Finite Group Databasebeta

← #12132346

Subgroups — #12132346

Order 1280.

1338 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
1C2×C5×D4×Q8C211yesmaximal, characteristic
2C22×#3967C211yesmaximal, characteristic, complemented
3C2×#65332C211yesmaximal, characteristic, complemented
4#10614104C211yesmaximal, characteristic
5#10602925C211yesmaximal, characteristic, complemented
6#10602925C212yesmaximal, complemented
7#10602574C212yesmaximal, complemented
8#10602489C211yesmaximal, characteristic, complemented
9#10602487C212yesmaximal, complemented
10#10602422C212yesmaximal, complemented
11#10602403C211yesmaximal, characteristic
12C23×C5×Q8C2211yescharacteristic
13C23×C5×Q8C2211yescharacteristic
14C5×D4×Q8—24no—
15C5×D4×Q8—24no—
16C2×C5×C4⋊Q8—21no—
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×C22⋊Q8—21no—
19C2×C5×C22⋊Q8—21no—
20C2×C5×C22⋊Q8C2211yescharacteristic
21C2×C5×C22⋊Q8C2211yescharacteristic
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×D4—21no—
24C2×C4×C5×D4C2211yescharacteristic
25C2×#4081—21no—
26C22×C4⋊Dic5C2211yescharacteristic
27C23×#168C2211yescharacteristic
28C2×#3967—24no—
29C2×#3967C2212yes—
30C2×#3967C2212yes—
31C2×#3967C2212yes—
32C2×#3967C2212yes—
33C2×#3967C2212yes—
34C2×#3967C2212yes—
35#65501C2212yes—
36#65501C2211yescharacteristic
37C2×C22⋊#168C2211yescharacteristic
38C2×C22⋊#168C2212yes—
39#65332—24no—
40#65272—21no—
41C2×#3940C2212yes—
42#65242—21no—
43#65232—21no—
44#65217C2212yes—
45#64749C2212yes—
46#64744C2212yes—
47#24041—51nomaximal, Sylow
48C22×C5×Q8—22no—
49C22×C5×Q8—22no—
50C22×C5×Q8—21no—
51C22×C5×Q8—21no—
52C22×C5×Q8—22no—
53C22×C5×Q8—24no—
54C22×C5×Q8—24no—
55C22×C5×Q8—42no—
56C22×C5×Q8—42no—
57C22×C5×Q8D414yes—
58C22×C5×Q8C2311yescharacteristic
59C22×C5×Q8C2311yescharacteristic
60C22×C5×Q8C2311yescharacteristic
61C22×C5×D4D411yescharacteristic
62C23×C4×C5—21no—
63C23×C4×C5—21no—
64C23×C4×C5C2311yescharacteristic
65C23×C4×C5C2311yescharacteristic
66C23×Dic5—21no—
67C5×C4⋊Q8—28no—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8—22no—
70C5×C22⋊Q8—28no—
71C5×C22⋊Q8—28no—
72C5×C22⋊Q8—22no—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8—22no—
75C5×C22⋊Q8D414yes—
76C4×C5×Q8—22no—
77C4×C5×Q8—22no—
78C4×C5×D4—28no—
79C4×C5×D4—22no—
80C4×C5×D4—22no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4D411yescharacteristic
91C2×C5×C22⋊C4D411yescharacteristic
92C2×C42×C5—21no—
93C2×C42×C5C2311yescharacteristic
94C2×C22⋊Dic5—21no—
95C2×C22⋊Dic5—22no—
96#4081—44no—
97C2×C4⋊Dic5—22no—
98C2×C4⋊Dic5—22no—
99C2×C4⋊Dic5—21no—
100C2×C4⋊Dic5C2312yes—
101C2×C4⋊Dic5C2312yes—
102C2×#683—22no—
103C22×#168—22no—
104C22×#168—42no—
105C22×#168C2312yes—
106C22×#168C2312yes—
107#3967—24no—
108#3967—24no—
109#3967—24no—
110#3967—24no—
111#3967—42no—
112#3967—42no—
113#3963—22no—
114#3963—22no—
115#3963—21no—
116C22⋊#168—22no—
117C22⋊#168—42no—
118#3940—24no—
119C2×D4×Q8D511yescharacteristic
120C2×#483—51no—
121C22×#103—51no—
122#2486—51no—
123#2043—52no—
124#2035—52no—
125#1892—51no—
126#1869—51no—
127#1865—52no—
128#1787—52no—
129#1787—51no—
130C24×C5C2×D411yescharacteristic
131C24×C5C2×D411yescharacteristic
132C2×C5×Q8—24no—
133C2×C5×Q8—24no—
134C2×C5×Q8—21no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—22no—
138C2×C5×Q8—28no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—22no—
142C2×C5×Q8—22no—
143C2×C5×Q8—44no—
144C2×C5×Q8—44no—
145C2×C5×Q8—44no—
146C2×C5×Q8—48no—
147C2×C5×Q8—48no—
148C2×C5×Q8—44no—
149C2×C5×Q8—44no—
150C2×C5×Q8C2×D412yes—
151C2×C5×Q8C2×D412yes—
152C2×C5×Q8C2×D412yes—
153C2×C5×Q8#4114yes—
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4SD1614yes—
157C22×C4×C5—21no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—24no—
161C22×C4×C5—24no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—44no—
170C22×C4×C5—44no—
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2×D412yes—
175C22×C4×C5C2×D412yes—
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2411yescharacteristic
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—42no—
181C22×Dic5—42no—
182C22×Dic5—42no—
183C5×C4⋊C4—24no—
184C5×C4⋊C4—24no—
185C5×C4⋊C4—24no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4C2×D412yes—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—22no—
195C42×C5—24no—
196C42×C5—21no—
197C42×C5—21no—
198C22⋊Dic5—42no—
199C22⋊Dic5—44no—
200C4⋊Dic5—22no—
201C4⋊Dic5—22no—
202C4⋊Dic5—22no—
203C4⋊Dic5—22no—
204C4⋊Dic5—42no—
205#683—44no—
206C2×#168—22no—
207C2×#168—22no—
208C2×#168—22no—
209C2×#168—22no—
210C2×#168—22no—
211C2×#168—22no—
212C2×#168—44no—
213C23×Q8C2×D511yescharacteristic
214C23×Q8C2×D511yescharacteristic
215C23×C8—51no—
216D4×Q8—24no—
217D4×Q8—24no—
218C2×C4⋊Q8—21no—
219C2×C4⋊Q8C2×D511yescharacteristic
220C2×#123—101no—
221C2×C22⋊Q8—21no—
222C2×C22⋊Q8—21no—
223C2×C22⋊Q8C2×D511yescharacteristic
224C2×C22⋊Q8C2×D511yescharacteristic
225C2×C4×Q8C2×D511yescharacteristic
226C2×C4×D4—21no—
227C2×C4×D4C2×D511yescharacteristic
228C22×C4⋊C4—51no—
229#483—104no—
230C2×#106—52no—
231C2×#103—52no—
232C2×#103—52no—
233C2×#103—52no—
234C2×#103—52no—
235C2×#103—52no—
236C2×#103—52no—
237C2×#103—104no—
238C2×C22⋊C8—51no—
239C2×C22⋊C8—52no—
240#398—101no—
241#397—101no—
242#389—52no—
243#381—101no—
244#380—51no—
245#380—52no—
246#340—52no—
247#336—52no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—21no—
251C23×C5—22no—
252C23×C5—21no—
253C23×C5—22no—
254C23×C5#12012yes—
255C23×C5C2×SD1612yes—
256C23×C5C2×SD1612yes—
257C23×C5#13712yes—
258C23×C5C22×D411yescharacteristic
259C5×Q8—22no—
260C5×Q8—24no—
261C5×Q8—22no—
262C5×Q8—22no—
263C5×Q8—24no—
264C5×Q8—48no—
265C5×Q8—48no—
266C5×Q8—41no—
267C5×Q8—41no—
268C5×Q8—42no—
269C5×Q8#12314yescomplemented
270C5×D4—28no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—24no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—24no—
279C2×C4×C5—21no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—21no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—48no—
287C2×C4×C5—48no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—42no—
294C2×C4×C5#12012yes—
295C2×C4×C5C2×SD1612yes—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C2×#4112yes—
299C2×C4×C5H(5,𝔽2)12yes—
300C2×Dic5—42no—
301C2×Dic5—42no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—84no—
307#168—42no—
308#168—42no—
309C22×Q8—22no—
310C22×Q8—22no—
311C22×Q8—21no—
312C22×Q8—21no—
313C22×Q8—22no—
314C22×Q8—24no—
315C22×Q8—24no—
316C22×Q8—42no—
317C22×Q8—42no—
318C22×Q8C22⋊D514yes—
319C22×Q8C22×D511yescharacteristic
320C22×Q8C22×D511yescharacteristic
321C22×Q8C22×D511yescharacteristic
322C22×D4C22⋊D511yescharacteristic
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—101no—
326C23×C4C22×D511yescharacteristic
327C23×C4C22×D511yescharacteristic
328C22×C8—52no—
329C22×C8—52no—
330C22×C8—102no—
331C22×C8—202no—
332C4⋊Q8—28no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335#123—204no—
336C22⋊Q8—28no—
337C22⋊Q8—28no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8C22⋊D514yes—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×D4—28no—
345C4×D4—22no—
346C4×D4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—52no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—102no—
355C2×C4⋊C4—102no—
356C2×C4⋊C4—101no—
357C2×C4⋊C4—102no—
358C2×C4⋊C4C22⋊D511yescharacteristic
359C2×C4⋊C4C22×D511yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—101no—
363C2×C22⋊C4—102no—
364C2×C22⋊C4C22⋊D511yescharacteristic
365C2×C22⋊C4C22⋊D511yescharacteristic
366C2×C42—21no—
367C2×C42C22×D511yescharacteristic
368#106—104no—
369#103—104no—
370#103—104no—
371#103—104no—
372#103—104no—
373#103—202no—
374#103—202no—
375C22⋊C8—102no—
376C22⋊C8—202no—
377#95—102no—
378#95—101no—
379#95—102no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—24no—
383C22×C5—24no—
384C22×C5—22no—
385C22×C5—24no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—42no—
389C22×C5#44814yes—
390C22×C5C2×#12011yescharacteristic
391C22×C5#54612yes—
392C22×C5C22×SD1611yescharacteristic
393C22×C5C2×#13711yescharacteristic
394C22×C5#57812yes—
395C4×C5—22no—
396C4×C5—24no—
397C4×C5—22no—
398C4×C5—41no—
399C4×C5—42no—
400C4×C5—42no—
401C4×C5—41no—
402C4×C5—42no—
403C4×C5C2×#12311yescharacteristic
404C4×C5#53411yescharacteristic
405C4×C5#54612yes—
406Dic5—82no—
407Dic5—82no—
408C24C2×C22⋊D511yescharacteristic
409C24C2×C22⋊D511yescharacteristic
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8#71114yes—
429C2×Q8C2×C22⋊D512yes—
430C2×Q8C2×C22⋊D512yes—
431C2×Q8C2×C22⋊D512yes—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4#68814yes—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—202no—
452C22×C4—202no—
453C22×C4—202no—
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C2×C22⋊D511yescharacteristic
456C22×C4C2×C22⋊D511yescharacteristic
457C22×C4C2×C22⋊D512yes—
458C22×C4C2×C22⋊D512yes—
459C22×C4C2×C22⋊D511yescharacteristic
460C22×C4C23×D511yescharacteristic
461C2×C8—102no—
462C2×C8—102no—
463C2×C8—102no—
464C2×C8—102no—
465C2×C8—102no—
466C2×C8—102no—
467C2×C8—204no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—102no—
476C4⋊C4—102no—
477C4⋊C4—102no—
478C4⋊C4—102no—
479C4⋊C4—202no—
480C4⋊C4—204no—
481C4⋊C4C2×C22⋊D512yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—202no—
487C22⋊C4—204no—
488C42—24no—
489C42—21no—
490C42—21no—
491C2×C5—24no—
492C2×C5—42no—
493C2×C5#240511yescharacteristic
494C2×C5C2×#44811yescharacteristic
495C2×C5#301111yescharacteristic
496C2×C5#329812yes—
497C2×C5#331212yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23C2×#68812yes—
505C23C2×#68812yes—
506C23#408812yes—
507C23#409912yes—
508C23C22×C22⋊D511yescharacteristic
509Q8—22no—
510Q8—24no—
511Q8—22no—
512Q8—22no—
513Q8—24no—
514Q8—48no—
515Q8—48no—
516Q8—41no—
517Q8—41no—
518Q8—42no—
519Q8#408114yescomplemented
520D4—28no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—202no—
545C2×C4—202no—
546C2×C4—202no—
547C2×C4—202no—
548C2×C4—202no—
549C2×C4—202no—
550C2×C4—404no—
551C2×C4C2×#68812yes—
552C2×C4#409912yes—
553C2×C4C2×#71112yes—
554C2×C4#414412yes—
555C2×C4C22×C22⋊D511yescharacteristic
556C2×C4C22×C22⋊D511yescharacteristic
557C8—202no—
558C8—202no—
559C5#2404111yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—22no—
565C22—24no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22#6551614yes—
570C22#6614012yes—
571C22C22×#68811yescharacteristic
572C22C2×#408811yescharacteristic
573C22#6617212yes—
574C22C2×#409911yescharacteristic
575C4—22no—
576C4—24no—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—42no—
583C4—402no—
584C4—402no—
585C4C2×#408111yescharacteristic
586C4#6613811yescharacteristic
587C4#6614012yes—
588C2—24no—
589C2—42no—
590C2#1061528112yes—
591C2#1061528312yes—
592C2#1061577811yescharacteristic
593C2C2×#6551611yescharacteristic
594C2#1061584811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×D4×Q8C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×D4×Q8D5
hypocenterC5#24041
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#3967C2semidirect
3. C2×#65332C2semidirect
5. #10602925C2semidirect
6. #10602925C2semidirect
7. #10602574C2semidirect
8. #10602489C2semidirect
9. #10602487C2semidirect
10. #10602422C2semidirect
269. C5×Q8#123semidirect
519. Q8#4081semidirect
559. C5#24041semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Q82
C22×#39672
C2×#653322
#106141042
#106029252
#106025742
#106024892
#106024872
#106024222
#106024032
#240415