Finite Group Databasebeta

← #14907758

Subgroups — #14907758

Order 2816.

1104 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622806C211yesmaximal, characteristic, complemented
2C2×#10622806C211yesmaximal, characteristic, complemented
3C2×#10622755C211yesmaximal, characteristic, complemented
4#13070580C214yesmaximal, complemented
5#13070573C214yesmaximal, complemented
6#13070365C211yesmaximal, characteristic, complemented
7#13070344C211yesmaximal, characteristic, complemented
8#13070244C211yesmaximal, characteristic, complemented
9C11×#3584C211yesmaximal, characteristic, complemented
10C11×#585—21no—
11C11×#578—21no—
12C11×#578C2212yescomplemented
13C11×#577—22no—
14C11×#577—22no—
15C11×#576—21no—
16C11×#576C2212yescomplemented
17C11×#575—22no—
18C11×#575—21no—
19C2×C11×#135—22no—
20C2×C11×#135C2211yescharacteristic
21C2×C11×#135C2211yescharacteristic
22C2×C11×#131C2211yescharacteristic
23C2×#67002—21no—
24C22×D88C2211yescharacteristic, complemented
25#10622806—22no—
26#10622806—22no—
27#10622806—22no—
28#10622806C2214yescomplemented
29C2×#66892—21no—
30C2×#66892C2212yescomplemented
31C2×#66890—21no—
32C2×#66890C2212yescomplemented
33#10622755—22no—
34#10622755C2214yescomplemented
35C2×#66887—21no—
36C2×#66883C2211yescharacteristic, complemented
37C2×#66876C2211yescharacteristic, complemented
38C2×#66876C2211yescharacteristic, complemented
39#10622205C2214yescomplemented
40#10622203C2212yescomplemented
41#10622201C2212yescomplemented
42#10622199C2214yescomplemented
43C11×2−1+4—22no—
44C11×2−1+4—22no—
45C11×2−1+4—41no—
46C11×H(5,𝔽2)—22no—
47C11×H(5,𝔽2)—22no—
48C11×H(5,𝔽2)—42no—
49C11×H(5,𝔽2)—41no—
50C2×C11×#41—24no—
51C2×C11×#41—21no—
52C2×C11×#41—21no—
53C2×C11×#41—42no—
54C2×C11×#41—42no—
55C2×C11×#41D411yescharacteristic
56C22×D44—21no—
57C11×#137—24no—
58C11×#137—22no—
59C11×#137—24no—
60C11×#137—22no—
61C11×AGL(1,ℤ/8)—24no—
62C11×AGL(1,ℤ/8)—22no—
63C11×AGL(1,ℤ/8)—24no—
64C11×AGL(1,ℤ/8)—22no—
65C11×#135—24no—
66C11×#135—24no—
67C11×#135—22no—
68C11×#135—22no—
69C11×#135—21no—
70C11×#135—21no—
71C11×#135—21no—
72C11×#135—21no—
73C11×#135D412yes—
74C11×#135D412yes—
75C11×#135D412yes—
76C11×#135C2312yes—
77C2×C11×Q16—22no—
78C2×C11×Q16D411yescharacteristic
79C2×C11×Q16C2311yescharacteristic
80C2×C11×SD16—24no—
81C2×C11×SD16—21no—
82C2×C11×SD16—21no—
83C2×C11×D8—22no—
84C2×C11×D8D411yescharacteristic
85C2×C11×D8C2311yescharacteristic
86C11×#131—21no—
87C11×#131—21no—
88C11×#131D412yes—
89C11×#131C2312yes—
90C2×C11×M16—21no—
91C2×C11×M16D411yescharacteristic
92C22×C8×C11—21no—
93C22×C8×C11C2311yescharacteristic
94#67002—24no—
95#67002—42no—
96C2×#4325—41no—
97C2×#4323—41no—
98C2×#4318—21no—
99C2×D88—21no—
100C2×D88—21no—
101C2×D88—21no—
102C2×D88—21no—
103C2×D88—21no—
104C2×D88C2312yes—
105#66892—24no—
106#66892—24no—
107#66890—24no—
108#66890—24no—
109#66887—24no—
110#66883—21no—
111#66883C2312yes—
112#66876—21no—
113#66876—21no—
114#66876—21no—
115#66876C2312yes—
116C2×C11⋊C16—21no—
117C2×C11⋊C16C2312yes—
118#33541—111nomaximal, Sylow
119C23×D11—41no—
120C11×#41—21no—
121C11×#41—21no—
122C11×#41—24no—
123C11×#41—24no—
124C11×#41—22no—
125C11×#41—22no—
126C11×#41—22no—
127C11×#41—22no—
128C11×#41—44no—
129C11×#41—44no—
130C11×#41—44no—
131C11×#41—44no—
132C11×#41—42no—
133C11×#41—42no—
134C11×#41—42no—
135C11×#41—42no—
136C11×#41—41no—
137C11×#41—41no—
138C11×#41—41no—
139C11×#41—41no—
140C11×#41D812yes—
141C11×#41C2×D412yes—
142C2×Q8×C11—24no—
143C2×Q8×C11—21no—
144C2×Q8×C11—21no—
145C2×Q8×C11—42no—
146C2×Q8×C11—42no—
147C2×Q8×C11D811yescharacteristic
148C2×D4×C11—21no—
149C2×D4×C11—24no—
150C2×D4×C11—21no—
151C2×D4×C11—21no—
152C2×D4×C11—44no—
153C2×D4×C11—44no—
154C2×D4×C11—42no—
155C2×D4×C11—44no—
156C2×D4×C11—41no—
157C2×D4×C11—41no—
158C2×D4×C11—42no—
159C2×D4×C11D811yescharacteristic
160C22×C4×C11—21no—
161C22×C4×C11—44no—
162C22×C4×C11—41no—
163C22×C4×C11—41no—
164C22×C4×C11C2×D411yescharacteristic
165C2×D44—22no—
166C2×D44—41no—
167C2×D44—41no—
168C2×D44—41no—
169C2×D44—41no—
170C2×D44—41no—
171C11×Q16—24no—
172C11×Q16—21no—
173C11×Q16—21no—
174C11×Q16C2×D412yes—
175C11×Q16C2×D412yes—
176C11×SD16—24no—
177C11×SD16—24no—
178C11×SD16—22no—
179C11×SD16—22no—
180C11×SD16—22no—
181C11×SD16—22no—
182C11×D8—24no—
183C11×D8—21no—
184C11×D8—21no—
185C11×D8C2×D412yes—
186C11×D8C2×D412yes—
187C11×M16—22no—
188C11×M16—22no—
189C11×M16—21no—
190C11×M16C2×D412yes—
191C2×C8×C11—22no—
192C2×C8×C11—22no—
193C2×C8×C11—21no—
194C2×C8×C11C2×D411yescharacteristic
195C2×C8×C11C2×D411yescharacteristic
196C2×C8×C11C2×D411yescharacteristic
197C2×C8×C11C2×D411yescharacteristic
198C2×C8×C11C2×D411yescharacteristic
199C2×C8×C11C2411yescharacteristic
200#4325—44no—
201#4323—44no—
202#4318—22no—
203#4318—41no—
204C2×C11⋊C8—41no—
205D88—22no—
206D88—22no—
207D88—41no—
208D88—41no—
209C11⋊C16—22no—
210C11⋊C16—22no—
211#3584D1111yescharacteristic, complemented
212C2×#509—111no—
213C2×#509—111no—
214C2×#472—111no—
215#2193—114no—
216#2192—114no—
217#2148—111no—
218#2143—111no—
219#2117—111no—
220C23×C11—21no—
221C23×C11—44no—
222C23×C11—41no—
223C23×C11—41no—
224C23×C11AGL(1,ℤ/8)11yescharacteristic
225C22×D11—42no—
226C22×D11—81no—
227C22×D11—81no—
228C22×D11—162no—
229Q8×C11—21no—
230Q8×C11—22no—
231Q8×C11—22no—
232Q8×C11—44no—
233Q8×C11—41no—
234Q8×C11—41no—
235Q8×C11C2×D812yes—
236D4×C11—22no—
237D4×C11—22no—
238D4×C11—22no—
239D4×C11—22no—
240D4×C11—21no—
241D4×C11—44no—
242D4×C11—44no—
243D4×C11—44no—
244D4×C11—42no—
245D4×C11—42no—
246D4×C11—42no—
247D4×C11—42no—
248D4×C11—41no—
249D4×C11—41no—
250D4×C11C2×D812yes—
251C2×C4×C11—21no—
252C2×C4×C11—22no—
253C2×C4×C11—22no—
254C2×C4×C11—44no—
255C2×C4×C11—44no—
256C2×C4×C11—44no—
257C2×C4×C11—41no—
258C2×C4×C11—41no—
259C2×C4×C11—42no—
260C2×C4×C11—42no—
261C2×C4×C11—42no—
262C2×C4×C11—42no—
263C2×C4×C11#12011yescharacteristic
264C2×C4×C11#12011yescharacteristic
265C2×C4×C11C2×D811yescharacteristic
266C2×C4×C11C2×D811yescharacteristic
267C2×C4×C11AGL(1,ℤ/8)11yescharacteristic
268C2×C4×C11C22×D411yescharacteristic
269D44—42no—
270D44—42no—
271D44—81no—
272D44—81no—
273C8×C11—22no—
274C8×C11#12011yescharacteristic
275C8×C11#12011yescharacteristic
276C8×C11C22×D411yescharacteristic
277C8×C11C22×D411yescharacteristic
278C11⋊C8—42no—
279#585—21no—
280#578—21no—
281#578C2×D1112yescomplemented
282#577—22no—
283#577—22no—
284#576—21no—
285#576C2×D1112yescomplemented
286#575—22no—
287#575—21no—
288C2×AGL(1,ℤ/8)—221no—
289C2×#135—22no—
290C2×#135C2×D1111yescharacteristic
291C2×#135C2×D1111yescharacteristic
292C22×D8—111no—
293C2×#131C2×D1111yescharacteristic
294#509—114no—
295#509—222no—
296#509—222no—
297#509—222no—
298C2×SD32—112no—
299C2×SD32—221no—
300C2×D16—112no—
301C2×D16—221no—
302C2×M32—111no—
303C2×M32—111no—
304#472—114no—
305#472—222no—
306C2×#108—111no—
307C2×#100—221no—
308#362—112no—
309#361—112no—
310#359—114no—
311#350—114no—
312C22×C11—22no—
313C22×C11—41no—
314C22×C11—42no—
315C22×C11—42no—
316C22×C11—81no—
317C22×C11—161no—
318C22×C11—161no—
319C22×C11—164no—
320C22×C11#44711yescharacteristic
321C22×C11#44711yescharacteristic
322C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
323C2×D11—82no—
324C2×D11—162no—
325C2×D11—162no—
326C2×D11—162no—
327C4×C11—22no—
328C4×C11—42no—
329C4×C11—42no—
330C4×C11#44711yescharacteristic
331C4×C11#44711yescharacteristic
332C4×C11C2×#12011yescharacteristic
333C4×C11C22×D811yescharacteristic
3342−1+4—22no—
3352−1+4—22no—
3362−1+4—41no—
337H(5,𝔽2)—22no—
338H(5,𝔽2)—22no—
339H(5,𝔽2)—42no—
340H(5,𝔽2)—41no—
341C2×#41—24no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—42no—
345C2×#41—42no—
346C2×#41C22⋊D1111yescharacteristic
347C22×D4—221no—
348#137—22no—
349#137—24no—
350#137—24no—
351#137—22no—
352AGL(1,ℤ/8)—22no—
353AGL(1,ℤ/8)—24no—
354AGL(1,ℤ/8)—24no—
355AGL(1,ℤ/8)—22no—
356AGL(1,ℤ/8)—224no—
357AGL(1,ℤ/8)—442no—
358#135—24no—
359#135—24no—
360#135—22no—
361#135—22no—
362#135—21no—
363#135—21no—
364#135—21no—
365#135—21no—
366#135C22⋊D1112yes—
367#135C22⋊D1112yes—
368#135C22⋊D1112yes—
369#135C22×D1112yes—
370C2×Q16—22no—
371C2×Q16C22⋊D1111yescharacteristic
372C2×Q16C22×D1111yescharacteristic
373C2×SD16—24no—
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—441no—
377C2×D8—22no—
378C2×D8—112no—
379C2×D8—221no—
380C2×D8—221no—
381C2×D8—221no—
382C2×D8—221no—
383C2×D8—221no—
384C2×D8—441no—
385C2×D8C22⋊D1111yescharacteristic
386C2×D8C22×D1111yescharacteristic
387#131—21no—
388#131—21no—
389#131C22⋊D1112yes—
390#131C22×D1112yes—
391C2×M16—21no—
392C2×M16—221no—
393C2×M16C22⋊D1111yescharacteristic
394C22×C8—21no—
395C22×C8C22×D1111yescharacteristic
396SD32—224no—
397SD32—224no—
398D16—224no—
399D16—224no—
400M32—112no—
401M32—221no—
402M32—221no—
403M32—221no—
404C2×C16—112no—
405C2×C16—221no—
406#108—112no—
407#108—221no—
408#100—224no—
409C2×C11—21no—
410C2×C11—21no—
411C2×C11—21no—
412C2×C11—42no—
413C2×C11—82no—
414C2×C11—82no—
415C2×C11C2×#44711yescharacteristic
416D11—162no—
417C24—441no—
418#41—21no—
419#41—21no—
420#41—24no—
421#41—24no—
422#41—22no—
423#41—22no—
424#41—22no—
425#41—22no—
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—41no—
435#41—41no—
436#41—41no—
437#41—41no—
438#41#432312yes—
439#41C2×C22⋊D1112yes—
440C2×Q8—24no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8#432311yescharacteristic
446C2×D4—21no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—42no—
457C2×D4—222no—
458C2×D4—441no—
459C2×D4—441no—
460C2×D4—441no—
461C2×D4—441no—
462C2×D4—441no—
463C2×D4#432311yescharacteristic
464C22×C4—21no—
465C22×C4—44no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4C2×C22⋊D1111yescharacteristic
469Q16—24no—
470Q16—21no—
471Q16—21no—
472Q16C2×C22⋊D1112yes—
473Q16C2×C22⋊D1112yes—
474SD16—24no—
475SD16—24no—
476SD16—22no—
477SD16—22no—
478SD16—22no—
479SD16—22no—
480SD16—444no—
481D8—24no—
482D8—21no—
483D8—21no—
484D8—222no—
485D8—222no—
486D8—444no—
487D8—441no—
488D8—441no—
489D8C2×C22⋊D1112yes—
490D8C2×C22⋊D1112yes—
491M16—21no—
492M16—22no—
493M16—22no—
494M16—222no—
495M16—441no—
496M16C2×C22⋊D1112yes—
497C2×C8—21no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—441no—
501C2×C8C2×C22⋊D1111yescharacteristic
502C2×C8C2×C22⋊D1111yescharacteristic
503C2×C8C2×C22⋊D1111yescharacteristic
504C2×C8C2×C22⋊D1111yescharacteristic
505C2×C8C2×C22⋊D1111yescharacteristic
506C2×C8C23×D1111yescharacteristic
507C16—222no—
508C16—222no—
509C11#3354111yescharacteristic, complemented, Sylow
510C23—21no—
511C23—44no—
512C23—41no—
513C23—41no—
514C23—442no—
515C23—881no—
516C23—881no—
517C23—1762no—
518C23#6698511yescharacteristic
519Q8—21no—
520Q8—22no—
521Q8—22no—
522Q8—44no—
523Q8—41no—
524Q8—41no—
525Q8C2×#432312yes—
526D4—22no—
527D4—22no—
528D4—21no—
529D4—22no—
530D4—22no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—41no—
539D4—41no—
540D4—442no—
541D4—442no—
542D4—881no—
543D4—881no—
544D4C2×#432312yes—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4C2×#432311yescharacteristic
558C2×C4C2×#432311yescharacteristic
559C2×C4#6698511yescharacteristic
560C2×C4#6700611yescharacteristic
561C2×C4#6700611yescharacteristic
562C2×C4C22×C22⋊D1111yescharacteristic
563C8—22no—
564C8—442no—
565C8#6700611yescharacteristic
566C8#6700611yescharacteristic
567C8C22×C22⋊D1111yescharacteristic
568C8C22×C22⋊D1111yescharacteristic
569C22—22no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—161no—
575C22—161no—
576C22—164no—
577C22—882no—
578C22—1762no—
579C22—1762no—
580C22—1762no—
581C22#1062283011yescharacteristic
582C22#1062283011yescharacteristic
583C22C2×#6698511yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4#1062283011yescharacteristic
588C4#1062283011yescharacteristic
589C4C22×#432311yescharacteristic
590C4C2×#6700611yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—42no—
595C2—82no—
596C2—82no—
597C2—1762no—
598C2C2×#1062283011yescharacteristic
subgroupG/H
centerC2C2×#10622830
commutator, cosocleC2×C8×C11C24
FrattiniC2×C8C23×D11
FittingC11×#3584C2
socleC2×C11C2×#447
hypercenter#3584D11
hypocenterC11#33541
LCSG ▷ C2×C8×C11 ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. C2×#10622806C2semidirect
2. C2×#10622806C2semidirect
3. C2×#10622755C2semidirect
4. #13070580C2semidirect
5. #13070573C2semidirect
6. #13070365C2semidirect
7. #13070344C2semidirect
8. #13070244C2semidirect
9. C11×#3584C2semidirect
12. C11×#578C22semidirect
16. C11×#576C22semidirect
24. C22×D88C22semidirect
28. #10622806C22semidirect
30. C2×#66892C22semidirect
32. C2×#66890C22semidirect
34. #10622755C22semidirect
36. C2×#66883C22semidirect
37. C2×#66876C22semidirect
38. C2×#66876C22semidirect
39. #10622205C22semidirect
40. #10622203C22semidirect
41. #10622201C22semidirect
42. #10622199C22semidirect
211. #3584D11semidirect
281. #578C2×D11semidirect
285. #576C2×D11semidirect
509. C11#33541semidirect

Maximal subgroups

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

subgroupindex
C2×#106228062
C2×#106227552
#130705802
#130705732
#130703652
#130703442
#130702442
C11×#35842
#3354111