Finite Group Databasebeta

← #27182613

Subgroups — #27182613

Order 12928.

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×#15692280C211yesmaximal, characteristic, complemented
2C22×#13108059C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265567C211yesmaximal, characteristic, complemented
5#20265428C211yesmaximal, characteristic, complemented
6#20265408C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265372C211yesmaximal, characteristic, complemented
9#20265327C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265287C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#546×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×D404C2211yescharacteristic, complemented
18C2×D4×D101—21no—
19#15692280—24no—
20C2×#13108061C2211yescharacteristic, complemented
21C2×#13108061C2211yescharacteristic, complemented
22C2×#13108061C2211yescharacteristic, complemented
23C2×#13108059—24no—
24C2×#13108059C2211yescharacteristic
25C2×#13108059C2211yescharacteristic
26C2×#13108059C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108059C2211yescharacteristic, complemented
30C2×#13108058C2211yescharacteristic, complemented
31C22×#11717579C2211yescharacteristic, complemented
32#15692252C2211yescharacteristic, complemented
33#15692230—21no—
34#15692229C2211yescharacteristic
35#15692229C2211yescharacteristic, complemented
36#15692225—21no—
37#15692225—21no—
38C4×D404C2211yescharacteristic, complemented
39#15692186C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692176C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic, complemented
44#15692175C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic, complemented
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic, complemented
51C22×D4×C101C2211yescharacteristic, complemented
52#124×C101C2211yescharacteristic, complemented
53#123×C101—21no—
54C22⋊Q8×C101C2211yescharacteristic, complemented
55#121×C101—21no—
56#121×C101C2211yescharacteristic, complemented
57#120×C101—21no—
58C4×D4×C101C2211yescharacteristic
59C4×D4×C101C2211yescharacteristic, complemented
60C2×C22⋊C4×C101—21no—
61C23×D101—21no—
62C23×D101—21no—
63C2×D404—22no—
64C2×D404—22no—
65C2×D404—21no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—41no—
70C2×D404—41no—
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×D404C2311yescharacteristic
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76D4×D101—44no—
77C2×C4×D101—21no—
78#13108061—22no—
79#13108061—22no—
80#13108061—41no—
81#13108059—22no—
82#13108059—22no—
83#13108059—22no—
84#13108059—22no—
85#13108059—41no—
86#13108059D414yescomplemented
87#13108058—22no—
88C2×#11717579—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C2×#11717579C2311yescharacteristic
92C2×#11717579C2311yescharacteristic
93C4⋊Dic101C2311yescharacteristic
94#13108051—21no—
95#13108051—21no—
96#13108051—21no—
97C24×C101—21no—
98#41×C101—22no—
99#41×C101—41no—
100C2×Q8×C101C2311yescharacteristic
101C2×D4×C101—22no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101—41no—
106C2×D4×C101D411yescharacteristic
107C2×D4×C101C2311yescharacteristic
108C2×D4×C101C2311yescharacteristic
109C22×C4×C101—21no—
110C22×C4×C101—21no—
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C4⋊C4×C101—21no—
114C4⋊C4×C101D411yescharacteristic
115C4⋊C4×C101C2311yescharacteristic
116C22⋊C4×C101—21no—
117C22⋊C4×C101—21no—
118C22⋊C4×C101—21no—
119C22⋊C4×C101—21no—
120C22⋊C4×C101—21no—
121C22⋊C4×C101D411yescharacteristic
122C22⋊C4×C101D411yescharacteristic
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—42no—
127C23×C101—42no—
128C23×C101—81no—
129C23×C101C2×D411yescharacteristic
130C23×C101C2×D411yescharacteristic
131C22×D101—21no—
132C22×D101—21no—
133C22×D101—21no—
134C22×D101—42no—
135C22×D101—42no—
136C22×D101—42no—
137C22×D101—42no—
138C22×D101—81no—
139C22×D101—81no—
140Q8×C101—21no—
141Q8×C101—21no—
142D4×C101—22no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—21no—
146D4×C101—42no—
147D4×C101—42no—
148D4×C101—81no—
149D4×C101C2×D412yescomplemented
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101—21no—
153C2×C4×C101—21no—
154C2×C4×C101—21no—
155C2×C4×C101—21no—
156C2×C4×C101—42no—
157C2×C4×C101—81no—
158C2×C4×C101C2×D411yescharacteristic
159C2×C4×C101C2×D411yescharacteristic
160C2×C4×C101C2×D411yescharacteristic
161C2×C4×C101C2×D411yescharacteristic
162C2×C4×C101C2411yescharacteristic
163C22⋊D101—42no—
164C22⋊D101—42no—
165C2×Dic101—21no—
166D404—21no—
167D404—21no—
168D404—21no—
169D404—21no—
170D404—21no—
171D404—42no—
172D404—42no—
173D404—42no—
174D404—81no—
175D404C2×D412yes—
176C4×D101—42no—
177#11717579—21no—
178#11717579—21no—
179#11717579—21no—
180#11717579C2×D412yescomplemented
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—21no—
184C22×C101—22no—
185C22×C101—21no—
186C22×C101—21no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101—41no—
191C22×C101—41no—
192C22×C101—84no—
193C22×C101AGL(1,ℤ/8)12yescomplemented
194C22×C101C22×D411yescharacteristic
195C2×D101—22no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C2×D101—84no—
205C2×D101—84no—
206C4×C101—21no—
207C4×C101—21no—
208C4×C101—21no—
209C4×C101—21no—
210C4×C101—41no—
211C4×C101—41no—
212C4×C101C22×D411yescharacteristic
213C4×C101H(5,𝔽2)11yescharacteristic
214Dic101—41no—
215C2×C101—22no—
216C2×C101—41no—
217C2×C101—42no—
218C2×C101—81no—
219C2×C101D4211yescharacteristic
220C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C101#57611yescharacteristic
222D101—42no—
223D101—81no—
224D101—81no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8—1011no—
229#546D10111yescharacteristic, complemented
230D42—1011no—
231#496—1011no—
232#469—1011no—
233#465—1011no—
234#463—1011no—
235#459—1011no—
236#450—1011no—
237#449—1011no—
238#447—1011no—
239#447—1011no—
240#442—1011no—
241#435—1011no—
242C2×#41C2×D10111yescharacteristic, complemented
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—2021no—
246C22×D4C2×D10111yescharacteristic, complemented
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16—1011no—
250C2×SD16—1011no—
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—1011no—
256C2×D8—1011no—
257C2×D8—2024no—
258C2×M16—1011no—
259C22×C8—1011no—
260#127—1011no—
261#124C2×D10111yescharacteristic, complemented
262#123—21no—
263C22⋊Q8C2×D10111yescharacteristic, complemented
264#121—21no—
265#121—1011no—
266#121—1011no—
267#121—2021no—
268#121C2×D10111yescharacteristic, complemented
269#120—21no—
270#120—2021no—
271#120—2021no—
272C4×D4—1011no—
273C4×D4C2×D10111yescharacteristic
274C4×D4C2×D10111yescharacteristic, complemented
275C2×C22⋊C4—21no—
276#106—1011no—
277C4⋊C8—1011no—
278#103—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102—1011no—
283#102—1011no—
284C22⋊C8—1011no—
285C22⋊C8—1011no—
286#97—1011no—
287C24—21no—
288C24—2021no—
289C24—2021no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D10111yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—1011no—
299C2×D4—1011no—
300C2×D4—1011no—
301C2×D4—4041no—
302C2×D4—4041no—
303C2×D4—4044no—
304C2×D4—2022no—
305C2×D4—2022no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4—2021no—
312C2×D4C22⋊D10111yescharacteristic
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2021no—
318C22×C4C22×D10111yescharacteristic
319C22×C4C22×D10111yescharacteristic
320SD16—4041no—
321SD16—2022no—
322SD16—2022no—
323D8—1014no—
324D8—4041no—
325D8—2022no—
326D8—2022no—
327D8—2022no—
328D8—2022no—
329M16—2022no—
330C2×C8—1011no—
331C2×C8—1011no—
332C2×C8—1011no—
333C2×C8—1011no—
334C2×C8—2022no—
335C4⋊C4—21no—
336C4⋊C4—1011no—
337C4⋊C4C22⋊D10111yescharacteristic
338C4⋊C4C22×D10111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4C22⋊D10111yescharacteristic
348C22⋊C4C22⋊D10111yescharacteristic
349C42C22×D10111yescharacteristic
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—8081no—
356C23—8081no—
357C23—4042no—
358C23—4042no—
359C23—4042no—
360C23—4042no—
361C23—2021no—
362C23—2021no—
363C23—2021no—
364C23C2×C22⋊D10111yescharacteristic
365C23C2×C22⋊D10111yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—8081no—
376D4—1012no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—4042no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387D4D4×D10112yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—4042no—
397C2×C4—2021no—
398C2×C4C2×C22⋊D10111yescharacteristic
399C2×C4C2×C22⋊D10111yescharacteristic
400C2×C4C2×C22⋊D10111yescharacteristic
401C2×C4C2×C22⋊D10111yescharacteristic
402C2×C4C23×D10111yescharacteristic
403C8—1012no—
404C8—2021no—
405C8—2021no—
406C8—2021no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—84no—
419C22—8084no—
420C22—8084no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—4041no—
426C22—4041no—
427C22—4041no—
428C22—4041no—
429C22—2022no—
430C22#1569228012yescomplemented
431C22C22×C22⋊D10111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—4041no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—8081no—
446C2—8081no—
447C2—4042no—
448C2D4×C22⋊D10111yescharacteristic
449C2C2×#1569228011yescharacteristic
450C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#546×C101C2
socleC22×C101C22×D4
hypercenter#546D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C22×#13108059C2semidirect
3. D4×D404C2semidirect
4. #20265567C2semidirect
5. #20265428C2semidirect
6. #20265408C2semidirect
7. #20265389C2semidirect
8. #20265372C2semidirect
9. #20265327C2semidirect
10. #20265322C2semidirect
11. #20265311C2semidirect
12. #20265310C2semidirect
13. #20265287C2semidirect
14. #20265249C2semidirect
15. #546×C101C2semidirect
17. C22×D404C22semidirect
20. C2×#13108061C22semidirect
21. C2×#13108061C22semidirect
22. C2×#13108061C22semidirect
26. C2×#13108059C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108059C22semidirect
30. C2×#13108058C22semidirect
31. C22×#11717579C22semidirect
32. #15692252C22semidirect
35. #15692229C22semidirect
38. C4×D404C22semidirect
39. #15692186C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692176C22semidirect
43. #15692176C22semidirect
44. #15692175C22semidirect
45. #15692175C22semidirect
46. #15692175C22semidirect
47. C22⋊#11717579C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
50. C2×#41×C101C22semidirect
51. C22×D4×C101C22semidirect
52. #124×C101C22semidirect
54. C22⋊Q8×C101C22semidirect
56. #121×C101C22semidirect
59. C4×D4×C101C22semidirect
86. #13108059D4semidirect
149. D4×C101C2×D4semidirect
180. #11717579C2×D4semidirect
193. C22×C101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
229. #546D101semidirect
242. C2×#41C2×D101semidirect
246. C22×D4C2×D101semidirect
261. #124C2×D101semidirect
263. C22⋊Q8C2×D101semidirect
268. #121C2×D101semidirect
274. C4×D4C2×D101semidirect
387. D4D4×D101semidirect
430. C22#15692280semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C22×#131080592
D4×D4042
#202655672
#202654282
#202654082
#202653892
#202653722
#202653272
#202653222
#202653112
#202653102
#202652872
#202652492
#546×C1012
#3264101