Finite Group Databasebeta

← #27180381

Subgroups — #27180381

Order 12928.

806 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D101C211yesmaximal, characteristic, complemented
2C2×#15692282C211yesmaximal, characteristic, complemented
3C2×#15692279C211yesmaximal, characteristic, complemented
4#121×D101C211yesmaximal, characteristic, complemented
5#20265324C211yesmaximal, characteristic, complemented
6#20265322C211yesmaximal, characteristic, complemented
7#20265320C211yesmaximal, characteristic, complemented
8#20265317C211yesmaximal, characteristic, complemented
9#20265314C211yesmaximal, characteristic, complemented
10#20265312C211yesmaximal, characteristic, complemented
11#20265309C211yesmaximal, characteristic, complemented
12#20265177C211yesmaximal, characteristic, complemented
13#20265175C211yesmaximal, characteristic, complemented
14#20265139C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074—21no—
18C2×#13108074C2211yescharacteristic, complemented
19C2×#13108073—21no—
20C2×#13108073C2211yescharacteristic
21#41×D101—24no—
22#41×D101—24no—
23C2×Q8×D101C2211yescharacteristic, complemented
24C2×D4×D101—21no—
25C2×D4×D101—21no—
26C2×D4×D101C2211yescharacteristic, complemented
27C2×D4×D101C2211yescharacteristic, complemented
28C22×C4×D101—21no—
29C22×C4×D101C2211yescharacteristic
30#15692282—24no—
31#15692279—24no—
32C2×#13108063C2211yescharacteristic, complemented
33C2×#13108062C2211yescharacteristic, complemented
34C2×#13108061C2211yescharacteristic, complemented
35C2×D808C2211yescharacteristic, complemented
36C2×#13108059C2211yescharacteristic, complemented
37C2×#13108059C2211yescharacteristic, complemented
38C2×#13108057C2211yescharacteristic, complemented
39C2×#13108057C2211yescharacteristic, complemented
40#15692230C2211yescharacteristic
41#15692229C2211yescharacteristic, complemented
42#15692228—21no—
43#15692227—21no—
44#15692226C2211yescharacteristic
45C4⋊C4×D101C2211yescharacteristic, complemented
46C22⋊C4×D101—21no—
47#15692180C2211yescharacteristic, complemented
48#15692179C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, complemented
52#15692175C2211yescharacteristic, complemented
53C22⋊#11717579C2211yescharacteristic, complemented
54#15692167C2211yescharacteristic, complemented
55#15692167C2211yescharacteristic, complemented
56C2×#41×C101C2211yescharacteristic, complemented
57C2×SD16×C101C2211yescharacteristic, complemented
58C2×D8×C101C2211yescharacteristic, complemented
59#121×C101C2211yescharacteristic, complemented
60#103×C101C2211yescharacteristic, complemented
61#102×C101C2211yescharacteristic, complemented
62C22⋊C8×C101C2211yescharacteristic, complemented
63C23×D101—21no—
64C23×D101—21no—
65C23×D101D411yescharacteristic
66#13108075—22no—
67#13108075D414yescomplemented
68#13108074—28no—
69#13108074—22no—
70#13108074—22no—
71#13108073—28no—
72#13108073—22no—
73#13108073D414yescomplemented
74C2×Dic202—21no—
75C2×Dic202C2311yescharacteristic
76Q8×D101—22no—
77Q8×D101—22no—
78C2×D404—21no—
79C2×D404—21no—
80C2×D404C2311yescharacteristic
81C2×D404C2311yescharacteristic
82C2×C22⋊D101—21no—
83C2×C22⋊D101—21no—
84C2×C22⋊D101—21no—
85C2×C22⋊D101—21no—
86C2×C22⋊D101—21no—
87C2×C22⋊D101—21no—
88D4×D101—28no—
89D4×D101—22no—
90D4×D101—22no—
91D4×D101—22no—
92D4×D101—22no—
93D4×D101—44no—
94C22×Dic101—21no—
95C22×Dic101D411yescharacteristic
96C2×C4×D101—28no—
97C2×C4×D101—21no—
98C2×C4×D101—21no—
99C2×C4×D101—21no—
100C2×C4×D101—22no—
101C2×C4×D101—22no—
102C2×C4×D101—21no—
103C2×C4×D101D411yescharacteristic
104C2×C4×D101D411yescharacteristic
105C2×C4×D101C2311yescharacteristic
106C2×C4×D101C2311yescharacteristic
107#13108063—22no—
108#13108062—22no—
109#13108061—22no—
110D808—22no—
111#13108059—22no—
112#13108059—22no—
113#13108057—22no—
114#13108057—22no—
115C2×#11717579C2311yescharacteristic
116C2×#11717579C2311yescharacteristic
117C4⋊Dic101C2311yescharacteristic
118#13108053—21no—
119C22⋊Dic101—21no—
120#13108051—21no—
121#13108051—21no—
122#41×C101—22no—
123#41×C101—22no—
124C2×Q8×C101C2311yescharacteristic
125C2×D4×C101—21no—
126C2×D4×C101—21no—
127C2×D4×C101C2311yescharacteristic
128C2×D4×C101C2311yescharacteristic
129C22×C4×C101—21no—
130C22×C4×C101C2311yescharacteristic
131SD16×C101—22no—
132D8×C101—22no—
133C2×C8×C101C2311yescharacteristic
134C2×C8×C101C2311yescharacteristic
135C4⋊C4×C101C2311yescharacteristic
136C22⋊C4×C101—21no—
137C23×C101—21no—
138C23×C101—21no—
139C23×C101C2×D411yescharacteristic
140C22×D101—24no—
141C22×D101—21no—
142C22×D101—22no—
143C22×D101—22no—
144C22×D101—21no—
145C22×D101—41no—
146C22×D101—42no—
147C22×D101—42no—
148C22×D101—42no—
149C22×D101—42no—
150C22×D101—42no—
151C22×D101C2×D411yescharacteristic
152C22×D101C2×D411yescharacteristic
153Q8×C101—21no—
154Q8×C101C2×D412yescomplemented
155D4×C101—24no—
156D4×C101—21no—
157D4×C101—21no—
158D4×C101—21no—
159D4×C101—42no—
160D4×C101C2×D412yescomplemented
161C2×C4×C101—24no—
162C2×C4×C101—21no—
163C2×C4×C101—21no—
164C2×C4×C101—21no—
165C2×C4×C101—21no—
166C2×C4×C101C2×D411yescharacteristic
167C2×C4×C101C2411yescharacteristic
168C22⋊D101—24no—
169C22⋊D101—24no—
170C22⋊D101—24no—
171C22⋊D101—42no—
172C22⋊D101—42no—
173C22⋊D101—42no—
174C2×Dic101—21no—
175C2×Dic101—24no—
176C2×Dic101—22no—
177C2×Dic101—21no—
178C2×Dic101C2×D411yescharacteristic
179C2×Dic101C2×D411yescharacteristic
180D404—24no—
181D404—21no—
182D404—21no—
183D404—21no—
184D404—42no—
185D404C2×D412yescomplemented
186C4×D101—24no—
187C4×D101—24no—
188C4×D101—22no—
189C4×D101—21no—
190C4×D101—21no—
191C4×D101—22no—
192C4×D101—24no—
193C4×D101—42no—
194C4×D101C2×D412yes—
195C4×D101C2×D412yes—
196Dic202—24no—
197Dic202—21no—
198Dic202C2×D412yescomplemented
199C8×C101—21no—
200C8×C101—21no—
201#11717579—21no—
202#11717579—21no—
203C22×C101—21no—
204C22×C101—22no—
205C22×C101—21no—
206C22×C101—21no—
207C22×C101—41no—
208C22×C101—41no—
209C22×C101—41no—
210C22×C101—41no—
211C22×C101—41no—
212C22×C101C22×D411yescharacteristic
213C2×D101—21no—
214C2×D101—21no—
215C2×D101—21no—
216C2×D101—21no—
217C2×D101—22no—
218C2×D101—44no—
219C2×D101—48no—
220C2×D101—41no—
221C2×D101—41no—
222C2×D101—41no—
223C2×D101—41no—
224C2×D101—41no—
225C2×D101—84no—
226C2×D101#12012yes—
227C2×D101AGL(1,ℤ/8)12yescomplemented
228C4×C101—22no—
229C4×C101—21no—
230C4×C101—41no—
231C4×C101C22×D411yescharacteristic
232C4×C101C22×D411yescharacteristic
233Dic101—21no—
234Dic101—22no—
235Dic101—41no—
236Dic101#12012yes—
237C2×C101—41no—
238C2×C101—42no—
239C2×C101—81no—
240C2×C101C2×#12011yescharacteristic
241C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
242C2×C101#57511yescharacteristic
243D101—22no—
244D101—22no—
245D101—42no—
246D101—81no—
247#3006—1011nomaximal, Sylow
248C101#300611yescharacteristic, complemented, Sylow
249C22×#41—1011no—
250C2×AGL(1,ℤ/8)—1011no—
251C2×AGL(1,ℤ/8)—1011no—
252C2×#121—1011no—
253#449—1011no—
254#449—1011no—
255#449—1011no—
256#449D10111yescharacteristic, complemented
257#448—1011no—
258#448—1011no—
259#447—1011no—
260#447—1011no—
261#417—1011no—
262#417—1011no—
263#407—1011no—
264C2×#41—1011no—
265C2×#41—1011no—
266C2×#41—1011no—
267C2×#41—2021no—
268C2×#41—2021no—
269C2×#41—2024no—
270C2×#41—2024no—
271C2×#41C2×D10111yescharacteristic, complemented
272C22×Q8—1011no—
273C22×D4—1011no—
274C22×D4—1011no—
275C22×D4—2021no—
276C22×D4—2021no—
277C23×C4—1011no—
278C23×C4—2021no—
279AGL(1,ℤ/8)—2024no—
280AGL(1,ℤ/8)—2024no—
281C2×SD16—1011no—
282C2×SD16—1011no—
283C2×SD16—1011no—
284C2×SD16C2×D10111yescharacteristic, complemented
285C2×D8—1011no—
286C2×D8—1011no—
287C2×D8—1011no—
288C2×D8C2×D10111yescharacteristic, complemented
289C2×M16—1011no—
290C2×M16—1011no—
291#121—1011no—
292#121—1011no—
293#121—1011no—
294#121—2021no—
295#121—2021no—
296#121C2×D10111yescharacteristic, complemented
297C2×C4⋊C4—1011no—
298C2×C22⋊C4—2021no—
299#103—1011no—
300#103—1011no—
301#103—1011no—
302#103C2×D10111yescharacteristic, complemented
303#102—1011no—
304#102—1011no—
305#102—1011no—
306#102C2×D10111yescharacteristic, complemented
307C22⋊C8—1011no—
308C22⋊C8—1011no—
309C22⋊C8—1011no—
310C22⋊C8C2×D10111yescharacteristic, complemented
311C24—1011no—
312C24—2021no—
313C24—2021no—
314#41—22no—
315#41—22no—
316#41—1014no—
317#41—1014no—
318#41—2022no—
319#41—2022no—
320#41—2028no—
321#41—2022no—
322#41—2022no—
323#41—2028no—
324C2×Q8—1011no—
325C2×Q8—2021no—
326C2×Q8—2022no—
327C2×Q8—2022no—
328C2×Q8C22×D10111yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—1011no—
332C2×D4—1011no—
333C2×D4—4044no—
334C2×D4—2021no—
335C2×D4—2021no—
336C2×D4—2022no—
337C2×D4—2022no—
338C2×D4—2021no—
339C2×D4—2021no—
340C2×D4—2022no—
341C2×D4—2028no—
342C2×D4—2021no—
343C2×D4—2021no—
344C2×D4—2021no—
345C2×D4—2021no—
346C2×D4—2022no—
347C2×D4C22×D10111yescharacteristic
348C2×D4C22×D10111yescharacteristic
349C22×C4—21no—
350C22×C4—1011no—
351C22×C4—1011no—
352C22×C4—1011no—
353C22×C4—1011no—
354C22×C4—1011no—
355C22×C4—2022no—
356C22×C4—2022no—
357C22×C4—2021no—
358C22×C4—2021no—
359C22×C4—2021no—
360C22×C4—2028no—
361C22×C4—2021no—
362C22×C4—2021no—
363C22×C4C22×D10111yescharacteristic
364SD16—22no—
365SD16—2022no—
366SD16—2022no—
367SD16—2022no—
368D8—22no—
369D8—2022no—
370D8—2022no—
371D8—2022no—
372M16—2022no—
373M16—2022no—
374C2×C8—1011no—
375C2×C8—1011no—
376C2×C8C22×D10111yescharacteristic
377C2×C8C22×D10111yescharacteristic
378C4⋊C4—1011no—
379C4⋊C4—2021no—
380C4⋊C4C22×D10111yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—2021no—
383C22⋊C4—2021no—
384C22⋊C4—2021no—
385C23—21no—
386C23—21no—
387C23—1011no—
388C23—1011no—
389C23—4041no—
390C23—4042no—
391C23—4042no—
392C23—4042no—
393C23—4042no—
394C23—4042no—
395C23—2022no—
396C23—2022no—
397C23—2024no—
398C23—2021no—
399C23—2021no—
400C23D4×D10111yescharacteristic
401Q8—21no—
402Q8—1012no—
403Q8—2024no—
404Q8—2021no—
405Q8D4×D10112yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—1012no—
412D4—4042no—
413D4—4042no—
414D4—4042no—
415D4—4042no—
416D4—2024no—
417D4—2024no—
418D4—2024no—
419D4—2024no—
420D4—2021no—
421D4—2021no—
422D4—2021no—
423D4D4×D10112yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—1011no—
430C2×C4—1012no—
431C2×C4—1012no—
432C2×C4—1011no—
433C2×C4—4042no—
434C2×C4—2022no—
435C2×C4—2024no—
436C2×C4—2022no—
437C2×C4—2024no—
438C2×C4—2024no—
439C2×C4—2022no—
440C2×C4—2024no—
441C2×C4—2021no—
442C2×C4—2021no—
443C2×C4—2021no—
444C2×C4—2021no—
445C2×C4D4×D10111yescharacteristic
446C2×C4C23×D10111yescharacteristic
447C8—21no—
448C8—21no—
449C8—2021no—
450C8—2021no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—8084no—
461C22—1012no—
462C22—1012no—
463C22—4048no—
464C22—4044no—
465C22—4041no—
466C22—4041no—
467C22—4041no—
468C22—4041no—
469C22—4041no—
470C22—2022no—
471C22—2021no—
472C22—2021no—
473C22—2021no—
474C22—2021no—
475C22C2×D4×D10111yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—1012no—
480C4—4041no—
481C4—2021no—
482C4—2022no—
483C4C2×D4×D10111yescharacteristic
484C4C2×D4×D10111yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—8081no—
489C2—4042no—
490C2—2022no—
491C2—2022no—
492C2#120×D10111yescharacteristic
493C2AGL(1,ℤ/8)×D10111yescharacteristic
494C2#2026597411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3006
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D101C2semidirect
2. C2×#15692282C2semidirect
3. C2×#15692279C2semidirect
4. #121×D101C2semidirect
5. #20265324C2semidirect
6. #20265322C2semidirect
7. #20265320C2semidirect
8. #20265317C2semidirect
9. #20265314C2semidirect
10. #20265312C2semidirect
11. #20265309C2semidirect
12. #20265177C2semidirect
13. #20265175C2semidirect
14. #20265139C2semidirect
15. #449×C101C2semidirect
18. C2×#13108074C22semidirect
23. C2×Q8×D101C22semidirect
26. C2×D4×D101C22semidirect
27. C2×D4×D101C22semidirect
32. C2×#13108063C22semidirect
33. C2×#13108062C22semidirect
34. C2×#13108061C22semidirect
35. C2×D808C22semidirect
36. C2×#13108059C22semidirect
37. C2×#13108059C22semidirect
38. C2×#13108057C22semidirect
39. C2×#13108057C22semidirect
41. #15692229C22semidirect
45. C4⋊C4×D101C22semidirect
47. #15692180C22semidirect
48. #15692179C22semidirect
49. #15692178C22semidirect
50. #15692177C22semidirect
51. #15692176C22semidirect
52. #15692175C22semidirect
53. C22⋊#11717579C22semidirect
54. #15692167C22semidirect
55. #15692167C22semidirect
56. C2×#41×C101C22semidirect
57. C2×SD16×C101C22semidirect
58. C2×D8×C101C22semidirect
59. #121×C101C22semidirect
60. #103×C101C22semidirect
61. #102×C101C22semidirect
62. C22⋊C8×C101C22semidirect
67. #13108075D4semidirect
73. #13108073D4semidirect
154. Q8×C101C2×D4semidirect
160. D4×C101C2×D4semidirect
185. D404C2×D4semidirect
198. Dic202C2×D4semidirect
227. C2×D101AGL(1,ℤ/8)semidirect
248. C101#3006semidirect
256. #449D101semidirect
271. C2×#41C2×D101semidirect
284. C2×SD16C2×D101semidirect
288. C2×D8C2×D101semidirect
296. #121C2×D101semidirect
302. #103C2×D101semidirect
306. #102C2×D101semidirect
310. C22⋊C8C2×D101semidirect
405. Q8D4×D101semidirect
423. D4D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1012
C2×#156922822
C2×#156922792
#121×D1012
#202653242
#202653222
#202653202
#202653172
#202653142
#202653122
#202653092
#202651772
#202651752
#202651392
#449×C1012
#3006101