Finite Group Databasebeta

← #27182725

Subgroups — #27182725

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
2C2×#15692279C211yesmaximal, characteristic, complemented
3#20265826C211yesmaximal, characteristic, complemented
4#20265811C211yesmaximal, characteristic, complemented
5#20265548C211yesmaximal, characteristic, complemented
6#20265426C211yesmaximal, characteristic, complemented
7#20265421C211yesmaximal, characteristic, complemented
8#20265381C211yesmaximal, characteristic, complemented
9#20265370C211yesmaximal, characteristic, complemented
10#20265330C211yesmaximal, characteristic, complemented
11#20265326C211yesmaximal, characteristic, complemented
12#20265323C211yesmaximal, characteristic, complemented
13#20265257C211yesmaximal, characteristic, complemented
14#20265252C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic, complemented
17C22×D404C2211yescharacteristic
18C2×D4×D101—21no—
19C2×D4×D101C2211yescharacteristic
20#15692280—24no—
21#15692279—24no—
22C2×#13108063C2211yescharacteristic
23C2×#13108063C2211yescharacteristic, complemented
24C2×#13108062C2211yescharacteristic, complemented
25C2×#13108061C2211yescharacteristic
26C2×#13108061C2211yescharacteristic, complemented
27C2×#13108059C2211yescharacteristic
28C2×#13108059C2211yescharacteristic, complemented
29C2×#13108058C2211yescharacteristic, complemented
30C2×#13108057C2211yescharacteristic, complemented
31#15692253C2211yescharacteristic, complemented
32#15692251C2211yescharacteristic
33#15692238—21no—
34#15692236—21no—
35#15692234C2211yescharacteristic, complemented
36#15692231C2211yescharacteristic, complemented
37#15692230C2211yescharacteristic
38#15692228—21no—
39#15692227—21no—
40#15692225—21no—
41#15692223—21no—
42D4×Dic101C2211yescharacteristic, complemented
43#15692217C2211yescharacteristic
44C4×C22⋊D101C2211yescharacteristic
45C2×#13108053—21no—
46#15692188C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692178C2211yescharacteristic, complemented
49#15692177C2211yescharacteristic
50#15692177C2211yescharacteristic, complemented
51#15692176C2211yescharacteristic, 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#13108073—22no—
64#13108073—41no—
65C2×Dic202C2311yescharacteristic
66C2×D404—22no—
67C2×D404—21no—
68C2×D404—21no—
69C2×D404—41no—
70C2×D404C2311yescharacteristic
71C2×D404C2311yescharacteristic
72C2×C22⋊D101—21no—
73C2×C22⋊D101—21no—
74C2×C22⋊D101—21no—
75C2×C22⋊D101—21no—
76C2×C22⋊D101D411yescharacteristic
77D4×D101—22no—
78D4×D101—41no—
79D4×D101—44no—
80C22×Dic101—21no—
81C22×Dic101—21no—
82C2×C4×D101—21no—
83C2×C4×D101C2311yescharacteristic
84#13108063—22no—
85#13108063—41no—
86#13108062—22no—
87#13108061—22no—
88#13108061—22no—
89#13108059—22no—
90#13108059D414yescomplemented
91#13108058—22no—
92#13108057—22no—
93C2×#11717579C2311yescharacteristic
94C2×#11717579C2311yescharacteristic
95C4⋊Dic101C2311yescharacteristic
96#13108053—21no—
97#13108053—21no—
98#13108053—21no—
99#13108053—21no—
100#13108053D411yescharacteristic
101C22⋊Dic101—21no—
102C22⋊Dic101—21no—
103C22⋊Dic101D411yescharacteristic
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051D411yescharacteristic
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—41no—
119D8×C101—22no—
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—42no—
130C22×D101—42no—
131C22×D101—42no—
132C22×D101—42no—
133C22×D101—81no—
134C22×D101—81no—
135C22×D101C2×D411yescharacteristic
136Q8×C101—21no—
137Q8×C101—21no—
138D4×C101—21no—
139D4×C101—21no—
140D4×C101—21no—
141D4×C101—42no—
142D4×C101—42no—
143D4×C101C2×D412yescomplemented
144C2×C4×C101—21no—
145C2×C4×C101—21no—
146C2×C4×C101—21no—
147C2×C4×C101—21no—
148C2×C4×C101—42no—
149C2×C4×C101C2×D411yescharacteristic
150C2×C4×C101C2411yescharacteristic
151C22⋊D101—22no—
152C22⋊D101—42no—
153C22⋊D101—42no—
154C22⋊D101—42no—
155C22⋊D101—81no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—42no—
159C2×Dic101—42no—
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162C2×Dic101C2×D411yescharacteristic
163D404—21no—
164D404—21no—
165D404—21no—
166D404—42no—
167D404—42no—
168D404C2×D412yes—
169C4×D101—21no—
170C4×D101—21no—
171C4×D101—42no—
172Dic202—21no—
173Dic202—21no—
174C8×C101—21no—
175C8×C101—21no—
176#11717579—21no—
177#11717579C2×D412yescomplemented
178C22×C101—22no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—41no—
185C22×C101—41no—
186C22×C101—41no—
187C22×C101C22×D411yescharacteristic
188C2×D101—21no—
189C2×D101—22no—
190C2×D101—21no—
191C2×D101—21no—
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—41no—
201C4×C101—41no—
202C4×C101C22×D411yescharacteristic
203C4×C101H(5,𝔽2)11yescharacteristic
204Dic101—21no—
205Dic101—21no—
206Dic101—41no—
207Dic101—41no—
208Dic101AGL(1,ℤ/8)12yescomplemented
209C2×C101—41no—
210C2×C101—42no—
211C2×C101—81no—
212C2×C101D4211yescharacteristic
213C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C101#57711yescharacteristic
215D101—41no—
216D101—42no—
217D101—81no—
218#3272—1011nomaximal, Sylow
219C101#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C2×AGL(1,ℤ/8)—1011no—
222#547—1011no—
223D42—1011no—
224#492—1011no—
225#468—1011no—
226#468—1011no—
227#461—1011no—
228#459—1011no—
229#450—1011no—
230#450—1011no—
231#449—1011no—
232#449D10111yescharacteristic, complemented
233C4×D8—1011no—
234#436—1011no—
235C2×#41—1011no—
236C2×#41C2×D10111yescharacteristic, complemented
237C22×D4—1011no—
238C22×D4—1011no—
239C22×D4—2021no—
240AGL(1,ℤ/8)—2024no—
241AGL(1,ℤ/8)—2024no—
242C2×SD16—1011no—
243C2×SD16—1011no—
244C2×SD16—1011no—
245C2×SD16—1011no—
246C2×SD16—1011no—
247C2×SD16C2×D10111yescharacteristic, complemented
248C2×D8—1011no—
249C2×D8—1011no—
250C2×D8C2×D10111yescharacteristic, complemented
251C2×M16—1011no—
252C2×M16—1011no—
253C4⋊Q8—1011no—
254#127—1011no—
255#123—2021no—
256#123—2021no—
257C22⋊Q8—1011no—
258C22⋊Q8—1011no—
259#121—1011no—
260#121—2021no—
261#121—2021no—
262#121C2×D10111yescharacteristic
263#120—2021no—
264#120—2021no—
265C4×D4—1011no—
266C4×D4—1011no—
267C4×D4—1011no—
268C2×C4⋊C4—2021no—
269#107—1011no—
270C4⋊C8—1011no—
271#103—1011no—
272#103C2×D10111yescharacteristic, complemented
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102C2×D10111yescharacteristic, complemented
277C22⋊C8—1011no—
278C22⋊C8C2×D10111yescharacteristic, complemented
279C4×C8—1011no—
280C24—2021no—
281C24—2021no—
282#41—22no—
283#41—41no—
284#41—4041no—
285#41—2022no—
286C2×Q8—1011no—
287C2×Q8C22×D10111yescharacteristic
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—1011no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—4041no—
294C2×D4—4044no—
295C2×D4—4041no—
296C2×D4—2022no—
297C2×D4—2021no—
298C2×D4—2021no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2022no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4C22×D10111yescharacteristic
305C2×D4C22×D10111yescharacteristic
306C22×C4—21no—
307C22×C4—1011no—
308C22×C4—2021no—
309C22×C4—2021no—
310C22×C4—2021no—
311C22×C4C22×D10111yescharacteristic
312SD16—41no—
313SD16—4041no—
314SD16—2022no—
315SD16—2022no—
316SD16—2022no—
317SD16—2022no—
318D8—22no—
319D8—1014no—
320D8—2022no—
321M16—2022no—
322M16—2022no—
323C2×C8—1011no—
324C2×C8—1011no—
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—1011no—
328C4⋊C4—1011no—
329C4⋊C4—2021no—
330C4⋊C4—2021no—
331C4⋊C4—2021no—
332C4⋊C4—2021no—
333C4⋊C4C22×D10111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—1011no—
336C22⋊C4—1011no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4—2021no—
340C22⋊C4—2021no—
341C22⋊C4—2021no—
342C42—1011no—
343C23—21no—
344C23—21no—
345C23—8081no—
346C23—8081no—
347C23—1011no—
348C23—4042no—
349C23—4042no—
350C23—4042no—
351C23—4042no—
352C23—2021no—
353C23—2021no—
354C23D4×D10111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2021no—
358Q8—2021no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—42no—
364D4—8081no—
365D4—1012no—
366D4—4042no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—4042no—
371D4—2022no—
372D4—2021no—
373D4—2021no—
374D4—2021no—
375D4D4×D10112yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—1011no—
382C2×C4—1011no—
383C2×C4—1011no—
384C2×C4—4042no—
385C2×C4—4042no—
386C2×C4—4042no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4D4×D10111yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—21no—
395C8—1012no—
396C8—2021no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—8084no—
407C22—8084no—
408C22—4041no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—2021no—
414C22—2021no—
415C22—2022no—
416C22—2021no—
417C22C2×D4×D10111yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—1012no—
422C4—4041no—
423C4—4041no—
424C4—2021no—
425C4—2021no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230011yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—8081no—
432C2—4042no—
433C2—4041no—
434C2#2026580911yescharacteristic
435C2AGL(1,ℤ/8)×D10111yescharacteristic
436C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3272
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×#15692280C2semidirect
2. C2×#15692279C2semidirect
3. #20265826C2semidirect
4. #20265811C2semidirect
5. #20265548C2semidirect
6. #20265426C2semidirect
7. #20265421C2semidirect
8. #20265381C2semidirect
9. #20265370C2semidirect
10. #20265330C2semidirect
11. #20265326C2semidirect
12. #20265323C2semidirect
13. #20265257C2semidirect
14. #20265252C2semidirect
15. #449×C101C2semidirect
16. C2×#13108073C22semidirect
23. C2×#13108063C22semidirect
24. C2×#13108062C22semidirect
26. C2×#13108061C22semidirect
28. C2×#13108059C22semidirect
29. C2×#13108058C22semidirect
30. C2×#13108057C22semidirect
31. #15692253C22semidirect
35. #15692234C22semidirect
36. #15692231C22semidirect
42. D4×Dic101C22semidirect
46. #15692188C22semidirect
47. #15692184C22semidirect
48. #15692178C22semidirect
50. #15692177C22semidirect
51. #15692176C22semidirect
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
90. #13108059D4semidirect
143. D4×C101C2×D4semidirect
177. #11717579C2×D4semidirect
208. Dic101AGL(1,ℤ/8)semidirect
219. C101#3272semidirect
232. #449D101semidirect
236. C2×#41C2×D101semidirect
247. C2×SD16C2×D101semidirect
250. C2×D8C2×D101semidirect
272. #103C2×D101semidirect
276. #102C2×D101semidirect
278. C22⋊C8C2×D101semidirect
375. D4D4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922802
C2×#156922792
#202658262
#202658112
#202655482
#202654262
#202654212
#202653812
#202653702
#202653302
#202653262
#202653232
#202652572
#202652522
#449×C1012
#3272101