Finite Group Databasebeta

← #27182628

Subgroups — #27182628

Order 12928.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692280C211yesmaximal, characteristic, complemented
2C22×#13108061C211yesmaximal, characteristic, complemented
3D4×D404C211yesmaximal, characteristic, complemented
4#20265567C211yesmaximal, characteristic, complemented
5#20265425C211yesmaximal, characteristic, complemented
6#20265411C211yesmaximal, characteristic, complemented
7#20265389C211yesmaximal, characteristic, complemented
8#20265377C211yesmaximal, characteristic, complemented
9#20265327C211yesmaximal, characteristic, complemented
10#20265322C211yesmaximal, characteristic, complemented
11#20265316C211yesmaximal, characteristic, complemented
12#20265310C211yesmaximal, characteristic, complemented
13#20265275C211yesmaximal, characteristic, complemented
14#20265249C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×D404C2211yescharacteristic
18C2×D4×D101—21no—
19#15692280—24no—
20C2×#13108061—24no—
21C2×#13108061C2211yescharacteristic
22C2×#13108061C2211yescharacteristic
23C2×#13108061C2211yescharacteristic, complemented
24C2×#13108061C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic, complemented
26C2×#13108061C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108059C2211yescharacteristic, complemented
30C2×#13108058C2211yescharacteristic, complemented
31C22×#11717579C2211yescharacteristic, complemented
32#15692252C2211yescharacteristic
33#15692230—21no—
34#15692229C2211yescharacteristic
35#15692229C2211yescharacteristic
36#15692225—21no—
37#15692225—21no—
38C4×D404C2211yescharacteristic
39#15692188C2211yescharacteristic, complemented
40C4⋊#11717579C2211yescharacteristic, complemented
41#15692179C2211yescharacteristic, complemented
42#15692179C2211yescharacteristic, complemented
43#15692176C2211yescharacteristic, complemented
44#15692175C2211yescharacteristic
45#15692175C2211yescharacteristic, complemented
46#15692175C2211yescharacteristic, complemented
47C22⋊#11717579C2211yescharacteristic, complemented
48C22⋊#11717579C2211yescharacteristic, complemented
49#15692166C2211yescharacteristic, complemented
50C2×#41×C101C2211yescharacteristic, complemented
51C22×Q8×C101C2211yescharacteristic, complemented
52#124×C101—21no—
53#124×C101C2211yescharacteristic, complemented
54C22⋊Q8×C101—21no—
55C22⋊Q8×C101C2211yescharacteristic, complemented
56#121×C101—21no—
57#121×C101C2211yescharacteristic, complemented
58C4×Q8×C101C2211yescharacteristic, complemented
59C4×D4×C101—21no—
60C4×D4×C101C2211yescharacteristic
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—22no—
81#13108061—22no—
82#13108061—41no—
83#13108061D414yescomplemented
84#13108059—22no—
85#13108059—22no—
86#13108059—41no—
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—
97#41×C101—22no—
98#41×C101—41no—
99C2×Q8×C101—21no—
100C2×Q8×C101—22no—
101C2×Q8×C101—41no—
102C2×Q8×C101C2311yescharacteristic
103C2×Q8×C101C2311yescharacteristic
104C2×D4×C101—21no—
105C2×D4×C101—21no—
106C2×D4×C101D411yescharacteristic
107C2×D4×C101C2311yescharacteristic
108C22×C4×C101—21no—
109C22×C4×C101—21no—
110C22×C4×C101C2311yescharacteristic
111C22×C4×C101C2311yescharacteristic
112C4⋊C4×C101—21no—
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×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C42×C101—21no—
123C42×C101C2311yescharacteristic
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C23×C101C2×D411yescharacteristic
127C22×D101—21no—
128C22×D101—21no—
129C22×D101—21no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—42no—
134C22×D101—81no—
135C22×D101—81no—
136Q8×C101—21no—
137Q8×C101—21no—
138Q8×C101—21no—
139Q8×C101—42no—
140Q8×C101C2×D412yescomplemented
141D4×C101—22no—
142D4×C101—21no—
143D4×C101—21no—
144D4×C101—42no—
145D4×C101—42no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—21no—
150C2×C4×C101—21no—
151C2×C4×C101—21no—
152C2×C4×C101—21no—
153C2×C4×C101—42no—
154C2×C4×C101—42no—
155C2×C4×C101C2×D411yescharacteristic
156C2×C4×C101C2×D411yescharacteristic
157C2×C4×C101C2×D411yescharacteristic
158C2×C4×C101C2×D411yescharacteristic
159C2×C4×C101C2411yescharacteristic
160C22⋊D101—42no—
161C22⋊D101—42no—
162C2×Dic101—21no—
163D404—21no—
164D404—21no—
165D404—21no—
166D404—21no—
167D404—21no—
168D404—42no—
169D404—42no—
170D404—42no—
171D404—81no—
172D404C2×D412yes—
173C4×D101—42no—
174#11717579—21no—
175#11717579—21no—
176#11717579—21no—
177#11717579C2×D412yescomplemented
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101AGL(1,ℤ/8)12yescomplemented
187C22×C101C22×D411yescharacteristic
188C2×D101—22no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—84no—
198C2×D101—84no—
199C4×C101—21no—
200C4×C101—21no—
201C4×C101—22no—
202C4×C101—21no—
203C4×C101—21no—
204C4×C101—41no—
205C4×C101—41no—
206C4×C101C22×D411yescharacteristic
207C4×C101H(5,𝔽2)11yescharacteristic
208Dic101—41no—
209C2×C101—22no—
210C2×C101—41no—
211C2×C101—81no—
212C2×C101D4211yescharacteristic
213C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C101#57711yescharacteristic
215D101—42no—
216D101—81no—
217D101—81no—
218#3265—1011nomaximal, Sylow
219C101#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C22×SD16—1011no—
222#548D10111yescharacteristic, complemented
223D42—1011no—
224#496—1011no—
225#468—1011no—
226#466—1011no—
227#463—1011no—
228#460—1011no—
229#450—1011no—
230#449—1011no—
231#448—1011no—
232#447—1011no—
233#440—1011no—
234#435—1011no—
235C2×#41C2×D10111yescharacteristic, complemented
236C22×Q8C2×D10111yescharacteristic, complemented
237C22×D4—1011no—
238C22×D4—1011no—
239C22×D4—2021no—
240AGL(1,ℤ/8)—2024no—
241C2×SD16—1011no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—1011no—
246C2×SD16—1011no—
247C2×SD16—2024no—
248C2×D8—1011no—
249C2×D8—1011no—
250C2×D8—1011no—
251C2×M16—1011no—
252C22×C8—1011no—
253#127—1011no—
254#124—21no—
255#124C2×D10111yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D10111yescharacteristic, complemented
258#121—21no—
259#121—1011no—
260#121—1011no—
261#121—2021no—
262#121C2×D10111yescharacteristic, complemented
263#120—2021no—
264#120—2021no—
265C4×Q8C2×D10111yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1011no—
268C4×D4C2×D10111yescharacteristic
269#107—1011no—
270C4⋊C8—1011no—
271#103—1011no—
272#103—1011no—
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102—1011no—
277C22⋊C8—1011no—
278C22⋊C8—1011no—
279#97—1011no—
280C24—2021no—
281C24—2021no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D10111yescharacteristic
288C2×Q8C22×D10111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—1011no—
294C2×D4—4041no—
295C2×D4—4041no—
296C2×D4—4044no—
297C2×D4—2022no—
298C2×D4—2022no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4C22⋊D10111yescharacteristic
306C2×D4C22×D10111yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—2021no—
310C22×C4C22×D10111yescharacteristic
311C22×C4C22×D10111yescharacteristic
312SD16—1014no—
313SD16—4041no—
314SD16—2022no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—4041no—
319D8—2022no—
320D8—2022no—
321M16—2022no—
322C2×C8—1011no—
323C2×C8—1011no—
324C2×C8—1011no—
325C2×C8—1011no—
326C2×C8—2022no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1011no—
330C4⋊C4C22⋊D10111yescharacteristic
331C4⋊C4C22×D10111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4C22⋊D10111yescharacteristic
340C22⋊C4C22⋊D10111yescharacteristic
341C42—21no—
342C42C22×D10111yescharacteristic
343C23—21no—
344C23—8081no—
345C23—8081no—
346C23—4042no—
347C23—4042no—
348C23—4042no—
349C23—4042no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23C2×C22⋊D10111yescharacteristic
354C23C2×C22⋊D10111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D10112yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—8081no—
366D4—1012no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—4042no—
371D4—4042no—
372D4—2021no—
373D4—2021no—
374D4—2021no—
375D4—2021no—
376D4—2021no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—4042no—
387C2×C4—2021no—
388C2×C4C2×C22⋊D10111yescharacteristic
389C2×C4C2×C22⋊D10111yescharacteristic
390C2×C4C2×C22⋊D10111yescharacteristic
391C2×C4C2×C22⋊D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—1012no—
394C8—2021no—
395C8—2021no—
396C8—2021no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—8084no—
406C22—8084no—
407C22—4041no—
408C22—4041no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—4041no—
415C22—2022no—
416C22#1569228012yescomplemented
417C22C22×C22⋊D10111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—4041no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—8081no—
432C2—8081no—
433C2—4042no—
434C2D4×C22⋊D10111yescharacteristic
435C2C2×#1569228011yescharacteristic
436C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#548×C101C2
socleC22×C101C22×D4
hypercenter#548D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#15692280C2semidirect
2. C22×#13108061C2semidirect
3. D4×D404C2semidirect
4. #20265567C2semidirect
5. #20265425C2semidirect
6. #20265411C2semidirect
7. #20265389C2semidirect
8. #20265377C2semidirect
9. #20265327C2semidirect
10. #20265322C2semidirect
11. #20265316C2semidirect
12. #20265310C2semidirect
13. #20265275C2semidirect
14. #20265249C2semidirect
15. #548×C101C2semidirect
23. C2×#13108061C22semidirect
24. C2×#13108061C22semidirect
25. C2×#13108061C22semidirect
26. C2×#13108061C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108059C22semidirect
30. C2×#13108058C22semidirect
31. C22×#11717579C22semidirect
39. #15692188C22semidirect
40. C4⋊#11717579C22semidirect
41. #15692179C22semidirect
42. #15692179C22semidirect
43. #15692176C22semidirect
45. #15692175C22semidirect
46. #15692175C22semidirect
47. C22⋊#11717579C22semidirect
48. C22⋊#11717579C22semidirect
49. #15692166C22semidirect
50. C2×#41×C101C22semidirect
51. C22×Q8×C101C22semidirect
53. #124×C101C22semidirect
55. C22⋊Q8×C101C22semidirect
57. #121×C101C22semidirect
58. C4×Q8×C101C22semidirect
83. #13108061D4semidirect
140. Q8×C101C2×D4semidirect
177. #11717579C2×D4semidirect
186. C22×C101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
222. #548D101semidirect
235. C2×#41C2×D101semidirect
236. C22×Q8C2×D101semidirect
255. #124C2×D101semidirect
257. C22⋊Q8C2×D101semidirect
262. #121C2×D101semidirect
265. C4×Q8C2×D101semidirect
359. Q8D4×D101semidirect
416. 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×#131080612
D4×D4042
#202655672
#202654252
#202654112
#202653892
#202653772
#202653272
#202653222
#202653162
#202653102
#202652752
#202652492
#548×C1012
#3265101