Finite Group Databasebeta

← #25079440

Subgroups — #25079440

Order 11008.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144989C211yesmaximal, characteristic, complemented
2#19144595C211yesmaximal, characteristic
3C2×#14583104C211yesmaximal, characteristic, complemented
4#19134156C212yesmaximal, complemented
5#19134142C212yesmaximal, complemented
6#19134032C212yesmaximal, complemented
7#19134018C212yesmaximal, complemented
8#19133347C211yesmaximal, characteristic, complemented
9#19133281C211yesmaximal, characteristic, complemented
10#19133273C211yesmaximal, characteristic, complemented
11D4×M16×C43C211yesmaximal, characteristic, complemented
12C22×#10621883—21no—
13#14583449C2211yescharacteristic
14C2×#13063325C2211yescharacteristic
15#14583259—22no—
16#14583250C2212yescomplemented
17#14583246C2212yescomplemented
18#14583212—21no—
19#14583179C2211yescharacteristic
20#14583179C2211yescharacteristic
21#14583104—21no—
22#14583104—21no—
23#14583104C2212yes—
24#14583104C2212yes—
25#14583094C2211yescharacteristic
26#14583094C2211yescharacteristic
27#14583033—21no—
28C4×#10621882—21no—
29C4×#10621882—21no—
30#14582952C2211yescharacteristic
31C2×C4⋊(C43⋊C8)C2211yescharacteristic
32#14582901—21no—
33#14582901—21no—
34C2×#13063248—21no—
35#14582652C2212yescomplemented
36#14582650C2212yescomplemented
37#14582646C2212yescomplemented
38#14582644C2212yescomplemented
39#14582644C2212yescomplemented
40#14582642C2212yescomplemented
41C22×M16×C43—21no—
42C2×C4×D4×C43C2211yescharacteristic
43C43×#436C2212yes—
44C43×#435—22no—
45C8×D4×C43C2212yes—
46C43×#423C2211yescharacteristic
47C43×#407—21no—
48C4×M16×C43C2211yescharacteristic
49C2×#10621898—21no—
50C22×Dic86—21no—
51C2×#10621883—22no—
52C2×#10621883—22no—
53C2×#10621883—44no—
54C2×#10621882—21no—
55C2×#10621882—21no—
56C22×C43⋊C8—21no—
57#13063325—21no—
58#13063325—21no—
59#13063325—21no—
60#13063325C2312yes—
61#13063325C2311yescharacteristic
62#13063322C2311yescharacteristic
63#13063313—21no—
64#13063299—21no—
65#13063299—21no—
66C4×D172—21no—
67C4×D172—21no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70#13063258—21no—
71#13063258—21no—
72C4⋊(C43⋊C8)—21no—
73C4⋊(C43⋊C8)C2311yescharacteristic
74C4⋊(C43⋊C8)C2311yescharacteristic
75C4⋊(C43⋊C8)C2312yes—
76#13063250—22no—
77#13063250—22no—
78#13063248—22no—
79#13063248—22no—
80#13063247—22no—
81#13063247—22no—
82#13063246—21no—
83#13063246—21no—
84#13063245—21no—
85#13063245—21no—
86C22⋊(C43⋊C8)—21no—
87C4×C43⋊C8—21no—
88C22×D4×C43D411yescharacteristic
89C23×C4×C43—21no—
90C2×M16×C43—21no—
91C2×M16×C43—22no—
92C2×M16×C43—44no—
93C2×M16×C43D411yescharacteristic
94C2×M16×C43D411yescharacteristic
95C22×C8×C43—22no—
96C4×D4×C43—21no—
97C4×D4×C43—21no—
98C4×D4×C43D411yescharacteristic
99C4×D4×C43D411yescharacteristic
100C4×D4×C43C2311yescharacteristic
101C4×D4×C43C2311yescharacteristic
102C2×C4⋊C4×C43D411yescharacteristic
103C2×C22⋊C4×C43—21no—
104C2×C42×C43C2311yescharacteristic
105C4⋊C8×C43—21no—
106C4⋊C8×C43C2312yes—
107C22⋊C8×C43—22no—
108C22⋊C8×C43—22no—
109#97×C43D412yes—
110C4×C8×C43C2312yes—
111C24×C43—21no—
112C2×D4×C43—21no—
113C2×D4×C43—22no—
114C2×D4×C43—41no—
115C2×D4×C43C2×D411yescharacteristic
116C2×D4×C43C2×D411yescharacteristic
117C22×C4×C43—21no—
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—21no—
122C22×C4×C43—42no—
123C22×C4×C43—42no—
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43#4111yescharacteristic
127C2×C22⋊D43—41no—
128C22×Dic43—41no—
129#10621898—42no—
130#10621898—42no—
131C2×D172—21no—
132C2×D172—21no—
133C2×D172—21no—
134C2×C4×D43—41no—
135C2×Dic86—22no—
136C2×Dic86—21no—
137C2×Dic86—42no—
138C2×Dic86—42no—
139C2×Dic86—41no—
140M16×C43—22no—
141M16×C43—22no—
142M16×C43—44no—
143M16×C43—42no—
144C2×C8×C43—22no—
145C2×C8×C43—44no—
146C2×C8×C43C2×D412yes—
147C2×C8×C43C2×D412yes—
148C4⋊C4×C43—21no—
149C4⋊C4×C43C2×D411yescharacteristic
150C4⋊C4×C43C2×D411yescharacteristic
151C22⋊C4×C43—21no—
152C22⋊C4×C43—21no—
153C22⋊C4×C43—21no—
154C22⋊C4×C43—21no—
155C42×C43C2×D411yescharacteristic
156C42×C43C2×D411yescharacteristic
157C42×C43#4111yescharacteristic
158C42×C43C2411yescharacteristic
159#10621883—44no—
160#10621883—44no—
161#10621882—42no—
162#10621882—42no—
163(C2×D43)⋊C4—41no—
164(C2×D43)⋊C4—41no—
165C4⋊Dic43—21no—
166C4⋊Dic43—22no—
167C4⋊Dic43—22no—
168C4⋊Dic43—21no—
169C4⋊Dic43—41no—
170C4⋊Dic43—41no—
171C2×C43⋊C8—21no—
172C2×C43⋊C8—21no—
173C2×C43⋊C8—22no—
174C2×C43⋊C8—42no—
175C23×C43—21no—
176C23×C43—21no—
177C23×C43—42no—
178C23×C43—42no—
179C23×C43—81no—
180C23×C43#12111yescharacteristic
181C22×D43—41no—
182D4×C43—21no—
183D4×C43—21no—
184D4×C43—21no—
185D4×C43—42no—
186D4×C43AGL(1,ℤ/8)12yes—
187C2×C4×C43—21no—
188C2×C4×C43—21no—
189C2×C4×C43—21no—
190C2×C4×C43—21no—
191C2×C4×C43—41no—
192C2×C4×C43—42no—
193C2×C4×C43—41no—
194C2×C4×C43—41no—
195C2×C4×C43—41no—
196C2×C4×C43—41no—
197C2×C4×C43—42no—
198C2×C4×C43—42no—
199C2×C4×C43#12111yescharacteristic
200C2×C4×C43#12111yescharacteristic
201C2×C4×C43#12111yescharacteristic
202C2×C4×C43#13512yes—
203C2×C4×C43AGL(1,ℤ/8)12yes—
204C2×C4×C43C22×D411yescharacteristic
205C2×C4×C43C22×D411yescharacteristic
206C2×C4×C43C2×#4111yescharacteristic
207C22⋊D43—82no—
208C2×Dic43—42no—
209C2×Dic43—41no—
210C2×Dic43—82no—
211D172—41no—
212D172—41no—
213D172—41no—
214D172—41no—
215D172—81no—
216C4×D43—82no—
217Dic86—41no—
218Dic86—42no—
219Dic86—41no—
220Dic86—42no—
221Dic86—82no—
222C8×C43—22no—
223C8×C43—22no—
224C8×C43—42no—
225C43⋊C8—42no—
226C43⋊C8—41no—
227C43⋊C8—41no—
228#19169—431nomaximal, Sylow
229C22×C43—21no—
230C22×C43—21no—
231C22×C43—22no—
232C22×C43—41no—
233C22×C43—41no—
234C22×C43—41no—
235C22×C43—41no—
236C22×C43—41no—
237C22×C43—84no—
238C22×C43#46312yes—
239C22×C43C2×#12111yescharacteristic
240C2×D43—81no—
241C2×D43—81no—
242C2×D43—81no—
243C4×C43—21no—
244C4×C43—21no—
245C4×C43—21no—
246C4×C43—41no—
247C4×C43—41no—
248C4×C43#46312yes—
249C4×C43C2×#13511yescharacteristic
250C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C43#57511yescharacteristic
253Dic43—81no—
254Dic43—82no—
255#3141—431no—
256#3100—431no—
257C2×#461—431no—
258D4×M16D4311yescharacteristic, complemented
259#1701—432no—
260#1699—432no—
261#1681—432no—
262#1679—432no—
263#1565—431no—
264#1549—431no—
265#1547—431no—
266C2×C43—22no—
267C2×C43—42no—
268C2×C43—81no—
269C2×C43C2×#46311yescharacteristic
270C2×C43#305511yescharacteristic
271C2×C43#306911yescharacteristic
272D43—161no—
273C22×SD16—861no—
274C22×M16—21no—
275#532—431no—
276C2×C4⋊Q8—431no—
277C2×C4×D4C2×D4311yescharacteristic
278#489—862no—
279#487—432no—
280#486—432no—
281#480—861no—
282#474—431no—
283#474—431no—
284#461—432no—
285#461—432no—
286#461—861no—
287#461—861no—
288#459—431no—
289#459—431no—
290#447—861no—
291C4×D8—861no—
292C4×D8—861no—
293#436C2×D4312yes—
294#435—22no—
295C8×D4C2×D4312yes—
296#431—431no—
297#423C2×D4311yescharacteristic
298C2×C4⋊C8—431no—
299#417—861no—
300#417—861no—
301C2×#103—861no—
302#407—21no—
303C4×M16C2×D4311yescharacteristic
304#335—432no—
305#334—432no—
306#332—432no—
307#332—432no—
308#332—432no—
309#331—432no—
310C43#1916911yescharacteristic, complemented, Sylow
311C2×#41—861no—
312C22×Q8—861no—
313C22×D4D17211yescharacteristic
314C23×C4—21no—
315C2×SD16—862no—
316C2×SD16—862no—
317C2×SD16—1724no—
318C2×D8—861no—
319C2×D8—861no—
320C2×M16—21no—
321C2×M16—22no—
322C2×M16—44no—
323C2×M16C22⋊D4311yescharacteristic
324C2×M16C22⋊D4311yescharacteristic
325C22×C8—22no—
326C22×C8—861no—
327C4⋊Q8—432no—
328C4⋊Q8—431no—
329C4⋊Q8—861no—
330C4⋊Q8—861no—
331C4⋊Q8—861no—
332#127—431no—
333#124—861no—
334#121—861no—
335#121—861no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—861no—
339C4×D4—861no—
340C4×D4D17211yescharacteristic
341C4×D4D17211yescharacteristic
342C4×D4C22×D4311yescharacteristic
343C4×D4C22×D4311yescharacteristic
344C2×C4⋊C4—861no—
345C2×C4⋊C4—861no—
346C2×C4⋊C4D17211yescharacteristic
347C2×C22⋊C4—21no—
348C2×C42C22×D4311yescharacteristic
349#107—861no—
350#107—861no—
351C4⋊C8—21no—
352C4⋊C8—432no—
353C4⋊C8—431no—
354C4⋊C8—431no—
355C4⋊C8—861no—
356C4⋊C8C22×D4312yes—
357#103—862no—
358#103—862no—
359#103—862no—
360#103—862no—
361#102—861no—
362#102—861no—
363#102—862no—
364#102—862no—
365#102—861no—
366#102—861no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—861no—
370#97C22⋊D4312yes—
371C4×C8—861no—
372C4×C8C22×D4312yes—
373C24—21no—
374#41—1722no—
375#41—1722no—
376C2×Q8—861no—
377C2×Q8—862no—
378C2×Q8—1722no—
379C2×Q8—1722no—
380C2×Q8—1721no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—861no—
385C2×D4—861no—
386C2×D4—861no—
387C2×D4—1721no—
388C2×D4C2×D17211yescharacteristic
389C2×D4C2×D17211yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—1721no—
398C22×C4—1721no—
399C22×C4C2×D17211yescharacteristic
400C22×C4#1062189811yescharacteristic
401C22×C4C2×C22⋊D4311yescharacteristic
402SD16—1724no—
403SD16—1724no—
404D8—1722no—
405D8—1722no—
406M16—22no—
407M16—22no—
408M16—44no—
409M16—42no—
410C2×C8—22no—
411C2×C8—44no—
412C2×C8—861no—
413C2×C8—861no—
414C2×C8—862no—
415C2×C8—1722no—
416C2×C8C2×C22⋊D4312yes—
417C2×C8C2×C22⋊D4312yes—
418C4⋊C4—21no—
419C4⋊C4—862no—
420C4⋊C4—861no—
421C4⋊C4—862no—
422C4⋊C4—861no—
423C4⋊C4—1721no—
424C4⋊C4—1721no—
425C4⋊C4C2×D17211yescharacteristic
426C4⋊C4C2×D17211yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—1721no—
432C22⋊C4—1721no—
433C42C2×D17211yescharacteristic
434C42#1062189811yescharacteristic
435C42C2×C22⋊D4311yescharacteristic
436C42C23×D4311yescharacteristic
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—81no—
442C23—1721no—
443C23#1306329911yescharacteristic
444Q8—3442no—
445Q8—1721no—
446Q8—1721no—
447Q8—1722no—
448Q8—1722no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—3441no—
454D4—3442no—
455D4—1721no—
456D4—1721no—
457D4—1721no—
458D4—1721no—
459D4#1306335312yes—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—3442no—
473C2×C4—3442no—
474C2×C4—1721no—
475C2×C4—1722no—
476C2×C4#1306329911yescharacteristic
477C2×C4#1306329911yescharacteristic
478C2×C4#1306329911yescharacteristic
479C2×C4#1306334512yes—
480C2×C4#1306335112yes—
481C2×C4C22×C22⋊D4311yescharacteristic
482C2×C4C22×D17211yescharacteristic
483C2×C4C2×#1062189811yescharacteristic
484C8—22no—
485C8—22no—
486C8—42no—
487C8—1722no—
488C8—1721no—
489C8—1721no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—3441no—
500C22—3441no—
501C22—3441no—
502C22#1458311412yes—
503C22C2×#1306329911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—3442no—
510C4—3441no—
511C4#1458311412yes—
512C4C2×#1306334511yescharacteristic
513C4C2×#1306335111yescharacteristic
514C4C2×#1306335311yescharacteristic
515C4#1458369911yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—6881no—
520C2C2×#1458311411yescharacteristic
521C2#1914421411yescharacteristic
522C2#1914433211yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingD4×M16×C43C2
socleC22×C43C2×#121
prime radical#19144595C2
hypercenterD4×M16D43
hypocenterC43#19169
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #19144989C2semidirect
3. C2×#14583104C2semidirect
4. #19134156C2semidirect
5. #19134142C2semidirect
6. #19134032C2semidirect
7. #19134018C2semidirect
8. #19133347C2semidirect
9. #19133281C2semidirect
10. #19133273C2semidirect
11. D4×M16×C43C2semidirect
16. #14583250C22semidirect
17. #14583246C22semidirect
35. #14582652C22semidirect
36. #14582650C22semidirect
37. #14582646C22semidirect
38. #14582644C22semidirect
39. #14582644C22semidirect
40. #14582642C22semidirect
258. D4×M16D43semidirect
310. C43#19169semidirect

Maximal subgroups

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

subgroupindex
#191449892
#191445952
C2×#145831042
#191341562
#191341422
#191340322
#191340182
#191333472
#191332812
#191332732
D4×M16×C432
#1916943