Finite Group Databasebeta

← #14424104

Subgroups — #14424104

Order 1920.

1508 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6817C211yesmaximal, characteristic
2C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#91831C212yesmaximal, complemented
4#11751308C211yesmaximal, characteristic, complemented
5#11751151C212yesmaximal, complemented
6#11751148C214yesmaximal, complemented
7#11751147C212yesmaximal, complemented
8#11751067C212yesmaximal, complemented
9#10614466—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic, complemented
11C23×C3×C4×C5C2211yescharacteristic
12C22×#1220—21no—
13C2×#6817—21no—
14C2×#6817—21no—
15C2×#6817—28no—
16C2×#6817C2212yes—
17C2×#6817C2212yes—
18C22×D60C2211yescharacteristic
19C22×C4×D15—21no—
20C22×Dic30C2211yescharacteristic
21C3×C5×C22⋊Q8—21no—
22C3×C5×C22⋊Q8—21no—
23C3×C5×C22⋊Q8C2212yes—
24C3×C5×C22⋊Q8C2212yescomplemented
25C2×C3×C5×C4⋊C4—21no—
26C2×C3×C5×C4⋊C4C2211yescharacteristic, complemented
27C2×C3×C5×C22⋊C4—21no—
28C2×#6718C2214yescomplemented
29#91831—28no—
30C2×#6717C2214yescomplemented
31C2×#6710C2212yescomplemented
32#91114C2212yes—
33#91114C2212yescomplemented
34#91094C2214yescomplemented
35#91093C2214yescomplemented
36#82735—51nomaximal, Hall
37C22×#709—31no—
38C2×C5×C22⋊Q8D311yescharacteristic, complemented
39C2×#4088—32no—
40#65496—31no—
41#65339—32no—
42#65336—34no—
43#65335—32no—
44#65255—32no—
45C24×C3×C5D411yescharacteristic
46C23×D15—21no—
47C2×C3×C5×Q8—21no—
48C2×C3×C5×Q8—24no—
49C2×C3×C5×Q8C2312yes—
50C22×C3×C4×C5—24no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5D411yescharacteristic
54C22×C3×C4×C5D411yescharacteristic
55C22×C3×C4×C5D411yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—28no—
60C2×#1220—22no—
61C2×#1220—22no—
62C22×Dic15—21no—
63#6817—28no—
64#6817—28no—
65#6817—24no—
66#6817—24no—
67#6817—24no—
68#6817D418yes—
69C2×D60—21no—
70C2×D60—24no—
71C2×D60C2312yes—
72C2×C4×D15—28no—
73C2×C4×D15—22no—
74C2×C4×D15—22no—
75C2×Dic30—21no—
76C2×Dic30—24no—
77C2×Dic30C2312yes—
78C3×C5×C4⋊C4—22no—
79C3×C5×C4⋊C4—22no—
80C3×C5×C4⋊C4—21no—
81C3×C5×C4⋊C4C2312yes—
82C3×C5×C22⋊C4—22no—
83C3×C5×C22⋊C4—22no—
84#6718—28no—
85#6717—28no—
86#6710—24no—
87C2×C3⋊#168C2314yes—
88C22×#235—51no—
89C2×C3×C22⋊Q8D511yescharacteristic, complemented
90C2×#942—52no—
91#5207—51no—
92#5050—52no—
93#5047—54no—
94#5046—52no—
95#4966—52no—
96C22×C5×Q8C2×D311yescharacteristic, complemented
97C23×C4×C5C2×D311yescharacteristic
98C22×C22⋊D5—61no—
99C2×#709—32no—
100C2×#709—32no—
101C2×#709—68no—
102C2×#709—61no—
103C2×#709—61no—
104C22×D20—31no—
105C22×C4×D5—61no—
106C22×Dic10—31no—
107C5×C22⋊Q8—21no—
108C5×C22⋊Q8—21no—
109C5×C22⋊Q8C2×D312yes—
110C5×C22⋊Q8C2×D312yescomplemented
111C2×C5×C4⋊C4—21no—
112C2×C5×C4⋊C4C2×D311yescharacteristic, complemented
113C2×C5×C22⋊C4—21no—
114C2×#689—34no—
115#4088—68no—
116C2×#688—34no—
117C2×#681—32no—
118C22⋊#168—32no—
119C22⋊#168—32no—
120#3942—34no—
121#3941—34no—
122#3007—151noSylow
123C23×C3×C5—22no—
124C23×C3×C5—22no—
125C23×C3×C5—41no—
126C23×C3×C5C2×D411yescharacteristic
127C23×C3×C5C2×D411yescharacteristic
128C23×C3×C5C2×D411yescharacteristic
129C22×D15—24no—
130C22×D15—22no—
131C22×D15—42no—
132C22×D15—42no—
133C3×C5×Q8—24no—
134C3×C5×Q8—42no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—42no—
143C2×C3×C4×C5—42no—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D414yes—
148C2×C3×C4×C5C2411yescharacteristic
149#1220—28no—
150#1220—28no—
151C2×Dic15—24no—
152C2×Dic15—22no—
153D60—24no—
154D60—22no—
155D60C2×D414yes—
156C4×D15—28no—
157C4×D15—28no—
158Dic30—24no—
159Dic30—22no—
160Dic30C2×D414yes—
161C3⋊#168—24no—
162C22×C3×Q8C2×D511yescharacteristic, complemented
163C23×C3×C4C2×D511yescharacteristic
164C22×#67—101no—
165C2×#235—52no—
166C2×#235—52no—
167C2×#235—108no—
168C2×#235—101no—
169C2×#235—101no—
170C22×D12—51no—
171C22×C4×D3—101no—
172C22×Dic6—51no—
173C3×C22⋊Q8—21no—
174C3×C22⋊Q8—21no—
175C3×C22⋊Q8C2×D512yes—
176C3×C22⋊Q8C2×D512yescomplemented
177C2×C3×C4⋊C4—21no—
178C2×C3×C4⋊C4C2×D511yescharacteristic, complemented
179C2×C3×C22⋊C4—21no—
180C2×#216—54no—
181#942—108no—
182C2×#215—54no—
183C2×#208—52no—
184#830—52no—
185#830—52no—
186#810—54no—
187#809—54no—
188C24×C5#6711yescharacteristic
189C23×D5—61no—
190C2×C5×Q8—21no—
191C2×C5×Q8—24no—
192C2×C5×Q8C22×D312yes—
193C22×C4×C5—24no—
194C22×C4×C5—21no—
195C22×C4×C5—21no—
196C22×C4×C5#6711yescharacteristic
197C22×C4×C5#6711yescharacteristic
198C22×C4×C5#6711yescharacteristic
199C22×C4×C5C22×D311yescharacteristic
200C22×C4×C5C22×D311yescharacteristic
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—68no—
203C2×C22⋊D5—62no—
204C2×C22⋊D5—62no—
205C22×Dic5—61no—
206#709—38no—
207#709—64no—
208#709—64no—
209#709—68no—
210#709—68no—
211#709—64no—
212C2×D20—32no—
213C2×D20—64no—
214C2×D20—61no—
215C2×C4×D5—68no—
216C2×C4×D5—62no—
217C2×C4×D5—62no—
218C2×Dic10—32no—
219C2×Dic10—64no—
220C2×Dic10—61no—
221C5×C4⋊C4—22no—
222C5×C4⋊C4—22no—
223C5×C4⋊C4—21no—
224C5×C4⋊C4C22×D312yes—
225C5×C22⋊C4—22no—
226C5×C22⋊C4—22no—
227#689—68no—
228#688—68no—
229#681—64no—
230C2×#168—34no—
231C22×#41—151no—
232C2×#137—152no—
233C2×C22⋊Q8D1511yescharacteristic, complemented
234#452—154no—
235#451—152no—
236#450—152no—
237#417—152no—
238#407—151no—
239C22×C3×C5—21no—
240C22×C3×C5—21no—
241C22×C3×C5—21no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5—21no—
245C22×C3×C5—21no—
246C22×C3×C5—44no—
247C22×C3×C5#12012yes—
248C22×C3×C5#13712yescomplemented
249C22×C3×C5C22×D411yescharacteristic
250C2×D15—24no—
251C2×D15—48no—
252C2×D15—42no—
253C2×D15—42no—
254C3×C4×C5—21no—
255C3×C4×C5—21no—
256C3×C4×C5—42no—
257C3×C4×C5—42no—
258C3×C4×C5#12012yes—
259C3×C4×C5C22×D412yes—
260Dic15—24no—
261C24×C3C22⋊D511yescharacteristic
262C23×D3—101no—
263C2×C3×Q8—21no—
264C2×C3×Q8—24no—
265C2×C3×Q8C22×D512yes—
266C22×C3×C4—24no—
267C22×C3×C4—21no—
268C22×C3×C4—21no—
269C22×C3×C4C22⋊D511yescharacteristic
270C22×C3×C4C22⋊D511yescharacteristic
271C22×C3×C4C22⋊D511yescharacteristic
272C22×C3×C4C22×D511yescharacteristic
273C22×C3×C4C22×D511yescharacteristic
274C22×C3×C4C22×D511yescharacteristic
275C2×#67—108no—
276C2×#67—102no—
277C2×#67—102no—
278C22×Dic3—101no—
279#235—58no—
280#235—104no—
281#235—104no—
282#235—108no—
283#235—108no—
284#235—104no—
285C2×D12—52no—
286C2×D12—104no—
287C2×D12—101no—
288C2×C4×D3—108no—
289C2×C4×D3—102no—
290C2×C4×D3—102no—
291C2×Dic6—52no—
292C2×Dic6—104no—
293C2×Dic6—101no—
294C3×C4⋊C4—22no—
295C3×C4⋊C4—22no—
296C3×C4⋊C4—21no—
297C3×C4⋊C4C22×D512yes—
298C3×C22⋊C4—22no—
299C3×C22⋊C4—22no—
300#216—108no—
301#215—108no—
302#208—104no—
303C2×C3⋊C8—54no—
304C23×C5—22no—
305C23×C5—22no—
306C23×C5—41no—
307C23×C5C2×#6711yescharacteristic
308C23×C5C2×#6711yescharacteristic
309C23×C5C2×#6711yescharacteristic
310C22×D5—64no—
311C22×D5—62no—
312C22×D5—122no—
313C22×D5—122no—
314C5×Q8—24no—
315C5×Q8—42no—
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—22no—
319C2×C4×C5—22no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—42no—
324C2×C4×C5—42no—
325C2×C4×C5D3×D414yes—
326C2×C4×C5C2×#6711yescharacteristic
327C2×C4×C5C2×#6711yescharacteristic
328C2×C4×C5C2×#6711yescharacteristic
329C2×C4×C5C23×D311yescharacteristic
330C22⋊D5—68no—
331C22⋊D5—68no—
332C2×Dic5—64no—
333C2×Dic5—62no—
334D20—34no—
335D20—64no—
336D20—62no—
337C4×D5—68no—
338C4×D5—68no—
339Dic10—34no—
340Dic10—64no—
341Dic10—62no—
342#168—64no—
343C2×#41—152no—
344C2×#41—152no—
345C2×#41—308no—
346C2×#41—301no—
347C2×#41—301no—
348C22×Q8—151no—
349C22×Q8C2×D1511yescharacteristic, complemented
350C22×D4—151no—
351C22×D4—301no—
352C23×C4—301no—
353C23×C4C2×D1511yescharacteristic
354#137—308no—
355C2×Q16—154no—
356C2×SD16—154no—
357C2×M16—152no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8C2×D1512yes—
361C22⋊Q8C2×D1512yescomplemented
362C2×C4⋊C4—21no—
363C2×C4⋊C4C2×D1511yescharacteristic, complemented
364C2×C22⋊C4—21no—
365#103—154no—
366#102—154no—
367C22⋊C8—152no—
368C22⋊C8—152no—
369C2×C3×C5—22no—
370C2×C3×C5—22no—
371C2×C3×C5—41no—
372C2×C3×C5C2×#12011yescharacteristic
373C2×C3×C5C2×#13711yescharacteristic
374C2×C3×C5#57511yescharacteristic
375D15—44no—
376C23×C3—22no—
377C23×C3—22no—
378C23×C3—41no—
379C23×C3C2×C22⋊D511yescharacteristic
380C23×C3C2×C22⋊D511yescharacteristic
381C23×C3C2×C22⋊D511yescharacteristic
382C22×D3—104no—
383C22×D3—102no—
384C22×D3—202no—
385C22×D3—202no—
386C3×Q8—24no—
387C3×Q8—42no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4D4×D514yes—
398C2×C3×C4C2×C22⋊D511yescharacteristic
399C2×C3×C4C2×C22⋊D511yescharacteristic
400C2×C3×C4C2×C22⋊D511yescharacteristic
401C2×C3×C4C23×D511yescharacteristic
402#67—108no—
403#67—108no—
404C2×Dic3—104no—
405C2×Dic3—102no—
406D12—54no—
407D12—104no—
408D12—102no—
409C4×D3—108no—
410C4×D3—108no—
411Dic6—54no—
412Dic6—104no—
413Dic6—102no—
414C3⋊C8—104no—
415C22×C5—21no—
416C22×C5—21no—
417C22×C5—21no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5—21no—
422C22×C5—44no—
423C22×C5#93712yes—
424C22×C5#94212yescomplemented
425C22×C5C22×#6711yescharacteristic
426C2×D5—64no—
427C2×D5—128no—
428C2×D5—122no—
429C2×D5—122no—
430C4×C5—21no—
431C4×C5—21no—
432C4×C5—42no—
433C4×C5—42no—
434C4×C5#93712yes—
435C4×C5C2×D3×D412yes—
436Dic5—64no—
437C24—301no—
438C24#122011yescharacteristic
439#41—158no—
440#41—304no—
441#41—304no—
442#41—308no—
443#41—308no—
444#41—304no—
445C2×Q8—21no—
446C2×Q8—24no—
447C2×Q8—152no—
448C2×Q8—304no—
449C2×Q8—301no—
450C2×Q8C22×D1512yes—
451C2×D4—152no—
452C2×D4—304no—
453C2×D4—308no—
454C2×D4—302no—
455C2×D4—302no—
456C2×D4—301no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—308no—
461C22×C4—301no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4#122011yescharacteristic
465C22×C4#122011yescharacteristic
466C22×C4#122011yescharacteristic
467C22×C4C22×D1511yescharacteristic
468C22×C4C22×D1511yescharacteristic
469C22×C4C22×D1511yescharacteristic
470Q16—308no—
471SD16—308no—
472M16—304no—
473C2×C8—154no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4C22×D1512yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C3×C5#300711yescharacteristic, complemented, Hall
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—44no—
489C22×C3#408312yes—
490C22×C3#408812yescomplemented
491C22×C3C22×C22⋊D511yescharacteristic
492C2×D3—104no—
493C2×D3—208no—
494C2×D3—202no—
495C2×D3—202no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4#408312yes—
501C3×C4C2×D4×D512yes—
502Dic3—104no—
503C2×C5—22no—
504C2×C5—22no—
505C2×C5—41no—
506C2×C5C2×#93711yescharacteristic
507C2×C5C2×#94211yescharacteristic
508C2×C5#582211yescharacteristic
509D5—124no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—304no—
514C23—302no—
515C23—602no—
516C23—602no—
517C23C2×#122011yescharacteristic
518C23C2×#122011yescharacteristic
519C23C2×#122011yescharacteristic
520Q8—24no—
521Q8—42no—
522Q8—154no—
523Q8—304no—
524Q8—302no—
525D4—154no—
526D4—304no—
527D4—308no—
528D4—308no—
529D4—302no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—308no—
540C2×C4—304no—
541C2×C4—302no—
542C2×C4—308no—
543C2×C4D4×D1514yes—
544C2×C4C2×#122011yescharacteristic
545C2×C4C2×#122011yescharacteristic
546C2×C4C2×#122011yescharacteristic
547C2×C4C23×D1511yescharacteristic
548C8—304no—
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—41no—
552C2×C3C2×#408311yescharacteristic
553C2×C3C2×#408811yescharacteristic
554C2×C3#6615611yescharacteristic
555D3—204no—
556C5#8273511yescharacteristic, complemented, Sylow
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—44no—
565C22—304no—
566C22—608no—
567C22—602no—
568C22—602no—
569C22#9182612yes—
570C22#9183112yescomplemented
571C22C22×#122011yescharacteristic
572C4—21no—
573C4—21no—
574C4—42no—
575C4—42no—
576C4—304no—
577C4#9182612yes—
578C4C2×D4×D1512yes—
579C3#1061446611yescharacteristic, complemented, Sylow
580C2—22no—
581C2—22no—
582C2—41no—
583C2—604no—
584C2C2×#9182611yescharacteristic
585C2C2×#9183111yescharacteristic
586C2#1175658111yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC22×C3×C5C22×D4
prime radicalC22×#6817C2
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5#3007
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×C22⋊Q8C2semidirect
3. C2×#91831C2semidirect
4. #11751308C2semidirect
5. #11751151C2semidirect
6. #11751148C2semidirect
7. #11751147C2semidirect
8. #11751067C2semidirect
10. C22×C3×C5×Q8C22semidirect
24. C3×C5×C22⋊Q8C22semidirect
26. C2×C3×C5×C4⋊C4C22semidirect
28. C2×#6718C22semidirect
30. C2×#6717C22semidirect
31. C2×#6710C22semidirect
33. #91114C22semidirect
34. #91094C22semidirect
35. #91093C22semidirect
38. C2×C5×C22⋊Q8D3semidirect
89. C2×C3×C22⋊Q8D5semidirect
96. C22×C5×Q8C2×D3semidirect
110. C5×C22⋊Q8C2×D3semidirect
112. C2×C5×C4⋊C4C2×D3semidirect
162. C22×C3×Q8C2×D5semidirect
176. C3×C22⋊Q8C2×D5semidirect
178. C2×C3×C4⋊C4C2×D5semidirect
233. C2×C22⋊Q8D15semidirect
248. C22×C3×C5#137semidirect
349. C22×Q8C2×D15semidirect
361. C22⋊Q8C2×D15semidirect
363. C2×C4⋊C4C2×D15semidirect
424. C22×C5#942semidirect
480. C3×C5#3007semidirect
490. C22×C3#4088semidirect
556. C5#82735semidirect
570. C22#91831semidirect
579. C3#10614466semidirect

Maximal subgroups

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

subgroupindex
C22×#68172
C2×C3×C5×C22⋊Q82
C2×#918312
#117513082
#117511512
#117511482
#117511472
#117510672
#106144663
#827355