Finite Group Databasebeta

← #12098881

Subgroups — #12098881

Order 1280.

1590 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×M16C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C2×#65410C211yesmaximal, characteristic, complemented
4#10613998C211yesmaximal, characteristic
5C2×#65276C211yesmaximal, characteristic
6#10602374C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10601424C212yesmaximal, complemented
9#10601423C212yesmaximal, complemented
10#10601371C212yesmaximal, complemented
11#10601370C212yesmaximal, complemented
12C23×C4×D5C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×#685—21no—
17C2×#4072C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C2×D5×M16—24no—
21C2×D5×M16—24no—
22C2×D5×M16C418yescomplemented
23C22×#676C2212yes—
24C22×C8×D5C2212yes—
25C5×#428—24no—
26C2×C5×#107C2212yescomplemented
27C2×C5×#106C2212yescomplemented
28C2×#3963—21no—
29#65410—24no—
30C2×#3950C2212yescomplemented
31C2×#3949C2212yescomplemented
32#65276—21no—
33#65276—21no—
34#65276C2211yescharacteristic
35#65276C2212yes—
36#65276C2211yescharacteristic
37#65276C2212yes—
38#65276C2211yescharacteristic
39#65276C2211yescharacteristic
40#65244—21no—
41#65232—21no—
42C4×#685—21no—
43C4×#685—21no—
44#64749C2212yescomplemented
45#64749C2212yescomplemented
46#64749C2212yescomplemented
47#64749C2212yescomplemented
48#20661—51nomaximal, Sylow
49C24×D5D411yescharacteristic
50C23×C4×C5—21no—
51C23×C4×C5C2311yescharacteristic
52C23×Dic5—21no—
53C23×Dic5D411yescharacteristic
54C22×C4×D5—22no—
55C22×C4×D5—22no—
56C22×C4×D5—22no—
57C22×C4×D5—22no—
58C22×C4×D5C2×C414yes—
59C22×C4×D5D411yescharacteristic
60C22×C4×D5D411yescharacteristic
61C22×C4×D5Q814yes—
62C22×C4×D5C2311yescharacteristic
63C22×C4×D5C2311yescharacteristic
64C2×C5×M16C2×C414yes—
65C2×C5×M16D414yes—
66C2×C5×M16D414yes—
67C22×C5×C8C2312yes—
68C5×#117—22no—
69C5×#117—22no—
70C2×C5×C4⋊C4—22no—
71C2×C5×C4⋊C4—22no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2312yes—
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C42×C5—21no—
78C2×C22⋊Dic5—21no—
79C2×#685—22no—
80C2×#685—22no—
81C2×#685—21no—
82C2×#685—21no—
83C2×#685—21no—
84C2×#685—21no—
85C2×#685—44no—
86#4072—22no—
87#4072—22no—
88C2×C4⋊Dic5—22no—
89C2×C4⋊Dic5—22no—
90C2×C4⋊Dic5—21no—
91C2×C4⋊Dic5C2312yes—
92C2×C4⋊Dic5C2311yescharacteristic
93C2×C4⋊Dic5C2311yescharacteristic
94C2×C4×Dic5—21no—
95C2×#681—22no—
96C2×#681—22no—
97C2×#681C2×C414yes—
98C22×#168C2312yes—
99D5×M16—232no—
100C2×#676—24no—
101C2×#676—24no—
102C2×#676C2×C418yescomplemented
103C2×C8×D5—24no—
104C2×C8×D5—24no—
105C2×C8×D5C2×C418yescomplemented
106C5×#107—24no—
107C5×#106—24no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—22no—
113#3963—22no—
114#3950—24no—
115#3949—24no—
116C23×M16—51no—
117C2×#428—51no—
118C2×#428D511yescharacteristic, complemented
119#2280—51no—
120C2×#380—51no—
121#1809—52no—
122#1809—52no—
123#1808—52no—
124#1807—52no—
125#1736—51no—
126#1736—51no—
127C24×C5C2×D411yescharacteristic
128C23×D5—22no—
129C23×D5—22no—
130C23×D5—22no—
131C23×D5—22no—
132C23×D5—41no—
133C23×D5—41no—
134C23×D5C4⋊C414yes—
135C23×D5C2×D411yescharacteristic
136C23×D5C2×D411yescharacteristic
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—22no—
140C22×C4×C5—42no—
141C22×C4×C5—42no—
142C22×C4×C5—42no—
143C22×C4×C5C22×C412yes—
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D412yes—
147C22×C4×C5C2×Q812yes—
148C22×C4×C5#4112yes—
149C22×C4×C5#4112yes—
150C22×C4×C5C2411yescharacteristic
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—22no—
156C22×Dic5—42no—
157C22×Dic5—42no—
158C22×Dic5—42no—
159C22×Dic5C4⋊C414yes—
160C22×Dic5C2×D411yescharacteristic
161C22×Dic5C2×D411yescharacteristic
162C2×C4×D5—28no—
163C2×C4×D5—28no—
164C2×C4×D5—22no—
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—28no—
169C2×C4×D5—28no—
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C2×C4×D5—21no—
173C2×C4×D5—21no—
174C2×C4×D5—21no—
175C2×C4×D5—21no—
176C2×C4×D5—21no—
177C2×C4×D5—21no—
178C2×C4×D5C4⋊C414yes—
179C2×C4×D5C4⋊C414yes—
180C2×C4×D5C22×C412yes—
181C2×C4×D5C22×C412yes—
182C2×C4×D5C2×Q812yes—
183C2×C4×D5C2×Q812yes—
184C5×M16C22⋊C4116yescomplemented
185C2×C5×C8C22×C414yes—
186C2×C5×C8C2×D414yes—
187C2×C5×C8C2×D414yes—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—42no—
195C5×C22⋊C4—42no—
196C42×C5—42no—
197C22⋊Dic5—42no—
198#685—42no—
199#685—42no—
200#685—44no—
201#685—42no—
202#685—44no—
203#685—42no—
204C4⋊Dic5—21no—
205C4⋊Dic5—22no—
206C4⋊Dic5—21no—
207C4⋊Dic5—22no—
208C4⋊Dic5—21no—
209C4⋊Dic5—21no—
210C4⋊Dic5—42no—
211C4×Dic5—42no—
212#681—28no—
213C2×#168—22no—
214C2×#168—22no—
215C2×#168C22×C414yes—
216#676—216no—
217C8×D5—216no—
218C24×C4—51no—
219C22×M16—51no—
220C22×M16—58no—
221C22×M16—52no—
222C22×M16—104no—
223C22×M16—104no—
224C22×M16C2×D511yescharacteristic
225C23×C8—52no—
226C2×#117—51no—
227C2×#117C2×D511yescharacteristic
228C22×C4⋊C4—51no—
229C22×C4⋊C4C2×D511yescharacteristic
230C22×C22⋊C4—101no—
231#428—24no—
232#428—104no—
233C2×#107—52no—
234C2×#107C2×D512yescomplemented
235C2×#106—52no—
236C2×#106C2×D512yescomplemented
237#381—101no—
238#380—51no—
239#380—52no—
240#380—51no—
241#380—52no—
242#380—51no—
243#380—51no—
244#380—101no—
245#380—101no—
246#379—101no—
247C4×C22⋊C4—101no—
248C4×C22⋊C4—101no—
249C2×#95—101no—
250#340—52no—
251#340—52no—
252#340—52no—
253#340—52no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5C2×C4⋊C412yes—
259C23×C5#12112yes—
260C23×C5#12112yes—
261C23×C5C22×D411yescharacteristic
262C22×D5—21no—
263C22×D5—21no—
264C22×D5—21no—
265C22×D5—21no—
266C22×D5—21no—
267C22×D5—21no—
268C22×D5—21no—
269C22×D5—21no—
270C22×D5—21no—
271C22×D5—21no—
272C22×D5—28no—
273C22×D5—28no—
274C22×D5—44no—
275C22×D5—44no—
276C22×D5—44no—
277C22×D5—44no—
278C22×D5—44no—
279C22×D5C2×C4⋊C412yes—
280C22×D5C2×C4⋊C412yes—
281C22×D5AGL(1,ℤ/8)12yes—
282C22×D5#13712yes—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—42no—
286C2×C4×C5—42no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—42no—
292C2×C4×C5—84no—
293C2×C4×C5C2×C22⋊C414yes—
294C2×C4×C5C2×C4⋊C412yes—
295C2×C4×C5#11714yes—
296C2×C4×C5#12112yes—
297C2×C4×C5#12112yes—
298C2×C4×C5C22⋊Q814yes—
299C2×C4×C5C22⋊Q814yes—
300C2×C4×C5C23×C411yescharacteristic
301C2×C4×C5C22×D411yescharacteristic
302C2×C4×C5C22×D411yescharacteristic
303C2×C4×C5C22×Q811yescharacteristic
304C2×C4×C5C2×#4111yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—28no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—42no—
320C2×Dic5—84no—
321C2×Dic5C2×C4⋊C412yes—
322C2×Dic5C2×C4⋊C412yes—
323C4×D5—22no—
324C4×D5—22no—
325C4×D5—22no—
326C4×D5—22no—
327C4×D5—22no—
328C4×D5—22no—
329C4×D5—22no—
330C4×D5—22no—
331C4×D5—28no—
332C4×D5—28no—
333C5×C8C2×C22⋊C418yescomplemented
334#168—24no—
335C25—51no—
336C23×C4—21no—
337C23×C4—51no—
338C23×C4—54no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—54no—
342C23×C4—51no—
343C23×C4—51no—
344C23×C4—102no—
345C23×C4—102no—
346C23×C4—102no—
347C23×C4—102no—
348C23×C4—101no—
349C23×C4C22×D511yescharacteristic
350C2×M16—54no—
351C2×M16—58no—
352C2×M16—102no—
353C2×M16—102no—
354C2×M16—1032no—
355C2×M16—104no—
356C2×M16—104no—
357C2×M16C4×D514yes—
358C2×M16D2014yes—
359C2×M16C22⋊D514yes—
360C22×C8—52no—
361C22×C8—58no—
362C22×C8—104no—
363C22×C8—104no—
364C22×C8C22×D512yes—
365#117—22no—
366#117—22no—
367#117—102no—
368#117—102no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—52no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—51no—
375C2×C4⋊C4—102no—
376C2×C4⋊C4—102no—
377C2×C4⋊C4—101no—
378C2×C4⋊C4C22×D512yes—
379C2×C4⋊C4C22×D511yescharacteristic
380C2×C4⋊C4C22×D511yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—102no—
383C2×C22⋊C4—102no—
384C2×C22⋊C4—101no—
385C2×C22⋊C4—101no—
386C2×C22⋊C4—101no—
387C2×C22⋊C4—101no—
388C2×C22⋊C4—101no—
389C2×C22⋊C4—204no—
390C2×C42—21no—
391C2×C42—101no—
392#107—24no—
393#107—104no—
394#106—24no—
395#106—104no—
396#95—102no—
397#95—102no—
398#95—101no—
399#95—101no—
400#95—101no—
401#95—101no—
402C22×C5—22no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—22no—
408C22×C5#38014yes—
409C22×C5C22×C4⋊C411yescharacteristic
410C22×C5C2×#12111yescharacteristic
411C22×C5C2×#12111yescharacteristic
412C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C5C2×#13711yescharacteristic
414C22×C5#57511yescharacteristic
415C22×C5#57511yescharacteristic
416C2×D5—22no—
417C2×D5—22no—
418C2×D5—22no—
419C2×D5—22no—
420C2×D5—22no—
421C2×D5—22no—
422C2×D5—22no—
423C2×D5—22no—
424C2×D5—22no—
425C2×D5—416no—
426C2×D5—41no—
427C2×D5—41no—
428C2×D5—41no—
429C2×D5—41no—
430C2×D5#42814yescomplemented
431C4×C5—81no—
432C4×C5—82no—
433C4×C5—81no—
434C4×C5#38014yes—
435C4×C5C22×C22⋊C411yescharacteristic
436C4×C5C2×#11711yescharacteristic
437C4×C5C2×C22⋊Q811yescharacteristic
438C4×C5C2×C22⋊Q811yescharacteristic
439Dic5—22no—
440Dic5—22no—
441Dic5—81no—
442Dic5—82no—
443Dic5—81no—
444C24—51no—
445C24—51no—
446C24—54no—
447C24—102no—
448C24—102no—
449C24—102no—
450C24—102no—
451C24—201no—
452C24—201no—
453C24D4×D511yescharacteristic
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—51no—
461C22×C4—51no—
462C22×C4—54no—
463C22×C4—52no—
464C22×C4—52no—
465C22×C4—52no—
466C22×C4—52no—
467C22×C4—54no—
468C22×C4—54no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—101no—
472C22×C4—101no—
473C22×C4—101no—
474C22×C4—101no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4—101no—
479C22×C4—108no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—108no—
483C22×C4—108no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—108no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—102no—
490C22×C4—202no—
491C22×C4—202no—
492C22×C4—202no—
493C22×C4C2×C4×D512yes—
494C22×C4C2×D2012yes—
495C22×C4D4×D511yescharacteristic
496C22×C4#71112yes—
497C22×C4Q8×D512yes—
498C22×C4#71312yes—
499C22×C4C2×C22⋊D512yes—
500C22×C4C23×D511yescharacteristic
501M16—108no—
502M16—1016no—
503M16#685116yescomplemented
504C2×C8—54no—
505C2×C8—102no—
506C2×C8—102no—
507C2×C8—1016no—
508C2×C8C2×C4×D514yes—
509C2×C8C2×D2014yes—
510C2×C8C2×C22⋊D514yes—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—101no—
519C4⋊C4—102no—
520C4⋊C4—101no—
521C4⋊C4—102no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—202no—
525C22⋊C4—42no—
526C22⋊C4—204no—
527C22⋊C4—202no—
528C22⋊C4—202no—
529C22⋊C4—204no—
530C22⋊C4—202no—
531C22⋊C4—202no—
532C22⋊C4—202no—
533C42—42no—
534C42—202no—
535C2×C5—24no—
536C2×C5C2×#38011yescharacteristic
537C2×C5C2×#42811yescharacteristic
538C2×C5#291111yescharacteristic
539C2×C5#305511yescharacteristic
540C2×C5#305511yescharacteristic
541C2×C5#305611yescharacteristic
542C2×C5#305611yescharacteristic
543D5—24no—
544D5—42no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—52no—
550C23—52no—
551C23—52no—
552C23—52no—
553C23—108no—
554C23—101no—
555C23—101no—
556C23—101no—
557C23—101no—
558C23—101no—
559C23—101no—
560C23—101no—
561C23—101no—
562C23—101no—
563C23—101no—
564C23—108no—
565C23—204no—
566C23—204no—
567C23—204no—
568C23—204no—
569C23—204no—
570C23D5×C4⋊C412yes—
571C23#404112yes—
572C23#408412yes—
573C23C2×D4×D511yescharacteristic
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—52no—
584C2×C4—52no—
585C2×C4—84no—
586C2×C4—101no—
587C2×C4—101no—
588C2×C4—101no—
589C2×C4—101no—
590C2×C4—108no—
591C2×C4—102no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—102no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—108no—
600C2×C4—108no—
601C2×C4—201no—
602C2×C4—201no—
603C2×C4—202no—
604C2×C4—202no—
605C2×C4—201no—
606C2×C4—201no—
607C2×C4—201no—
608C2×C4—201no—
609C2×C4—202no—
610C2×C4—404no—
611C2×C4D5×C4⋊C412yes—
612C2×C4#403814yes—
613C2×C4#404112yes—
614C2×C4#404314yes—
615C2×C4C2×#68514yes—
616C2×C4#408412yes—
617C2×C4#409214yes—
618C2×C4C22×C4×D511yescharacteristic
619C2×C4C22×D2011yescharacteristic
620C2×C4C2×#71111yescharacteristic
621C2×C4C2×Q8×D511yescharacteristic
622C2×C4C2×#71311yescharacteristic
623C2×C4C22×C22⋊D511yescharacteristic
624C8—104no—
625C8C2×#68518yescomplemented
626C5#2066111yescharacteristic, complemented, Sylow
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—54no—
634C22—102no—
635C22—102no—
636C22—102no—
637C22—102no—
638C22—102no—
639C22—102no—
640C22—102no—
641C22—102no—
642C22—102no—
643C22—2016no—
644C22—201no—
645C22—201no—
646C22—201no—
647C22—201no—
648C22#6527614yes—
649C22C2×D5×C4⋊C411yescharacteristic
650C22C2×#404111yescharacteristic
651C22D5×AGL(1,ℤ/8)11yescharacteristic
652C22#6610711yescharacteristic
653C22D5×#13711yescharacteristic
654C22#6611111yescharacteristic
655C22C2×#408411yescharacteristic
656C4—82no—
657C4—81no—
658C4—81no—
659C4—102no—
660C4—102no—
661C4—402no—
662C4—401no—
663C4—401no—
664C4#6527614yes—
665C4C2×#403811yescharacteristic
666C4C2×#404311yescharacteristic
667C4C22×#68511yescharacteristic
668C4C2×#409211yescharacteristic
669C2—24no—
670C2—104no—
671C2—202no—
672C2D5×#42811yescharacteristic
673C2#1061093911yescharacteristic
674C2#1061094211yescharacteristic
675C2#1061094311yescharacteristic
676C2C2×#6527611yescharacteristic
677C2#1061568111yescharacteristic
678C2#1061571611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC24×D5D4
cosocleC22×C4×C5C24
hypercenterC2×#428D5
hypocenterC5#20661
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
3. C2×#65410C2semidirect
6. #10602374C2semidirect
7. #10602351C2semidirect
8. #10601424C2semidirect
9. #10601423C2semidirect
10. #10601371C2semidirect
11. #10601370C2semidirect
22. C2×D5×M16C4semidirect
26. C2×C5×#107C22semidirect
27. C2×C5×#106C22semidirect
30. C2×#3950C22semidirect
31. C2×#3949C22semidirect
44. #64749C22semidirect
45. #64749C22semidirect
46. #64749C22semidirect
47. #64749C22semidirect
102. C2×#676C2×C4semidirect
105. C2×C8×D5C2×C4semidirect
118. C2×#428D5semidirect
184. C5×M16C22⋊C4semidirect
234. C2×#107C2×D5semidirect
236. C2×#106C2×D5semidirect
333. C5×C8C2×C22⋊C4semidirect
430. C2×D5#428semidirect
503. M16#685semidirect
625. C8C2×#685semidirect
626. C5#20661semidirect

Maximal subgroups

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

subgroupindex
C22×D5×M162
C2×C5×#4282
C2×#654102
#106139982
C2×#652762
#106023742
#106023512
#106014242
#106014232
#106013712
#106013702
#206615