Finite Group Databasebeta

← #12027810

Subgroups — #12027810

Order 1280.

1614 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4090C211yesmaximal, characteristic
2C2×C5×#391C211yesmaximal, characteristic
3C2×#65519C211yesmaximal, characteristic
4#10602359C212yesmaximal
5#10602329C212yesmaximal
6#10600217C212yesmaximal
7#10600203C212yesmaximal
8#10600198C212yesmaximal
9#10600195C212yesmaximal
10C23×C5×Q8C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C42×C5C2211yescharacteristic
14C2×#4090—24no—
15C2×#4090—24no—
16C2×#4090C418yescomplemented
17C22×C4⋊Dic5C2211yescharacteristic
18C22×#683C2212yes—
19C22×#683C2212yes—
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C5×#391—24no—
23C2×C5×#95C2212yes—
24C2×C5×#95C2212yes—
25#65519—24no—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36#14450—51nomaximal, Sylow
37C22×C5×Q8—41no—
38C22×C5×Q8—41no—
39C22×C5×Q8—41no—
40C22×C5×Q8—41no—
41C22×C5×Q8C2×C414yes—
42C22×C5×Q8D414yes—
43C22×C5×Q8D414yes—
44C23×C4×C5C2312yes—
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×Dic5C2312yes—
50C23×Dic5C2312yes—
51C23×Dic5C2312yes—
52C23×Dic5C2312yes—
53C22×Dic10—22no—
54C22×Dic10—22no—
55C22×Dic10—41no—
56C22×Dic10—41no—
57C22×Dic10—41no—
58C22×Dic10—41no—
59C22×Dic10C2×C414yes—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4—41no—
62C2×C5×C4⋊C4—41no—
63C2×C42×C5—22no—
64C2×C42×C5—41no—
65C2×C42×C5—41no—
66#4090—232no—
67C2×C4⋊Dic5—22no—
68C2×C4⋊Dic5—41no—
69C2×C4⋊Dic5—41no—
70C2×#683—24no—
71C2×#683—24no—
72C2×#683—24no—
73C2×#683—24no—
74C2×#683C2×C418yes—
75C2×#683C2×C418yes—
76C2×C4×Dic5—22no—
77C2×C4×Dic5—22no—
78C2×C4×Dic5—22no—
79C2×C4×Dic5—41no—
80C2×C4×Dic5—41no—
81C2×C4×Dic5C2×C414yes—
82C5×#95—24no—
83C5×#95—24no—
84#3963—24no—
85#3963—24no—
86#3963—24no—
87#3963—24no—
88#3963—24no—
89#3963—42no—
90#3963—42no—
91#3963—42no—
92#3963—42no—
93#3963Q818yescomplemented
94C22×C4⋊Q8—51no—
95C2×#391—51no—
96C2×#391D511yescharacteristic
97#1446—52no—
98#1441—52no—
99#1441—52no—
100#1436—52no—
101#1436—52no—
102#1434—52no—
103C24×C5C2411yescharacteristic
104C2×C5×Q8—44no—
105C2×C5×Q8—44no—
106C2×C5×Q8—44no—
107C2×C5×Q8—44no—
108C2×C5×Q8—44no—
109C2×C5×Q8—44no—
110C2×C5×Q8C22⋊C4116yes—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5C22×C412yes—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C22×C412yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×D412yes—
141C22×C4×C5C2×D412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D412yes—
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×Q814yes—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5C22×C414yes—
170C22×Dic5C22×C414yes—
171C22×Dic5C2×Q814yes—
172C22×Dic5C2×Q814yes—
173C22×Dic5#4114yes—
174C2×Dic10—28no—
175C2×Dic10—44no—
176C2×Dic10—44no—
177C2×Dic10—44no—
178C2×Dic10—44no—
179C2×Dic10—44no—
180C2×Dic10—44no—
181C5×C4⋊C4—44no—
182C42×C5—44no—
183C4⋊Dic5—44no—
184#683—216no—
185#683—216no—
186C4×Dic5—28no—
187C4×Dic5—44no—
188C23×Q8—51no—
189C23×Q8C2×D511yescharacteristic
190C2×C4⋊Q8—58no—
191C2×C4⋊Q8—104no—
192C2×C4⋊Q8—104no—
193C22×C4⋊C4—52no—
194C22×C4⋊C4—52no—
195C22×C4⋊C4—51no—
196C22×C4⋊C4C2×D511yescharacteristic
197C22×C42—51no—
198C22×C42—51no—
199C22×C42C2×D511yescharacteristic
200#391—24no—
201#391—104no—
202C2×#95—52no—
203C2×#95—52no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—52no—
207C2×#95—52no—
208C2×#95—52no—
209C2×#95—52no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95C2×D512yes—
213C2×#95C2×D512yes—
214C23×C5C23×C411yescharacteristic
215C23×C5C22×D411yescharacteristic
216C23×C5C22×D411yescharacteristic
217C23×C5C22×Q812yes—
218C23×C5C2×#4112yes—
219C23×C5H(5,𝔽2)11yescharacteristic
220C23×C5H(5,𝔽2)11yescharacteristic
221C23×C5H(5,𝔽2)11yescharacteristic
222C23×C5H(5,𝔽2)11yescharacteristic
223C23×C52−1+412yes—
224C23×C52−1+412yes—
225C5×Q8—416no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—22no—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—22no—
234C2×C4×C5—22no—
235C2×C4×C5—22no—
236C2×C4×C5—22no—
237C2×C4×C5—22no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—24no—
245C2×C4×C5—24no—
246C2×C4×C5—24no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5C2×C22⋊C414yes—
259C2×C4×C5C2×C22⋊C414yes—
260C2×C4×C5C2×C22⋊C414yes—
261C2×Dic5—24no—
262C2×Dic5—24no—
263C2×Dic5—24no—
264C2×Dic5—24no—
265C2×Dic5—24no—
266C2×Dic5—24no—
267C2×Dic5—24no—
268C2×Dic5—24no—
269C2×Dic5—24no—
270C2×Dic5—24no—
271C2×Dic5—24no—
272C2×Dic5—24no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284C2×Dic5—42no—
285C2×Dic5—42no—
286C2×Dic5—42no—
287C2×Dic5—42no—
288C2×Dic5C4×Q818yes—
289Dic10—416no—
290C22×Q8—41no—
291C22×Q8—41no—
292C22×Q8—41no—
293C22×Q8—41no—
294C22×Q8—54no—
295C22×Q8—102no—
296C22×Q8—102no—
297C22×Q8—201no—
298C22×Q8—201no—
299C22×Q8—201no—
300C22×Q8—201no—
301C22×Q8C4×D514yes—
302C22×Q8D2014yes—
303C22×Q8C22⋊D514yes—
304C23×C4—52no—
305C23×C4—52no—
306C23×C4—52no—
307C23×C4—52no—
308C23×C4C22×D512yes—
309C23×C4C22×D512yes—
310C23×C4C22×D511yescharacteristic
311C23×C4C22×D511yescharacteristic
312C23×C4C22×D511yescharacteristic
313C4⋊Q8—1032no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—41no—
316C2×C4⋊C4—41no—
317C2×C4⋊C4—58no—
318C2×C4⋊C4—58no—
319C2×C4⋊C4—104no—
320C2×C4⋊C4—104no—
321C2×C4⋊C4—104no—
322C2×C4⋊C4—104no—
323C2×C4⋊C4—102no—
324C2×C4⋊C4—201no—
325C2×C4⋊C4—201no—
326C2×C42—22no—
327C2×C42—41no—
328C2×C42—41no—
329C2×C42—54no—
330C2×C42—102no—
331C2×C42—102no—
332C2×C42—102no—
333C2×C42—201no—
334C2×C42—201no—
335#95—24no—
336#95—24no—
337#95—58no—
338#95—104no—
339#95—104no—
340#95—104no—
341#95—104no—
342#95—104no—
343#95—202no—
344#95—202no—
345#95—202no—
346#95—202no—
347C22×C5C22×C22⋊C411yescharacteristic
348C22×C5C2×C4×Q812yes—
349C22×C5#51811yescharacteristic
350C22×C5#51811yescharacteristic
351C22×C5#51912yes—
352C22×C5#53411yescharacteristic
353C22×C5#53411yescharacteristic
354C22×C5#53411yescharacteristic
355C22×C5#53411yescharacteristic
356C22×C5#54212yes—
357C22×C5#54212yes—
358C22×C5#54412yes—
359C22×C5#54412yes—
360C22×C5#54812yes—
361C22×C5#54812yes—
362C22×C5D4×Q812yes—
363C22×C5D4×Q812yes—
364C22×C5#55412yes—
365C22×C5#55511yescharacteristic
366C22×C5Q8211yescharacteristic
367C22×C5#56311yescharacteristic
368C22×C5#56312yes—
369C22×C5#56311yescharacteristic
370C4×C5—24no—
371C4×C5—24no—
372C4×C5—24no—
373C4×C5—44no—
374C4×C5—82no—
375Dic5—28no—
376Dic5—44no—
377Dic5—44no—
378Dic5—82no—
379C24C23×D511yescharacteristic
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—108no—
387C2×Q8—204no—
388C2×Q8—204no—
389C2×Q8—204no—
390C2×Q8—204no—
391C2×Q8—204no—
392C2×Q8—204no—
393C2×Q8#685116yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—54no—
420C22×C4—54no—
421C22×C4—54no—
422C22×C4—54no—
423C22×C4—54no—
424C22×C4—102no—
425C22×C4—102no—
426C22×C4—102no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—102no—
430C22×C4—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4—102no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—102no—
444C22×C4—102no—
445C22×C4—102no—
446C22×C4—102no—
447C22×C4C2×C4×D512yes—
448C22×C4C2×C4×D512yes—
449C22×C4C2×C4×D512yes—
450C22×C4C2×D2012yes—
451C22×C4C2×D2012yes—
452C22×C4C2×D2012yes—
453C22×C4Q8×D514yes—
454C22×C4C2×C22⋊D512yes—
455C22×C4C2×C22⋊D512yes—
456C22×C4C2×C22⋊D512yes—
457C4⋊C4—44no—
458C4⋊C4—1016no—
459C4⋊C4—1016no—
460C4⋊C4—204no—
461C42—44no—
462C42—108no—
463C42—204no—
464C2×C5#230811yescharacteristic
465C2×C5#233112yes—
466C2×C5Q8×C22⋊C412yes—
467C2×C5#236911yescharacteristic
468C2×C5#238111yescharacteristic
469C2×C5#272912yes—
470C2×C5#272912yes—
471C2×C5#273111yescharacteristic
472C2×C5#273111yescharacteristic
473C2×C5#273411yescharacteristic
474C2×C5#273411yescharacteristic
475C23C22×C4×D511yescharacteristic
476C23C22×D2011yescharacteristic
477C23#414411yescharacteristic
478C23#414411yescharacteristic
479C23#414411yescharacteristic
480C23#414411yescharacteristic
481C23C2×Q8×D512yes—
482C23#414712yes—
483C23D5×#4112yes—
484C23#415012yes—
485C23C22×C22⋊D511yescharacteristic
486Q8—416no—
487Q8—2016no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—58no—
521C2×C4—104no—
522C2×C4—104no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—104no—
526C2×C4—104no—
527C2×C4—104no—
528C2×C4—104no—
529C2×C4—104no—
530C2×C4—104no—
531C2×C4—104no—
532C2×C4—104no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—202no—
539C2×C4—202no—
540C2×C4—202no—
541C2×C4—202no—
542C2×C4—202no—
543C2×C4—202no—
544C2×C4—202no—
545C2×C4—202no—
546C2×C4—202no—
547C2×C4—202no—
548C2×C4C2×#68514yes—
549C2×C4C2×#68514yes—
550C2×C4C2×#68514yes—
551C5#1445011yescharacteristic, complemented, Sylow
552C22#6582011yescharacteristic
553C22#6582011yescharacteristic
554C22#6582711yescharacteristic
555C22#6582711yescharacteristic
556C22C4×Q8×D512yes—
557C22#6590612yes—
558C22Q8×D2012yes—
559C22#6591012yes—
560C22#6595712yes—
561C22#6597012yes—
562C22#6597412yes—
563C22#6598012yes—
564C22#6600312yes—
565C22#6600411yescharacteristic
566C22#6600611yescharacteristic
567C22#6605611yescharacteristic
568C22#6606312yes—
569C22#6606911yescharacteristic
570C22C22×#68511yescharacteristic
571C22#6613811yescharacteristic
572C22#6613811yescharacteristic
573C22Q8×C22⋊D512yes—
574C22#6616512yes—
575C4—24no—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—82no—
580C4—108no—
581C4—204no—
582C4—204no—
583C4—402no—
584C2#1060821712yes—
585C2#1060822511yescharacteristic
586C2#1060822811yescharacteristic
587C2#1060830311yescharacteristic
588C2#1060830511yescharacteristic
589C2#1060838612yes—
590C2#1061385211yescharacteristic
591C2Q8×#68512yes—
592C2#1061453612yes—
593C2#1061472111yescharacteristic
594C2#1061498111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#391C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×#391D5
hypocenterC5#14450
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
16. C2×#4090C4semidirect
93. #3963Q8semidirect
551. C5#14450semidirect

Maximal subgroups

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

subgroupindex
C22×#40902
C2×C5×#3912
C2×#655192
#106023592
#106023292
#106002172
#106002032
#106001982
#106001952
#144505