
82C2×D116—22no—
83C2×D116—21no—
84C2×D116—21no—
85C2×D116—41no—
86C2×D116C2311yescharacteristic
87C2×D116C2311yescharacteristic
88C2×D116C2311yescharacteristic
89C2×C4×D29—21no—
90C2×C4×D29—21no—
91C2×C4×D29—21no—
92C2×C4×D29C2311yescharacteristic
93C2×Dic58C2311yescharacteristic
94D8×C29—22no—
95M16×C29—22no—
96C2×C8×C29C2311yescharacteristic
97C2×C8×C29C2311yescharacteristic
98C4⋊C4×C29C2311yescharacteristic
99C4⋊C4×C29C2311yescharacteristic
100C22⋊C4×C29—21no—
101C22⋊C4×C29—21no—
102C22⋊Dic29—21no—
103C22⋊Dic29—21no—
104C22⋊Dic29D411yescharacteristic
105#90783—41no—
106#90782—22no—
107#90782—22no—
108#90782—41no—
109#90781—21no—
110#90781—21no—
111#90781—21no—
112#90781—21no—
113#90781D411yescharacteristic
114C4⋊Dic29—21no—
115#90779D411yescharacteristic
116C4×Dic29C2311yescharacteristic
117C2×#6597C2311yescharacteristic
118C2×#6597C2311yescharacteristic
119D232—22no—
120D232D414yescomplemented
121#90773—22no—
122#90773—22no—
123C8×D29—22no—
124C23×C29—21no—
125C23×C29—21no—
126C23×C29C2×D411yescharacteristic
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—21no—
130C22×D29—42no—
131C22×D29—42no—
132C22×D29—42no—
133C22×D29—42no—
134C22×D29—42no—
135C22×D29—42no—
136C22×D29—81no—
137C22×D29—81no—
138C22×D29—81no—
139C22×D29C2×D411yescharacteristic
140D4×C29—21no—
141D4×C29—21no—
142D4×C29—21no—
143D4×C29—21no—
144D4×C29—42no—
145D4×C29—81no—
146C2×C4×C29—21no—
147C2×C4×C29—21no—
148C2×C4×C29—21no—
149C2×C4×C29—21no—
150C2×C4×C29C2×D411yescharacteristic
151C2×C4×C29C2411yescharacteristic
152C22⋊D29—22no—
153C22⋊D29—42no—
154C22⋊D29—42no—
155C22⋊D29—42no—
156C22⋊D29—42no—
157C22⋊D29—81no—
158C2×Dic29—21no—
159C2×Dic29C2×D411yescharacteristic
160C2×Dic29C2×D411yescharacteristic
161C2×Dic29C2×D411yescharacteristic
162D116—21no—
163D116—21no—
164D116—21no—
165D116—21no—
166D116—42no—
167D116—42no—
168D116C2×D412yes—
169D116C2×D412yescomplemented
170C4×D29—21no—
171C4×D29—21no—
172C4×D29—42no—
173C4×D29—42no—
174C4×D29—81no—
175Dic58—21no—
176Dic58—21no—
177C8×C29—21no—
178C8×C29C2×D412yescomplemented
179#6597—21no—
180#6597—21no—
181#3264—291nomaximal, Sylow
182C22×C29—21no—
183C22×C29—21no—
184C22×C29—21no—
185C22×C29—41no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29—41no—
189C22×C29—41no—
190C22×C29—41no—
191C22×C29C22×D411yescharacteristic
192C2×D29—21no—
193C2×D29—21no—
194C2×D29—22no—
195C2×D29—22no—
196C2×D29—41no—
197C2×D29—41no—
198C2×D29—41no—
199C2×D29—41no—
200C2×D29—41no—
201C2×D29—41no—
202C2×D29—41no—
203C2×D29—84no—
204C2×D29—84no—
205C2×D29—84no—
206C2×D29AGL(1,ℤ/8)12yescomplemented
207C4×C29—21no—
208C4×C29—41no—
209C4×C29—41no—
210C4×C29C22×D411yescharacteristic
211C4×C29H(5,𝔽2)11yescharacteristic
212Dic29—21no—
213Dic29—21no—
214Dic29—21no—
215Dic29—41no—
216C2×AGL(1,ℤ/8)—291no—
217C22×D8—291no—
218#546—291no—
219D42—291no—
220#496—291no—
221#469D2911yescharacteristic, complemented
222#465—291no—
223#463—291no—
224#459—291no—
225#450—291no—
226#449—291no—
227#447—291no—
228#447—291no—
229#442—291no—
230#435—291no—
231C2×C29—41no—
232C2×C29—81no—
233C2×C29—81no—
234C2×C29D4211yescharacteristic
235C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C29#57611yescharacteristic
237D29—22no—
238D29—42no—
239D29—42no—
240D29—81no—
241C2×#41—291no—
242C22×D4—291no—
243C22×D4—291no—
244C22×D4—291no—
245C22×D4—581no—
246AGL(1,ℤ/8)—584no—
247C2×SD16—291no—
248C2×SD16—291no—
249C2×SD16—291no—
250C2×D8—291no—
251C2×D8—291no—
252C2×D8—291no—
253C2×D8—291no—
254C2×D8—291no—
255C2×D8—584no—
256C2×D8C2×D2911yescharacteristic, complemented
257C2×M16C2×D2911yescharacteristic, complemented
258C22×C8—291no—
259#127—291no—
260#124—291no—
261#123—581no—
262C22⋊Q8—291no—
263#121—291no—
264#121—581no—
265#121—581no—
266#121C2×D2911yescharacteristic, complemented
267#121C2×D2911yescharacteristic, complemented
268#120—581no—
269#120—581no—
270#120—581no—
271C4×D4—291no—
272C4×D4—291no—
273C4×D4—291no—
274C2×C22⋊C4—581no—
275#106C2×D2911yescharacteristic, complemented
276C4⋊C8—291no—
277#103—291no—
278#102—291no—
279#102—291no—
280#102—291no—
281#102C2×D2911yescharacteristic, complemented
282#102C2×D2911yescharacteristic, complemented
283C22⋊C8—291no—
284C22⋊C8—291no—
285#97—291no—
286C29#326411yescharacteristic, complemented, Sylow
287C24—581no—
288C24—581no—
289C24—581no—
290#41—582no—
291#41—1161no—
292C2×Q8—291no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—291no—
296C2×D4—291no—
297C2×D4—291no—
298C2×D4—291no—
299C2×D4—582no—
300C2×D4—581no—
301C2×D4—581no—
302C2×D4—581no—
303C2×D4—581no—
304C2×D4—581no—
305C2×D4—582no—
306C2×D4—582no—
307C2×D4—581no—
308C2×D4—581no—
309C2×D4—1161no—
310C2×D4—1161no—
311C2×D4—1161no—
312C2×D4—1164no—
313C2×D4C22×D2911yescharacteristic
314C2×D4C22×D2911yescharacteristic
315C22×C4—291no—
316C22×C4—581no—
317C22×C4—581no—
318C22×C4—581no—
319C22×C4C22×D2911yescharacteristic
320SD16—582no—
321SD16—582no—
322SD16—1161no—
323D8—22no—
324D8—294no—
325D8—582no—
326D8—582no—
327D8—582no—
328D8—1161no—
329M16—22no—
330C2×C8—291no—
331C2×C8—291no—
332C2×C8—582no—
333C2×C8C22×D2911yescharacteristic
334C2×C8C22×D2911yescharacteristic
335C4⋊C4—291no—
336C4⋊C4—581no—
337C4⋊C4C22×D2911yescharacteristic
338C4⋊C4C22×D2911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—291no—
342C22⋊C4—291no—
343C22⋊C4—581no—
344C22⋊C4—581no—
345C22⋊C4—581no—
346C22⋊C4—581no—
347C22⋊C4—581no—
348C22⋊C4—581no—
349C42—291no—
350C23—21no—
351C23—21no—
352C23—291no—
353C23—581no—
354C23—581no—
355C23—581no—
356C23—1162no—
357C23—1162no—
358C23—1162no—
359C23—1162no—
360C23—1162no—
361C23—1162no—
362C23—2321no—
363C23—2321no—
364C23—2321no—
365C23D4×D2911yescharacteristic
366Q8—581no—
367Q8—581no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—292no—
375D4—292no—
376D4—582no—
377D4—581no—
378D4—581no—
379D4—581no—
380D4—581no—
381D4—1162no—
382D4—1162no—
383D4—1162no—
384D4—1162no—
385D4—1162no—
386D4—1162no—
387D4—2321no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—291no—
393C2×C4—291no—
394C2×C4—291no—
395C2×C4—581no—
396C2×C4—581no—
397C2×C4—581no—
398C2×C4—1162no—
399C2×C4—1162no—
400C2×C4—2321no—
401C2×C4D4×D2911yescharacteristic
402C2×C4C23×D2911yescharacteristic
403C8—21no—
404C8—581no—
405C8—581no—
406C8D4×D2912yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—292no—
417C22—581no—
418C22—581no—
419C22—582no—
420C22—582no—
421C22—1161no—
422C22—1161no—
423C22—1161no—
424C22—1161no—
425C22—1161no—
426C22—1161no—
427C22—1161no—
428C22—2324no—
429C22—2324no—
430C22—2324no—
431C22C2×D4×D2911yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—581no—
436C4—581no—
437C4—581no—
438C4—1161no—
439C4C2×D4×D2911yescharacteristic
440C4#1174536411yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—582no—
445C2—1162no—
446C2—1162no—
447C2—2321no—
448C2#1427179711yescharacteristic
449C2AGL(1,ℤ/8)×D2911yescharacteristic
450C2#1427196711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#469C2
socleC22×C29C22×D4
hypercenter#469D29
hypocenterC29#3264
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#11745270C2semidirect
2. C2×D8×D29C2semidirect
3. #14271810C2semidirect
4. #14271797C2semidirect
5. #14271551C2semidirect
6. #14271393C2semidirect
7. #14271377C2semidirect
8. #14271357C2semidirect
9. #14271315C2semidirect
10. #14271306C2semidirect
11. #14271298C2semidirect
12. #14271296C2semidirect
13. #14271274C2semidirect
14. #14271232C2semidirect
15. C29×#469C2semidirect
18. C2×D4×D29C22semidirect
19. C2×#90805C22semidirect
20. C22×D116C22semidirect
21. C2×D8×C29C22semidirect
22. C2×M16×C29C22semidirect
23. C29×#121C22semidirect
24. C29×#121C22semidirect
25. #11745302C22semidirect
29. #11745298C22semidirect
30. C2×#90783C22semidirect
32. C2×#90782C22semidirect
33. C2×#90782C22semidirect
38. C2×D232C22semidirect
39. C2×#90773C22semidirect
40. C2×#90773C22semidirect
41. C2×C8×D29C22semidirect
42. #11745259C22semidirect
44. #11745255C22semidirect
45. #11745255C22semidirect
50. C29×#106C22semidirect
51. C29×#102C22semidirect
52. C29×#102C22semidirect
53. C22⋊#6597C22semidirect
54. #11745168C22semidirect
55. #11745163C22semidirect
56. #11745162C22semidirect
57. #11745158C22semidirect
58. #11745157C22semidirect
59. #11745157C22semidirect
60. #11745157C22semidirect
120. D232D4semidirect
169. D116C2×D4semidirect
178. C8×C29C2×D4semidirect
206. C2×D29AGL(1,ℤ/8)semidirect
221. #469D29semidirect
256. C2×D8C2×D29semidirect
257. C2×M16C2×D29semidirect
266. #121C2×D29semidirect
267. #121C2×D29semidirect
275. #106C2×D29semidirect
281. #102C2×D29semidirect
282. #102C2×D29semidirect
286. C29#3264semidirect
406. C8D4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452702
C2×D8×D292
#142718102
#142717972
#142715512
#142713932
#142713772
#142713572
#142713152
#142713062
#142712982
#142712962
#142712742
#142712322
C29×#4692
#326429