Finite Group Databasebeta

← #46845666

Subgroups — #46845666

Order 34688.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576206C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3C2×#22576181C211yesmaximal, characteristic, complemented
4#31774476C211yesmaximal, characteristic, complemented
5#31774062C211yesmaximal, characteristic, complemented
6#31774059C211yesmaximal, characteristic, complemented
7#31774050C211yesmaximal, characteristic, complemented
8#31774049C211yesmaximal, characteristic, complemented
9#31774041C211yesmaximal, characteristic, complemented
10#31774030C211yesmaximal, characteristic, complemented
11#31774029C211yesmaximal, characteristic, complemented
12#31773897C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16#22576206—24no—
17#22576206—24no—
18C2×#16864655C2211yescharacteristic
19C2×#16864655C2211yescharacteristic
20C2×#16864654—21no—
21C2×#16864654C2211yescharacteristic
22C2×#16864653C2211yescharacteristic, complemented
23C22×D1084—21no—
24C22×D1084C2211yescharacteristic
25C2×D4×D271—21no—
26C2×D4×D271—21no—
27C2×D4×D271C2211yescharacteristic
28C2×D4×D271C2211yescharacteristic
29#22576184—24no—
30#22576181—24no—
31C2×#16864644C2211yescharacteristic, complemented
32C2×#16864643C2211yescharacteristic, complemented
33C2×#16864643C2211yescharacteristic, complemented
34C2×#16864642C2211yescharacteristic, complemented
35C2×#16864639C2211yescharacteristic, complemented
36C2×#16864639C2211yescharacteristic, complemented
37C2×#16864637C2211yescharacteristic, complemented
38C2×C8×D271C2211yescharacteristic, complemented
39#22576140C2211yescharacteristic, complemented
40#22576139C2211yescharacteristic, complemented
41#22576137—21no—
42#22576133—21no—
43#22576132—21no—
44#22576131—21no—
45#22576131C2211yescharacteristic, complemented
46C4⋊C4×D271C2211yescharacteristic, complemented
47#22576084C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52#22576080C2211yescharacteristic, complemented
53C22⋊(C271⋊C8)C2211yescharacteristic, complemented
54(C2×D271)⋊C8C2211yescharacteristic, complemented
55(C2×D271)⋊C8C2211yescharacteristic, complemented
56C2×#41×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic, complemented
59#121×C271C2211yescharacteristic, complemented
60#103×C271C2211yescharacteristic, complemented
61#102×C271C2211yescharacteristic, complemented
62C22⋊C8×C271C2211yescharacteristic, complemented
63C23×D271—21no—
64C23×D271—21no—
65C23×D271—21no—
66#16864655—22no—
67#16864655—22no—
68#16864655—22no—
69#16864655D414yescomplemented
70#16864654—28no—
71#16864654—22no—
72#16864654—22no—
73#16864653—22no—
74#16864653—22no—
75C2×Dic542D411yescharacteristic
76C2×Dic542C2311yescharacteristic
77C2×D1084—28no—
78C2×D1084—21no—
79C2×D1084—21no—
80C2×D1084—22no—
81C2×D1084—22no—
82C2×D1084—21no—
83C2×D1084D411yescharacteristic
84C2×D1084C2311yescharacteristic
85C2×D1084C2311yescharacteristic
86C2×C22⋊D271—21no—
87C2×C22⋊D271—21no—
88C2×C22⋊D271—21no—
89C2×C22⋊D271—21no—
90C2×C22⋊D271D411yescharacteristic
91C2×C22⋊D271D411yescharacteristic
92D4×D271—28no—
93D4×D271—28no—
94D4×D271—22no—
95D4×D271—22no—
96D4×D271—22no—
97D4×D271D414yescomplemented
98C22×Dic271—21no—
99C2×C4×D271—21no—
100C2×C4×D271—21no—
101C2×C4×D271—21no—
102C2×C4×D271C2311yescharacteristic
103C2×C4×D271C2311yescharacteristic
104#16864644—22no—
105#16864643—22no—
106#16864643—22no—
107#16864642—22no—
108#16864639—22no—
109#16864639—22no—
110#16864637—22no—
111C2×C271⋊C8C2311yescharacteristic
112C2×C271⋊C8C2311yescharacteristic
113C8×D271—22no—
114C4⋊Dic271—21no—
115C4⋊Dic271C2311yescharacteristic
116Dic271⋊C4—21no—
117Dic271⋊C4—21no—
118C22⋊Dic271—21no—
119C22⋊Dic271—21no—
120(C2×D271)⋊C4—21no—
121#41×C271—22no—
122#41×C271—22no—
123C2×Q8×C271C2311yescharacteristic
124C2×D4×C271—21no—
125C2×D4×C271—21no—
126C2×D4×C271C2311yescharacteristic
127C2×D4×C271C2311yescharacteristic
128C22×C4×C271—21no—
129C22×C4×C271C2311yescharacteristic
130SD16×C271—22no—
131D8×C271—22no—
132C2×C8×C271C2311yescharacteristic
133C2×C8×C271C2311yescharacteristic
134C4⋊C4×C271C2311yescharacteristic
135C22⋊C4×C271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—24no—
139C22×D271—24no—
140C22×D271—24no—
141C22×D271—81no—
142C22×D271—81no—
143C22×D271—81no—
144C22×D271—81no—
145C22×D271—81no—
146C22×D271—81no—
147C22×D271C2×D411yescharacteristic
148C22×D271C2×D411yescharacteristic
149Dic542—22no—
150Dic542—21no—
151Dic542—21no—
152D1084—24no—
153D1084—24no—
154D1084—24no—
155D1084—22no—
156D1084—21no—
157D1084—21no—
158D1084C2×D412yes—
159D1084C2×D412yes—
160C22⋊D271—24no—
161C22⋊D271—22no—
162C22⋊D271—22no—
163C22⋊D271—24no—
164C22⋊D271—42no—
165C22⋊D271—81no—
166C2×Dic271—21no—
167C2×Dic271—21no—
168C2×Dic271—42no—
169C2×Dic271C2×D411yescharacteristic
170C2×Dic271C2×D411yescharacteristic
171C4×D271—24no—
172C4×D271—21no—
173C4×D271—21no—
174C4×D271—24no—
175C4×D271—81no—
176C4×D271C2×D412yes—
177C4×D271C2×D412yes—
178C271⋊C8—21no—
179C271⋊C8—21no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271C2×D411yescharacteristic
183Q8×C271—21no—
184Q8×C271C2×D412yes—
185D4×C271—24no—
186D4×C271—21no—
187D4×C271—21no—
188D4×C271—21no—
189D4×C271—81no—
190D4×C271C2×D412yes—
191C2×C4×C271—24no—
192C2×C4×C271—21no—
193C2×C4×C271—21no—
194C2×C4×C271—21no—
195C2×C4×C271—21no—
196C2×C4×C271C2×D411yescharacteristic
197C2×C4×C271C2411yescharacteristic
198C8×C271—21no—
199C8×C271—21no—
200C22×C271—21no—
201C22×C271—22no—
202C22×C271—21no—
203C22×C271—21no—
204C22×C271—41no—
205C22×C271—41no—
206C22×C271—41no—
207C22×C271—41no—
208C22×C271—41no—
209C22×C271C22×D411yescharacteristic
210C2×D271—21no—
211C2×D271—21no—
212C2×D271—21no—
213C2×D271—22no—
214C2×D271—22no—
215C2×D271—21no—
216C2×D271—21no—
217C2×D271—41no—
218C2×D271—41no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—84no—
222C2×D271—84no—
223C2×D271—84no—
224C2×D271#12012yes—
225C4×C271—21no—
226C4×C271—22no—
227C4×C271—41no—
228C4×C271C22×D411yescharacteristic
229C4×C271C22×D411yescharacteristic
230Dic271—21no—
231Dic271—41no—
232Dic271—41no—
233Dic271#12012yes—
234C2×C271—41no—
235C2×C271—42no—
236C2×C271—81no—
237C2×C271C2×#12011yescharacteristic
238C2×C271#57611yescharacteristic
239C2×C271#57711yescharacteristic
240D271—22no—
241D271—41no—
242D271—42no—
243D271—42no—
244C271#301211yescharacteristic, complemented, Sylow
245#3012—2711nomaximal, Sylow
246C2×H(5,𝔽2)—2711no—
247C2×AGL(1,ℤ/8)—2711no—
248C2×#135—2711no—
249#537—2711no—
250#452—2711no—
251#452—2711no—
252#450—2711no—
253#450—2711no—
254#449—2711no—
255#449D27111yescharacteristic, complemented
256#447—2711no—
257#447—2711no—
258#417—2711no—
259C2×#102—2711no—
260#408—2711no—
261H(5,𝔽2)—5424no—
262H(5,𝔽2)—5424no—
263C2×#41—2711no—
264C2×#41—2711no—
265C2×#41—2711no—
266C2×#41—2711no—
267C2×#41—5421no—
268C2×#41C2×D27111yescharacteristic
269C22×D4—2711no—
270C22×D4—2711no—
271C22×D4—2711no—
272C22×D4—5421no—
273C22×D4—5421no—
274C22×D4—5421no—
275AGL(1,ℤ/8)—5424no—
276#135—5424no—
277C2×Q16—2711no—
278C2×SD16—2711no—
279C2×SD16—2711no—
280C2×SD16—2711no—
281C2×SD16C2×D27111yescharacteristic, complemented
282C2×D8—2711no—
283C2×D8—2711no—
284C2×D8C2×D27111yescharacteristic, complemented
285C2×M16—2711no—
286C22×C8—2711no—
287C22⋊Q8—2711no—
288C22⋊Q8—2711no—
289C22⋊Q8—5421no—
290#121—2711no—
291#121—5421no—
292#121—5421no—
293#121—5421no—
294#121C2×D27111yescharacteristic, complemented
295C2×C4⋊C4—2711no—
296#103—2711no—
297#103C2×D27111yescharacteristic, complemented
298#102—2711no—
299#102—2711no—
300#102—2711no—
301#102—2711no—
302#102—2711no—
303#102C2×D27111yescharacteristic, complemented
304C22⋊C8—2711no—
305C22⋊C8—2711no—
306C22⋊C8—2711no—
307C22⋊C8C2×D27111yescharacteristic, complemented
308C24—5421no—
309C24—5421no—
310C24—5421no—
311#41—22no—
312#41—22no—
313#41—2714no—
314#41—5422no—
315#41—5422no—
316#41—5428no—
317#41—5422no—
318#41—5422no—
319#41—5422no—
320#41—5422no—
321#41—5422no—
322C2×Q8—2711no—
323C2×Q8—2711no—
324C2×Q8C22×D27111yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—2714no—
328C2×D4—2711no—
329C2×D4—2711no—
330C2×D4—2711no—
331C2×D4—2711no—
332C2×D4—2711no—
333C2×D4—5422no—
334C2×D4—5422no—
335C2×D4—5421no—
336C2×D4—5421no—
337C2×D4—5428no—
338C2×D4—5422no—
339C2×D4—5421no—
340C2×D4—5421no—
341C2×D4—5422no—
342C2×D4—5428no—
343C2×D4—5428no—
344C2×D4—5422no—
345C2×D4—5421no—
346C2×D4—5421no—
347C2×D4—5421no—
348C2×D4C22×D27111yescharacteristic
349C2×D4C22×D27111yescharacteristic
350C22×C4—21no—
351C22×C4—2711no—
352C22×C4—2711no—
353C22×C4—5421no—
354C22×C4—5421no—
355C22×C4—5421no—
356C22×C4—5421no—
357C22×C4C22×D27111yescharacteristic
358Q16—5422no—
359SD16—22no—
360SD16—5422no—
361SD16—5422no—
362SD16—5422no—
363D8—22no—
364D8—5422no—
365D8—5422no—
366M16—5422no—
367C2×C8—2711no—
368C2×C8—2711no—
369C2×C8—5422no—
370C2×C8C22×D27111yescharacteristic
371C2×C8C22×D27111yescharacteristic
372C4⋊C4—2711no—
373C4⋊C4—5421no—
374C4⋊C4—5421no—
375C4⋊C4—5421no—
376C4⋊C4C22×D27111yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—5421no—
379C22⋊C4—5421no—
380C22⋊C4—5421no—
381C23—21no—
382C23—21no—
383C23—2711no—
384C23—2711no—
385C23—5424no—
386C23—5424no—
387C23—5424no—
388C23—5421no—
389C23—5421no—
390C23—21681no—
391C23—21681no—
392C23—21681no—
393C23—21681no—
394C23—21681no—
395C23—21681no—
396C23D4×D27111yescharacteristic
397Q8—21no—
398Q8—5422no—
399Q8—5421no—
400Q8—5421no—
401Q8D4×D27112yes—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—81no—
407D4—2712no—
408D4—2712no—
409D4—5422no—
410D4—5424no—
411D4—5424no—
412D4—5424no—
413D4—5422no—
414D4—5424no—
415D4—5422no—
416D4—5424no—
417D4—5421no—
418D4—5421no—
419D4—10842no—
420D4—21681no—
421D4D4×D27112yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—2711no—
428C2×C4—2712no—
429C2×C4—2712no—
430C2×C4—2711no—
431C2×C4—5424no—
432C2×C4—5424no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—5421no—
436C2×C4—5421no—
437C2×C4—10842no—
438C2×C4—21681no—
439C2×C4D4×D27111yescharacteristic
440C2×C4C23×D27111yescharacteristic
441C8—21no—
442C8—21no—
443C8—5421no—
444C8—5421no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—2712no—
455C22—5421no—
456C22—5421no—
457C22—5422no—
458C22—5422no—
459C22—5421no—
460C22—5421no—
461C22—5421no—
462C22—10841no—
463C22—10841no—
464C22—10841no—
465C22—10841no—
466C22—21684no—
467C22—21684no—
468C22—21684no—
469C22C2×D4×D27111yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—2712no—
474C4—5421no—
475C4—10841no—
476C4—10841no—
477C4C2×D4×D27111yescharacteristic
478C4C2×D4×D27111yescharacteristic
479C2—41no—
480C2—42no—
481C2—81no—
482C2—5422no—
483C2—10841no—
484C2—10842no—
485C2—10842no—
486C2#120×D27111yescharacteristic
487C2#3177470111yescharacteristic
488C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3012
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22576206C2semidirect
2. C2×#22576184C2semidirect
3. C2×#22576181C2semidirect
4. #31774476C2semidirect
5. #31774062C2semidirect
6. #31774059C2semidirect
7. #31774050C2semidirect
8. #31774049C2semidirect
9. #31774041C2semidirect
10. #31774030C2semidirect
11. #31774029C2semidirect
12. #31773897C2semidirect
13. #102×D271C2semidirect
14. #31773862C2semidirect
15. #449×C271C2semidirect
22. C2×#16864653C22semidirect
31. C2×#16864644C22semidirect
32. C2×#16864643C22semidirect
33. C2×#16864643C22semidirect
34. C2×#16864642C22semidirect
35. C2×#16864639C22semidirect
36. C2×#16864639C22semidirect
37. C2×#16864637C22semidirect
38. C2×C8×D271C22semidirect
39. #22576140C22semidirect
40. #22576139C22semidirect
45. #22576131C22semidirect
46. C4⋊C4×D271C22semidirect
47. #22576084C22semidirect
48. #22576082C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576080C22semidirect
52. #22576080C22semidirect
53. C22⋊(C271⋊C8)C22semidirect
54. (C2×D271)⋊C8C22semidirect
55. (C2×D271)⋊C8C22semidirect
57. C2×SD16×C271C22semidirect
58. C2×D8×C271C22semidirect
59. #121×C271C22semidirect
60. #103×C271C22semidirect
61. #102×C271C22semidirect
62. C22⋊C8×C271C22semidirect
69. #16864655D4semidirect
97. D4×D271D4semidirect
244. C271#3012semidirect
255. #449D271semidirect
281. C2×SD16C2×D271semidirect
284. C2×D8C2×D271semidirect
294. #121C2×D271semidirect
297. #103C2×D271semidirect
303. #102C2×D271semidirect
307. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225762062
C2×#225761842
C2×#225761812
#317744762
#317740622
#317740592
#317740502
#317740492
#317740412
#317740302
#317740292
#317738972
#102×D2712
#317738622
#449×C2712
#3012271