Finite Group Databasebeta

← #16830974

Subgroups — #16830974

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×#11745273C211yesmaximal, characteristic, complemented
2C2×D8×D29C211yesmaximal, characteristic, complemented
3#14271808C211yesmaximal, characteristic, complemented
4D4×D116C211yesmaximal, characteristic, complemented
5#14271552C211yesmaximal, characteristic, complemented
6#14271417C211yesmaximal, characteristic, complemented
7#14271393C211yesmaximal, characteristic, complemented
8#14271378C211yesmaximal, characteristic, complemented
9#14271358C211yesmaximal, characteristic, complemented
10#14271315C211yesmaximal, characteristic, complemented
11#14271309C211yesmaximal, characteristic, complemented
12#14271298C211yesmaximal, characteristic, complemented
13#14271298C211yesmaximal, characteristic, complemented
14#14271236C211yesmaximal, characteristic, complemented
15C29×#442C211yesmaximal, characteristic, complemented
16C2×#90807C2211yescharacteristic, complemented
17C2×D4×D29C2211yescharacteristic
18C2×D4×D29C2211yescharacteristic, complemented
19C2×D4×D29C2211yescharacteristic, complemented
20C22×D116—21no—
21C2×D8×C29C2211yescharacteristic
22C4×D4×C29C2211yescharacteristic, complemented
23C4×D4×C29C2211yescharacteristic, complemented
24C2×#90783C2211yescharacteristic, complemented
25C2×#90782C2211yescharacteristic
26C2×#90782C2211yescharacteristic, complemented
27C2×#90782C2211yescharacteristic, complemented
28#11745292—21no—
29#11745292—21no—
30C2×#90781—21no—
31#11745273—24no—
32D8×D29—24no—
33C2×D232C2211yescharacteristic, complemented
34C2×D232C2211yescharacteristic, complemented
35C2×#90773C2211yescharacteristic, complemented
36C2×#90773C2211yescharacteristic, complemented
37C2×#90772C2211yescharacteristic, complemented
38C2×C8×D29C2211yescharacteristic, complemented
39#11745259C2211yescharacteristic, complemented
40#11745257C2211yescharacteristic
41#11745257C2211yescharacteristic, complemented
42#11745257C2211yescharacteristic, complemented
43#11745248—21no—
44#11745244—21no—
45#11745244—21no—
46#11745244—21no—
47#11745237C2211yescharacteristic, complemented
48#11745236C2211yescharacteristic, complemented
49C4×D116C2211yescharacteristic
50C29×#106C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic, complemented
52C29×#102C2211yescharacteristic, complemented
53C29×#97C2211yescharacteristic, complemented
54#11745168C2211yescharacteristic, complemented
55#11745168C2211yescharacteristic, complemented
56#11745158C2211yescharacteristic, complemented
57#11745157C2211yescharacteristic, complemented
58#11745157C2211yescharacteristic, complemented
59#11745157C2211yescharacteristic, complemented
60C4⋊#6597C2211yescharacteristic, complemented
61C23×D29—21no—
62C23×D29—21no—
63C23×D29—21no—
64C2×D4×C29C2311yescharacteristic
65C2×D4×C29C2311yescharacteristic
66C22×C4×C29—21no—
67C22×C4×C29—21no—
68C2×C22⋊D29—21no—
69C2×C22⋊D29—21no—
70C2×C22⋊D29—21no—
71C2×C22⋊D29—21no—
72C22×Dic29—21no—
73#90807—22no—
74#90807—41no—
75D4×D29—22no—
76D4×D29—22no—
77D4×D29—22no—
78D4×D29—41no—
79D4×D29—41no—
80D4×D29—41no—
81C2×D116—21no—
82C2×D116—21no—
83C2×D116—21no—
84C2×D116—21no—
85C2×D116—21no—
86C2×D116—44no—
87C2×D116D411yescharacteristic
88C2×D116C2311yescharacteristic
89C2×D116C2311yescharacteristic
90C2×D116C2311yescharacteristic
91C2×C4×D29C2311yescharacteristic
92C2×C4×D29C2311yescharacteristic
93C2×Dic58C2311yescharacteristic
94D8×C29D414yescomplemented
95C2×C8×C29C2311yescharacteristic
96C2×C8×C29C2311yescharacteristic
97C4⋊C4×C29C2311yescharacteristic
98C4⋊C4×C29C2311yescharacteristic
99C22⋊C4×C29—21no—
100C22⋊C4×C29—21no—
101C42×C29C2311yescharacteristic
102#90783—22no—
103#90782—22no—
104#90782—22no—
105#90782—22no—
106#90781—21no—
107#90781—21no—
108#90781—21no—
109#90781—21no—
110#90781—21no—
111#90781—21no—
112#90781D411yescharacteristic
113#90781D411yescharacteristic
114C4⋊Dic29—21no—
115C4⋊Dic29D411yescharacteristic
116C2×#6597C2311yescharacteristic
117C2×#6597C2311yescharacteristic
118D232—22no—
119D232—41no—
120#90773—22no—
121#90773—41no—
122#90772—22no—
123C8×D29—22no—
124C23×C29—21no—
125C23×C29—21no—
126C22×D29—21no—
127C22×D29—21no—
128C22×D29—21no—
129C22×D29—42no—
130C22×D29—42no—
131C22×D29—42no—
132C22×D29—42no—
133C22×D29—42no—
134C22×D29—42no—
135C22×D29—81no—
136C22×D29—81no—
137C22×D29—81no—
138C22×D29C2×D411yescharacteristic
139C22×D29C2×D411yescharacteristic
140D4×C29—21no—
141D4×C29—21no—
142D4×C29C2×D412yes—
143D4×C29C2×D412yescomplemented
144C2×C4×C29—21no—
145C2×C4×C29—21no—
146C2×C4×C29—42no—
147C2×C4×C29—81no—
148C2×C4×C29C2×D411yescharacteristic
149C2×C4×C29C2×D411yescharacteristic
150C2×C4×C29C2411yescharacteristic
151C22⋊D29—42no—
152C22⋊D29—42no—
153C22⋊D29—42no—
154C22⋊D29—42no—
155C2×Dic29—21no—
156C2×Dic29—42no—
157C2×Dic29C2×D411yescharacteristic
158C2×Dic29C2×D411yescharacteristic
159D116—22no—
160D116—21no—
161D116—21no—
162D116—21no—
163D116—21no—
164D116—21no—
165D116—21no—
166D116—42no—
167D116—42no—
168D116—42no—
169D116—81no—
170D116—81no—
171C4×D29—21no—
172C4×D29—21no—
173C4×D29—21no—
174C4×D29—21no—
175Dic58—21no—
176Dic58—21no—
177C8×C29—21no—
178C8×C29C2×D412yescomplemented
179#6597—21no—
180#6597—21no—
181#3264—291nomaximal, Sylow
182C22×C29—22no—
183C22×C29—22no—
184C22×C29—41no—
185C22×C29—41no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29C22×D411yescharacteristic
189C2×D29—21no—
190C2×D29—21no—
191C2×D29—21no—
192C2×D29—21no—
193C2×D29—21no—
194C2×D29—41no—
195C2×D29—41no—
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—21no—
209C4×C29—41no—
210C4×C29—41no—
211C4×C29C22×D411yescharacteristic
212C4×C29H(5,𝔽2)11yescharacteristic
213Dic29—21no—
214Dic29—21no—
215Dic29—41no—
216C2×AGL(1,ℤ/8)—291no—
217C22×D8—291no—
218#546—291no—
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#442D2911yescharacteristic, complemented
230#435—291no—
231C2×C29—42no—
232C2×C29—42no—
233C2×C29D4211yescharacteristic
234C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C29#57611yescharacteristic
236D29—22no—
237D29—41no—
238D29—81no—
239D29—81no—
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
257C2×M16—291no—
258C22×C8—291no—
259#127—291no—
260#124—291no—
261#123—581no—
262C22⋊Q8—291no—
263#121—291no—
264#121—291no—
265#121—291no—
266#121—581no—
267#121—581no—
268#120—581no—
269#120—581no—
270#120—581no—
271C4×D4—291no—
272C4×D4C2×D2911yescharacteristic, complemented
273C4×D4C2×D2911yescharacteristic, complemented
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#97C2×D2911yescharacteristic, complemented
286C29#326411yescharacteristic, complemented, Sylow
287C24—581no—
288C24—581no—
289C24—581no—
290#41—582no—
291#41—1161no—
292C2×Q8—291no—
293C2×D4—291no—
294C2×D4—291no—
295C2×D4—291no—
296C2×D4—291no—
297C2×D4—581no—
298C2×D4—581no—
299C2×D4—581no—
300C2×D4—581no—
301C2×D4—581no—
302C2×D4—581no—
303C2×D4—582no—
304C2×D4—582no—
305C2×D4—582no—
306C2×D4—581no—
307C2×D4—581no—
308C2×D4—581no—
309C2×D4—1164no—
310C2×D4—1161no—
311C2×D4—1161no—
312C2×D4—1161no—
313C2×D4C22×D2911yescharacteristic
314C2×D4C22×D2911yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—291no—
318C22×C4—291no—
319C22×C4—581no—
320SD16—582no—
321SD16—582no—
322SD16—1161no—
323D8—582no—
324D8—582no—
325D8—582no—
326D8—582no—
327D8—1161no—
328D8D11614yescomplemented
329M16—582no—
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—
349C42C22×D2911yescharacteristic
350C23—21no—
351C23—21no—
352C23—291no—
353C23—291no—
354C23—581no—
355C23—581no—
356C23—581no—
357C23—1162no—
358C23—1162no—
359C23—1162no—
360C23—1162no—
361C23—1162no—
362C23—1162no—
363C23—2321no—
364C23—2321no—
365C23—2321no—
366Q8—581no—
367Q8—581no—
368D4—21no—
369D4—21no—
370D4—582no—
371D4—581no—
372D4—581no—
373D4—581no—
374D4—581no—
375D4—581no—
376D4—581no—
377D4—1162no—
378D4—1162no—
379D4—1162no—
380D4—1162no—
381D4—1162no—
382D4—1162no—
383D4—1162no—
384D4—2321no—
385D4—2321no—
386D4C2×D11612yes—
387D4C2×D11612yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—291no—
393C2×C4—291no—
394C2×C4—581no—
395C2×C4—581no—
396C2×C4—581no—
397C2×C4—581no—
398C2×C4—581no—
399C2×C4—1162no—
400C2×C4D4×D2911yescharacteristic
401C2×C4D4×D2911yescharacteristic
402C2×C4C23×D2911yescharacteristic
403C8—21no—
404C8—581no—
405C8—581no—
406C8C2×D11612yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—292no—
414C22—581no—
415C22—581no—
416C22—581no—
417C22—581no—
418C22—581no—
419C22—1161no—
420C22—1161no—
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—21no—
434C4—41no—
435C4—41no—
436C4—581no—
437C4—581no—
438C4—1161no—
439C4C22×D11611yescharacteristic
440C4#1174536411yescharacteristic
441C2—42no—
442C2—42no—
443C2—582no—
444C2—1161no—
445C2—2321no—
446C2—2321no—
447C2—2321no—
448C2D4×D11611yescharacteristic
449C2AGL(1,ℤ/8)×D2911yescharacteristic
450C2#1427196711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#442C2
socleC22×C29C22×D4
hypercenter#442D29
hypocenterC29#3264
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. C2×D8×D29C2semidirect
3. #14271808C2semidirect
4. D4×D116C2semidirect
5. #14271552C2semidirect
6. #14271417C2semidirect
7. #14271393C2semidirect
8. #14271378C2semidirect
9. #14271358C2semidirect
10. #14271315C2semidirect
11. #14271309C2semidirect
12. #14271298C2semidirect
13. #14271298C2semidirect
14. #14271236C2semidirect
15. C29×#442C2semidirect
16. C2×#90807C22semidirect
18. C2×D4×D29C22semidirect
19. C2×D4×D29C22semidirect
22. C4×D4×C29C22semidirect
23. C4×D4×C29C22semidirect
24. C2×#90783C22semidirect
26. C2×#90782C22semidirect
27. C2×#90782C22semidirect
33. C2×D232C22semidirect
34. C2×D232C22semidirect
35. C2×#90773C22semidirect
36. C2×#90773C22semidirect
37. C2×#90772C22semidirect
38. C2×C8×D29C22semidirect
39. #11745259C22semidirect
41. #11745257C22semidirect
42. #11745257C22semidirect
47. #11745237C22semidirect
48. #11745236C22semidirect
50. C29×#106C22semidirect
51. C29×#102C22semidirect
52. C29×#102C22semidirect
53. C29×#97C22semidirect
54. #11745168C22semidirect
55. #11745168C22semidirect
56. #11745158C22semidirect
57. #11745157C22semidirect
58. #11745157C22semidirect
59. #11745157C22semidirect
60. C4⋊#6597C22semidirect
94. D8×C29D4semidirect
143. D4×C29C2×D4semidirect
178. C8×C29C2×D4semidirect
206. C2×D29AGL(1,ℤ/8)semidirect
229. #442D29semidirect
272. C4×D4C2×D29semidirect
273. C4×D4C2×D29semidirect
275. #106C2×D29semidirect
281. #102C2×D29semidirect
282. #102C2×D29semidirect
285. #97C2×D29semidirect
286. C29#3264semidirect
328. D8D116semidirect
387. D4C2×D116semidirect
406. C8C2×D116semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
C2×D8×D292
#142718082
D4×D1162
#142715522
#142714172
#142713932
#142713782
#142713582
#142713152
#142713092
#142712982
#142712362
C29×#4422
#326429