Finite Group Databasebeta

← #14424289

Subgroups — #14424289

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×#92017C211yesmaximal, characteristic, complemented
2C2×C5×#908C211yesmaximal, characteristic, complemented
3#11754606C211yesmaximal, characteristic, complemented
4C22×#6652C211yesmaximal, characteristic, complemented
5#11751230C211yesmaximal, characteristic, complemented
6C3×#65420C211yesmaximal, characteristic, complemented
7#11748382C211yesmaximal, characteristic, complemented
8#11748153C214yesmaximal, complemented
9#11748119C214yesmaximal, complemented
10#10615678—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C22×C5×D12C2211yescharacteristic, complemented
14C22×C3×D20C2211yescharacteristic
15C2×D5×#67—22no—
16#92017—24no—
17#92017—24no—
18C2×#6777C2212yescomplemented
19C2×D3×D20—21no—
20C2×#6766C2212yescomplemented
21C2×#6765—21no—
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C22×C3⋊#168C2211yescharacteristic, complemented
24C5×#908—24no—
25C2×C5×D24C2212yescomplemented
26C2×C5×#204C2212yescomplemented
27C3×#4072C2211yescharacteristic, complemented
28#91570—22no—
29#91466C2214yescomplemented
30#91451—21no—
31#91373—21no—
32C2×#6652—24no—
33C2×#6652C2214yescomplemented
34#91104C2214yescomplemented
35C3×#3953C2214yescomplemented
36#90972C2212yescomplemented
37#90966C2212yescomplemented
38#78928—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×AGL(1,ℤ/8)—31no—
41#66139—31no—
42C22×#686—31no—
43#65522—31no—
44#65461—34no—
45#65445—34no—
46#65420D311yescharacteristic, complemented
47#65418—31no—
48C23×C5×D3—21no—
49C23×C3×D5—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×#67—21no—
59C5×#235—22no—
60C5×#235D414yescomplemented
61C2×C5×D12—22no—
62C2×C5×D12—22no—
63C2×C5×D12—21no—
64C2×C5×D12C2312yes—
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
67C2×C5×Dic6C2311yescharacteristic
68C2×C3×D20—21no—
69C2×C3×D20—22no—
70C2×C3×D20D414yescomplemented
71C2×C3×D20C2312yes—
72D5×#67—216no—
73C2×#1203—24no—
74C2×#1202—22no—
75C2×#1201—22no—
76C2×D5×Dic3—22no—
77#6777—24no—
78#6777—24no—
79D3×D20—28no—
80#6766—24no—
81#6766—24no—
82#6765—28no—
83C3×C5×M16—22no—
84C2×C3×C5×C8C2312yes—
85C2×C3⋊#168—22no—
86C2×C3⋊#168C2312yes—
87C5×D24—24no—
88C5×#204—24no—
89C3×C22⋊Dic5—21no—
90C3×C4⋊Dic5C2312yes—
91C3×C4×Dic5—21no—
92(C2×D3)⋊Dic5—24no—
93#6652—28no—
94C2×#1004—51no—
95C22×#213—51no—
96C2×#908D511yescharacteristic, complemented
97#5525—51no—
98C3×#418—51no—
99#5129—51no—
100#5002—51no—
101#4781—54no—
102#4773—54no—
103C2×C5×#41—31no—
104C22×C5×D4—31no—
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—61no—
113C22×D20C2×D311yescharacteristic
114C5×AGL(1,ℤ/8)—64no—
115C2×C5×SD16—32no—
116C2×C5×D8—32no—
117C2×C5×M16C2×D311yescharacteristic, complemented
118#4099—62no—
119#4086—61no—
120#4084—34no—
121#4082—61no—
122C2×#686—34no—
123C2×#686—64no—
124#4072C2×D311yescharacteristic, complemented
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953C2×D314yescomplemented
129#3952—34no—
130#3008—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C22×C5×D3—22no—
135C22×C5×D3—42no—
136C22×C5×D3—82no—
137C22×C3×D5—22no—
138C22×C3×D5—24no—
139C22×C3×D5—82no—
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×#67—24no—
153C2×C5×Dic3—21no—
154C5×D12—21no—
155C5×D12—22no—
156C5×D12—22no—
157C5×D12—42no—
158C5×D12C2×D412yes—
159C4×C5×D3—24no—
160C5×Dic6—21no—
161C5×Dic6C2×D412yes—
162C2×C3×Dic5—22no—
163C3×D20—24no—
164C3×D20—22no—
165C3×D20C2×D414yes—
166#1203—84no—
167#1202—28no—
168#1201—28no—
169D5×Dic3—28no—
170C3×C5×C8—22no—
171C3⋊#168—22no—
172C22×C3×D4—51no—
173C22×#67—102no—
174#1004—104no—
175#1004—104no—
176C2×#237—52no—
177C2×#237—101no—
178C2×D3×D4—52no—
179C2×D3×D4—101no—
180C2×#235—51no—
181C2×#235C2×D511yescharacteristic
182C22×D12C2×D511yescharacteristic, complemented
183C2×C3×M16C2×D511yescharacteristic, complemented
184C3×#117—51no—
185C2×#213—54no—
186C2×#213—104no—
187C22×C3⋊C8—51no—
188#908—24no—
189C2×D24C2×D512yescomplemented
190C2×#204C2×D512yescomplemented
191#895—54no—
192#882—102no—
193#875—101no—
194#874—101no—
195C3×#102—54no—
196(C2×D3)⋊C8—54no—
197#810—52no—
198#809—52no—
199C24×C5—61no—
200C23×D5—21no—
201C23×D5—62no—
202C5×#41—34no—
203C5×#41—62no—
204C2×C5×Q8—31no—
205C2×C5×D4—31no—
206C2×C5×D4—32no—
207C2×C5×D4—61no—
208C2×C5×D4—62no—
209C2×C5×D4—62no—
210C2×C5×D4—61no—
211C22×C4×C5—61no—
212C22×C4×C5C22×D311yescharacteristic
213C2×C22⋊D5—32no—
214C2×C22⋊D5—64no—
215C2×C22⋊D5—62no—
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—21no—
225C2×D20—22no—
226C2×D20—31no—
227C2×D20—68no—
228C2×D20—62no—
229C2×D20D1214yescomplemented
230C2×D20C22×D312yes—
231C2×C4×D5—32no—
232C2×C4×D5—62no—
233C2×Dic10—31no—
234C5×SD16—64no—
235C5×D8—64no—
236C5×M16—22no—
237C2×C5×C8C22×D312yes—
238C22⋊Dic5—21no—
239C22⋊Dic5—64no—
240#686—68no—
241C4⋊Dic5C22×D312yes—
242C4×Dic5—21no—
243C2×#168—32no—
244C2×#168—62no—
245C2×H(5,𝔽2)—151no—
246C2×AGL(1,ℤ/8)—31no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—51no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—24no—
259C2×C5×D3—22no—
260C2×C5×D3—42no—
261C2×C5×D3—42no—
262C2×C5×D3—44no—
263C2×C5×D3—84no—
264C2×C3×D5—24no—
265C2×C3×D5—44no—
266C2×C3×D5—84no—
267D3×D5—88no—
268C3×C4×C5#12012yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271Dic15—22no—
272C3×Dic5—42no—
273C5×Dic3—22no—
274C24×C3—101no—
275C23×D3—21no—
276C23×D3—102no—
277C2×C3×D4—54no—
278C2×C3×D4—52no—
279C2×C3×D4—101no—
280C2×C3×D4—102no—
281C22×C3×C4C22×D511yescharacteristic
282C2×#67—21no—
283C2×#67—52no—
284C2×#67—102no—
285C2×#67—102no—
286C2×#67—1016no—
287C22×Dic3—102no—
288#237—104no—
289#237—108no—
290#237—104no—
291D3×D4—108no—
292D3×D4—104no—
293D3×D4—104no—
294#235—22no—
295#235—102no—
296#235—102no—
297#235C22⋊D514yescomplemented
298C2×D12—22no—
299C2×D12—22no—
300C2×D12—21no—
301C2×D12—51no—
302C2×D12—104no—
303C2×D12C22×D512yes—
304C2×D12C22×D511yescharacteristic
305C2×C4×D3—21no—
306C2×C4×D3—52no—
307C2×Dic6—51no—
308C2×Dic6C22×D511yescharacteristic
309C3×M16—22no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4—52no—
312C3×C22⋊C4—101no—
313C3×C42—101no—
314#213—108no—
315(C2×D3)⋊C4—104no—
316C2×C3⋊C8—52no—
317C2×C3⋊C8—102no—
318D24—24no—
319#204—24no—
320C23×C5—61no—
321C23×C5—62no—
322C23×C5—122no—
323C23×C5—242no—
324C23×C5D3×D411yescharacteristic
325C22×D5—22no—
326C22×D5—24no—
327C22×D5—32no—
328C22×D5—68no—
329C22×D5—82no—
330C22×D5—244no—
331C5×Q8—32no—
332C5×Q8—61no—
333C5×D4—32no—
334C5×D4—64no—
335C5×D4—61no—
336C5×D4—62no—
337C5×D4—62no—
338C5×D4—122no—
339C2×C4×C5—61no—
340C2×C4×C5—64no—
341C2×C4×C5C2×D1212yes—
342C2×C4×C5D3×D411yescharacteristic
343C2×C4×C5D3×D412yes—
344C2×C4×C5C23×D311yescharacteristic
345C22⋊D5—68no—
346C22⋊D5—64no—
347C22⋊D5—244no—
348C2×Dic5—22no—
349C2×Dic5—32no—
350D20—24no—
351D20—22no—
352D20—68no—
353D20—62no—
354D20C2×D1214yes—
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—31no—
364C2×#41—151no—
365C2×#41—152no—
366C2×#41—301no—
367C22×D4—31no—
368C22×D4—51no—
369C22×D4—152no—
370C22×D4—301no—
371C22×D4—302no—
372AGL(1,ℤ/8)—64no—
373C2×SD16—32no—
374C2×D8—32no—
375C2×D8—154no—
376C2×D8—304no—
377C2×M16D3×D511yescharacteristic, complemented
378C22×C8—151no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—51no—
384#103—152no—
385#102—54no—
386#102—152no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5C2×#12011yescharacteristic
390C2×C3×C5#57612yes—
391D15—42no—
392C3×D5—44no—
393C5×D3—42no—
394C5×D3—82no—
395C23×C3—102no—
396C23×C3—104no—
397C23×C3—402no—
398C23×C3D4×D511yescharacteristic
399C22×D3—21no—
400C22×D3—22no—
401C22×D3—42no—
402C22×D3—52no—
403C22×D3—82no—
404C22×D3—108no—
405C22×D3—404no—
406C3×D4—54no—
407C3×D4—102no—
408C3×D4—104no—
409C2×C3×C4—102no—
410C2×C3×C4D4×D512yes—
411C2×C3×C4D4×D511yescharacteristic
412C2×C3×C4C2×C22⋊D512yes—
413C2×C3×C4C23×D511yescharacteristic
414#67—24no—
415#67—108no—
416#67—108no—
417#67—104no—
418C2×Dic3—21no—
419C2×Dic3—52no—
420C2×Dic3—108no—
421D12—21no—
422D12—22no—
423D12—22no—
424D12—42no—
425D12—102no—
426D12—404no—
427D12C2×C22⋊D512yes—
428C4×D3—24no—
429C4×D3—102no—
430C4×D3—102no—
431Dic6—21no—
432Dic6—102no—
433Dic6C2×C22⋊D512yes—
434C3×C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—62no—
438C22×C5—122no—
439C22×C5—122no—
440C22×C5—124no—
441C22×C5—244no—
442C22×C5#88212yes—
443C22×C5C2×D3×D411yescharacteristic
444C2×D5—24no—
445C2×D5—44no—
446C2×D5—62no—
447C2×D5—64no—
448C2×D5—84no—
449C2×D5—248no—
450C4×C5—62no—
451C4×C5#88212yes—
452C4×C5C22×D1211yescharacteristic
453C4×C5C2×D3×D411yescharacteristic
454Dic5—42no—
455Dic5—62no—
456C24—61no—
457C24—101no—
458C24—302no—
459#41—34no—
460#41—62no—
461#41—308no—
462#41—304no—
463#41—304no—
464#41—302no—
465#41—302no—
466C2×Q8—31no—
467C2×Q8—151no—
468C2×D4—31no—
469C2×D4—32no—
470C2×D4—54no—
471C2×D4—52no—
472C2×D4—61no—
473C2×D4—62no—
474C2×D4—61no—
475C2×D4—62no—
476C2×D4—101no—
477C2×D4—102no—
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—61no—
488C22×C4—152no—
489C22×C4—302no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—64no—
492D8—64no—
493D8—308no—
494M16—22no—
495C2×C8—152no—
496C2×C8—302no—
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—104no—
505C22×C3—204no—
506C22×C3—404no—
507C22×C3#408312yes—
508C22×C3C2×D4×D511yescharacteristic
509C2×D3—22no—
510C2×D3—42no—
511C2×D3—44no—
512C2×D3—42no—
513C2×D3—84no—
514C2×D3—104no—
515C2×D3—102no—
516C2×D3—408no—
517C3×C4—202no—
518C3×C4#408312yes—
519C3×C4C2×D4×D511yescharacteristic
520C3×C4C22×C22⋊D511yescharacteristic
521Dic3—22no—
522Dic3—102no—
523C2×C5—22no—
524C2×C5—122no—
525C2×C5—242no—
526C2×C5C2×#88211yescharacteristic
527C2×C5#577412yes—
528D5—44no—
529D5—122no—
530C23—61no—
531C23—62no—
532C23—102no—
533C23—104no—
534C23—122no—
535C23—152no—
536C23—242no—
537C23—308no—
538C23—402no—
539C23—1204no—
540C23#677711yescharacteristic
541Q8—32no—
542Q8—61no—
543Q8—302no—
544D4—32no—
545D4—54no—
546D4—64no—
547D4—61no—
548D4—62no—
549D4—62no—
550D4—104no—
551D4—102no—
552D4—122no—
553D4—308no—
554D4—308no—
555D4—304no—
556D4—302no—
557D4—1204no—
558C2×C4—61no—
559C2×C4—64no—
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—204no—
573C2×C3#6610812yes—
574C2×C3C2×#408311yescharacteristic
575D3—42no—
576D3—82no—
577D3—202no—
578C5#7892811yescharacteristic, complemented, Sylow
579C22—22no—
580C22—62no—
581C22—104no—
582C22—122no—
583C22—124no—
584C22—122no—
585C22—204no—
586C22—244no—
587C22—304no—
588C22—302no—
589C22—404no—
590C22—1208no—
591C22#9147512yes—
592C22C2×#677711yescharacteristic
593C4—62no—
594C4—202no—
595C4—302no—
596C4#9147512yes—
597C4C2×D5×D1211yescharacteristic
598C4C2×D3×C22⋊D511yescharacteristic
599C3#1061567811yescharacteristic, complemented, Sylow
600C2—22no—
601C2—122no—
602C2—204no—
603C2—242no—
604C2—602no—
605C2#1175403812yes—
606C2C2×#9147511yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
hypercenterC2×M16D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#92017C2semidirect
2. C2×C5×#908C2semidirect
3. #11754606C2semidirect
4. C22×#6652C2semidirect
5. #11751230C2semidirect
6. C3×#65420C2semidirect
7. #11748382C2semidirect
8. #11748153C2semidirect
9. #11748119C2semidirect
13. C22×C5×D12C22semidirect
18. C2×#6777C22semidirect
20. C2×#6766C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C22×C3⋊#168C22semidirect
25. C2×C5×D24C22semidirect
26. C2×C5×#204C22semidirect
27. C3×#4072C22semidirect
29. #91466C22semidirect
33. C2×#6652C22semidirect
34. #91104C22semidirect
35. C3×#3953C22semidirect
36. #90972C22semidirect
37. #90966C22semidirect
46. #65420D3semidirect
60. C5×#235D4semidirect
70. C2×C3×D20D4semidirect
96. C2×#908D5semidirect
117. C2×C5×M16C2×D3semidirect
124. #4072C2×D3semidirect
128. #3953C2×D3semidirect
182. C22×D12C2×D5semidirect
183. C2×C3×M16C2×D5semidirect
189. C2×D24C2×D5semidirect
190. C2×#204C2×D5semidirect
229. C2×D20D12semidirect
297. #235C22⋊D5semidirect
377. C2×M16D3×D5semidirect
502. C3×C5#3008semidirect
578. C5#78928semidirect
599. C3#10615678semidirect

Maximal subgroups

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

subgroupindex
C2×#920172
C2×C5×#9082
#117546062
C22×#66522
#117512302
C3×#654202
#117483822
#117481532
#117481192
#106156783
#789285