Finite Group Databasebeta

← #14424268

Subgroups — #14424268

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×#92018C211yesmaximal, characteristic, complemented
2C22×C5×D24C211yesmaximal, characteristic, complemented
3#11754606C211yesmaximal, characteristic, complemented
4C2×#91299C211yesmaximal, characteristic, complemented
5#11751211C211yesmaximal, characteristic, complemented
6C3×#65402C211yesmaximal, characteristic, complemented
7#11748386C211yesmaximal, characteristic, complemented
8#11748159C214yesmaximal, complemented
9#11748119C214yesmaximal, complemented
10#10615564—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic
13C22×C5×D12C2211yescharacteristic, complemented
14C2×C3×#709C2211yescharacteristic
15C2×D3×C22⋊D5—22no—
16#92018—24no—
17#92018—24no—
18C2×#6777C2212yescomplemented
19C2×D5×D12—21no—
20C2×#6772—21no—
21C2×#6768C2212yescomplemented
22C22×C3×C5×C8C2211yescharacteristic, complemented
23C2×#6710C2211yescharacteristic, complemented
24C2×C5×D24—24no—
25C2×C5×D24C2214yescomplemented
26C3×#4072C2211yescharacteristic, complemented
27#91570—22no—
28#91466C2214yescomplemented
29#91451—21no—
30#91373—21no—
31C2×#6656C2212yescomplemented
32#91299—24no—
33C2×#6652C2212yescomplemented
34#91104C2214yescomplemented
35C3×#3953C2212yescomplemented
36C3×#3948C2212yescomplemented
37#90966C2214yescomplemented
38#78834—51nomaximal, Hall
39C2×#4144—31no—
40C22×C5×D8—31no—
41#66139—31no—
42C2×#4078—31no—
43#65504—31no—
44#65447—34no—
45#65445—34no—
46#65402D311yescharacteristic, complemented
47#65399—31no—
48C23×C5×D3—21no—
49C23×C5×D3—21no—
50C22×D3×D5—22no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×#1220D412yes—
53#6817—22no—
54#6817—22no—
55C2×D60D411yescharacteristic
56C2×C4×D15C2312yes—
57C2×Dic30D411yescharacteristic
58C2×C5×D12—21no—
59C2×C5×D12—22no—
60C2×C5×D12—22no—
61C2×C5×D12—22no—
62C2×C5×D12—21no—
63C2×C5×D12D414yescomplemented
64C2×C5×D12C2312yes—
65C2×C5×D12C2312yes—
66C2×C3×C22⋊D5—21no—
67C3×#709—22no—
68C3×#709D414yescomplemented
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72D3×C22⋊D5—216no—
73C2×#1203—22no—
74C2×#1203—24no—
75C2×#1201—22no—
76C2×D3×Dic5—22no—
77#6777—24no—
78#6777—24no—
79D5×D12—28no—
80#6772—28no—
81#6768—24no—
82#6768—24no—
83C2×C3×C5×C8—22no—
84C2×C3×C5×C8C2312yes—
85#6710—22no—
86C2×C3⋊#168C2312yes—
87C5×D24—28no—
88C3×C22⋊Dic5—21no—
89C3×C4⋊Dic5C2312yes—
90C3×C4×Dic5—21no—
91(C2×D3)⋊Dic5—24no—
92#6656—24no—
93#6652—24no—
94C2×#1009—51no—
95C2×#949—51no—
96C22×D24D511yescharacteristic, complemented
97#5525—51no—
98C3×#416—51no—
99#5110—51no—
100#5001—51no—
101#4810—54no—
102#4773—54no—
103C22×C5×D4—31no—
104C22×C5×D4—31no—
105C22×C22⋊D5—62no—
106#4144—64no—
107#4144—64no—
108C2×#711—32no—
109C2×#711—61no—
110C2×D4×D5—32no—
111C2×D4×D5—61no—
112C2×#709—31no—
113C2×#709C2×D311yescharacteristic
114C2×C5×D8—34no—
115C2×C5×D8—64no—
116C22×C5×C8C2×D311yescharacteristic, complemented
117#4099—62no—
118#4086—61no—
119#4084—34no—
120#4082—61no—
121C2×#687—32no—
122#4078—64no—
123C2×#686—32no—
124#4072C2×D311yescharacteristic, complemented
125C2×#681—31no—
126#3964—34no—
127#3953C2×D312yescomplemented
128#3952—34no—
129#3948C2×D312yescomplemented
130#3008—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—22no—
134C22×C5×D3—24no—
135C22×C5×D3—22no—
136C22×C5×D3—42no—
137C22×C5×D3—82no—
138C22×C5×D3—82no—
139C22×C3×D5—21no—
140C2×D3×D5—28no—
141C2×D3×D5—84no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2411yescharacteristic
146#1220—24no—
147C2×Dic15C2×D412yes—
148D60—22no—
149C4×D15—22no—
150C4×D15—22no—
151Dic30—22no—
152C5×D12—24no—
153C5×D12—22no—
154C5×D12—22no—
155C5×D12—22no—
156C5×D12—42no—
157C5×D12C2×D414yes—
158C3×C22⋊D5—24no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—22no—
161C3×D20—21no—
162C3×D20C2×D412yes—
163C3×C4×D5—24no—
164C3×Dic10—21no—
165C3×Dic10C2×D412yes—
166#1203—28no—
167#1203—84no—
168#1201—28no—
169D3×Dic5—28no—
170C3×C5×C8—22no—
171C3⋊#168—22no—
172C2×C3×#41—51no—
173#1009—104no—
174#1009—104no—
175C2×#239—52no—
176C2×D3×D4—52no—
177C2×D3×D4—102no—
178C2×#235—51no—
179C2×#235—101no—
180C22×D12—101no—
181C22×D12C2×D511yescharacteristic
182C22×D12C2×D511yescharacteristic, complemented
183C22×C3×C8C2×D511yescharacteristic, complemented
184C3×#117—51no—
185#949—104no—
186C2×#215—52no—
187C2×#213—52no—
188C2×#208—51no—
189C2×D24—24no—
190C2×D24C2×D514yescomplemented
191#895—54no—
192#882—102no—
193#875—101no—
194#874—101no—
195C3×#103—52no—
196C3×#102—52no—
197(C2×D3)⋊C8—54no—
198#809—54no—
199C24×C5—61no—
200C24×C5—61no—
201C23×D5—62no—
202C2×C5×D4—34no—
203C2×C5×D4—32no—
204C2×C5×D4—32no—
205C2×C5×D4—62no—
206C2×C5×D4—62no—
207C2×C5×D4—61no—
208C2×C5×D4—62no—
209C2×C5×D4—61no—
210C22×C4×C5C22×D311yescharacteristic
211C2×C22⋊D5—21no—
212C2×C22⋊D5—32no—
213C2×C22⋊D5—62no—
214C2×C22⋊D5—64no—
215C2×C22⋊D5—62no—
216C2×C22⋊D5—616no—
217C22×Dic5—62no—
218#711—64no—
219#711—68no—
220#711—64no—
221D4×D5—68no—
222D4×D5—64no—
223D4×D5—64no—
224#709—22no—
225#709—62no—
226#709—62no—
227#709D1214yescomplemented
228C2×D20—31no—
229C2×D20C22×D311yescharacteristic
230C2×C4×D5—21no—
231C2×C4×D5—32no—
232C2×Dic10—31no—
233C2×Dic10C22×D311yescharacteristic
234C5×D8—68no—
235C2×C5×C8—22no—
236C2×C5×C8C22×D312yes—
237C22⋊Dic5—21no—
238C22⋊Dic5—64no—
239#687—64no—
240#686—64no—
241C4⋊Dic5C22×D312yes—
242C4×Dic5—21no—
243#681—62no—
244C2×#168—32no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—31no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—151no—
252#416—51no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—24no—
259C2×C5×D3—24no—
260C2×C5×D3—44no—
261C2×C5×D3—42no—
262C2×C5×D3—44no—
263C2×C5×D3—84no—
264C2×C5×D3—84no—
265C2×C3×D5—22no—
266C2×C3×D5—42no—
267D3×D5—88no—
268C3×C4×C5#12012yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271Dic15—22no—
272C3×Dic5—22no—
273C3×Dic5—42no—
274C23×D3—21no—
275C23×D3—21no—
276C23×D3—102no—
277C3×#41—54no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×D4—51no—
281C2×C3×D4—101no—
282C22×C3×C4—101no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—52no—
285C2×#67—102no—
286#239—104no—
287#239—104no—
288D3×D4—104no—
289D3×D4—1016no—
290D3×D4—104no—
291#235—108no—
292#235—102no—
293#235—102no—
294C2×D12—21no—
295C2×D12—22no—
296C2×D12—22no—
297C2×D12—22no—
298C2×D12—21no—
299C2×D12—51no—
300C2×D12—108no—
301C2×D12—102no—
302C2×D12—104no—
303C2×D12C22⋊D514yescomplemented
304C2×D12C22×D512yes—
305C2×D12C22×D512yes—
306C2×C4×D3—52no—
307C2×C4×D3—102no—
308C2×Dic6—51no—
309C2×C3×C8—22no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4—52no—
312C3×C22⋊C4—101no—
313C3×C42—101no—
314#215—104no—
315#213—104no—
316(C2×D3)⋊C4—104no—
317#208—102no—
318C2×C3⋊C8—52no—
319D24—28no—
320C23×C5—62no—
321C23×C5—64no—
322C23×C5—62no—
323C23×C5—122no—
324C23×C5—242no—
325C23×C5—242no—
326C23×C5D3×D411yescharacteristic
327C22×D5—21no—
328C22×D5—32no—
329C22×D5—68no—
330C22×D5—244no—
331C5×D4—34no—
332C5×D4—64no—
333C5×D4—62no—
334C5×D4—62no—
335C5×D4—62no—
336C5×D4—122no—
337C2×C4×C5C2×D1212yes—
338C2×C4×C5D3×D411yescharacteristic
339C2×C4×C5D3×D412yes—
340C2×C4×C5C23×D311yescharacteristic
341C22⋊D5—24no—
342C22⋊D5—68no—
343C22⋊D5—68no—
344C22⋊D5—64no—
345C22⋊D5—244no—
346C2×Dic5—21no—
347C2×Dic5—22no—
348C2×Dic5—32no—
349C2×Dic5—68no—
350D20—21no—
351D20—62no—
352D20C2×D1212yes—
353C4×D5—24no—
354C4×D5—62no—
355C4×D5—62no—
356Dic10—21no—
357Dic10—62no—
358Dic10C2×D1212yes—
359C5×C8—22no—
360#168—62no—
361H(5,𝔽2)—304no—
362H(5,𝔽2)—304no—
363C2×#41—51no—
364C2×#41—151no—
365C2×#41—152no—
366C2×#41—301no—
367C22×D4—31no—
368C22×D4—31no—
369C22×D4—152no—
370C22×D4—301no—
371C22×D4—302no—
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—34no—
375C2×D8—64no—
376C2×D8—152no—
377C2×M16—151no—
378C22×C8D3×D511yescharacteristic, complemented
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—51no—
384#103—52no—
385#102—52no—
386#102—154no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5C2×#12011yescharacteristic
390C2×C3×C5#57612yes—
391D15—42no—
392C3×D5—42no—
393C5×D3—44no—
394C5×D3—82no—
395C23×C3—101no—
396C23×C3D4×D511yescharacteristic
397C22×D3—22no—
398C22×D3—24no—
399C22×D3—22no—
400C22×D3—42no—
401C22×D3—52no—
402C22×D3—82no—
403C22×D3—82no—
404C22×D3—108no—
405C22×D3—404no—
406C3×Q8—52no—
407C3×Q8—101no—
408C3×D4—52no—
409C3×D4—104no—
410C3×D4—101no—
411C2×C3×C4—101no—
412C2×C3×C4—104no—
413C2×C3×C4—102no—
414C2×C3×C4D4×D512yes—
415C2×C3×C4D4×D511yescharacteristic
416C2×C3×C4C2×C22⋊D512yes—
417C2×C3×C4C23×D511yescharacteristic
418#67—108no—
419#67—104no—
420C2×Dic3—52no—
421D12—22no—
422D12—22no—
423D12—22no—
424D12—24no—
425D12—42no—
426D12—108no—
427D12—102no—
428D12—404no—
429D12C2×C22⋊D514yes—
430C4×D3—108no—
431C4×D3—102no—
432C4×D3—102no—
433Dic6—102no—
434C3×C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—64no—
438C22×C5—124no—
439C22×C5—122no—
440C22×C5—124no—
441C22×C5—244no—
442C22×C5—244no—
443C22×C5#88212yes—
444C22×C5C2×D3×D411yescharacteristic
445C2×D5—22no—
446C2×D5—42no—
447C2×D5—62no—
448C2×D5—64no—
449C2×D5—248no—
450C4×C5#88212yes—
451C4×C5C22×D1211yescharacteristic
452C4×C5C2×D3×D411yescharacteristic
453Dic5—22no—
454Dic5—42no—
455Dic5—62no—
456C24—61no—
457C24—61no—
458C24—302no—
459#41—54no—
460#41—102no—
461#41—308no—
462#41—304no—
463#41—304no—
464#41—302no—
465#41—302no—
466C2×Q8—51no—
467C2×Q8—151no—
468C2×D4—34no—
469C2×D4—32no—
470C2×D4—32no—
471C2×D4—51no—
472C2×D4—62no—
473C2×D4—62no—
474C2×D4—61no—
475C2×D4—61no—
476C2×D4—62no—
477C2×D4—101no—
478C2×D4—152no—
479C2×D4—151no—
480C2×D4—308no—
481C2×D4—302no—
482C2×D4—304no—
483C2×D4—302no—
484C2×D4—304no—
485C2×D4—304no—
486C2×D4—3016no—
487C22×C4—101no—
488C22×C4—152no—
489C22×C4—302no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—304no—
492D8—68no—
493D8—304no—
494M16—302no—
495C2×C8—22no—
496C2×C8—152no—
497C2×C8C2×D3×D512yes—
498C4⋊C4—52no—
499C22⋊C4—101no—
500C22⋊C4—304no—
501C42—101no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—102no—
505C22×C3—202no—
506C22×C3#408312yes—
507C22×C3C2×D4×D511yescharacteristic
508C2×D3—24no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—42no—
512C2×D3—84no—
513C2×D3—84no—
514C2×D3—104no—
515C2×D3—102no—
516C2×D3—408no—
517C3×C4—102no—
518C3×C4—202no—
519C3×C4#408312yes—
520C3×C4C2×D4×D511yescharacteristic
521C3×C4C22×C22⋊D511yescharacteristic
522Dic3—102no—
523C2×C5—22no—
524C2×C5—124no—
525C2×C5—242no—
526C2×C5C2×#88211yescharacteristic
527C2×C5#576912yes—
528D5—42no—
529D5—122no—
530C23—62no—
531C23—64no—
532C23—62no—
533C23—101no—
534C23—122no—
535C23—152no—
536C23—242no—
537C23—242no—
538C23—308no—
539C23—1204no—
540C23#677711yescharacteristic
541Q8—52no—
542Q8—101no—
543Q8—302no—
544D4—34no—
545D4—52no—
546D4—64no—
547D4—62no—
548D4—62no—
549D4—62no—
550D4—104no—
551D4—101no—
552D4—122no—
553D4—308no—
554D4—308no—
555D4—304no—
556D4—302no—
557D4—1204no—
558C2×C4—101no—
559C2×C4—104no—
560C2×C4—102no—
561C2×C4—152no—
562C2×C4—308no—
563C2×C4—302no—
564C2×C4—302no—
565C2×C4D5×D1212yes—
566C2×C4#677711yescharacteristic
567C2×C4D3×C22⋊D512yes—
568C2×C4C22×D3×D511yescharacteristic
569C8—22no—
570C8—302no—
571C2×C3—22no—
572C2×C3—202no—
573C2×C3#6609112yes—
574C2×C3C2×#408311yescharacteristic
575D3—44no—
576D3—82no—
577D3—202no—
578C5#7883411yescharacteristic, complemented, Sylow
579C22—22no—
580C22—64no—
581C22—102no—
582C22—124no—
583C22—122no—
584C22—124no—
585C22—202no—
586C22—244no—
587C22—244no—
588C22—302no—
589C22—304no—
590C22—1208no—
591C22#9147512yes—
592C22C2×#677711yescharacteristic
593C4—102no—
594C4—202no—
595C4—302no—
596C4#9147512yes—
597C4C2×D5×D1211yescharacteristic
598C4C2×D3×C22⋊D511yescharacteristic
599C3#1061556411yescharacteristic, complemented, Sylow
600C2—22no—
601C2—124no—
602C2—202no—
603C2—242no—
604C2—602no—
605C2#1175398312yes—
606C2C2×#9147511yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
hypercenterC22×C8D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#92018C2semidirect
2. C22×C5×D24C2semidirect
3. #11754606C2semidirect
4. C2×#91299C2semidirect
5. #11751211C2semidirect
6. C3×#65402C2semidirect
7. #11748386C2semidirect
8. #11748159C2semidirect
9. #11748119C2semidirect
13. C22×C5×D12C22semidirect
18. C2×#6777C22semidirect
21. C2×#6768C22semidirect
22. C22×C3×C5×C8C22semidirect
23. C2×#6710C22semidirect
25. C2×C5×D24C22semidirect
26. C3×#4072C22semidirect
28. #91466C22semidirect
31. C2×#6656C22semidirect
33. C2×#6652C22semidirect
34. #91104C22semidirect
35. C3×#3953C22semidirect
36. C3×#3948C22semidirect
37. #90966C22semidirect
46. #65402D3semidirect
63. C2×C5×D12D4semidirect
68. C3×#709D4semidirect
96. C22×D24D5semidirect
116. C22×C5×C8C2×D3semidirect
124. #4072C2×D3semidirect
127. #3953C2×D3semidirect
129. #3948C2×D3semidirect
182. C22×D12C2×D5semidirect
183. C22×C3×C8C2×D5semidirect
190. C2×D24C2×D5semidirect
227. #709D12semidirect
303. C2×D12C22⋊D5semidirect
378. C22×C8D3×D5semidirect
502. C3×C5#3008semidirect
578. C5#78834semidirect
599. C3#10615564semidirect

Maximal subgroups

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

subgroupindex
C2×#920182
C22×C5×D242
#117546062
C2×#912992
#117512112
C3×#654022
#117483862
#117481592
#117481192
#106155643
#788345