Finite Group Databasebeta

← #16831106

Subgroups — #16831106

Order 3712.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745276C211yesmaximal, characteristic, complemented
2#14271832C211yesmaximal, characteristic, complemented
3D4×C22⋊D29C211yesmaximal, characteristic, complemented
4#14271538C211yesmaximal, characteristic, complemented
5#14271416C211yesmaximal, characteristic, complemented
6#14271413C211yesmaximal, characteristic, complemented
7#14271361C211yesmaximal, characteristic, complemented
8#14271356C211yesmaximal, characteristic, complemented
9#14271308C211yesmaximal, characteristic, complemented
10#14271307C211yesmaximal, characteristic, complemented
11#14271297C211yesmaximal, characteristic, complemented
12#14271295C211yesmaximal, characteristic, complemented
13SD16×Dic29C211yesmaximal, characteristic, complemented
14#14271239C211yesmaximal, characteristic, complemented
15C2×C29×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic, complemented
17C22×D4×C29C2211yescharacteristic
18C22×C22⋊D29—21no—
19C2×#90809C2211yescharacteristic, complemented
20C2×D4×D29C2211yescharacteristic
21C29×AGL(1,ℤ/8)—24no—
22C2×SD16×C29C2211yescharacteristic
23C2×SD16×C29C2211yescharacteristic, complemented
24C2×D8×C29C2211yescharacteristic
25C2×D8×C29C2211yescharacteristic, complemented
26C2×M16×C29C2211yescharacteristic, complemented
27#11745315—21no—
28Q8×Dic29C2211yescharacteristic, complemented
29C2×#90784C2211yescharacteristic, complemented
30#11745302—21no—
31#11745302C2211yescharacteristic
32#11745302C2211yescharacteristic, complemented
33#11745301—21no—
34#11745300—21no—
35#11745300C2211yescharacteristic
36#11745300C2211yescharacteristic, complemented
37#11745299—21no—
38D4×Dic29C2211yescharacteristic
39C2×#90782C2211yescharacteristic
40C2×#90782C2211yescharacteristic, complemented
41C2×#90782C2211yescharacteristic, complemented
42#11745292—21no—
43#11745292C2211yescharacteristic
44#11745292C2211yescharacteristic, complemented
45C4×C22⋊D29—21no—
46C4×C22⋊D29C2211yescharacteristic
47#11745276—24no—
48C2×D232C2211yescharacteristic, complemented
49C2×#90772C2211yescharacteristic, complemented
50#11745183C2211yescharacteristic, complemented
51#11745180C2211yescharacteristic
52#11745180C2211yescharacteristic, complemented
53#11745180C2211yescharacteristic, complemented
54C22⋊#6597C2211yescharacteristic, complemented
55#11745169C2211yescharacteristic, complemented
56#11745169C2211yescharacteristic, complemented
57#11745168C2211yescharacteristic, complemented
58#11745165C2211yescharacteristic, complemented
59#11745162C2211yescharacteristic, complemented
60C8×Dic29C2211yescharacteristic, complemented
61C24×C29—21no—
62C23×D29—21no—
63#41×C29—22no—
64#41×C29—41no—
65C2×Q8×C29C2311yescharacteristic
66C2×D4×C29—22no—
67C2×D4×C29—21no—
68C2×D4×C29—21no—
69C2×D4×C29—41no—
70C2×D4×C29C2311yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C2×D4×C29C2311yescharacteristic
73C22×C4×C29—21no—
74C22×C4×C29C2311yescharacteristic
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⋊D29—21no—
82C2×C22⋊D29—21no—
83C2×C22⋊D29—21no—
84C2×C22⋊D29—44no—
85C2×C22⋊D29D411yescharacteristic
86C22×Dic29—21no—
87#90809—22no—
88#90809—41no—
89D4×D29—22no—
90D4×D29—41no—
91C2×D116—21no—
92C2×D116C2311yescharacteristic
93C2×D116C2311yescharacteristic
94C2×C4×D29—21no—
95C2×C4×D29C2311yescharacteristic
96SD16×C29—22no—
97SD16×C29D414yescomplemented
98D8×C29—22no—
99D8×C29—22no—
100M16×C29—22no—
101C2×C8×C29C2311yescharacteristic
102C2×C8×C29C2311yescharacteristic
103C22⋊Dic29—21no—
104C22⋊Dic29—21no—
105C22⋊Dic29—21no—
106C22⋊Dic29D411yescharacteristic
107#90784—22no—
108#90782—22no—
109#90782—41no—
110#90782—41no—
111#90781—21no—
112#90781—21no—
113#90781D411yescharacteristic
114C4⋊Dic29—21no—
115C4⋊Dic29C2311yescharacteristic
116C4⋊Dic29C2311yescharacteristic
117#90779D411yescharacteristic
118C4×Dic29—21no—
119C4×Dic29C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122D232—22no—
123#90772—22no—
124C23×C29—21no—
125C23×C29—21no—
126C23×C29—21no—
127C23×C29—42no—
128C23×C29—42no—
129C23×C29—81no—
130C23×C29C2×D411yescharacteristic
131C22×D29—21no—
132C22×D29—21no—
133C22×D29—42no—
134C22×D29—42no—
135C22×D29—81no—
136C22×D29C2×D411yescharacteristic
137Q8×C29—21no—
138Q8×C29C2×D412yescomplemented
139D4×C29—21no—
140D4×C29—21no—
141D4×C29—21no—
142D4×C29—21no—
143D4×C29—21no—
144D4×C29—42no—
145D4×C29—42no—
146D4×C29C2×D412yes—
147C2×C4×C29—21no—
148C2×C4×C29—21no—
149C2×C4×C29—21no—
150C2×C4×C29—42no—
151C2×C4×C29C2×D411yescharacteristic
152C2×C4×C29C2411yescharacteristic
153C22⋊D29—22no—
154C22⋊D29—42no—
155C22⋊D29—42no—
156C22⋊D29—42no—
157C22⋊D29—42no—
158C22⋊D29—42no—
159C22⋊D29—81no—
160C22⋊D29—81no—
161C22⋊D29—81no—
162C22⋊D29—81no—
163C2×Dic29—21no—
164C2×Dic29—21no—
165C2×Dic29—42no—
166C2×Dic29C2×D411yescharacteristic
167C2×Dic29C2×D411yescharacteristic
168C2×Dic29C2×D411yescharacteristic
169D116—21no—
170D116—21no—
171D116—21no—
172D116—21no—
173D116—42no—
174C4×D29—21no—
175C4×D29—21no—
176C4×D29—42no—
177C8×C29—21no—
178C8×C29C2×D412yescomplemented
179#6597—21no—
180#6597—21no—
181#3274—291nomaximal, Sylow
182C22×C29—22no—
183C22×C29—21no—
184C22×C29—21no—
185C22×C29—21no—
186C22×C29—41no—
187C22×C29—41no—
188C22×C29—41no—
189C22×C29—41no—
190C22×C29—41no—
191C22×C29—41no—
192C22×C29—41no—
193C22×C29—41no—
194C22×C29—84no—
195C22×C29C22×D411yescharacteristic
196C2×D29—21no—
197C2×D29—21no—
198C2×D29—21no—
199C2×D29—41no—
200C2×D29—41no—
201C2×D29—41no—
202C2×D29—41no—
203C2×D29—41no—
204C2×D29—41no—
205C2×D29—84no—
206C4×C29—22no—
207C4×C29—21no—
208C4×C29C22×D411yescharacteristic
209C4×C29H(5,𝔽2)11yescharacteristic
210Dic29—21no—
211Dic29—21no—
212Dic29—41no—
213Dic29—41no—
214Dic29AGL(1,ℤ/8)12yescomplemented
215C2×AGL(1,ℤ/8)—291no—
216C2×AGL(1,ℤ/8)D2911yescharacteristic, complemented
217#550—291no—
218D42—291no—
219#493—291no—
220#469—291no—
221#469—291no—
222#460—291no—
223#459—291no—
224#449—291no—
225#449—291no—
226#447—291no—
227#447—291no—
228C4×SD16—291no—
229#436—291no—
230C2×C29—42no—
231C2×C29—41no—
232C2×C29—81no—
233C2×C29—81no—
234C2×C29D4211yescharacteristic
235C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C29#57611yescharacteristic
237D29—41no—
238D29—81no—
239D29—81no—
240C2×#41—291no—
241C2×#41C2×D2911yescharacteristic, complemented
242C22×D4—291no—
243C22×D4—581no—
244C22×D4C2×D2911yescharacteristic
245AGL(1,ℤ/8)—24no—
246AGL(1,ℤ/8)—584no—
247C2×SD16—291no—
248C2×SD16C2×D2911yescharacteristic
249C2×SD16C2×D2911yescharacteristic, complemented
250C2×D8—291no—
251C2×D8—291no—
252C2×D8—291no—
253C2×D8—291no—
254C2×D8C2×D2911yescharacteristic
255C2×D8C2×D2911yescharacteristic, complemented
256C2×M16—291no—
257C2×M16C2×D2911yescharacteristic, complemented
258#127—291no—
259#127—291no—
260#127—581no—
261#121—291no—
262#121—291no—
263#121—291no—
264#121—291no—
265#121—581no—
266#121—581no—
267#121—581no—
268#120—581no—
269#120—581no—
270C4×Q8—291no—
271C4×D4—291no—
272C4×D4—291no—
273C4×D4—581no—
274#106—291no—
275C4⋊C8—291no—
276#103—291no—
277#102—291no—
278#102—291no—
279#102—291no—
280#102—291no—
281#102—291no—
282C22⋊C8—291no—
283C22⋊C8—291no—
284C4×C8—291no—
285C29#327411yescharacteristic, complemented, Sylow
286C24—21no—
287C24—581no—
288#41—22no—
289#41—41no—
290#41—582no—
291#41—1161no—
292C2×Q8C22×D2911yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—291no—
298C2×D4—291no—
299C2×D4—291no—
300C2×D4—581no—
301C2×D4—581no—
302C2×D4—581no—
303C2×D4—581no—
304C2×D4—581no—
305C2×D4—581no—
306C2×D4—582no—
307C2×D4—581no—
308C2×D4—581no—
309C2×D4—581no—
310C2×D4—581no—
311C2×D4—1161no—
312C2×D4—1164no—
313C2×D4C22×D2911yescharacteristic
314C2×D4C22×D2911yescharacteristic
315C2×D4C22×D2911yescharacteristic
316C22×C4—21no—
317C22×C4—291no—
318C22×C4—581no—
319C22×C4—581no—
320C22×C4C22×D2911yescharacteristic
321SD16—22no—
322SD16—582no—
323SD16C22⋊D2914yescomplemented
324D8—22no—
325D8—22no—
326D8—582no—
327D8—582no—
328D8—1161no—
329D8—1161no—
330M16—22no—
331M16—582no—
332C2×C8—291no—
333C2×C8—291no—
334C2×C8C22×D2911yescharacteristic
335C2×C8C22×D2911yescharacteristic
336C4⋊C4—291no—
337C4⋊C4—291no—
338C4⋊C4—291no—
339C4⋊C4—581no—
340C22⋊C4—291no—
341C22⋊C4—291no—
342C22⋊C4—581no—
343C22⋊C4—581no—
344C22⋊C4—581no—
345C22⋊C4—581no—
346C22⋊C4—581no—
347C42—291no—
348C42—581no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—291no—
356C23—581no—
357C23—581no—
358C23—1162no—
359C23—1162no—
360C23—2321no—
361C23D4×D2911yescharacteristic
362Q8—21no—
363Q8C2×C22⋊D2912yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—42no—
371D4—582no—
372D4—581no—
373D4—581no—
374D4—581no—
375D4—581no—
376D4—1162no—
377D4—1162no—
378D4—1162no—
379D4—1162no—
380D4—1162no—
381D4—1162no—
382D4—2321no—
383D4—2321no—
384D4—2321no—
385D4—2321no—
386D4C2×C22⋊D2912yes—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—291no—
392C2×C4—291no—
393C2×C4—291no—
394C2×C4—581no—
395C2×C4—581no—
396C2×C4—581no—
397C2×C4—581no—
398C2×C4—1162no—
399C2×C4—1162no—
400C2×C4D4×D2911yescharacteristic
401C2×C4C23×D2911yescharacteristic
402C8—21no—
403C8—581no—
404C8—581no—
405C8C2×C22⋊D2912yescomplemented
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—581no—
420C22—581no—
421C22—581no—
422C22—1161no—
423C22—1161no—
424C22—1161no—
425C22—1161no—
426C22—1161no—
427C22—1161no—
428C22—2324no—
429C22C2×D4×D2911yescharacteristic
430C4—21no—
431C4—22no—
432C4—292no—
433C4—581no—
434C4—581no—
435C4—1161no—
436C4—1161no—
437C4#1174535911yescharacteristic
438C4C22×C22⋊D2911yescharacteristic
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—1161no—
444C2—2321no—
445C2—2321no—
446C2D4×C22⋊D2911yescharacteristic
447C2AGL(1,ℤ/8)×D2911yescharacteristic
448C2#1427196711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×AGL(1,ℤ/8)C2
socleC22×C29C22×D4
hypercenterC2×AGL(1,ℤ/8)D29
hypocenterC29#3274
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11745276C2semidirect
2. #14271832C2semidirect
3. D4×C22⋊D29C2semidirect
4. #14271538C2semidirect
5. #14271416C2semidirect
6. #14271413C2semidirect
7. #14271361C2semidirect
8. #14271356C2semidirect
9. #14271308C2semidirect
10. #14271307C2semidirect
11. #14271297C2semidirect
12. #14271295C2semidirect
13. SD16×Dic29C2semidirect
14. #14271239C2semidirect
15. C2×C29×AGL(1,ℤ/8)C2semidirect
16. C2×#41×C29C22semidirect
19. C2×#90809C22semidirect
23. C2×SD16×C29C22semidirect
25. C2×D8×C29C22semidirect
26. C2×M16×C29C22semidirect
28. Q8×Dic29C22semidirect
29. C2×#90784C22semidirect
32. #11745302C22semidirect
36. #11745300C22semidirect
40. C2×#90782C22semidirect
41. C2×#90782C22semidirect
44. #11745292C22semidirect
48. C2×D232C22semidirect
49. C2×#90772C22semidirect
50. #11745183C22semidirect
52. #11745180C22semidirect
53. #11745180C22semidirect
54. C22⋊#6597C22semidirect
55. #11745169C22semidirect
56. #11745169C22semidirect
57. #11745168C22semidirect
58. #11745165C22semidirect
59. #11745162C22semidirect
60. C8×Dic29C22semidirect
97. SD16×C29D4semidirect
138. Q8×C29C2×D4semidirect
178. C8×C29C2×D4semidirect
214. Dic29AGL(1,ℤ/8)semidirect
216. C2×AGL(1,ℤ/8)D29semidirect
241. C2×#41C2×D29semidirect
249. C2×SD16C2×D29semidirect
255. C2×D8C2×D29semidirect
257. C2×M16C2×D29semidirect
285. C29#3274semidirect
323. SD16C22⋊D29semidirect
363. Q8C2×C22⋊D29semidirect
405. C8C2×C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452762
#142718322
D4×C22⋊D292
#142715382
#142714162
#142714132
#142713612
#142713562
#142713082
#142713072
#142712972
#142712952
SD16×Dic292
#142712392
C2×C29×AGL(1,ℤ/8)2
#327429