Finite Group Databasebeta

← #25079409

Subgroups — #25079409

Order 11008.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144970C211yesmaximal, characteristic, complemented
2#19144603C211yesmaximal, characteristic, complemented
3C2×#14583094C211yesmaximal, characteristic
4#19134156C212yesmaximal, complemented
5#19134128C212yesmaximal, complemented
6#19134032C212yesmaximal, complemented
7#19134004C212yesmaximal, complemented
8#19133347C211yesmaximal, characteristic, complemented
9#19133286C211yesmaximal, characteristic, complemented
10#19133264C211yesmaximal, characteristic, complemented
11D4×M16×C43C211yesmaximal, characteristic, complemented
12C22×#10621882—21no—
13#14583456C2211yescharacteristic
14C2×#13063322C2211yescharacteristic
15#14583264—22no—
16#14583246C2212yescomplemented
17#14583246C2212yescomplemented
18#14583229—21no—
19#14583189C2211yescharacteristic, complemented
20#14583189C2211yescharacteristic, complemented
21#14583104C2211yescharacteristic, complemented
22#14583104C2211yescharacteristic, complemented
23#14583094—21no—
24#14583094—21no—
25#14583094C2212yes—
26#14583094C2212yes—
27#14583050—21no—
28C4×#10621883—21no—
29C4×#10621883—21no—
30#14582952C2211yescharacteristic
31C2×C4⋊(C43⋊C8)C2211yescharacteristic
32#14582905—21no—
33#14582905—21no—
34C2×#13063245—21no—
35#14582652C2212yescomplemented
36#14582650C2212yescomplemented
37#14582647C2212yescomplemented
38#14582644C2212yescomplemented
39#14582642C2212yescomplemented
40#14582642C2212yescomplemented
41C22×M16×C43—21no—
42C2×C4×D4×C43C2211yescharacteristic
43C43×#436C2212yescomplemented
44C43×#435—22no—
45C8×D4×C43C2212yescomplemented
46C43×#423C2211yescharacteristic
47C43×#407—21no—
48C4×M16×C43C2211yescharacteristic
49C22×D172—21no—
50C22×D172—21no—
51C22×D172—21no—
52C2×#10621883—21no—
53C2×#10621883—21no—
54C2×#10621882—22no—
55C2×#10621882—22no—
56C2×#10621882—44no—
57C22×C43⋊C8—21no—
58#13063325C2311yescharacteristic
59#13063325C2311yescharacteristic
60#13063322—21no—
61#13063322—21no—
62#13063322—21no—
63#13063322C2312yes—
64#13063317—21no—
65#13063304—21no—
66#13063304—21no—
67C4×Dic86—21no—
68C4×Dic86—21no—
69#13063284—21no—
70#13063256—21no—
71#13063256—21no—
72C4⋊(C43⋊C8)—21no—
73C4⋊(C43⋊C8)C2312yes—
74C4⋊(C43⋊C8)C2311yescharacteristic
75C4⋊(C43⋊C8)C2311yescharacteristic
76#13063248—21no—
77#13063248—21no—
78#13063247—22no—
79#13063247—22no—
80#13063247—22no—
81#13063247—22no—
82#13063246—21no—
83#13063246—21no—
84#13063245—22no—
85#13063245—22no—
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—
112C23×D43—41no—
113C2×D4×C43—21no—
114C2×D4×C43—22no—
115C2×D4×C43—41no—
116C2×D4×C43C2×D411yescharacteristic
117C2×D4×C43C2×D411yescharacteristic
118C22×C4×C43—21no—
119C22×C4×C43—21no—
120C22×C4×C43—21no—
121C22×C4×C43—21no—
122C22×C4×C43—21no—
123C22×C4×C43—42no—
124C22×C4×C43—42no—
125C22×C4×C43C2×D411yescharacteristic
126C22×C4×C43C2×D411yescharacteristic
127C22×C4×C43#4111yescharacteristic
128C2×D172—22no—
129C2×D172—22no—
130C2×D172—22no—
131C2×D172—41no—
132C2×D172—41no—
133C2×D172—41no—
134C2×D172—42no—
135C2×D172—42no—
136C2×D172—42no—
137C2×D172—42no—
138C2×D172—82no—
139C2×Dic86—21no—
140C2×Dic86—21no—
141M16×C43—22no—
142M16×C43—22no—
143M16×C43—44no—
144M16×C43—42no—
145C2×C8×C43—22no—
146C2×C8×C43—44no—
147C2×C8×C43C2×D412yes—
148C2×C8×C43C2×D412yes—
149C4⋊C4×C43—21no—
150C4⋊C4×C43C2×D411yescharacteristic
151C4⋊C4×C43C2×D411yescharacteristic
152C22⋊C4×C43—21no—
153C22⋊C4×C43—21no—
154C22⋊C4×C43—21no—
155C22⋊C4×C43—21no—
156C42×C43C2×D411yescharacteristic
157C42×C43C2×D411yescharacteristic
158C42×C43#4111yescharacteristic
159C42×C43C2411yescharacteristic
160C22⋊Dic43—41no—
161#10621883—42no—
162#10621883—42no—
163#10621882—44no—
164#10621882—44no—
165C4⋊Dic43—21no—
166C4⋊Dic43—21no—
167C4⋊Dic43—21no—
168C4⋊Dic43—21no—
169Dic43⋊C4—41no—
170Dic43⋊C4—41no—
171C4×Dic43—41no—
172C2×C43⋊C8—21no—
173C2×C43⋊C8—21no—
174C2×C43⋊C8—22no—
175C2×C43⋊C8—42no—
176C23×C43—21no—
177C23×C43—21no—
178C23×C43—42no—
179C23×C43—42no—
180C23×C43—81no—
181C23×C43#12111yescharacteristic
182C22×D43—42no—
183C22×D43—82no—
184C22×D43—82no—
185C22×D43—82no—
186D4×C43—21no—
187D4×C43—21no—
188D4×C43—21no—
189D4×C43—42no—
190D4×C43AGL(1,ℤ/8)12yes—
191C2×C4×C43—21no—
192C2×C4×C43—21no—
193C2×C4×C43—21no—
194C2×C4×C43—21no—
195C2×C4×C43—41no—
196C2×C4×C43—42no—
197C2×C4×C43—41no—
198C2×C4×C43—41no—
199C2×C4×C43—41no—
200C2×C4×C43—41no—
201C2×C4×C43—42no—
202C2×C4×C43—42no—
203C2×C4×C43#12111yescharacteristic
204C2×C4×C43#12111yescharacteristic
205C2×C4×C43#12111yescharacteristic
206C2×C4×C43#13512yes—
207C2×C4×C43AGL(1,ℤ/8)12yes—
208C2×C4×C43C22×D411yescharacteristic
209C2×C4×C43C22×D411yescharacteristic
210C2×C4×C43C2×#4111yescharacteristic
211C2×Dic43—41no—
212C2×Dic43—41no—
213D172—42no—
214D172—42no—
215D172—42no—
216D172—42no—
217D172—82no—
218D172—82no—
219D172—82no—
220D172—82no—
221Dic86—41no—
222Dic86—41no—
223Dic86—81no—
224C8×C43—22no—
225C8×C43—22no—
226C8×C43—42no—
227C43⋊C8—42no—
228C43⋊C8—41no—
229C43⋊C8—41no—
230#19166—431nomaximal, Sylow
231C22×C43—21no—
232C22×C43—21no—
233C22×C43—22no—
234C22×C43—41no—
235C22×C43—41no—
236C22×C43—41no—
237C22×C43—41no—
238C22×C43—41no—
239C22×C43—84no—
240C22×C43#46312yes—
241C22×C43C2×#12111yescharacteristic
242C2×D43—82no—
243C2×D43—82no—
244C2×D43—82no—
245C2×D43—164no—
246C4×C43—21no—
247C4×C43—21no—
248C4×C43—21no—
249C4×C43—41no—
250C4×C43—41no—
251C4×C43#46312yes—
252C4×C43C2×#13511yescharacteristic
253C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C43#57511yescharacteristic
256Dic43—81no—
257Dic43—81no—
258#3138—431no—
259#3101—431no—
260C2×#459—431no—
261D4×M16D4311yescharacteristic, complemented
262#1701—432no—
263#1697—432no—
264#1681—432no—
265#1677—432no—
266#1565—431no—
267#1550—431no—
268#1545—431no—
269C2×C43—22no—
270C2×C43—42no—
271C2×C43—81no—
272C2×C43C2×#46311yescharacteristic
273C2×C43#305511yescharacteristic
274C2×C43#306811yescharacteristic
275D43—162no—
276C22×D8—861no—
277C22×M16—21no—
278#533—431no—
279C2×#127—431no—
280C2×C4×D4C2×D4311yescharacteristic
281#490—862no—
282#486—432no—
283#486—432no—
284#483—861no—
285#476—431no—
286#476—431no—
287#461—431no—
288#461—431no—
289#459—432no—
290#459—432no—
291#459—861no—
292#459—861no—
293#450—861no—
294C4×SD16—861no—
295C4×SD16—861no—
296#436C2×D4312yescomplemented
297#435—22no—
298C8×D4C2×D4312yescomplemented
299#431—431no—
300#423C2×D4311yescharacteristic
301C2×C4⋊C8—431no—
302#418—861no—
303#418—861no—
304C2×#102—861no—
305#407—21no—
306C4×M16C2×D4311yescharacteristic
307#335—432no—
308#334—432no—
309#333—432no—
310#332—432no—
311#331—432no—
312#331—432no—
313C43#1916611yescharacteristic, complemented, Sylow
314C22×D4—861no—
315C22×D4—861no—
316C22×D4—861no—
317C22×D4D17211yescharacteristic
318C23×C4—21no—
319C2×SD16—861no—
320C2×SD16—861no—
321C2×D8—862no—
322C2×D8—862no—
323C2×D8—1724no—
324C2×M16—21no—
325C2×M16—22no—
326C2×M16—44no—
327C2×M16C22⋊D4311yescharacteristic
328C2×M16C22⋊D4311yescharacteristic
329C22×C8—22no—
330C22×C8—861no—
331C4⋊Q8—431no—
332C4⋊Q8—431no—
333#127—432no—
334#127—861no—
335#127—861no—
336#127—861no—
337#125—861no—
338C22⋊Q8—861no—
339C22⋊Q8—861no—
340C4×Q8—861no—
341C4×Q8—861no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4D17211yescharacteristic
345C4×D4D17211yescharacteristic
346C4×D4C22×D4311yescharacteristic
347C4×D4C22×D4311yescharacteristic
348#117—861no—
349C2×C4⋊C4D17211yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D4311yescharacteristic
352#106—861no—
353#106—861no—
354C4⋊C8—21no—
355C4⋊C8—431no—
356C4⋊C8—431no—
357C4⋊C8—432no—
358C4⋊C8—861no—
359C4⋊C8C22×D4312yes—
360#103—861no—
361#103—861no—
362#102—861no—
363#102—861no—
364#102—862no—
365#102—862no—
366#102—862no—
367#102—862no—
368#102—862no—
369#102—862no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—861no—
373#97C22⋊D4312yes—
374C4×C8—861no—
375C4×C8C22×D4312yes—
376C24—21no—
377C24—1721no—
378C2×Q8—861no—
379C2×Q8—861no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—862no—
384C2×D4—862no—
385C2×D4—862no—
386C2×D4—3442no—
387C2×D4—1722no—
388C2×D4—1722no—
389C2×D4—1721no—
390C2×D4—1721no—
391C2×D4—1721no—
392C2×D4—1722no—
393C2×D4—1722no—
394C2×D4C2×D17211yescharacteristic
395C2×D4C2×D17211yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×D17211yescharacteristic
404C22×C4#1062189811yescharacteristic
405C22×C4C2×C22⋊D4311yescharacteristic
406SD16—1722no—
407SD16—1722no—
408D8—1724no—
409D8—1724no—
410M16—22no—
411M16—22no—
412M16—44no—
413M16—42no—
414C2×C8—22no—
415C2×C8—44no—
416C2×C8—861no—
417C2×C8—861no—
418C2×C8—862no—
419C2×C8—1722no—
420C2×C8C2×C22⋊D4312yes—
421C2×C8C2×C22⋊D4312yes—
422C4⋊C4—21no—
423C4⋊C4—861no—
424C4⋊C4—861no—
425C4⋊C4—861no—
426C4⋊C4—861no—
427C4⋊C4—1721no—
428C4⋊C4—1721no—
429C4⋊C4C2×D17211yescharacteristic
430C4⋊C4C2×D17211yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—1721no—
436C42—1721no—
437C42C2×D17211yescharacteristic
438C42#1062189811yescharacteristic
439C42C2×C22⋊D4311yescharacteristic
440C42C23×D4311yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—3442no—
447C23—3442no—
448C23—3442no—
449C23—1722no—
450C23#1306329911yescharacteristic
451Q8—3441no—
452Q8—1721no—
453Q8—1721no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—3442no—
459D4—3442no—
460D4—3442no—
461D4—3442no—
462D4—1722no—
463D4—1722no—
464D4—1722no—
465D4—1722no—
466D4#1306335312yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—1721no—
480C2×C4—1721no—
481C2×C4#1306329911yescharacteristic
482C2×C4#1306329911yescharacteristic
483C2×C4#1306329911yescharacteristic
484C2×C4#1306334512yes—
485C2×C4#1306335112yes—
486C2×C4C22×C22⋊D4311yescharacteristic
487C2×C4C22×D17211yescharacteristic
488C2×C4C2×#1062189811yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—1722no—
493C8—1721no—
494C8—1721no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—3442no—
505C22—3442no—
506C22—3442no—
507C22—6884no—
508C22#1458311412yes—
509C22C2×#1306329911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—3441no—
516C4—3441no—
517C4#1458311412yes—
518C4C2×#1306334511yescharacteristic
519C4C2×#1306335111yescharacteristic
520C4C2×#1306335311yescharacteristic
521C4#1458369911yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—6882no—
526C2C2×#1458311411yescharacteristic
527C2#1914421411yescharacteristic
528C2#1914432311yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingD4×M16×C43C2
socleC22×C43C2×#121
prime radicalC2×#14583094C2
hypercenterD4×M16D43
hypocenterC43#19166
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. #19144970C2semidirect
2. #19144603C2semidirect
4. #19134156C2semidirect
5. #19134128C2semidirect
6. #19134032C2semidirect
7. #19134004C2semidirect
8. #19133347C2semidirect
9. #19133286C2semidirect
10. #19133264C2semidirect
11. D4×M16×C43C2semidirect
16. #14583246C22semidirect
17. #14583246C22semidirect
19. #14583189C22semidirect
20. #14583189C22semidirect
21. #14583104C22semidirect
22. #14583104C22semidirect
35. #14582652C22semidirect
36. #14582650C22semidirect
37. #14582647C22semidirect
38. #14582644C22semidirect
39. #14582642C22semidirect
40. #14582642C22semidirect
43. C43×#436C22semidirect
45. C8×D4×C43C22semidirect
261. D4×M16D43semidirect
296. #436C2×D43semidirect
298. C8×D4C2×D43semidirect
313. C43#19166semidirect

Maximal subgroups

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

subgroupindex
#191449702
#191446032
C2×#145830942
#191341562
#191341282
#191340322
#191340042
#191333472
#191332862
#191332642
D4×M16×C432
#1916643