Finite Group Databasebeta

← #27182632

Subgroups — #27182632

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×#15692283C211yesmaximal, characteristic, complemented
2C22×#13108063C211yesmaximal, characteristic, complemented
3#20265836C211yesmaximal, characteristic, complemented
4D4×D404C211yesmaximal, characteristic, complemented
5#20265566C211yesmaximal, characteristic, complemented
6#20265423C211yesmaximal, characteristic, complemented
7#20265413C211yesmaximal, characteristic, complemented
8#20265393C211yesmaximal, characteristic, complemented
9#20265378C211yesmaximal, characteristic, complemented
10#20265330C211yesmaximal, characteristic, complemented
11#20265324C211yesmaximal, characteristic, complemented
12#20265318C211yesmaximal, characteristic, complemented
13#20265313C211yesmaximal, characteristic, complemented
14#20265279C211yesmaximal, characteristic, complemented
15#435×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic, complemented
17C22×Dic202C2211yescharacteristic, complemented
18C22×D404C2211yescharacteristic
19C22×D404C2211yescharacteristic
20C2×D4×D101—21no—
21#15692283—24no—
22C2×#13108063—24no—
23C2×#13108063C2211yescharacteristic
24C2×#13108063C2211yescharacteristic
25C2×#13108063C2211yescharacteristic, complemented
26C2×#13108063C2211yescharacteristic, complemented
27C2×#13108063C2211yescharacteristic, complemented
28C2×#13108063C2211yescharacteristic, complemented
29C2×D808C2211yescharacteristic
30C2×D808C2211yescharacteristic, complemented
31C2×D808C2211yescharacteristic, complemented
32#15692252C2211yescharacteristic
33#15692243C2211yescharacteristic, complemented
34#15692241—21no—
35#15692234C2211yescharacteristic, complemented
36#15692233—21no—
37#15692230—21no—
38#15692229C2211yescharacteristic
39#15692229C2211yescharacteristic
40#15692229C2211yescharacteristic, complemented
41#15692227—21no—
42#15692225—21no—
43#15692225—21no—
44C4×Dic202C2211yescharacteristic, complemented
45C4×D404C2211yescharacteristic
46#15692215—21no—
47#15692189C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692180C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic
51#15692177C2211yescharacteristic, complemented
52#15692177C2211yescharacteristic, complemented
53#15692177C2211yescharacteristic, complemented
54C2×M16×C101C2211yescharacteristic, complemented
55C22×C8×C101C2211yescharacteristic, complemented
56C4×D4×C101C2211yescharacteristic
57C4⋊C8×C101C2211yescharacteristic, complemented
58C22⋊C8×C101C2211yescharacteristic, complemented
59C22⋊C8×C101C2211yescharacteristic, complemented
60#97×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62C23×D101—21no—
63#13108074—22no—
64#13108074—41no—
65C2×Dic202—22no—
66C2×Dic202—21no—
67C2×Dic202—41no—
68C2×Dic202C2311yescharacteristic
69C2×Dic202C2311yescharacteristic
70C2×D404—22no—
71C2×D404—22no—
72C2×D404—21no—
73C2×D404—21no—
74C2×D404—21no—
75C2×D404—21no—
76C2×D404—41no—
77C2×D404—41no—
78C2×D404C2311yescharacteristic
79C2×D404C2311yescharacteristic
80C2×D404C2311yescharacteristic
81C2×D404C2311yescharacteristic
82C2×C22⋊D101—21no—
83C2×C22⋊D101—21no—
84C2×C22⋊D101—21no—
85C2×C22⋊D101—21no—
86D4×D101—44no—
87C22×Dic101—21no—
88C2×C4×D101—21no—
89C2×C4×D101—21no—
90#13108063—22no—
91#13108063—22no—
92#13108063—22no—
93#13108063—22no—
94#13108063—41no—
95#13108063D414yescomplemented
96D808—22no—
97D808—22no—
98D808—41no—
99C4⋊Dic101C2311yescharacteristic
100C4⋊Dic101C2311yescharacteristic
101#13108053—21no—
102#13108053—21no—
103C22⋊Dic101—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051—21no—
107#13108051—21no—
108#13108051—21no—
109#13108051—21no—
110C4×Dic101—21no—
111C2×D4×C101D411yescharacteristic
112C22×C4×C101C2311yescharacteristic
113C22×C4×C101C2311yescharacteristic
114M16×C101—22no—
115C2×C8×C101—22no—
116C2×C8×C101C2311yescharacteristic
117C2×C8×C101C2311yescharacteristic
118C2×C8×C101C2311yescharacteristic
119C2×C8×C101C2311yescharacteristic
120C4⋊C4×C101D411yescharacteristic
121C22⋊C4×C101D411yescharacteristic
122C22⋊C4×C101D411yescharacteristic
123C42×C101C2311yescharacteristic
124C23×C101C2×D411yescharacteristic
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
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—
136D4×C101—22no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101C2×D411yescharacteristic
142C2×C4×C101C2×D411yescharacteristic
143C2×C4×C101C2×D411yescharacteristic
144C2×C4×C101C2×D411yescharacteristic
145C2×C4×C101C2411yescharacteristic
146C22⋊D101—42no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—42no—
155D404—21no—
156D404—21no—
157D404—21no—
158D404—21no—
159D404—21no—
160D404—21no—
161D404—21no—
162D404—42no—
163D404—42no—
164D404—42no—
165D404—81no—
166D404C2×D412yes—
167C4×D101—42no—
168C4×D101—42no—
169Dic202—21no—
170Dic202—21no—
171Dic202—21no—
172Dic202—42no—
173Dic202C2×D412yescomplemented
174C8×C101—21no—
175C8×C101—21no—
176C8×C101—21no—
177C8×C101C2×D412yescomplemented
178C22×C101—21no—
179C22×C101—21no—
180C22×C101—21no—
181C22×C101—21no—
182C22×C101—21no—
183C22×C101AGL(1,ℤ/8)12yescomplemented
184C22×C101C22×D411yescharacteristic
185C2×D101—22no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—41no—
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—21no—
202C4×C101—21no—
203C4×C101C22×D411yescharacteristic
204C4×C101H(5,𝔽2)11yescharacteristic
205Dic101—22no—
206Dic101—41no—
207Dic101—41no—
208Dic101—41no—
209C2×C101—22no—
210C2×C101—41no—
211C2×C101D4211yescharacteristic
212C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C101#57711yescharacteristic
214D101—42no—
215D101—81no—
216D101—81no—
217D101—81no—
218#3265—1011nomaximal, Sylow
219C101#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1011no—
221C22×SD16—1011no—
222#548—1011no—
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#435D10111yescharacteristic, complemented
235C2×#41—1011no—
236C22×Q8—1011no—
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×M16C2×D10111yescharacteristic, complemented
252C22×C8C2×D10111yescharacteristic, complemented
253#127—1011no—
254#124—1011no—
255#124—2021no—
256C22⋊Q8—1011no—
257C22⋊Q8—2021no—
258#121—1011no—
259#121—1011no—
260#121—1011no—
261#121—2021no—
262#121—2021no—
263#120—2021no—
264#120—2021no—
265C4×Q8—1011no—
266C4×D4—1011no—
267C4×D4—2021no—
268C4×D4C2×D10111yescharacteristic
269#107—1011no—
270C4⋊C8C2×D10111yescharacteristic, complemented
271#103—1011no—
272#103—1011no—
273#102—1011no—
274#102—1011no—
275#102—1011no—
276#102—1011no—
277C22⋊C8C2×D10111yescharacteristic, complemented
278C22⋊C8C2×D10111yescharacteristic, complemented
279#97C2×D10111yescharacteristic, complemented
280C24—2021no—
281C24—2021no—
282#41—4041no—
283#41—2022no—
284C2×Q8—1011no—
285C2×Q8—1011no—
286C2×Q8—4041no—
287C2×Q8—2022no—
288C2×Q8—2021no—
289C2×D4—1011no—
290C2×D4—1011no—
291C2×D4—1011no—
292C2×D4—1011no—
293C2×D4—4041no—
294C2×D4—4041no—
295C2×D4—4044no—
296C2×D4—2022no—
297C2×D4—2022no—
298C2×D4—2021no—
299C2×D4—2021no—
300C2×D4—2021no—
301C2×D4—2021no—
302C2×D4—2021no—
303C2×D4—2021no—
304C2×D4—2021no—
305C2×D4—2021no—
306C2×D4D40411yescharacteristic
307C22×C4—2021no—
308C22×C4—2021no—
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—22no—
322C2×C8—22no—
323C2×C8C22×D10111yescharacteristic
324C2×C8C22×D10111yescharacteristic
325C2×C8C22×D10111yescharacteristic
326C2×C8C22×D10111yescharacteristic
327C4⋊C4—1011no—
328C4⋊C4—1011no—
329C4⋊C4—2021no—
330C4⋊C4—2021no—
331C4⋊C4D40411yescharacteristic
332C22⋊C4—2021no—
333C22⋊C4—2021no—
334C22⋊C4—2021no—
335C22⋊C4—2021no—
336C22⋊C4—2021no—
337C22⋊C4—2021no—
338C22⋊C4—2021no—
339C22⋊C4D40411yescharacteristic
340C22⋊C4D40411yescharacteristic
341C42—2021no—
342C42C22×D10111yescharacteristic
343C23—8081no—
344C23—8081no—
345C23—4042no—
346C23—4042no—
347C23—4042no—
348C23—4042no—
349C23—2021no—
350C23—2021no—
351C23—2021no—
352C23—2021no—
353C23C2×D40411yescharacteristic
354C23C2×D40411yescharacteristic
355Q8—1012no—
356Q8—4042no—
357Q8—2021no—
358Q8—2021no—
359Q8—2021no—
360D4—22no—
361D4—8081no—
362D4—1012no—
363D4—4042no—
364D4—4042no—
365D4—4042no—
366D4—4042no—
367D4—4042no—
368D4—4042no—
369D4—4042no—
370D4—2021no—
371D4—2021no—
372D4—2021no—
373D4—2021no—
374D4—2021no—
375D4—2021no—
376D4—2021no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—4042no—
382C2×C4—4042no—
383C2×C4—4042no—
384C2×C4—2021no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4C2×D40411yescharacteristic
389C2×C4C2×D40411yescharacteristic
390C2×C4C2×D40411yescharacteristic
391C2×C4C2×D40411yescharacteristic
392C2×C4C23×D10111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D10112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—8084no—
403C22—8084no—
404C22—4041no—
405C22—4041no—
406C22—4041no—
407C22—4041no—
408C22—4041no—
409C22—4041no—
410C22—4041no—
411C22—4041no—
412C22—4041no—
413C22—4041no—
414C22—4041no—
415C22—2022no—
416C22#1569228312yescomplemented
417C22C22×D40411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—4041no—
423C4—4041no—
424C4—4041no—
425C4—2022no—
426C4C2×D4×D10111yescharacteristic
427C4#1569230111yescharacteristic
428C2—22no—
429C2—41no—
430C2—8081no—
431C2—8081no—
432C2—8081no—
433C2—4042no—
434C2D4×D40411yescharacteristic
435C2C2×#1569228311yescharacteristic
436C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#435×C101C2
socleC22×C101C22×D4
hypercenter#435D101
hypocenterC101#3265
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#15692283C2semidirect
2. C22×#13108063C2semidirect
3. #20265836C2semidirect
4. D4×D404C2semidirect
5. #20265566C2semidirect
6. #20265423C2semidirect
7. #20265413C2semidirect
8. #20265393C2semidirect
9. #20265378C2semidirect
10. #20265330C2semidirect
11. #20265324C2semidirect
12. #20265318C2semidirect
13. #20265313C2semidirect
14. #20265279C2semidirect
15. #435×C101C2semidirect
16. C2×#13108074C22semidirect
17. C22×Dic202C22semidirect
25. C2×#13108063C22semidirect
26. C2×#13108063C22semidirect
27. C2×#13108063C22semidirect
28. C2×#13108063C22semidirect
30. C2×D808C22semidirect
31. C2×D808C22semidirect
33. #15692243C22semidirect
35. #15692234C22semidirect
40. #15692229C22semidirect
44. C4×Dic202C22semidirect
47. #15692189C22semidirect
48. #15692180C22semidirect
49. #15692180C22semidirect
51. #15692177C22semidirect
52. #15692177C22semidirect
53. #15692177C22semidirect
54. C2×M16×C101C22semidirect
55. C22×C8×C101C22semidirect
57. C4⋊C8×C101C22semidirect
58. C22⋊C8×C101C22semidirect
59. C22⋊C8×C101C22semidirect
60. #97×C101C22semidirect
95. #13108063D4semidirect
173. Dic202C2×D4semidirect
177. C8×C101C2×D4semidirect
183. C22×C101AGL(1,ℤ/8)semidirect
219. C101#3265semidirect
234. #435D101semidirect
251. C2×M16C2×D101semidirect
252. C22×C8C2×D101semidirect
270. C4⋊C8C2×D101semidirect
277. C22⋊C8C2×D101semidirect
278. C22⋊C8C2×D101semidirect
279. #97C2×D101semidirect
396. C8D4×D101semidirect
416. C22#15692283semidirect

Maximal subgroups

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

subgroupindex
C2×#156922832
C22×#131080632
#202658362
D4×D4042
#202655662
#202654232
#202654132
#202653932
#202653782
#202653302
#202653242
#202653182
#202653132
#202652792
#435×C1012
#3265101