Finite Group Databasebeta

← #16830973

Subgroups — #16830973

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×#11745311C211yesmaximal, characteristic, complemented
2C22×#90782C211yesmaximal, characteristic, complemented
3D4×D116C211yesmaximal, characteristic, complemented
4#14271553C211yesmaximal, characteristic, complemented
5#14271414C211yesmaximal, characteristic, complemented
6#14271394C211yesmaximal, characteristic, complemented
7#14271375C211yesmaximal, characteristic, complemented
8#14271358C211yesmaximal, characteristic, complemented
9#14271313C211yesmaximal, characteristic, complemented
10#14271308C211yesmaximal, characteristic, complemented
11#14271297C211yesmaximal, characteristic, complemented
12#14271296C211yesmaximal, characteristic, complemented
13#14271273C211yesmaximal, characteristic, complemented
14#14271235C211yesmaximal, characteristic, complemented
15C29×#546C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic, complemented
17C22×D4×C29C2211yescharacteristic, complemented
18C2×D4×D29—21no—
19C22×D116C2211yescharacteristic
20C22×D116C2211yescharacteristic, complemented
21C29×#124C2211yescharacteristic, complemented
22C29×#123—21no—
23C29×C22⋊Q8C2211yescharacteristic, complemented
24C29×#121—21no—
25C29×#121C2211yescharacteristic, complemented
26C29×#120—21no—
27C4×D4×C29C2211yescharacteristic
28C4×D4×C29C2211yescharacteristic, complemented
29C2×C22⋊C4×C29—21no—
30#11745311—24no—
31C2×#90784C2211yescharacteristic, complemented
32C2×#90784C2211yescharacteristic, complemented
33C2×#90784C2211yescharacteristic, complemented
34C2×#90782—24no—
35C2×#90782C2211yescharacteristic
36C2×#90782C2211yescharacteristic
37C2×#90782C2211yescharacteristic, complemented
38C2×#90782C2211yescharacteristic, complemented
39C2×#90782C2211yescharacteristic, complemented
40C2×#90782C2211yescharacteristic, complemented
41#11745292C2211yescharacteristic
42#11745292C2211yescharacteristic, complemented
43C2×#90777C2211yescharacteristic, complemented
44C22×#6597C2211yescharacteristic, complemented
45#11745257—21no—
46#11745244—21no—
47#11745244—21no—
48#11745236C2211yescharacteristic, complemented
49C4×D116C2211yescharacteristic, complemented
50#11745183C2211yescharacteristic, complemented
51#11745180C2211yescharacteristic, complemented
52#11745180C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic, complemented
54C22⋊#6597C2211yescharacteristic, complemented
55#11745157C2211yescharacteristic, complemented
56#11745157C2211yescharacteristic, complemented
57#11745157C2211yescharacteristic, complemented
58#11745156C2211yescharacteristic, complemented
59C4⋊#6597C2211yescharacteristic, complemented
60#11745151C2211yescharacteristic, complemented
61C24×C29—21no—
62C23×D29—21no—
63C23×D29—21no—
64#41×C29—22no—
65#41×C29—41no—
66C2×Q8×C29C2311yescharacteristic
67C2×D4×C29—22no—
68C2×D4×C29—21no—
69C2×D4×C29—21no—
70C2×D4×C29—21no—
71C2×D4×C29—41no—
72C2×D4×C29D411yescharacteristic
73C2×D4×C29C2311yescharacteristic
74C2×D4×C29C2311yescharacteristic
75C22×C4×C29—21no—
76C22×C4×C29—21no—
77C22×C4×C29C2311yescharacteristic
78C22×C4×C29C2311yescharacteristic
79C2×C22⋊D29—21no—
80C2×C22⋊D29—21no—
81D4×D29—44no—
82C2×D116—22no—
83C2×D116—22no—
84C2×D116—21no—
85C2×D116—21no—
86C2×D116—21no—
87C2×D116—21no—
88C2×D116—41no—
89C2×D116—41no—
90C2×D116C2311yescharacteristic
91C2×D116C2311yescharacteristic
92C2×D116C2311yescharacteristic
93C2×C4×D29—21no—
94C4⋊C4×C29—21no—
95C4⋊C4×C29D411yescharacteristic
96C4⋊C4×C29C2311yescharacteristic
97C22⋊C4×C29—21no—
98C22⋊C4×C29—21no—
99C22⋊C4×C29—21no—
100C22⋊C4×C29—21no—
101C22⋊C4×C29—21no—
102C22⋊C4×C29D411yescharacteristic
103C22⋊C4×C29D411yescharacteristic
104C42×C29C2311yescharacteristic
105#90784—22no—
106#90784—22no—
107#90784—41no—
108#90782—22no—
109#90782—22no—
110#90782—22no—
111#90782—22no—
112#90782—41no—
113#90782D414yescomplemented
114#90781—21no—
115#90781—21no—
116#90781—21no—
117C4⋊Dic29C2311yescharacteristic
118#90777—22no—
119C2×#6597—22no—
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122C2×#6597C2311yescharacteristic
123C2×#6597C2311yescharacteristic
124C23×C29—21no—
125C23×C29—21no—
126C23×C29—42no—
127C23×C29—42no—
128C23×C29—81no—
129C23×C29C2×D411yescharacteristic
130C23×C29C2×D411yescharacteristic
131C22×D29—21no—
132C22×D29—21no—
133C22×D29—21no—
134C22×D29—42no—
135C22×D29—42no—
136C22×D29—42no—
137C22×D29—42no—
138C22×D29—81no—
139C22×D29—81no—
140Q8×C29—21no—
141Q8×C29—21no—
142D4×C29—22no—
143D4×C29—21no—
144D4×C29—21no—
145D4×C29—21no—
146D4×C29—42no—
147D4×C29—42no—
148D4×C29—81no—
149D4×C29C2×D412yescomplemented
150C2×C4×C29—21no—
151C2×C4×C29—21no—
152C2×C4×C29—21no—
153C2×C4×C29—21no—
154C2×C4×C29—21no—
155C2×C4×C29—21no—
156C2×C4×C29—42no—
157C2×C4×C29—81no—
158C2×C4×C29C2×D411yescharacteristic
159C2×C4×C29C2×D411yescharacteristic
160C2×C4×C29C2×D411yescharacteristic
161C2×C4×C29C2×D411yescharacteristic
162C2×C4×C29C2411yescharacteristic
163C22⋊D29—42no—
164C22⋊D29—42no—
165C2×Dic29—21no—
166D116—21no—
167D116—21no—
168D116—21no—
169D116—21no—
170D116—21no—
171D116—42no—
172D116—42no—
173D116—42no—
174D116—81no—
175D116C2×D412yes—
176C4×D29—42no—
177#6597—21no—
178#6597—21no—
179#6597—21no—
180#6597C2×D412yescomplemented
181#3264—291nomaximal, Sylow
182C22×C29—21no—
183C22×C29—21no—
184C22×C29—21no—
185C22×C29—22no—
186C22×C29—21no—
187C22×C29—21no—
188C22×C29—41no—
189C22×C29—41no—
190C22×C29—41no—
191C22×C29—41no—
192C22×C29—41no—
193C22×C29—84no—
194C22×C29AGL(1,ℤ/8)12yescomplemented
195C22×C29C22×D411yescharacteristic
196C2×D29—22no—
197C2×D29—41no—
198C2×D29—41no—
199C2×D29—41no—
200C2×D29—41no—
201C2×D29—41no—
202C2×D29—41no—
203C2×D29—41no—
204C2×D29—41no—
205C2×D29—84no—
206C2×D29—84no—
207C4×C29—21no—
208C4×C29—21no—
209C4×C29—21no—
210C4×C29—21no—
211C4×C29—41no—
212C4×C29—41no—
213C4×C29C22×D411yescharacteristic
214C4×C29H(5,𝔽2)11yescharacteristic
215Dic29—41no—
216C2×AGL(1,ℤ/8)—291no—
217C22×D8—291no—
218#546D2911yescharacteristic, complemented
219D42—291no—
220#496—291no—
221#469—291no—
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—22no—
232C2×C29—41no—
233C2×C29—42no—
234C2×C29—81no—
235C2×C29D4211yescharacteristic
236C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
237C2×C29#57611yescharacteristic
238D29—42no—
239D29—81no—
240D29—81no—
241C2×#41C2×D2911yescharacteristic, complemented
242C22×D4—291no—
243C22×D4—291no—
244C22×D4—581no—
245C22×D4C2×D2911yescharacteristic, complemented
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—291no—
256C2×D8—584no—
257C2×M16—291no—
258C22×C8—291no—
259#127—291no—
260#124C2×D2911yescharacteristic, complemented
261#123—21no—
262C22⋊Q8C2×D2911yescharacteristic, complemented
263#121—21no—
264#121—291no—
265#121—291no—
266#121—581no—
267#121C2×D2911yescharacteristic, complemented
268#120—21no—
269#120—581no—
270#120—581no—
271C4×D4—291no—
272C4×D4C2×D2911yescharacteristic
273C4×D4C2×D2911yescharacteristic, complemented
274C2×C22⋊C4—21no—
275#106—291no—
276C4⋊C8—291no—
277#103—291no—
278#102—291no—
279#102—291no—
280#102—291no—
281#102—291no—
282#102—291no—
283C22⋊C8—291no—
284C22⋊C8—291no—
285#97—291no—
286C29#326411yescharacteristic, complemented, Sylow
287C24—21no—
288C24—581no—
289C24—581no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D2911yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—291no—
299C2×D4—291no—
300C2×D4—291no—
301C2×D4—582no—
302C2×D4—582no—
303C2×D4—581no—
304C2×D4—581no—
305C2×D4—581no—
306C2×D4—581no—
307C2×D4—581no—
308C2×D4—581no—
309C2×D4—1161no—
310C2×D4—1161no—
311C2×D4—1164no—
312C2×D4C22⋊D2911yescharacteristic
313C2×D4C22×D2911yescharacteristic
314C2×D4C22×D2911yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—581no—
318C22×C4C22×D2911yescharacteristic
319C22×C4C22×D2911yescharacteristic
320SD16—582no—
321SD16—582no—
322SD16—1161no—
323D8—294no—
324D8—582no—
325D8—582no—
326D8—582no—
327D8—582no—
328D8—1161no—
329M16—582no—
330C2×C8—291no—
331C2×C8—291no—
332C2×C8—291no—
333C2×C8—291no—
334C2×C8—582no—
335C4⋊C4—21no—
336C4⋊C4—291no—
337C4⋊C4C22⋊D2911yescharacteristic
338C4⋊C4C22×D2911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—581no—
345C22⋊C4—581no—
346C22⋊C4—581no—
347C22⋊C4C22⋊D2911yescharacteristic
348C22⋊C4C22⋊D2911yescharacteristic
349C42C22×D2911yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—581no—
356C23—581no—
357C23—581no—
358C23—1162no—
359C23—1162no—
360C23—1162no—
361C23—1162no—
362C23—2321no—
363C23—2321no—
364C23C2×C22⋊D2911yescharacteristic
365C23C2×C22⋊D2911yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—292no—
376D4—581no—
377D4—581no—
378D4—581no—
379D4—581no—
380D4—581no—
381D4—1162no—
382D4—1162no—
383D4—1162no—
384D4—1162no—
385D4—1162no—
386D4—2321no—
387D4D4×D2912yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—581no—
397C2×C4—1162no—
398C2×C4C2×C22⋊D2911yescharacteristic
399C2×C4C2×C22⋊D2911yescharacteristic
400C2×C4C2×C22⋊D2911yescharacteristic
401C2×C4C2×C22⋊D2911yescharacteristic
402C2×C4C23×D2911yescharacteristic
403C8—292no—
404C8—581no—
405C8—581no—
406C8—581no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—582no—
420C22—1161no—
421C22—1161no—
422C22—1161no—
423C22—1161no—
424C22—1161no—
425C22—1161no—
426C22—1161no—
427C22—1161no—
428C22—2324no—
429C22—2324no—
430C22#1174531112yescomplemented
431C22C22×C22⋊D2911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—1161no—
439C4C2×D4×D2911yescharacteristic
440C4#1174535911yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—1162no—
446C2—2321no—
447C2—2321no—
448C2D4×C22⋊D2911yescharacteristic
449C2C2×#1174531111yescharacteristic
450C2#1427196611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#546C2
socleC22×C29C22×D4
hypercenter#546D29
hypocenterC29#3264
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#11745311C2semidirect
2. C22×#90782C2semidirect
3. D4×D116C2semidirect
4. #14271553C2semidirect
5. #14271414C2semidirect
6. #14271394C2semidirect
7. #14271375C2semidirect
8. #14271358C2semidirect
9. #14271313C2semidirect
10. #14271308C2semidirect
11. #14271297C2semidirect
12. #14271296C2semidirect
13. #14271273C2semidirect
14. #14271235C2semidirect
15. C29×#546C2semidirect
16. C2×#41×C29C22semidirect
17. C22×D4×C29C22semidirect
20. C22×D116C22semidirect
21. C29×#124C22semidirect
23. C29×C22⋊Q8C22semidirect
25. C29×#121C22semidirect
28. C4×D4×C29C22semidirect
31. C2×#90784C22semidirect
32. C2×#90784C22semidirect
33. C2×#90784C22semidirect
37. C2×#90782C22semidirect
38. C2×#90782C22semidirect
39. C2×#90782C22semidirect
40. C2×#90782C22semidirect
42. #11745292C22semidirect
43. C2×#90777C22semidirect
44. C22×#6597C22semidirect
48. #11745236C22semidirect
49. C4×D116C22semidirect
50. #11745183C22semidirect
51. #11745180C22semidirect
52. #11745180C22semidirect
53. C22⋊#6597C22semidirect
54. C22⋊#6597C22semidirect
55. #11745157C22semidirect
56. #11745157C22semidirect
57. #11745157C22semidirect
58. #11745156C22semidirect
59. C4⋊#6597C22semidirect
60. #11745151C22semidirect
113. #90782D4semidirect
149. D4×C29C2×D4semidirect
180. #6597C2×D4semidirect
194. C22×C29AGL(1,ℤ/8)semidirect
218. #546D29semidirect
241. C2×#41C2×D29semidirect
245. C22×D4C2×D29semidirect
260. #124C2×D29semidirect
262. C22⋊Q8C2×D29semidirect
267. #121C2×D29semidirect
273. C4×D4C2×D29semidirect
286. C29#3264semidirect
387. D4D4×D29semidirect
430. C22#11745311semidirect

Maximal subgroups

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

subgroupindex
C2×#117453112
C22×#907822
D4×D1162
#142715532
#142714142
#142713942
#142713752
#142713582
#142713132
#142713082
#142712972
#142712962
#142712732
#142712352
C29×#5462
#326429