Finite Group Databasebeta

← #37581822

Subgroups — #37581822

Order 24448.

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×#20259368C211yesmaximal, characteristic, complemented
2C2×#20259346C211yesmaximal, characteristic, complemented
3C2×#20259343C211yesmaximal, characteristic, complemented
4#27133242C211yesmaximal, characteristic, complemented
5#27132828C211yesmaximal, characteristic, complemented
6#27132825C211yesmaximal, characteristic, complemented
7#27132816C211yesmaximal, characteristic, complemented
8#27132815C211yesmaximal, characteristic, complemented
9#27132807C211yesmaximal, characteristic, complemented
10#27132796C211yesmaximal, characteristic, complemented
11#27132795C211yesmaximal, characteristic, complemented
12#27132663C211yesmaximal, characteristic, complemented
13#102×D191C211yesmaximal, characteristic, complemented
14#27132628C211yesmaximal, characteristic, complemented
15#449×C191C211yesmaximal, characteristic, complemented
16#20259368—24no—
17#20259368—24no—
18C2×#15690288C2211yescharacteristic
19C2×#15690288C2211yescharacteristic
20C2×#15690287—21no—
21C2×#15690287C2211yescharacteristic
22C2×#15690286C2211yescharacteristic, complemented
23C22×D764—21no—
24C22×D764C2211yescharacteristic
25C2×D4×D191—21no—
26C2×D4×D191—21no—
27C2×D4×D191C2211yescharacteristic
28C2×D4×D191C2211yescharacteristic
29#20259346—24no—
30#20259343—24no—
31C2×#15690277C2211yescharacteristic, complemented
32C2×#15690276C2211yescharacteristic, complemented
33C2×#15690276C2211yescharacteristic, complemented
34C2×#15690275C2211yescharacteristic, complemented
35C2×#15690272C2211yescharacteristic, complemented
36C2×#15690272C2211yescharacteristic, complemented
37C2×#15690270C2211yescharacteristic, complemented
38C2×C8×D191C2211yescharacteristic, complemented
39#20259302C2211yescharacteristic, complemented
40#20259301C2211yescharacteristic, complemented
41#20259299—21no—
42#20259295—21no—
43#20259294—21no—
44#20259293—21no—
45#20259293C2211yescharacteristic, complemented
46C4⋊C4×D191C2211yescharacteristic, complemented
47#20259246C2211yescharacteristic, complemented
48#20259244C2211yescharacteristic, complemented
49#20259244C2211yescharacteristic, complemented
50#20259243C2211yescharacteristic, complemented
51#20259242C2211yescharacteristic, complemented
52#20259242C2211yescharacteristic, complemented
53C22⋊(C191⋊C8)C2211yescharacteristic, complemented
54(C2×D191)⋊C8C2211yescharacteristic, complemented
55(C2×D191)⋊C8C2211yescharacteristic, complemented
56C2×#41×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic, complemented
58C2×D8×C191C2211yescharacteristic, complemented
59#121×C191C2211yescharacteristic, complemented
60#103×C191C2211yescharacteristic, complemented
61#102×C191C2211yescharacteristic, complemented
62C22⋊C8×C191C2211yescharacteristic, complemented
63C23×D191—21no—
64C23×D191—21no—
65C23×D191—21no—
66#15690288—22no—
67#15690288—22no—
68#15690288—22no—
69#15690288D414yescomplemented
70#15690287—28no—
71#15690287—22no—
72#15690287—22no—
73#15690286—22no—
74#15690286—22no—
75C2×Dic382D411yescharacteristic
76C2×Dic382C2311yescharacteristic
77C2×D764—28no—
78C2×D764—21no—
79C2×D764—21no—
80C2×D764—22no—
81C2×D764—22no—
82C2×D764—21no—
83C2×D764D411yescharacteristic
84C2×D764C2311yescharacteristic
85C2×D764C2311yescharacteristic
86C2×C22⋊D191—21no—
87C2×C22⋊D191—21no—
88C2×C22⋊D191—21no—
89C2×C22⋊D191—21no—
90C2×C22⋊D191D411yescharacteristic
91C2×C22⋊D191D411yescharacteristic
92D4×D191—28no—
93D4×D191—28no—
94D4×D191—22no—
95D4×D191—22no—
96D4×D191—22no—
97D4×D191D414yescomplemented
98C22×Dic191—21no—
99C2×C4×D191—21no—
100C2×C4×D191—21no—
101C2×C4×D191—21no—
102C2×C4×D191C2311yescharacteristic
103C2×C4×D191C2311yescharacteristic
104#15690277—22no—
105#15690276—22no—
106#15690276—22no—
107#15690275—22no—
108#15690272—22no—
109#15690272—22no—
110#15690270—22no—
111C2×C191⋊C8C2311yescharacteristic
112C2×C191⋊C8C2311yescharacteristic
113C8×D191—22no—
114C4⋊Dic191—21no—
115C4⋊Dic191C2311yescharacteristic
116Dic191⋊C4—21no—
117Dic191⋊C4—21no—
118C22⋊Dic191—21no—
119C22⋊Dic191—21no—
120(C2×D191)⋊C4—21no—
121#41×C191—22no—
122#41×C191—22no—
123C2×Q8×C191C2311yescharacteristic
124C2×D4×C191—21no—
125C2×D4×C191—21no—
126C2×D4×C191C2311yescharacteristic
127C2×D4×C191C2311yescharacteristic
128C22×C4×C191—21no—
129C22×C4×C191C2311yescharacteristic
130SD16×C191—22no—
131D8×C191—22no—
132C2×C8×C191C2311yescharacteristic
133C2×C8×C191C2311yescharacteristic
134C4⋊C4×C191C2311yescharacteristic
135C22⋊C4×C191—21no—
136C22×D191—21no—
137C22×D191—21no—
138C22×D191—24no—
139C22×D191—24no—
140C22×D191—24no—
141C22×D191—81no—
142C22×D191—81no—
143C22×D191—81no—
144C22×D191—81no—
145C22×D191—81no—
146C22×D191—81no—
147C22×D191C2×D411yescharacteristic
148C22×D191C2×D411yescharacteristic
149Dic382—22no—
150Dic382—21no—
151Dic382—21no—
152D764—24no—
153D764—24no—
154D764—24no—
155D764—22no—
156D764—21no—
157D764—21no—
158D764C2×D412yes—
159D764C2×D412yes—
160C22⋊D191—24no—
161C22⋊D191—22no—
162C22⋊D191—22no—
163C22⋊D191—24no—
164C22⋊D191—42no—
165C22⋊D191—81no—
166C2×Dic191—21no—
167C2×Dic191—21no—
168C2×Dic191—42no—
169C2×Dic191C2×D411yescharacteristic
170C2×Dic191C2×D411yescharacteristic
171C4×D191—24no—
172C4×D191—21no—
173C4×D191—21no—
174C4×D191—24no—
175C4×D191—81no—
176C4×D191C2×D412yes—
177C4×D191C2×D412yes—
178C191⋊C8—21no—
179C191⋊C8—21no—
180C23×C191—21no—
181C23×C191—21no—
182C23×C191C2×D411yescharacteristic
183Q8×C191—21no—
184Q8×C191C2×D412yes—
185D4×C191—24no—
186D4×C191—21no—
187D4×C191—21no—
188D4×C191—21no—
189D4×C191—81no—
190D4×C191C2×D412yes—
191C2×C4×C191—24no—
192C2×C4×C191—21no—
193C2×C4×C191—21no—
194C2×C4×C191—21no—
195C2×C4×C191—21no—
196C2×C4×C191C2×D411yescharacteristic
197C2×C4×C191C2411yescharacteristic
198C8×C191—21no—
199C8×C191—21no—
200C22×C191—21no—
201C22×C191—22no—
202C22×C191—21no—
203C22×C191—21no—
204C22×C191—41no—
205C22×C191—41no—
206C22×C191—41no—
207C22×C191—41no—
208C22×C191—41no—
209C22×C191C22×D411yescharacteristic
210C2×D191—21no—
211C2×D191—21no—
212C2×D191—21no—
213C2×D191—22no—
214C2×D191—22no—
215C2×D191—21no—
216C2×D191—21no—
217C2×D191—41no—
218C2×D191—41no—
219C2×D191—41no—
220C2×D191—41no—
221C2×D191—84no—
222C2×D191—84no—
223C2×D191—84no—
224C2×D191#12012yes—
225C4×C191—21no—
226C4×C191—22no—
227C4×C191—41no—
228C4×C191C22×D411yescharacteristic
229C4×C191C22×D411yescharacteristic
230Dic191—21no—
231Dic191—41no—
232Dic191—41no—
233Dic191#12012yes—
234C2×C191—41no—
235C2×C191—42no—
236C2×C191—81no—
237C2×C191C2×#12011yescharacteristic
238C2×C191#57611yescharacteristic
239C2×C191#57711yescharacteristic
240D191—22no—
241D191—41no—
242D191—42no—
243D191—42no—
244C191#301211yescharacteristic, complemented, Sylow
245#3012—1911nomaximal, Sylow
246C2×H(5,𝔽2)—1911no—
247C2×AGL(1,ℤ/8)—1911no—
248C2×#135—1911no—
249#537—1911no—
250#452—1911no—
251#452—1911no—
252#450—1911no—
253#450—1911no—
254#449—1911no—
255#449D19111yescharacteristic, complemented
256#447—1911no—
257#447—1911no—
258#417—1911no—
259C2×#102—1911no—
260#408—1911no—
261H(5,𝔽2)—3824no—
262H(5,𝔽2)—3824no—
263C2×#41—3821no—
264C2×#41—1911no—
265C2×#41—1911no—
266C2×#41—1911no—
267C2×#41—1911no—
268C2×#41C2×D19111yescharacteristic
269C22×D4—3821no—
270C22×D4—3821no—
271C22×D4—3821no—
272C22×D4—1911no—
273C22×D4—1911no—
274C22×D4—1911no—
275AGL(1,ℤ/8)—3824no—
276#135—3824no—
277C2×Q16—1911no—
278C2×SD16—1911no—
279C2×SD16—1911no—
280C2×SD16—1911no—
281C2×SD16C2×D19111yescharacteristic, complemented
282C2×D8—1911no—
283C2×D8—1911no—
284C2×D8C2×D19111yescharacteristic, complemented
285C2×M16—1911no—
286C22×C8—1911no—
287C22⋊Q8—3821no—
288C22⋊Q8—1911no—
289C22⋊Q8—1911no—
290#121—3821no—
291#121—3821no—
292#121—3821no—
293#121—1911no—
294#121C2×D19111yescharacteristic, complemented
295C2×C4⋊C4—1911no—
296#103—1911no—
297#103C2×D19111yescharacteristic, complemented
298#102—1911no—
299#102—1911no—
300#102—1911no—
301#102—1911no—
302#102—1911no—
303#102C2×D19111yescharacteristic, complemented
304C22⋊C8—1911no—
305C22⋊C8—1911no—
306C22⋊C8—1911no—
307C22⋊C8C2×D19111yescharacteristic, complemented
308C24—3821no—
309C24—3821no—
310C24—3821no—
311#41—22no—
312#41—22no—
313#41—3828no—
314#41—3822no—
315#41—3822no—
316#41—3822no—
317#41—3822no—
318#41—3822no—
319#41—3822no—
320#41—3822no—
321#41—1914no—
322C2×Q8—1911no—
323C2×Q8—1911no—
324C2×Q8C22×D19111yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—3828no—
328C2×D4—3821no—
329C2×D4—3821no—
330C2×D4—3822no—
331C2×D4—3822no—
332C2×D4—3828no—
333C2×D4—3828no—
334C2×D4—3822no—
335C2×D4—3822no—
336C2×D4—3821no—
337C2×D4—3821no—
338C2×D4—3822no—
339C2×D4—3821no—
340C2×D4—3821no—
341C2×D4—3821no—
342C2×D4—1911no—
343C2×D4—1911no—
344C2×D4—1914no—
345C2×D4—1911no—
346C2×D4—1911no—
347C2×D4—1911no—
348C2×D4C22×D19111yescharacteristic
349C2×D4C22×D19111yescharacteristic
350C22×C4—21no—
351C22×C4—3821no—
352C22×C4—3821no—
353C22×C4—3821no—
354C22×C4—3821no—
355C22×C4—1911no—
356C22×C4—1911no—
357C22×C4C22×D19111yescharacteristic
358Q16—3822no—
359SD16—22no—
360SD16—3822no—
361SD16—3822no—
362SD16—3822no—
363D8—22no—
364D8—3822no—
365D8—3822no—
366M16—3822no—
367C2×C8—3822no—
368C2×C8—1911no—
369C2×C8—1911no—
370C2×C8C22×D19111yescharacteristic
371C2×C8C22×D19111yescharacteristic
372C4⋊C4—3821no—
373C4⋊C4—3821no—
374C4⋊C4—3821no—
375C4⋊C4—1911no—
376C4⋊C4C22×D19111yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—3821no—
379C22⋊C4—3821no—
380C22⋊C4—3821no—
381C23—21no—
382C23—21no—
383C23—3824no—
384C23—3824no—
385C23—3824no—
386C23—3821no—
387C23—3821no—
388C23—1911no—
389C23—1911no—
390C23—15281no—
391C23—15281no—
392C23—15281no—
393C23—15281no—
394C23—15281no—
395C23—15281no—
396C23D4×D19111yescharacteristic
397Q8—21no—
398Q8—3822no—
399Q8—3821no—
400Q8—3821no—
401Q8D4×D19112yes—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—81no—
407D4—3824no—
408D4—3824no—
409D4—3824no—
410D4—3822no—
411D4—3824no—
412D4—3822no—
413D4—3822no—
414D4—3824no—
415D4—3821no—
416D4—3821no—
417D4—1912no—
418D4—1912no—
419D4—15281no—
420D4—7642no—
421D4D4×D19112yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—3824no—
428C2×C4—3824no—
429C2×C4—3821no—
430C2×C4—3821no—
431C2×C4—3821no—
432C2×C4—3821no—
433C2×C4—1911no—
434C2×C4—1911no—
435C2×C4—1912no—
436C2×C4—1912no—
437C2×C4—15281no—
438C2×C4—7642no—
439C2×C4D4×D19111yescharacteristic
440C2×C4C23×D19111yescharacteristic
441C8—21no—
442C8—21no—
443C8—3821no—
444C8—3821no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—3821no—
455C22—3821no—
456C22—3821no—
457C22—3821no—
458C22—3821no—
459C22—3822no—
460C22—3822no—
461C22—1912no—
462C22—15284no—
463C22—15284no—
464C22—15284no—
465C22—7641no—
466C22—7641no—
467C22—7641no—
468C22—7641no—
469C22C2×D4×D19111yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—3821no—
474C4—1912no—
475C4—7641no—
476C4—7641no—
477C4C2×D4×D19111yescharacteristic
478C4C2×D4×D19111yescharacteristic
479C2—41no—
480C2—42no—
481C2—81no—
482C2—3822no—
483C2—7641no—
484C2—7642no—
485C2—7642no—
486C2#120×D19111yescharacteristic
487C2#2713346711yescharacteristic
488C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#449×C191C2
socleC22×C191C22×D4
hypercenter#449D191
hypocenterC191#3012
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#20259368C2semidirect
2. C2×#20259346C2semidirect
3. C2×#20259343C2semidirect
4. #27133242C2semidirect
5. #27132828C2semidirect
6. #27132825C2semidirect
7. #27132816C2semidirect
8. #27132815C2semidirect
9. #27132807C2semidirect
10. #27132796C2semidirect
11. #27132795C2semidirect
12. #27132663C2semidirect
13. #102×D191C2semidirect
14. #27132628C2semidirect
15. #449×C191C2semidirect
22. C2×#15690286C22semidirect
31. C2×#15690277C22semidirect
32. C2×#15690276C22semidirect
33. C2×#15690276C22semidirect
34. C2×#15690275C22semidirect
35. C2×#15690272C22semidirect
36. C2×#15690272C22semidirect
37. C2×#15690270C22semidirect
38. C2×C8×D191C22semidirect
39. #20259302C22semidirect
40. #20259301C22semidirect
45. #20259293C22semidirect
46. C4⋊C4×D191C22semidirect
47. #20259246C22semidirect
48. #20259244C22semidirect
49. #20259244C22semidirect
50. #20259243C22semidirect
51. #20259242C22semidirect
52. #20259242C22semidirect
53. C22⋊(C191⋊C8)C22semidirect
54. (C2×D191)⋊C8C22semidirect
55. (C2×D191)⋊C8C22semidirect
57. C2×SD16×C191C22semidirect
58. C2×D8×C191C22semidirect
59. #121×C191C22semidirect
60. #103×C191C22semidirect
61. #102×C191C22semidirect
62. C22⋊C8×C191C22semidirect
69. #15690288D4semidirect
97. D4×D191D4semidirect
244. C191#3012semidirect
255. #449D191semidirect
281. C2×SD16C2×D191semidirect
284. C2×D8C2×D191semidirect
294. #121C2×D191semidirect
297. #103C2×D191semidirect
303. #102C2×D191semidirect
307. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
C2×#202593462
C2×#202593432
#271332422
#271328282
#271328252
#271328162
#271328152
#271328072
#271327962
#271327952
#271326632
#102×D1912
#271326282
#449×C1912
#3012191