Finite Group Databasebeta

← #27182608

Subgroups — #27182608

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×#15692281C211yesmaximal, characteristic, complemented
2C22×#13108059C211yesmaximal, characteristic, complemented
3#20265823C211yesmaximal, characteristic, complemented
4#20265564C211yesmaximal, characteristic, complemented
5#20265428C211yesmaximal, characteristic, complemented
6#20265404C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265369C211yesmaximal, characteristic, complemented
9#20265328C211yesmaximal, characteristic, complemented
10#20265320C211yesmaximal, characteristic, complemented
11#20265311C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265287C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15D42×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic, complemented
18#15692281—24no—
19C2×#13108062C2211yescharacteristic, complemented
20C2×#13108062C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic, complemented
22C2×#13108059—24no—
23C2×#13108059C2211yescharacteristic
24C2×#13108059C2211yescharacteristic
25C2×#13108059C2211yescharacteristic, complemented
26C2×#13108059C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108058C2211yescharacteristic, complemented
30C22×#11717579C2211yescharacteristic, complemented
31#15692243C2211yescharacteristic, complemented
32#15692240—21no—
33#15692234C2211yescharacteristic, complemented
34#15692229C2211yescharacteristic, complemented
35#15692228—21no—
36#15692223—21no—
37C4×D404C2211yescharacteristic, complemented
38C22⋊C4×D101—21no—
39#15692186C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692178C2211yescharacteristic, complemented
42#15692176C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic, complemented
44#15692176C2211yescharacteristic, complemented
45#15692175C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic, complemented
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C22×D4×C101—21no—
51C22×D4×C101C2211yescharacteristic
52C22×D4×C101C2211yescharacteristic, complemented
53#127×C101C2211yescharacteristic, complemented
54#121×C101—21no—
55#121×C101C2211yescharacteristic
56#121×C101C2211yescharacteristic, complemented
57#120×C101—21no—
58#120×C101—21no—
59C4×D4×C101C2211yescharacteristic
60C4×D4×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—41no—
68C2×D404C2311yescharacteristic
69C2×D404C2311yescharacteristic
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101—21no—
74#13108062—22no—
75#13108062—22no—
76#13108062—41no—
77#13108059—22no—
78#13108059—22no—
79#13108059—22no—
80#13108059—22no—
81#13108059—41no—
82#13108059D414yescomplemented
83#13108058—22no—
84C2×#11717579—22no—
85C2×#11717579C2311yescharacteristic
86C2×#11717579C2311yescharacteristic
87C2×#11717579C2311yescharacteristic
88C2×#11717579C2311yescharacteristic
89C4⋊Dic101C2311yescharacteristic
90#13108053—21no—
91C22⋊Dic101—21no—
92#13108051—21no—
93#13108051—21no—
94#13108051—21no—
95#13108051—21no—
96C24×C101—21no—
97C24×C101—21no—
98C2×D4×C101—22no—
99C2×D4×C101—22no—
100C2×D4×C101—21no—
101C2×D4×C101—21no—
102C2×D4×C101—21no—
103C2×D4×C101—21no—
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101—41no—
107C2×D4×C101—41no—
108C2×D4×C101—44no—
109C2×D4×C101D411yescharacteristic
110C2×D4×C101C2311yescharacteristic
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114C22×C4×C101C2311yescharacteristic
115C22×C4×C101C2311yescharacteristic
116C4⋊C4×C101D411yescharacteristic
117C4⋊C4×C101C2311yescharacteristic
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—21no—
127C23×C101—42no—
128C23×C101—42no—
129C23×C101—42no—
130C23×C101—42no—
131C23×C101—81no—
132C23×C101—81no—
133C23×C101C2×D411yescharacteristic
134C23×C101C2×D411yescharacteristic
135C22×D101—21no—
136C22×D101—21no—
137C22×D101—42no—
138C22×D101—42no—
139C22×D101—81no—
140D4×C101—22no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—21no—
145D4×C101—21no—
146D4×C101—42no—
147D4×C101—42no—
148D4×C101—42no—
149D4×C101—42no—
150D4×C101—42no—
151D4×C101—81no—
152D4×C101C2×D412yes—
153C2×C4×C101—21no—
154C2×C4×C101—21no—
155C2×C4×C101—21no—
156C2×C4×C101—21no—
157C2×C4×C101—21no—
158C2×C4×C101—42no—
159C2×C4×C101C2×D411yescharacteristic
160C2×C4×C101C2×D411yescharacteristic
161C2×C4×C101C2×D411yescharacteristic
162C2×C4×C101C2×D411yescharacteristic
163C2×C4×C101C2411yescharacteristic
164C22⋊D101—42no—
165C22⋊D101—81no—
166C2×Dic101—21no—
167C2×Dic101—21no—
168D404—21no—
169D404—21no—
170D404—21no—
171D404—42no—
172D404C2×D412yescomplemented
173C4×D101—42no—
174C4×D101—81no—
175Dic202—21no—
176Dic202—21no—
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—41no—
193C22×C101—41no—
194C22×C101—41no—
195C22×C101—84no—
196C22×C101—84no—
197C22×C101AGL(1,ℤ/8)12yescomplemented
198C22×C101C22×D411yescharacteristic
199C2×D101—22no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—41no—
204C2×D101—41no—
205C2×D101—84no—
206C4×C101—21no—
207C4×C101—21no—
208C4×C101—21no—
209C4×C101—21no—
210C4×C101—41no—
211C4×C101C22×D411yescharacteristic
212C4×C101H(5,𝔽2)11yescharacteristic
213Dic101—41no—
214Dic101—41no—
215C2×C101—22no—
216C2×C101—41no—
217C2×C101—42no—
218C2×C101—81no—
219C2×C101—81no—
220C2×C101D4211yescharacteristic
221C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C101#57611yescharacteristic
223D101—42no—
224D101—81no—
225#3264—1011nomaximal, Sylow
226C101#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1011no—
228C22×D8—1011no—
229#546—1011no—
230D42D10111yescharacteristic, complemented
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×#41—1011no—
243C22×D4—21no—
244C22×D4—1011no—
245C22×D4C2×D10111yescharacteristic
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#127C2×D10111yescharacteristic, complemented
261#124—1011no—
262#123—2021no—
263C22⋊Q8—1011no—
264#121—21no—
265#121—1011no—
266#121—2021no—
267#121C2×D10111yescharacteristic
268#121C2×D10111yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—2021no—
272C4×D4—1011no—
273C4×D4C2×D10111yescharacteristic
274C4×D4C2×D10111yescharacteristic, complemented
275C2×C22⋊C4—2021no—
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—21no—
289C24—2021no—
290#41—4041no—
291#41—2022no—
292C2×Q8—1011no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—1011no—
305C2×D4—1011no—
306C2×D4—4041no—
307C2×D4—2022no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2021no—
311C2×D4C22⋊D10111yescharacteristic
312C2×D4C22×D10111yescharacteristic
313C2×D4C22×D10111yescharacteristic
314C2×D4C22×D10111yescharacteristic
315C22×C4—21no—
316C22×C4—2021no—
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—1011no—
336C4⋊C4—2021no—
337C4⋊C4C22⋊D10111yescharacteristic
338C4⋊C4C22×D10111yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2021no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C22⋊C4C22⋊D10111yescharacteristic
348C22⋊C4C22⋊D10111yescharacteristic
349C42C22×D10111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—8081no—
360C23—4042no—
361C23—4042no—
362C23—2021no—
363C23—2021no—
364C23C2×C22⋊D10111yescharacteristic
365C23C2×C22⋊D10111yescharacteristic
366Q8—2021no—
367Q8—2021no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—8081no—
381D4—1012no—
382D4—4042no—
383D4—4042no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387D4D4×D10112yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—8081no—
395C2×C4—4042no—
396C2×C4—2021no—
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—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—8084no—
424C22—4041no—
425C22—4041no—
426C22—4041no—
427C22—4041no—
428C22—4041no—
429C22—2022no—
430C22#1569228112yescomplemented
431C22C22×C22⋊D10111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—4041no—
438C4—4041no—
439C4C2×D4×D10111yescharacteristic
440C4#1569230011yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—8081no—
447C2—4042no—
448C2D4×C22⋊D10111yescharacteristic
449C2C2×#1569228111yescharacteristic
450C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingD42×C101C2
socleC22×C101C22×D4
hypercenterD42D101
hypocenterC101#3264
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C22×#13108059C2semidirect
3. #20265823C2semidirect
4. #20265564C2semidirect
5. #20265428C2semidirect
6. #20265404C2semidirect
7. #20265392C2semidirect
8. #20265369C2semidirect
9. #20265328C2semidirect
10. #20265320C2semidirect
11. #20265311C2semidirect
12. #20265310C2semidirect
13. #20265287C2semidirect
14. #20265249C2semidirect
15. D42×C101C2semidirect
16. C2×#13108074C22semidirect
17. C22×D404C22semidirect
19. C2×#13108062C22semidirect
20. C2×#13108062C22semidirect
21. C2×#13108062C22semidirect
25. C2×#13108059C22semidirect
26. C2×#13108059C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C22×#11717579C22semidirect
31. #15692243C22semidirect
33. #15692234C22semidirect
34. #15692229C22semidirect
37. C4×D404C22semidirect
39. #15692186C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692178C22semidirect
42. #15692176C22semidirect
43. #15692176C22semidirect
44. #15692176C22semidirect
45. #15692175C22semidirect
46. #15692175C22semidirect
47. C22⋊#11717579C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
52. C22×D4×C101C22semidirect
53. #127×C101C22semidirect
56. #121×C101C22semidirect
60. C4×D4×C101C22semidirect
82. #13108059D4semidirect
172. D404C2×D4semidirect
180. #11717579C2×D4semidirect
197. C22×C101AGL(1,ℤ/8)semidirect
226. C101#3264semidirect
230. D42D101semidirect
246. C22×D4C2×D101semidirect
260. #127C2×D101semidirect
268. #121C2×D101semidirect
274. C4×D4C2×D101semidirect
430. C22#15692281semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C22×#131080592
#202658232
#202655642
#202654282
#202654042
#202653922
#202653692
#202653282
#202653202
#202653112
#202653102
#202652872
#202652492
D42×C1012
#3264101