Finite Group Databasebeta

← #25079381

Subgroups — #25079381

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144599C211yesmaximal, characteristic
2C2×#14583101C211yesmaximal, characteristic
3M16×D172C211yesmaximal, characteristic
4#19134105C212yesmaximal, complemented
5#19134091C212yesmaximal, complemented
6#19133981C212yesmaximal, complemented
7#19133967C212yesmaximal, complemented
8#19133347C211yesmaximal, characteristic, complemented
9#19133285C211yesmaximal, characteristic, complemented
10#19133280C211yesmaximal, characteristic, complemented
11C43×#3165C211yesmaximal, characteristic, complemented
12C22×#10621884—21no—
13C2×M16×D43—21no—
14C2×C4×D172C2211yescharacteristic
15#14583227—21no—
16#14583186C2211yescharacteristic
17#14583186C2211yescharacteristic
18#14583101—21no—
19#14583101—21no—
20#14583101C2212yes—
21#14583101C2211yescharacteristic
22#14583101C2212yes—
23#14583101C2211yescharacteristic
24#14583049—21no—
25C4×#10621884—21no—
26C4×#10621884—21no—
27#14582978C2212yes—
28#14582972—22no—
29C8×D172C2212yes—
30#14582952C2211yescharacteristic
31#14582923C2211yescharacteristic
32C2×C4⋊(C43⋊C8)C2211yescharacteristic
33C2×#13063249—21no—
34#14582862—21no—
35#14582652C2212yescomplemented
36#14582650C2212yescomplemented
37#14582648C2212yescomplemented
38#14582645C2212yescomplemented
39#14582645C2212yescomplemented
40#14582645C2212yescomplemented
41C43×#533C2211yescharacteristic
42C2×C4⋊Q8×C43C2211yescharacteristic
43C43×#501—22no—
44C43×#500C2212yescomplemented
45C43×#498C2212yescomplemented
46C43×#428—21no—
47C43×#428—21no—
48C4×M16×C43C2211yescharacteristic
49C22×D172D411yescharacteristic
50C22×C4×D43—21no—
51C2×#10621884—21no—
52C2×#10621884—21no—
53C2×#10621884—22no—
54C2×#10621884—22no—
55C2×#10621884—44no—
56C2×#10621877—21no—
57C2×#10621872—22no—
58M16×D43—44no—
59C22×C43⋊C8—21no—
60C2×C8×D43—22no—
61C4×D172—21no—
62C4×D172—21no—
63C4×D172D411yescharacteristic
64C4×D172D411yescharacteristic
65C4×D172C2311yescharacteristic
66C4×D172C2311yescharacteristic
67C2×C4⋊Dic43D411yescharacteristic
68C2×(C2×D43)⋊C4—21no—
69#13063256—21no—
70#13063256—21no—
71C4⋊(C43⋊C8)—21no—
72C4⋊(C43⋊C8)—21no—
73C4⋊(C43⋊C8)C2312yes—
74C4⋊(C43⋊C8)C2311yescharacteristic
75C4⋊(C43⋊C8)C2311yescharacteristic
76C4⋊(C43⋊C8)C2312yes—
77#13063249—22no—
78#13063249—22no—
79#13063249—21no—
80#13063249—21no—
81#13063245—21no—
82#13063245—21no—
83C22⋊(C43⋊C8)—21no—
84(C2×D43)⋊C8—22no—
85(C2×D43)⋊C8—22no—
86C4×C43⋊C8—21no—
87C22×Q8×C43—21no—
88C2×M16×C43D411yescharacteristic
89C2×M16×C43D411yescharacteristic
90C4⋊Q8×C43—21no—
91C4⋊Q8×C43—21no—
92C4⋊Q8×C43—21no—
93C4⋊Q8×C43C2311yescharacteristic
94C4⋊Q8×C43C2312yes—
95C4⋊Q8×C43C2311yescharacteristic
96#125×C43—21no—
97C22⋊Q8×C43—21no—
98C22⋊Q8×C43—21no—
99C4×Q8×C43—21no—
100C4×Q8×C43—21no—
101#117×C43—21no—
102C2×C4⋊C4×C43—21no—
103C2×C4⋊C4×C43—21no—
104C2×C42×C43C2311yescharacteristic
105#107×C43—22no—
106#107×C43—22no—
107#106×C43—22no—
108#106×C43—22no—
109#97×C43D412yes—
110C4×C8×C43C2312yes—
111C23×D43—21no—
112C2×Q8×C43—21no—
113C2×Q8×C43—22no—
114C2×Q8×C43—21no—
115C2×Q8×C43—42no—
116C2×Q8×C43—42no—
117C2×Q8×C43—41no—
118C22×C4×C43—41no—
119C22×C4×C43C2×D411yescharacteristic
120C22×C4×C43C2×D411yescharacteristic
121C22×C4×C43#4111yescharacteristic
122C22×Dic43—21no—
123C2×D172—21no—
124C2×D172—22no—
125C2×D172—41no—
126C2×D172C2×D411yescharacteristic
127C2×D172C2×D411yescharacteristic
128C2×C4×D43—21no—
129C2×C4×D43—21no—
130C2×C4×D43—21no—
131C2×C4×D43—21no—
132C2×C4×D43—42no—
133C2×C4×D43—42no—
134M16×C43—22no—
135M16×C43SD1614yescomplemented
136C2×C8×C43C2×D412yes—
137C2×C8×C43C2×D412yes—
138C4⋊C4×C43—21no—
139C4⋊C4×C43—22no—
140C4⋊C4×C43—21no—
141C4⋊C4×C43—21no—
142C4⋊C4×C43—22no—
143C4⋊C4×C43—21no—
144C4⋊C4×C43—41no—
145C4⋊C4×C43—41no—
146C4⋊C4×C43—41no—
147C4⋊C4×C43—41no—
148C22⋊C4×C43—41no—
149C42×C43—41no—
150C42×C43C2×D411yescharacteristic
151C42×C43C2×D411yescharacteristic
152C42×C43#4111yescharacteristic
153C42×C43C2411yescharacteristic
154#10621884—42no—
155#10621884—44no—
156#10621884—42no—
157#10621884—44no—
158(C2×D43)⋊C4—21no—
159(C2×D43)⋊C4—21no—
160(C2×D43)⋊C4—21no—
161(C2×D43)⋊C4—21no—
162C4⋊Dic43—21no—
163C4⋊Dic43C2×D411yescharacteristic
164C4⋊Dic43C2×D411yescharacteristic
165#10621877—42no—
166C2×C43⋊C8—21no—
167C2×C43⋊C8—21no—
168C2×C43⋊C8—22no—
169C2×C43⋊C8—22no—
170C2×C43⋊C8—42no—
171#10621872—44no—
172C8×D43—44no—
173C23×C43#12111yescharacteristic
174C22×D43—21no—
175C22×D43—21no—
176C22×D43—42no—
177C22×D43—42no—
178C22×D43—81no—
179Q8×C43—42no—
180Q8×C43—41no—
181Q8×C43—41no—
182Q8×C43—42no—
183Q8×C43—81no—
184Q8×C43—82no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—41no—
188C2×C4×C43—41no—
189C2×C4×C43—41no—
190C2×C4×C43—42no—
191C2×C4×C43—82no—
192C2×C4×C43#12111yescharacteristic
193C2×C4×C43#12111yescharacteristic
194C2×C4×C43#12111yescharacteristic
195C2×C4×C43C2×SD1612yes—
196C2×C4×C43#13712yes—
197C2×C4×C43C22×D411yescharacteristic
198C2×C4×C43C22×D411yescharacteristic
199C2×C4×C43C2×#4111yescharacteristic
200C2×Dic43—21no—
201C2×Dic43—21no—
202C2×Dic43—42no—
203D172—21no—
204D172—21no—
205D172—21no—
206D172—42no—
207D172AGL(1,ℤ/8)12yes—
208C4×D43—41no—
209C4×D43—41no—
210C4×D43—41no—
211C4×D43—41no—
212C4×D43—42no—
213C4×D43—42no—
214C8×C43—22no—
215C8×C43C2×SD1614yescomplemented
216C43⋊C8—42no—
217C43⋊C8—42no—
218C43⋊C8—41no—
219C43⋊C8—41no—
220#19164—431nomaximal, Sylow
221C22×C43—21no—
222C22×C43—21no—
223C22×C43#46112yes—
224C22×C43C2×#12111yescharacteristic
225C2×D43—22no—
226C2×D43—41no—
227C2×D43—41no—
228C2×D43—41no—
229C2×D43—41no—
230C2×D43—41no—
231C2×D43—84no—
232C4×C43—21no—
233C4×C43—21no—
234C4×C43—21no—
235C4×C43—82no—
236C4×C43—81no—
237C4×C43—81no—
238C4×C43#46112yes—
239C4×C43C22×SD1611yescharacteristic
240C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C43C2×#13711yescharacteristic
242C4×C43#57511yescharacteristic
243Dic43—41no—
244Dic43—41no—
245#3165D4311yescharacteristic, complemented
246#3101—431no—
247C2×#461—431no—
248D4×M16—431no—
249#1693—432no—
250#1691—432no—
251#1673—432no—
252#1671—432no—
253#1565—431no—
254#1550—431no—
255#1549—431no—
256C2×C43—22no—
257C2×C43C2×#46111yescharacteristic
258C2×C43#305511yescharacteristic
259C2×C43#306911yescharacteristic
260D43—42no—
261D43—81no—
262C22×SD16—861no—
263C22×M16—861no—
264#533C2×D4311yescharacteristic
265C2×C4⋊Q8C2×D4311yescharacteristic
266C2×C4×D4—431no—
267#501—22no—
268#500C2×D4312yescomplemented
269#498C2×D4312yescomplemented
270#483—861no—
271#476—431no—
272#476—431no—
273#461—431no—
274#461—431no—
275#461—432no—
276#461—432no—
277#461—861no—
278#461—861no—
279#450—861no—
280C4×SD16—861no—
281C4×SD16—861no—
282#436—432no—
283#435—862no—
284C8×D4—432no—
285#431—431no—
286#428—21no—
287#428—21no—
288#423—431no—
289C2×C4⋊C8—431no—
290C2×#103—861no—
291#407—861no—
292C4×M16C2×D4311yescharacteristic
293#335—432no—
294#334—432no—
295#333—432no—
296#332—432no—
297#332—432no—
298#332—432no—
299C43#1916411yescharacteristic, complemented, Sylow
300C22×Q8—21no—
301C22×D4—431no—
302C23×C4—861no—
303C2×SD16—862no—
304C2×SD16—862no—
305C2×SD16—861no—
306C2×SD16—861no—
307C2×SD16—1724no—
308C2×M16—861no—
309C2×M16—862no—
310C2×M16—1724no—
311C2×M16C22⋊D4311yescharacteristic
312C2×M16C22⋊D4311yescharacteristic
313C22×C8—861no—
314C22×C8—862no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8C22×D4312yes—
319C4⋊Q8C22×D4311yescharacteristic
320C4⋊Q8C22×D4311yescharacteristic
321#125—21no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—431no—
327C4×D4—431no—
328C4×D4—431no—
329C4×D4—431no—
330C4×D4—861no—
331C4×D4—861no—
332#117—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—431no—
336C2×C22⋊C4—861no—
337C2×C42C22×D4311yescharacteristic
338#107—22no—
339#107—22no—
340#106—22no—
341#106—22no—
342#106—861no—
343#106—861no—
344C4⋊C8—432no—
345C4⋊C8—431no—
346C4⋊C8—431no—
347C4⋊C8—432no—
348C4⋊C8—861no—
349C4⋊C8—861no—
350#103—862no—
351#103—862no—
352#103—861no—
353#103—861no—
354#102—861no—
355#102—861no—
356C22⋊C8—861no—
357C22⋊C8—862no—
358C22⋊C8—862no—
359#97C22⋊D4312yes—
360C4×C8—861no—
361C4×C8C22×D4312yes—
362C24—861no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—41no—
369C2×D4—431no—
370C2×D4—431no—
371C2×D4—862no—
372C2×D4—861no—
373C2×D4—1721no—
374C22×C4—41no—
375C22×C4—861no—
376C22×C4—861no—
377C22×C4—861no—
378C22×C4—861no—
379C22×C4—861no—
380C22×C4—1722no—
381C22×C4—1722no—
382C22×C4D4×D4311yescharacteristic
383C22×C4#1062190011yescharacteristic
384C22×C4C2×C22⋊D4311yescharacteristic
385SD16—1724no—
386SD16—1722no—
387SD16—1724no—
388SD16—1722no—
389M16—22no—
390M16—1722no—
391M16—1724no—
392M16#1062188414yescomplemented
393C2×C8—861no—
394C2×C8—861no—
395C2×C8—862no—
396C2×C8—862no—
397C2×C8—1722no—
398C2×C8—1724no—
399C2×C8C2×C22⋊D4312yes—
400C2×C8C2×C22⋊D4312yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—431no—
412C4⋊C4—431no—
413C4⋊C4—861no—
414C22⋊C4—41no—
415C22⋊C4—861no—
416C22⋊C4—861no—
417C22⋊C4—861no—
418C22⋊C4—861no—
419C42—41no—
420C42D4×D4311yescharacteristic
421C42#1062190011yescharacteristic
422C42C2×C22⋊D4311yescharacteristic
423C42C23×D4311yescharacteristic
424C23—861no—
425C23—861no—
426C23—3441no—
427C23—1722no—
428C23—1722no—
429C23#1306329611yescharacteristic
430Q8—42no—
431Q8—41no—
432Q8—41no—
433Q8—42no—
434Q8—81no—
435Q8—82no—
436D4—432no—
437D4—861no—
438D4—861no—
439D4—861no—
440D4—1722no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—861no—
449C2×C4—861no—
450C2×C4—1722no—
451C2×C4—1722no—
452C2×C4—1722no—
453C2×C4—1721no—
454C2×C4—1721no—
455C2×C4—1721no—
456C2×C4—1721no—
457C2×C4#1306329611yescharacteristic
458C2×C4#1306329611yescharacteristic
459C2×C4#1306329611yescharacteristic
460C2×C4C2×#1062188412yes—
461C2×C4#1306335612yes—
462C2×C4C2×D4×D4311yescharacteristic
463C2×C4C22×C22⋊D4311yescharacteristic
464C2×C4C2×#1062190011yescharacteristic
465C8—22no—
466C8—1722no—
467C8—1722no—
468C8—1721no—
469C8—1721no—
470C8C2×#1062188414yescomplemented
471C22—21no—
472C22—21no—
473C22—862no—
474C22—3444no—
475C22—1721no—
476C22—1721no—
477C22—1721no—
478C22—1721no—
479C22—1721no—
480C22#1458310112yes—
481C22C2×#1306329611yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—81no—
487C4—81no—
488C4—1721no—
489C4—1721no—
490C4#1458310112yes—
491C4C22×#1062188411yescharacteristic
492C4AGL(1,ℤ/8)×D4311yescharacteristic
493C4C2×#1306335611yescharacteristic
494C4#1458369511yescharacteristic
495C2—22no—
496C2—3441no—
497C2—1722no—
498C2C2×#1458310111yescharacteristic
499C2#1914420811yescharacteristic
500C2#1914432811yescharacteristic
subgroupG/H
centerC22C2×#13063296
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#3165C2
socleC22×C43C2×#121
prime radicalC22×D172D4
hypercenter#3165D43
hypocenterC43#19164
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #19134105C2semidirect
5. #19134091C2semidirect
6. #19133981C2semidirect
7. #19133967C2semidirect
8. #19133347C2semidirect
9. #19133285C2semidirect
10. #19133280C2semidirect
11. C43×#3165C2semidirect
35. #14582652C22semidirect
36. #14582650C22semidirect
37. #14582648C22semidirect
38. #14582645C22semidirect
39. #14582645C22semidirect
40. #14582645C22semidirect
44. C43×#500C22semidirect
45. C43×#498C22semidirect
135. M16×C43SD16semidirect
215. C8×C43C2×SD16semidirect
245. #3165D43semidirect
268. #500C2×D43semidirect
269. #498C2×D43semidirect
299. C43#19164semidirect
392. M16#10621884semidirect
470. C8C2×#10621884semidirect

Maximal subgroups

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

subgroupindex
#191445992
C2×#145831012
M16×D1722
#191341052
#191340912
#191339812
#191339672
#191333472
#191332852
#191332802
C43×#31652
#1916443