Finite Group Databasebeta

← #27182745

Subgroups — #27182745

Order 12928.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692282C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3#20265846C211yesmaximal, characteristic, complemented
4#20265811C211yesmaximal, characteristic, complemented
5#20265552C211yesmaximal, characteristic, complemented
6#20265429C211yesmaximal, characteristic, complemented
7#20265428C211yesmaximal, characteristic, complemented
8#20265375C211yesmaximal, characteristic, complemented
9#20265371C211yesmaximal, characteristic, complemented
10#20265325C211yesmaximal, characteristic, complemented
11#20265313C211yesmaximal, characteristic, complemented
12#20265309C211yesmaximal, characteristic, complemented
13#20265262C211yesmaximal, characteristic, complemented
14#20265252C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic
18C2×D4×D101—21no—
19C2×D4×D101C2211yescharacteristic
20#15692282—24no—
21#15692280—24no—
22C2×#13108061C2211yescharacteristic
23C2×#13108061C2211yescharacteristic, complemented
24C2×D808C2211yescharacteristic
25C2×D808C2211yescharacteristic, complemented
26C2×#13108059C2211yescharacteristic
27C2×#13108059C2211yescharacteristic, complemented
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108058C2211yescharacteristic, complemented
30C2×#13108057C2211yescharacteristic, complemented
31#15692251—21no—
32#15692251C2211yescharacteristic
33#15692251C2211yescharacteristic, complemented
34#15692230C2211yescharacteristic
35#15692229—21no—
36#15692229C2211yescharacteristic, complemented
37#15692228—21no—
38#15692226—21no—
39#15692226C2211yescharacteristic, complemented
40#15692225—21no—
41#15692223—21no—
42Q8×Dic101C2211yescharacteristic, complemented
43#15692217C2211yescharacteristic
44C4×C22⋊D101—21no—
45C4×C22⋊D101C2211yescharacteristic
46#15692186C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692177C2211yescharacteristic
49#15692177C2211yescharacteristic, complemented
50#15692176C2211yescharacteristic, complemented
51#15692175C2211yescharacteristic, complemented
52#15692167C2211yescharacteristic, complemented
53C8×Dic101C2211yescharacteristic, complemented
54C2×#41×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic, complemented
56C2×D8×C101C2211yescharacteristic, complemented
57#121×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108075—22no—
64#13108075—41no—
65C2×D404—22no—
66C2×D404—21no—
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—41no—
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×D404C2311yescharacteristic
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101—21no—
77C2×C22⋊D101—21no—
78C2×C22⋊D101—21no—
79C2×C22⋊D101—21no—
80C2×C22⋊D101D411yescharacteristic
81D4×D101—22no—
82D4×D101—41no—
83D4×D101—44no—
84C2×C4×D101—21no—
85C2×C4×D101—21no—
86C2×C4×D101C2311yescharacteristic
87#13108061—22no—
88#13108061D414yescomplemented
89D808—22no—
90D808—41no—
91#13108059—22no—
92#13108059—22no—
93#13108059—22no—
94#13108058—22no—
95#13108057—22no—
96C2×#11717579C2311yescharacteristic
97C2×#11717579C2311yescharacteristic
98C4⋊Dic101—21no—
99C4⋊Dic101C2311yescharacteristic
100#13108053D411yescharacteristic
101C22⋊Dic101—21no—
102C22⋊Dic101D411yescharacteristic
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051D411yescharacteristic
107C4×Dic101—21no—
108C4×Dic101C2311yescharacteristic
109#41×C101—22no—
110#41×C101—41no—
111C2×Q8×C101C2311yescharacteristic
112C2×D4×C101—21no—
113C2×D4×C101—21no—
114C2×D4×C101C2311yescharacteristic
115C2×D4×C101C2311yescharacteristic
116C22×C4×C101—21no—
117C22×C4×C101C2311yescharacteristic
118SD16×C101—22no—
119D8×C101—41no—
120C2×C8×C101C2311yescharacteristic
121C2×C8×C101C2311yescharacteristic
122C4⋊C4×C101C2311yescharacteristic
123C22⋊C4×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
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—
136C22×D101C2×D411yescharacteristic
137Q8×C101—21no—
138Q8×C101C2×D412yescomplemented
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—21no—
142D4×C101—21no—
143D4×C101—42no—
144D4×C101—42no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—21no—
149C2×C4×C101—42no—
150C2×C4×C101C2×D411yescharacteristic
151C2×C4×C101C2411yescharacteristic
152C22⋊D101—22no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—42no—
156C22⋊D101—81no—
157C22⋊D101—81no—
158C22⋊D101—81no—
159C22⋊D101—81no—
160C2×Dic101—21no—
161C2×Dic101C2×D411yescharacteristic
162C2×Dic101C2×D411yescharacteristic
163C2×Dic101C2×D411yescharacteristic
164D404—21no—
165D404—21no—
166D404—21no—
167D404—21no—
168D404—21no—
169D404—42no—
170D404—42no—
171D404—42no—
172D404C2×D412yes—
173C4×D101—21no—
174C4×D101—21no—
175C4×D101—42no—
176C4×D101—42no—
177C8×C101—21no—
178C8×C101—21no—
179#11717579—21no—
180#11717579C2×D412yescomplemented
181C22×C101—21no—
182C22×C101—21no—
183C22×C101—21no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101—41no—
188C22×C101—41no—
189C22×C101—41no—
190C22×C101C22×D411yescharacteristic
191C2×D101—21no—
192C2×D101—22no—
193C2×D101—21no—
194C2×D101—21no—
195C2×D101—41no—
196C2×D101—41no—
197C2×D101—41no—
198C2×D101—41no—
199C2×D101—41no—
200C2×D101—41no—
201C2×D101—41no—
202C2×D101—41no—
203C2×D101—84no—
204C2×D101—84no—
205C4×C101—22no—
206C4×C101—21no—
207C4×C101—41no—
208C4×C101C22×D411yescharacteristic
209C4×C101H(5,𝔽2)11yescharacteristic
210Dic101—21no—
211Dic101—21no—
212Dic101—41no—
213Dic101AGL(1,ℤ/8)12yescomplemented
214C2×C101—41no—
215C2×C101—81no—
216C2×C101—81no—
217C2×C101D4211yescharacteristic
218C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C101#57611yescharacteristic
220D101—41no—
221D101—42no—
222D101—81no—
223D101—81no—
224#3274—1011nomaximal, Sylow
225C101#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1011no—
227C2×AGL(1,ℤ/8)—1011no—
228#550—1011no—
229D42—1011no—
230#493—1011no—
231#469—1011no—
232#469—1011no—
233#460—1011no—
234#459—1011no—
235#449—1011no—
236#449D10111yescharacteristic, complemented
237#447—1011no—
238#447—1011no—
239C4×SD16—1011no—
240#436—1011no—
241C2×#41—1011no—
242C2×#41C2×D10111yescharacteristic, complemented
243C22×D4—1011no—
244C22×D4—1011no—
245C22×D4—2021no—
246AGL(1,ℤ/8)—2024no—
247AGL(1,ℤ/8)—2024no—
248C2×SD16—1011no—
249C2×SD16—1011no—
250C2×SD16C2×D10111yescharacteristic, complemented
251C2×D8—1011no—
252C2×D8—1011no—
253C2×D8—1011no—
254C2×D8—1011no—
255C2×D8—1011no—
256C2×D8C2×D10111yescharacteristic, complemented
257C2×M16—1011no—
258C2×M16—1011no—
259#127—1011no—
260#127—1011no—
261#127—2021no—
262#121—1011no—
263#121—1011no—
264#121—1011no—
265#121—2021no—
266#121—2021no—
267#121—2021no—
268#121C2×D10111yescharacteristic
269#120—2021no—
270#120—2021no—
271C4×Q8—1011no—
272C4×D4—1011no—
273C4×D4—1011no—
274C4×D4—2021no—
275#106—1011no—
276C4⋊C8—1011no—
277#103C2×D10111yescharacteristic, complemented
278#102—1011no—
279#102—1011no—
280#102—1011no—
281#102—1011no—
282#102C2×D10111yescharacteristic, complemented
283C22⋊C8—1011no—
284C22⋊C8C2×D10111yescharacteristic, complemented
285C4×C8—1011no—
286C24—2021no—
287C24—2021no—
288#41—22no—
289#41—41no—
290#41—4041no—
291#41—2022no—
292C2×Q8C22×D10111yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1011no—
296C2×D4—1011no—
297C2×D4—1011no—
298C2×D4—1011no—
299C2×D4—4041no—
300C2×D4—4044no—
301C2×D4—4041no—
302C2×D4—2022no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4—2021no—
307C2×D4—2021no—
308C2×D4—2021no—
309C2×D4—2021no—
310C2×D4—2022no—
311C2×D4—2021no—
312C2×D4—2021no—
313C2×D4—2021no—
314C2×D4C22×D10111yescharacteristic
315C2×D4C22×D10111yescharacteristic
316C22×C4—21no—
317C22×C4—1011no—
318C22×C4—2021no—
319C22×C4—2021no—
320C22×C4C22×D10111yescharacteristic
321SD16—22no—
322SD16—1014no—
323SD16—2022no—
324D8—41no—
325D8—4041no—
326D8—2022no—
327D8—2022no—
328D8—2022no—
329D8—2022no—
330M16—2022no—
331M16—2022no—
332C2×C8—1011no—
333C2×C8—1011no—
334C2×C8C22×D10111yescharacteristic
335C2×C8C22×D10111yescharacteristic
336C4⋊C4—1011no—
337C4⋊C4—1011no—
338C4⋊C4—2021no—
339C4⋊C4C22×D10111yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—1011no—
342C22⋊C4—1011no—
343C22⋊C4—2021no—
344C22⋊C4—2021no—
345C22⋊C4—2021no—
346C22⋊C4—2021no—
347C42—1011no—
348C42—2021no—
349C23—21no—
350C23—21no—
351C23—8081no—
352C23—8081no—
353C23—1011no—
354C23—4042no—
355C23—4042no—
356C23—4042no—
357C23—4042no—
358C23—2021no—
359C23—2021no—
360C23—2021no—
361C23D4×D10111yescharacteristic
362Q8—21no—
363Q8D4×D10112yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—8081no—
371D4—8081no—
372D4—8081no—
373D4—8081no—
374D4—1012no—
375D4—4042no—
376D4—4042no—
377D4—4042no—
378D4—4042no—
379D4—4042no—
380D4—4042no—
381D4—2022no—
382D4—2021no—
383D4—2021no—
384D4—2021no—
385D4—2021no—
386D4—2021no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—1011no—
393C2×C4—1011no—
394C2×C4—1011no—
395C2×C4—4042no—
396C2×C4—4042no—
397C2×C4—2021no—
398C2×C4—2021no—
399C2×C4—2021no—
400C2×C4D4×D10111yescharacteristic
401C2×C4C23×D10111yescharacteristic
402C8—21no—
403C8—21no—
404C8—1012no—
405C8—2021no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—8084no—
416C22—8084no—
417C22—4041no—
418C22—4041no—
419C22—4041no—
420C22—4041no—
421C22—4041no—
422C22—4041no—
423C22—4041no—
424C22—4041no—
425C22—2021no—
426C22—2022no—
427C22—2021no—
428C22—2021no—
429C22C2×D4×D10111yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—1012no—
434C4—4041no—
435C4—2021no—
436C4—2021no—
437C4C2×D4×D10111yescharacteristic
438C4#1569230011yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—8081no—
443C2—8081no—
444C2—4042no—
445C2—4041no—
446C2#2026580911yescharacteristic
447C2AGL(1,ℤ/8)×D10111yescharacteristic
448C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3274
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×#15692282C2semidirect
2. C2×#15692280C2semidirect
3. #20265846C2semidirect
4. #20265811C2semidirect
5. #20265552C2semidirect
6. #20265429C2semidirect
7. #20265428C2semidirect
8. #20265375C2semidirect
9. #20265371C2semidirect
10. #20265325C2semidirect
11. #20265313C2semidirect
12. #20265309C2semidirect
13. #20265262C2semidirect
14. #20265252C2semidirect
15. #449×C101C2semidirect
16. C2×#13108075C22semidirect
23. C2×#13108061C22semidirect
25. C2×D808C22semidirect
27. C2×#13108059C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C2×#13108057C22semidirect
33. #15692251C22semidirect
36. #15692229C22semidirect
39. #15692226C22semidirect
42. Q8×Dic101C22semidirect
46. #15692186C22semidirect
47. #15692184C22semidirect
49. #15692177C22semidirect
50. #15692176C22semidirect
51. #15692175C22semidirect
52. #15692167C22semidirect
53. C8×Dic101C22semidirect
54. C2×#41×C101C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×D8×C101C22semidirect
58. #103×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
88. #13108061D4semidirect
138. Q8×C101C2×D4semidirect
180. #11717579C2×D4semidirect
213. Dic101AGL(1,ℤ/8)semidirect
225. C101#3274semidirect
236. #449D101semidirect
242. C2×#41C2×D101semidirect
250. C2×SD16C2×D101semidirect
256. C2×D8C2×D101semidirect
277. #103C2×D101semidirect
282. #102C2×D101semidirect
284. C22⋊C8C2×D101semidirect
363. Q8D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922822
C2×#156922802
#202658462
#202658112
#202655522
#202654292
#202654282
#202653752
#202653712
#202653252
#202653132
#202653092
#202652622
#202652522
#449×C1012
#3274101