Finite Group Databasebeta

← #14424206

Subgroups — #14424206

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92100C211yesmaximal, characteristic, complemented
2C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11756564C211yesmaximal, characteristic, complemented
4C22×#6715C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11751273C214yesmaximal, complemented
7#11751257C214yesmaximal, complemented
8#11751232C211yesmaximal, characteristic, complemented
9#11751230C211yesmaximal, characteristic, complemented
10#10615678—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic, complemented
13#92100—24no—
14#92100—24no—
15C2×#6821C2212yescomplemented
16C2×D4×D15—22no—
17C2×D4×D15C2212yescomplemented
18C2×#6817—21no—
19C2×#6817C2211yescharacteristic
20C22×D60—21no—
21C22×D60C2211yescharacteristic
22C3×C5×AGL(1,ℤ/8)—24no—
23C2×C3×C5×SD16C2212yescomplemented
24C2×C3×C5×D8C2212yescomplemented
25C2×C3×C5×M16C2211yescharacteristic, complemented
26#91842—22no—
27#91829—21no—
28#91827C2214yescomplemented
29#91825—21no—
30C2×#6715—24no—
31C2×#6715C2214yescomplemented
32#91815C2211yescharacteristic, complemented
33C22×C3⋊#168C2211yescharacteristic, complemented
34#91119C2212yescomplemented
35#91116C2212yescomplemented
36#91105C2214yescomplemented
37#91104C2214yescomplemented
38#83947—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
41#66139—31no—
42C22×#686—31no—
43#65522—31no—
44#65461—34no—
45#65445—34no—
46#65420—31no—
47#65418—31no—
48C24×C3×C5—21no—
49C23×D15—21no—
50C23×D15—22no—
51C3×C5×#41—22no—
52C3×C5×#41D414yescomplemented
53C2×C3×C5×Q8C2311yescharacteristic
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4C2312yes—
59C2×C3×C5×D4C2311yescharacteristic
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5C2311yescharacteristic
62C2×#1220—24no—
63C2×#1220—22no—
64C2×#1220D412yes—
65#6821—24no—
66#6821—24no—
67D4×D15—216no—
68D4×D15—24no—
69D4×D15—24no—
70#6817—28no—
71#6817—22no—
72#6817—22no—
73C2×D60—28no—
74C2×D60—22no—
75C2×D60—21no—
76C2×D60—22no—
77C2×D60D411yescharacteristic
78C2×D60D414yescomplemented
79C2×D60C2312yes—
80C2×C4×D15—22no—
81C2×C4×D15C2312yes—
82C2×Dic30D411yescharacteristic
83C3×C5×SD16—24no—
84C3×C5×D8—24no—
85C3×C5×M16—22no—
86C2×C3×C5×C8C2312yes—
87#6719—21no—
88#6719—24no—
89#6715—28no—
90C4⋊Dic15C2312yes—
91C4×Dic15—21no—
92C2×C3⋊#168—22no—
93C2×C3⋊#168C2312yes—
94C2×#1009—51no—
95C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
96#5805—51no—
97C22×#213—51no—
98#5233—51no—
99#5172—54no—
100#5156—54no—
101#5131—51no—
102#5129—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×D4C2×D311yescharacteristic, complemented
105#4149—64no—
106#4149—64no—
107C2×#713—32no—
108C2×D4×D5—32no—
109C2×D4×D5—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×D20—31no—
113C22×D20—61no—
114C5×AGL(1,ℤ/8)—24no—
115C2×C5×SD16C2×D312yescomplemented
116C2×C5×D8C2×D312yescomplemented
117C2×C5×M16C2×D311yescharacteristic, complemented
118#4099—62no—
119#4086—61no—
120#4084—34no—
121#4082—61no—
122C2×#686—34no—
123C2×#686—64no—
124#4072—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953—34no—
129#3952—34no—
130#3008—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5—22no—
133C23×C3×C5—42no—
134C23×C3×C5—82no—
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—22no—
137C22×D15—28no—
138C22×D15—24no—
139C22×D15—84no—
140C22×D15—82no—
141C22×D15C2×D412yes—
142C3×C5×Q8—21no—
143C3×C5×Q8C2×D412yes—
144C3×C5×D4—24no—
145C3×C5×D4—21no—
146C3×C5×D4—22no—
147C3×C5×D4—22no—
148C3×C5×D4—42no—
149C3×C5×D4C2×D412yes—
150C2×C3×C4×C5—21no—
151C2×C3×C4×C5—24no—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D412yes—
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5C2411yescharacteristic
156#1220—28no—
157#1220—24no—
158#1220—84no—
159C2×Dic15—22no—
160C2×Dic15C2×D412yes—
161D60—28no—
162D60—22no—
163D60—24no—
164D60—22no—
165D60C2×D414yes—
166C4×D15—28no—
167C4×D15—22no—
168C4×D15—22no—
169Dic30—22no—
170C3×C5×C8—22no—
171C3⋊#168—22no—
172C2×C3×#41C2×D511yescharacteristic
173C22×C3×D4C2×D511yescharacteristic, complemented
174#1009—104no—
175#1009—104no—
176C2×#239—52no—
177C2×D3×D4—52no—
178C2×D3×D4—102no—
179C2×#235—51no—
180C2×#235—101no—
181C22×D12—51no—
182C22×D12—101no—
183C3×AGL(1,ℤ/8)—24no—
184C2×C3×SD16C2×D512yescomplemented
185C2×C3×D8C2×D512yescomplemented
186C2×C3×M16C2×D511yescharacteristic, complemented
187#953—102no—
188#940—101no—
189#938—54no—
190#936—101no—
191C2×#213—54no—
192C2×#213—104no—
193#926—51no—
194C22×C3⋊C8—51no—
195#835—52no—
196#832—52no—
197#821—54no—
198(C2×D3)⋊C8—54no—
199C24×C5—21no—
200C23×D5—62no—
201C23×D5—61no—
202C5×#41—22no—
203C5×#41#6714yescomplemented
204C2×C5×Q8C22×D311yescharacteristic
205C2×C5×D4—21no—
206C2×C5×D4—22no—
207C2×C5×D4—21no—
208C2×C5×D4—22no—
209C2×C5×D4C22×D312yes—
210C2×C5×D4C22×D311yescharacteristic
211C22×C4×C5—21no—
212C22×C4×C5C22×D311yescharacteristic
213C2×C22⋊D5—32no—
214C2×C22⋊D5—62no—
215C2×C22⋊D5—64no—
216#713—64no—
217#713—64no—
218D4×D5—64no—
219D4×D5—616no—
220D4×D5—64no—
221#709—68no—
222#709—62no—
223#709—62no—
224C2×D20—34no—
225C2×D20—32no—
226C2×D20—31no—
227C2×D20—62no—
228C2×D20—68no—
229C2×D20—61no—
230C2×D20—62no—
231C2×C4×D5—32no—
232C2×C4×D5—62no—
233C2×Dic10—31no—
234C5×SD16—24no—
235C5×D8—24no—
236C5×M16—22no—
237C2×C5×C8C22×D312yes—
238C22⋊Dic5—64no—
239C22⋊Dic5—61no—
240#686—68no—
241C4⋊Dic5—32no—
242C4×Dic5—61no—
243C2×#168—32no—
244C2×#168—62no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—151no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5—22no—
256C22×C3×C5—42no—
257C22×C3×C5—42no—
258C22×C3×C5—44no—
259C22×C3×C5—84no—
260C22×C3×C5#12012yes—
261C22×C3×C5C22×D411yescharacteristic
262C2×D15—22no—
263C2×D15—24no—
264C2×D15—24no—
265C2×D15—44no—
266C2×D15—84no—
267C2×D15—88no—
268C3×C4×C5—22no—
269C3×C4×C5#12012yes—
270C3×C4×C5C22×D411yescharacteristic
271C3×C4×C5C22×D411yescharacteristic
272Dic15—22no—
273Dic15—42no—
274C24×C3—21no—
275C23×D3—102no—
276C23×D3—101no—
277C3×#41—22no—
278C3×#41C22⋊D514yescomplemented
279C2×C3×Q8C22×D511yescharacteristic
280C2×C3×D4—21no—
281C2×C3×D4—22no—
282C2×C3×D4—21no—
283C2×C3×D4—22no—
284C2×C3×D4C22×D512yes—
285C2×C3×D4C22×D511yescharacteristic
286C22×C3×C4—21no—
287C22×C3×C4C22×D511yescharacteristic
288C2×#67—52no—
289C2×#67—102no—
290C2×#67—104no—
291#239—104no—
292#239—104no—
293D3×D4—104no—
294D3×D4—1016no—
295D3×D4—104no—
296#235—108no—
297#235—102no—
298#235—102no—
299C2×D12—54no—
300C2×D12—52no—
301C2×D12—51no—
302C2×D12—102no—
303C2×D12—108no—
304C2×D12—101no—
305C2×D12—102no—
306C2×C4×D3—52no—
307C2×C4×D3—102no—
308C2×Dic6—51no—
309C3×SD16—24no—
310C3×D8—24no—
311C3×M16—22no—
312C2×C3×C8C22×D512yes—
313#217—104no—
314#217—101no—
315#213—108no—
316C4⋊Dic3—52no—
317C4×Dic3—101no—
318C2×C3⋊C8—52no—
319C2×C3⋊C8—102no—
320C23×C5—21no—
321C23×C5—22no—
322C23×C5—42no—
323C23×C5—82no—
324C23×C5D3×D411yescharacteristic
325C22×D5—32no—
326C22×D5—62no—
327C22×D5—68no—
328C22×D5—64no—
329C22×D5—244no—
330C22×D5—242no—
331C5×Q8—21no—
332C5×Q8C2×#6712yes—
333C5×D4—24no—
334C5×D4—21no—
335C5×D4—22no—
336C5×D4—22no—
337C5×D4—42no—
338C5×D4C2×#6712yes—
339C2×C4×C5—21no—
340C2×C4×C5—24no—
341C2×C4×C5D3×D411yescharacteristic
342C2×C4×C5D3×D412yes—
343C2×C4×C5C2×#6712yes—
344C2×C4×C5C23×D311yescharacteristic
345C22⋊D5—68no—
346C22⋊D5—64no—
347C22⋊D5—244no—
348C2×Dic5—32no—
349C2×Dic5—62no—
350D20—34no—
351D20—68no—
352D20—64no—
353D20—62no—
354D20—62no—
355C4×D5—68no—
356C4×D5—62no—
357C4×D5—62no—
358Dic10—62no—
359C5×C8—22no—
360#168—62no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—151no—
364C2×#41—152no—
365C2×#41—301no—
366C2×#41C2×D1511yescharacteristic
367C22×D4—152no—
368C22×D4—151no—
369C22×D4—301no—
370C22×D4—302no—
371C22×D4C2×D1511yescharacteristic, complemented
372AGL(1,ℤ/8)—24no—
373C2×SD16C2×D1512yescomplemented
374C2×D8—154no—
375C2×D8—304no—
376C2×D8C2×D1512yescomplemented
377C2×M16C2×D1511yescharacteristic, complemented
378C22×C8—151no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—151no—
384#103—152no—
385#102—152no—
386#102—154no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5—42no—
390C2×C3×C5—82no—
391C2×C3×C5C2×#12011yescharacteristic
392C2×C3×C5#57612yes—
393D15—42no—
394D15—44no—
395C23×C3—21no—
396C23×C3—22no—
397C23×C3—42no—
398C23×C3—82no—
399C23×C3D4×D511yescharacteristic
400C22×D3—52no—
401C22×D3—102no—
402C22×D3—108no—
403C22×D3—104no—
404C22×D3—404no—
405C22×D3—402no—
406C3×Q8—21no—
407C3×Q8C2×C22⋊D512yes—
408C3×D4—24no—
409C3×D4—21no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—42no—
413C3×D4C2×C22⋊D512yes—
414C2×C3×C4—21no—
415C2×C3×C4—24no—
416C2×C3×C4D4×D512yes—
417C2×C3×C4D4×D511yescharacteristic
418C2×C3×C4C2×C22⋊D512yes—
419C2×C3×C4C23×D511yescharacteristic
420#67—108no—
421#67—104no—
422#67—404no—
423C2×Dic3—52no—
424C2×Dic3—102no—
425D12—54no—
426D12—108no—
427D12—102no—
428D12—104no—
429D12—102no—
430C4×D3—108no—
431C4×D3—102no—
432C4×D3—102no—
433Dic6—102no—
434C3×C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—22no—
438C22×C5—42no—
439C22×C5—44no—
440C22×C5—42no—
441C22×C5—84no—
442C22×C5#93712yes—
443C22×C5C2×D3×D411yescharacteristic
444C2×D5—64no—
445C2×D5—64no—
446C2×D5—62no—
447C2×D5—124no—
448C2×D5—244no—
449C2×D5—248no—
450C4×C5—22no—
451C4×C5#93712yes—
452C4×C5C2×D3×D411yescharacteristic
453C4×C5C22×#6711yescharacteristic
454Dic5—62no—
455Dic5—122no—
456C24—21no—
457C24—302no—
458C24—301no—
459#41—22no—
460#41—308no—
461#41—304no—
462#41—304no—
463#41—302no—
464#41—302no—
465#41#122014yescomplemented
466C2×Q8—151no—
467C2×Q8C22×D1511yescharacteristic
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—152no—
473C2×D4—154no—
474C2×D4—151no—
475C2×D4—152no—
476C2×D4—302no—
477C2×D4—308no—
478C2×D4—302no—
479C2×D4—304no—
480C2×D4—3016no—
481C2×D4—304no—
482C2×D4—304no—
483C2×D4—301no—
484C2×D4—302no—
485C2×D4C22×D1512yes—
486C2×D4C22×D1511yescharacteristic
487C22×C4—21no—
488C22×C4—152no—
489C22×C4—302no—
490C22×C4C22×D1511yescharacteristic
491SD16—24no—
492D8—24no—
493D8—308no—
494M16—22no—
495C2×C8—152no—
496C2×C8—302no—
497C2×C8C22×D1512yes—
498C4⋊C4—152no—
499C22⋊C4—304no—
500C22⋊C4—301no—
501C42—301no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—42no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—84no—
509C22×C3#408312yes—
510C22×C3C2×D4×D511yescharacteristic
511C2×D3—104no—
512C2×D3—104no—
513C2×D3—102no—
514C2×D3—204no—
515C2×D3—404no—
516C2×D3—408no—
517C3×C4—22no—
518C3×C4#408312yes—
519C3×C4C2×D4×D511yescharacteristic
520C3×C4C22×C22⋊D511yescharacteristic
521Dic3—102no—
522Dic3—202no—
523C2×C5—22no—
524C2×C5—42no—
525C2×C5—82no—
526C2×C5#577412yes—
527C2×C5C2×#93711yescharacteristic
528D5—122no—
529D5—124no—
530C23—21no—
531C23—22no—
532C23—42no—
533C23—82no—
534C23—152no—
535C23—308no—
536C23—302no—
537C23—304no—
538C23—1202no—
539C23—1204no—
540C23D4×D1511yescharacteristic
541Q8—21no—
542Q8—302no—
543Q8C2×#122012yes—
544D4—24no—
545D4—21no—
546D4—22no—
547D4—22no—
548D4—42no—
549D4—154no—
550D4—308no—
551D4—308no—
552D4—304no—
553D4—304no—
554D4—302no—
555D4—302no—
556D4—1204no—
557D4C2×#122012yes—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—152no—
561C2×C4—308no—
562C2×C4—302no—
563C2×C4—302no—
564C2×C4—302no—
565C2×C4D4×D1511yescharacteristic
566C2×C4D4×D1512yes—
567C2×C4C2×#122012yes—
568C2×C4C23×D1511yescharacteristic
569C8—22no—
570C8—302no—
571C2×C3—22no—
572C2×C3—42no—
573C2×C3—82no—
574C2×C3#6610812yes—
575C2×C3C2×#408311yescharacteristic
576D3—202no—
577D3—204no—
578C5#8394711yescharacteristic, complemented, Sylow
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—84no—
585C22—304no—
586C22—302no—
587C22—304no—
588C22—604no—
589C22—1204no—
590C22—1208no—
591C22#9182612yes—
592C22C2×D4×D1511yescharacteristic
593C4—22no—
594C4—302no—
595C4—602no—
596C4#9182612yes—
597C4C2×D4×D1511yescharacteristic
598C4C22×#122011yescharacteristic
599C3#1061567811yescharacteristic, complemented, Sylow
600C2—22no—
601C2—42no—
602C2—82no—
603C2—602no—
604C2—604no—
605C2#1175653312yes—
606C2C2×#9182611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5C22×D4
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#92100C2semidirect
2. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
3. #11756564C2semidirect
4. C22×#6715C2semidirect
5. #11751334C2semidirect
6. #11751273C2semidirect
7. #11751257C2semidirect
8. #11751232C2semidirect
9. #11751230C2semidirect
12. C22×C3×C5×D4C22semidirect
15. C2×#6821C22semidirect
17. C2×D4×D15C22semidirect
23. C2×C3×C5×SD16C22semidirect
24. C2×C3×C5×D8C22semidirect
25. C2×C3×C5×M16C22semidirect
28. #91827C22semidirect
31. C2×#6715C22semidirect
32. #91815C22semidirect
33. C22×C3⋊#168C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #91105C22semidirect
37. #91104C22semidirect
40. C2×C5×AGL(1,ℤ/8)D3semidirect
52. C3×C5×#41D4semidirect
78. C2×D60D4semidirect
95. C2×C3×AGL(1,ℤ/8)D5semidirect
104. C22×C5×D4C2×D3semidirect
115. C2×C5×SD16C2×D3semidirect
116. C2×C5×D8C2×D3semidirect
117. C2×C5×M16C2×D3semidirect
173. C22×C3×D4C2×D5semidirect
184. C2×C3×SD16C2×D5semidirect
185. C2×C3×D8C2×D5semidirect
186. C2×C3×M16C2×D5semidirect
203. C5×#41#67semidirect
246. C2×AGL(1,ℤ/8)D15semidirect
278. C3×#41C22⋊D5semidirect
371. C22×D4C2×D15semidirect
373. C2×SD16C2×D15semidirect
376. C2×D8C2×D15semidirect
377. C2×M16C2×D15semidirect
465. #41#1220semidirect
502. C3×C5#3008semidirect
578. C5#83947semidirect
599. C3#10615678semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C2×C3×C5×AGL(1,ℤ/8)2
#117565642
C22×#67152
#117513342
#117512732
#117512572
#117512322
#117512302
#106156783
#839475