Finite Group Databasebeta

← #14424428

Subgroups — #14424428

Order 1920.

1464 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92101C211yesmaximal, characteristic, complemented
2C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11756564C211yesmaximal, characteristic, complemented
4C22×#6716C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11751271C214yesmaximal, complemented
7#11751259C214yesmaximal, complemented
8#11751230C211yesmaximal, characteristic, complemented
9#11751221C211yesmaximal, characteristic, complemented
10#10615679—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13#92101—24no—
14#92101—24no—
15C2×Q8×D15C2212yescomplemented
16C2×#6819—22no—
17C2×#6819C2212yescomplemented
18C2×#6817—21no—
19C2×#6817C2211yescharacteristic
20C22×Dic30—21no—
21C22×Dic30C2211yescharacteristic
22C3×C5×AGL(1,ℤ/8)—24no—
23C2×C3×C5×SD16C2212yes—
24C2×C3×C5×D8C2212yes—
25C2×C3×C5×M16C2211yescharacteristic
26#91842—22no—
27#91829—21no—
28#91827C2214yes—
29#91825—21no—
30C2×#6716—24no—
31C2×#6716C2214yes—
32#91815C2211yescharacteristic
33C22×C3⋊#168C2211yescharacteristic
34#91119C2212yescomplemented
35#91116C2212yescomplemented
36#91104C2214yescomplemented
37#91100C2214yescomplemented
38#83948—51nomaximal, Hall
39C2×#4150—31no—
40C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
41#66139—31no—
42C22×#687—31no—
43#65522—31no—
44#65459—34no—
45#65447—34no—
46#65418—31no—
47#65409—31no—
48C24×C3×C5—21no—
49C3×C5×#41—22no—
50C3×C5×#41D414yescomplemented
51C2×C3×C5×Q8C2311yescharacteristic
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4C2312yes—
57C2×C3×C5×D4C2311yescharacteristic
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—22no—
61C2×#1220—24no—
62C2×#1220D412yes—
63C22×Dic15—21no—
64C22×Dic15—22no—
65Q8×D15—24no—
66Q8×D15—24no—
67#6819—216no—
68#6819—24no—
69#6819—24no—
70#6817—28no—
71#6817—22no—
72#6817—22no—
73C2×D60D411yescharacteristic
74C2×C4×D15—22no—
75C2×C4×D15C2312yes—
76C2×Dic30—28no—
77C2×Dic30—21no—
78C2×Dic30—22no—
79C2×Dic30—22no—
80C2×Dic30D411yescharacteristic
81C2×Dic30D414yescomplemented
82C2×Dic30C2312yes—
83C3×C5×SD16—24no—
84C3×C5×D8—24no—
85C3×C5×M16—22no—
86C2×C3×C5×C8C2312yes—
87#6719—21no—
88#6719—24no—
89#6716—28no—
90C4⋊Dic15C2312yes—
91C4×Dic15—21no—
92C2×C3⋊#168—22no—
93C2×C3⋊#168C2312yes—
94C2×#1010—51no—
95C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
96#5805—51no—
97C22×#214—51no—
98#5233—51no—
99#5170—54no—
100#5158—54no—
101#5129—51no—
102#5120—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×D4C2×D311yescharacteristic
105#4150—64no—
106#4150—64no—
107C2×Q8×D5—32no—
108C2×#711—32no—
109C2×#711—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×Dic10—31no—
113C22×Dic10—61no—
114C5×AGL(1,ℤ/8)—24no—
115C2×C5×SD16C2×D312yes—
116C2×C5×D8C2×D312yes—
117C2×C5×M16C2×D311yescharacteristic
118#4099—62no—
119#4086—61no—
120#4084—34no—
121#4082—61no—
122C2×#687—34no—
123C2×#687—64no—
124#4072—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3952—34no—
129#3948—34no—
130#3010—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5—22no—
133C23×C3×C5—42no—
134C23×C3×C5—82no—
135C23×C3×C5C2×D411yescharacteristic
136C22×D15C2×D412yes—
137C3×C5×Q8—21no—
138C3×C5×Q8C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—21no—
141C3×C5×D4—22no—
142C3×C5×D4—22no—
143C3×C5×D4—42no—
144C3×C5×D4C2×D412yes—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—24no—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2411yescharacteristic
151#1220—28no—
152#1220—24no—
153#1220—84no—
154C2×Dic15—28no—
155C2×Dic15—24no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15C2×D412yes—
159D60—22no—
160C4×D15—28no—
161C4×D15—22no—
162C4×D15—22no—
163Dic30—28no—
164Dic30—22no—
165Dic30—24no—
166Dic30—22no—
167Dic30C2×D414yes—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C2×C3×#41C2×D511yescharacteristic
171C22×C3×D4C2×D511yescharacteristic
172#1010—104no—
173#1010—104no—
174C2×D3×Q8—52no—
175C2×#237—52no—
176C2×#237—102no—
177C2×#235—51no—
178C2×#235—101no—
179C22×Dic6—51no—
180C22×Dic6—101no—
181C3×AGL(1,ℤ/8)—24no—
182C2×C3×SD16C2×D512yes—
183C2×C3×D8C2×D512yes—
184C2×C3×M16C2×D511yescharacteristic
185#953—102no—
186#940—101no—
187#938—54no—
188#936—101no—
189C2×#214—54no—
190C2×#214—104no—
191#926—51no—
192C22×C3⋊C8—51no—
193#835—52no—
194#832—52no—
195(C2×D3)⋊C8—54no—
196#816—54no—
197C24×C5—21no—
198C5×#41—22no—
199C5×#41#6714yescomplemented
200C2×C5×Q8C22×D311yescharacteristic
201C2×C5×D4—21no—
202C2×C5×D4—22no—
203C2×C5×D4—21no—
204C2×C5×D4—22no—
205C2×C5×D4C22×D312yes—
206C2×C5×D4C22×D311yescharacteristic
207C22×C4×C5—21no—
208C22×C4×C5C22×D311yescharacteristic
209C2×C22⋊D5—32no—
210C2×C22⋊D5—62no—
211C2×C22⋊D5—64no—
212C22×Dic5—62no—
213C22×Dic5—61no—
214Q8×D5—64no—
215Q8×D5—64no—
216#711—64no—
217#711—616no—
218#711—64no—
219#709—68no—
220#709—62no—
221#709—62no—
222C2×D20—31no—
223C2×C4×D5—32no—
224C2×C4×D5—62no—
225C2×Dic10—34no—
226C2×Dic10—32no—
227C2×Dic10—31no—
228C2×Dic10—62no—
229C2×Dic10—68no—
230C2×Dic10—61no—
231C2×Dic10—62no—
232C5×SD16—24no—
233C5×D8—24no—
234C5×M16—22no—
235C2×C5×C8C22×D312yes—
236C22⋊Dic5—64no—
237C22⋊Dic5—61no—
238#687—68no—
239C4⋊Dic5—32no—
240C4×Dic5—61no—
241C2×#168—32no—
242C2×#168—62no—
243C2×2−1+4—151no—
244C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
245C22×SD16—151no—
246#535—151no—
247#449—154no—
248#448—154no—
249#419—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—42no—
255C22×C3×C5—42no—
256C22×C3×C5—44no—
257C22×C3×C5—84no—
258C22×C3×C5#12012yes—
259C22×C3×C5C22×D411yescharacteristic
260C2×D15—22no—
261C2×D15—24no—
262C3×C4×C5—22no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—22no—
267Dic15—24no—
268Dic15—42no—
269C24×C3—21no—
270C3×#41—22no—
271C3×#41C22⋊D514yescomplemented
272C2×C3×Q8C22×D511yescharacteristic
273C2×C3×D4—21no—
274C2×C3×D4—22no—
275C2×C3×D4—21no—
276C2×C3×D4—22no—
277C2×C3×D4C22×D512yes—
278C2×C3×D4C22×D511yescharacteristic
279C22×C3×C4—21no—
280C22×C3×C4C22×D511yescharacteristic
281C2×#67—52no—
282C2×#67—102no—
283C2×#67—104no—
284C22×Dic3—102no—
285C22×Dic3—101no—
286D3×Q8—104no—
287D3×Q8—104no—
288#237—104no—
289#237—1016no—
290#237—104no—
291#235—108no—
292#235—102no—
293#235—102no—
294C2×D12—51no—
295C2×C4×D3—52no—
296C2×C4×D3—102no—
297C2×Dic6—54no—
298C2×Dic6—52no—
299C2×Dic6—51no—
300C2×Dic6—102no—
301C2×Dic6—108no—
302C2×Dic6—101no—
303C2×Dic6—102no—
304C3×SD16—24no—
305C3×D8—24no—
306C3×M16—22no—
307C2×C3×C8C22×D512yes—
308#217—104no—
309#217—101no—
310#214—108no—
311C4⋊Dic3—52no—
312C4×Dic3—101no—
313C2×C3⋊C8—52no—
314C2×C3⋊C8—102no—
315C23×C5—21no—
316C23×C5—22no—
317C23×C5—42no—
318C23×C5—82no—
319C23×C5D3×D411yescharacteristic
320C22×D5—32no—
321C5×Q8—21no—
322C5×Q8C2×#6712yes—
323C5×D4—24no—
324C5×D4—21no—
325C5×D4—22no—
326C5×D4—22no—
327C5×D4—42no—
328C5×D4C2×#6712yes—
329C2×C4×C5—21no—
330C2×C4×C5—24no—
331C2×C4×C5D3×D411yescharacteristic
332C2×C4×C5D3×D412yes—
333C2×C4×C5C2×#6712yes—
334C2×C4×C5C23×D311yescharacteristic
335C22⋊D5—68no—
336C22⋊D5—64no—
337C22⋊D5—244no—
338C2×Dic5—32no—
339C2×Dic5—68no—
340C2×Dic5—64no—
341C2×Dic5—62no—
342C2×Dic5—62no—
343D20—62no—
344C4×D5—68no—
345C4×D5—62no—
346C4×D5—62no—
347Dic10—34no—
348Dic10—68no—
349Dic10—64no—
350Dic10—62no—
351Dic10—62no—
352C5×C8—22no—
353#168—62no—
3542−1+4—304no—
3552−1+4—304no—
356C2×#41—151no—
357C2×#41—152no—
358C2×#41—301no—
359C2×#41—302no—
360C2×#41C2×D1511yescharacteristic
361C22×Q8—152no—
362C22×Q8—151no—
363C22×Q8—301no—
364C22×D4C2×D1511yescharacteristic
365AGL(1,ℤ/8)—24no—
366C2×SD16—154no—
367C2×SD16—304no—
368C2×SD16C2×D1512yes—
369C2×D8C2×D1512yes—
370C2×M16C2×D1511yescharacteristic
371C22×C8—151no—
372#127—301no—
373#124—301no—
374#121—154no—
375#120—302no—
376#117—151no—
377#103—152no—
378#103—154no—
379#102—152no—
380C22⋊C8—154no—
381C2×C3×C5—22no—
382C2×C3×C5—42no—
383C2×C3×C5—82no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—42no—
387C23×C3—21no—
388C23×C3—22no—
389C23×C3—42no—
390C23×C3—82no—
391C23×C3D4×D511yescharacteristic
392C22×D3—52no—
393C3×Q8—21no—
394C3×Q8C2×C22⋊D512yes—
395C3×D4—24no—
396C3×D4—21no—
397C3×D4—22no—
398C3×D4—22no—
399C3×D4—42no—
400C3×D4C2×C22⋊D512yes—
401C2×C3×C4—21no—
402C2×C3×C4—24no—
403C2×C3×C4D4×D512yes—
404C2×C3×C4D4×D511yescharacteristic
405C2×C3×C4C2×C22⋊D512yes—
406C2×C3×C4C23×D511yescharacteristic
407#67—108no—
408#67—104no—
409#67—404no—
410C2×Dic3—52no—
411C2×Dic3—108no—
412C2×Dic3—102no—
413C2×Dic3—104no—
414C2×Dic3—102no—
415D12—102no—
416C4×D3—108no—
417C4×D3—102no—
418C4×D3—102no—
419Dic6—54no—
420Dic6—108no—
421Dic6—102no—
422Dic6—104no—
423Dic6—102no—
424C3×C8—22no—
425C3⋊C8—102no—
426C22×C5—22no—
427C22×C5—22no—
428C22×C5—42no—
429C22×C5—44no—
430C22×C5—42no—
431C22×C5—84no—
432C22×C5#93712yes—
433C22×C5C2×D3×D411yescharacteristic
434C2×D5—64no—
435C2×D5—62no—
436C4×C5—22no—
437C4×C5#93712yes—
438C4×C5C2×D3×D411yescharacteristic
439C4×C5C22×#6711yescharacteristic
440Dic5—64no—
441Dic5—62no—
442Dic5—122no—
443C24—21no—
444#41—22no—
445#41—308no—
446#41—304no—
447#41—3016no—
448#41—304no—
449#41—302no—
450#41—302no—
451#41#122014yescomplemented
452C2×Q8—151no—
453C2×Q8—154no—
454C2×Q8—152no—
455C2×Q8—302no—
456C2×Q8—308no—
457C2×Q8—304no—
458C2×Q8—304no—
459C2×Q8—301no—
460C2×Q8—302no—
461C2×Q8C22×D1511yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—152no—
467C2×D4—151no—
468C2×D4—302no—
469C2×D4—304no—
470C2×D4C22×D1512yes—
471C2×D4C22×D1511yescharacteristic
472C22×C4—21no—
473C22×C4—152no—
474C22×C4—302no—
475C22×C4—302no—
476C22×C4—301no—
477C22×C4C22×D1511yescharacteristic
478SD16—24no—
479SD16—308no—
480D8—24no—
481M16—22no—
482C2×C8—152no—
483C2×C8—302no—
484C2×C8C22×D1512yes—
485C4⋊C4—152no—
486C22⋊C4—304no—
487C22⋊C4—301no—
488C42—301no—
489C3×C5#301011yescharacteristic, complemented, Hall
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—42no—
493C22×C3—44no—
494C22×C3—42no—
495C22×C3—84no—
496C22×C3#408312yes—
497C22×C3C2×D4×D511yescharacteristic
498C2×D3—104no—
499C2×D3—102no—
500C3×C4—22no—
501C3×C4#408312yes—
502C3×C4C2×D4×D511yescharacteristic
503C3×C4C22×C22⋊D511yescharacteristic
504Dic3—104no—
505Dic3—102no—
506Dic3—202no—
507C2×C5—22no—
508C2×C5—42no—
509C2×C5—82no—
510C2×C5#577512yes—
511C2×C5C2×#93711yescharacteristic
512D5—122no—
513C23—21no—
514C23—22no—
515C23—42no—
516C23—82no—
517C23—152no—
518C23D4×D1511yescharacteristic
519Q8—21no—
520Q8—154no—
521Q8—308no—
522Q8—302no—
523Q8—304no—
524Q8—302no—
525Q8C2×#122012yes—
526D4—24no—
527D4—21no—
528D4—22no—
529D4—22no—
530D4—42no—
531D4—308no—
532D4—304no—
533D4—302no—
534D4—1204no—
535D4C2×#122012yes—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—152no—
539C2×C4—308no—
540C2×C4—308no—
541C2×C4—304no—
542C2×C4—302no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4D4×D1511yescharacteristic
547C2×C4D4×D1512yes—
548C2×C4C2×#122012yes—
549C2×C4C23×D1511yescharacteristic
550C8—22no—
551C8—302no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—82no—
555C2×C3#6610912yes—
556C2×C3C2×#408311yescharacteristic
557D3—202no—
558C5#8394811yescharacteristic, complemented, Sylow
559C22—22no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—84no—
565C22—302no—
566C22—304no—
567C22#9182612yes—
568C22C2×D4×D1511yescharacteristic
569C4—22no—
570C4—302no—
571C4—304no—
572C4—602no—
573C4#9182612yes—
574C4C2×D4×D1511yescharacteristic
575C4C22×#122011yescharacteristic
576C3#1061567911yescharacteristic, complemented, Sylow
577C2—22no—
578C2—42no—
579C2—82no—
580C2—602no—
581C2#1175653412yes—
582C2C2×#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#3010
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×#92101C2semidirect
2. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
3. #11756564C2semidirect
4. C22×#6716C2semidirect
5. #11751334C2semidirect
6. #11751271C2semidirect
7. #11751259C2semidirect
8. #11751230C2semidirect
9. #11751221C2semidirect
15. C2×Q8×D15C22semidirect
17. C2×#6819C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #91104C22semidirect
37. #91100C22semidirect
40. C2×C5×AGL(1,ℤ/8)D3semidirect
50. C3×C5×#41D4semidirect
81. C2×Dic30D4semidirect
95. C2×C3×AGL(1,ℤ/8)D5semidirect
199. C5×#41#67semidirect
244. C2×AGL(1,ℤ/8)D15semidirect
271. C3×#41C22⋊D5semidirect
451. #41#1220semidirect
489. C3×C5#3010semidirect
558. C5#83948semidirect
576. C3#10615679semidirect

Maximal subgroups

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

subgroupindex
C2×#921012
C2×C3×C5×AGL(1,ℤ/8)2
#117565642
C22×#67162
#117513342
#117512712
#117512592
#117512302
#117512212
#106156793
#839485