Finite Group Databasebeta

← #12022954

Subgroups — #12022954

Order 1280.

2222 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#683C211yesmaximal, characteristic
2C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
3C2×#65235C211yesmaximal, characteristic
4C2×#65222C212yesmaximal, complemented
5#10602920C211yesmaximal, characteristic
6#10602337C211yesmaximal, characteristic
7#10602337C211yesmaximal, characteristic
8#10602337C211yesmaximal, characteristic
9#10602329C212yesmaximal, complemented
10#10600207C212yesmaximal, complemented
11#10600202C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×Dic5C2211yescharacteristic
14C24×Dic5C2211yescharacteristic
15C22×C5×C22⋊C4C2211yescharacteristic
16C22×C5×C22⋊C4C2211yescharacteristic
17C22×C42×C5C2211yescharacteristic
18C22×C42×C5C2211yescharacteristic
19C22×C22⋊Dic5C2211yescharacteristic
20C22×C22⋊Dic5C2211yescharacteristic
21C22×#683—28no—
22C22×#683—24no—
23C22×#683C2212yes—
24C22×#683C2212yes—
25C22×#683C2212yes—
26C4×C5×C22⋊C4—24no—
27C2×C5×#95C2211yescharacteristic
28C2×C5×#95C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38#65235C418yescomplemented
39#65222—28no—
40#14039—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—24no—
45C23×C4×C5—42no—
46C23×C4×C5D414yes—
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2311yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C23×Dic5—22no—
54C23×Dic5—22no—
55C23×Dic5—24no—
56C23×Dic5—24no—
57C23×Dic5—22no—
58C23×Dic5—24no—
59C23×Dic5—22no—
60C23×Dic5—22no—
61C23×Dic5—42no—
62C23×Dic5C2×C414yes—
63C23×Dic5C2312yes—
64C23×Dic5C2312yes—
65C23×Dic5C2312yes—
66C23×Dic5C2312yes—
67C2×C5×C22⋊C4—22no—
68C2×C5×C22⋊C4—22no—
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4—22no—
71C2×C5×C22⋊C4C2×C414yes—
72C2×C5×C22⋊C4Q814yes—
73C2×C42×C5—22no—
74C2×C42×C5—22no—
75C2×C42×C5—24no—
76C2×C42×C5—42no—
77C2×C22⋊Dic5—22no—
78C2×C22⋊Dic5—22no—
79C2×C22⋊Dic5—22no—
80C2×C22⋊Dic5—22no—
81C2×C22⋊Dic5—22no—
82C2×C22⋊Dic5C2×C414yes—
83C2×#683—232no—
84C2×#683—24no—
85C2×#683—24no—
86C2×#683—24no—
87C2×#683—24no—
88C2×#683—24no—
89C2×#683—24no—
90C2×#683—24no—
91C2×#683C2×C418yescomplemented
92C2×#683D418yes—
93C5×#95—22no—
94C5×#95—22no—
95#3963—24no—
96#3963—24no—
97#3963—24no—
98#3963—24no—
99#3963—24no—
100#3963—42no—
101#3963—42no—
102#3963—42no—
103#3963C2×C418yescomplemented
104C23×C4⋊C4—51no—
105C2×#387—52no—
106C2×#385—51no—
107C2×C4×C22⋊C4D511yescharacteristic
108#1442—52no—
109#1441—52no—
110#1436—52no—
111C22⋊#95—51no—
112C22⋊#95—51no—
113C22⋊#95—51no—
114C22⋊#95—51no—
115C24×C5—22no—
116C24×C5—22no—
117C24×C5—22no—
118C24×C5—22no—
119C24×C5C22×C412yes—
120C24×C5C2×D412yes—
121C24×C5C2×Q812yes—
122C24×C5#4112yes—
123C24×C5#4112yes—
124C24×C5#4112yes—
125C24×C5#4112yes—
126C24×C5C2411yescharacteristic
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—28no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—28no—
152C22×C4×C5—44no—
153C22×C4×C5—48no—
154C22×C4×C5—48no—
155C22×C4×C5C22×C412yes—
156C22×C4×C5C22×C412yes—
157C22×C4×C5C2×D414yes—
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×Q812yes—
161C22×C4×C5C2×Q812yes—
162C22×Dic5—216no—
163C22×Dic5—28no—
164C22×Dic5—28no—
165C22×Dic5—28no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—216no—
171C22×Dic5—28no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5—22no—
185C22×Dic5—22no—
186C22×Dic5—22no—
187C22×Dic5—22no—
188C22×Dic5—22no—
189C22×Dic5—22no—
190C22×Dic5—22no—
191C22×Dic5—48no—
192C22×Dic5—48no—
193C22×Dic5—44no—
194C22×Dic5—44no—
195C22×Dic5C22×C414yes—
196C22×Dic5C22×C414yes—
197C22×Dic5C2×D414yes—
198C22×Dic5C2×D414yes—
199C22×Dic5#4114yes—
200C5×C22⋊C4—28no—
201C5×C22⋊C4—28no—
202C42×C5—44no—
203C42×C5—44no—
204C22⋊Dic5—28no—
205C22⋊Dic5—44no—
206#683—216no—
207#683—216no—
208#683—216no—
209C24×C4—51no—
210C24×C4—51no—
211C24×C4C2×D511yescharacteristic
212C22×C4⋊C4—52no—
213C22×C4⋊C4—52no—
214C22×C4⋊C4—52no—
215C22×C4⋊C4—108no—
216C22×C4⋊C4—104no—
217C22×C22⋊C4—51no—
218C22×C22⋊C4—51no—
219C22×C22⋊C4C2×D511yescharacteristic
220C22×C22⋊C4C2×D511yescharacteristic
221C22×C42C2×D511yescharacteristic
222C22×C42C2×D511yescharacteristic
223#387—108no—
224#385—58no—
225C4×C22⋊C4—24no—
226C2×#95—52no—
227C2×#95—52no—
228C2×#95—52no—
229C2×#95—52no—
230C2×#95—52no—
231C2×#95—52no—
232C2×#95—52no—
233C2×#95—52no—
234C2×#95—52no—
235C2×#95C2×D511yescharacteristic
236C2×#95C2×D511yescharacteristic
237C23×C5—24no—
238C23×C5—24no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—24no—
242C23×C5—22no—
243C23×C5—22no—
244C23×C5—24no—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5—24no—
248C23×C5—24no—
249C23×C5—24no—
250C23×C5—24no—
251C23×C5—24no—
252C23×C5—24no—
253C23×C5#11714yes—
254C23×C5C4×D414yes—
255C23×C5C4×Q814yes—
256C23×C5C22⋊Q814yes—
257C23×C5#12314yes—
258C23×C5#12514yes—
259C23×C5#12614yes—
260C23×C5C23×C411yescharacteristic
261C23×C5C22×D412yes—
262C23×C5C22×D411yescharacteristic
263C23×C5C22×Q811yescharacteristic
264C23×C5C2×#4111yescharacteristic
265C23×C5C2×#4111yescharacteristic
266C23×C5C2×#4111yescharacteristic
267C23×C5C2×#4111yescharacteristic
268C23×C5C2×#4112yes—
269C23×C5H(5,𝔽2)12yes—
270C23×C52−1+412yes—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—24no—
274C2×C4×C5—24no—
275C2×C4×C5—24no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—24no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—416no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—42no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—42no—
297C2×C4×C5—42no—
298C2×C4×C5—42no—
299C2×C4×C5—42no—
300C2×C4×C5—42no—
301C2×C4×C5#12014yes—
302C2×Dic5—232no—
303C2×Dic5—24no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—24no—
307C2×Dic5—24no—
308C2×Dic5—24no—
309C2×Dic5—24no—
310C2×Dic5—24no—
311C2×Dic5—24no—
312C2×Dic5—24no—
313C2×Dic5—24no—
314C2×Dic5—24no—
315C2×Dic5—416no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—42no—
320C2×Dic5—42no—
321C2×Dic5—42no—
322C2×Dic5—42no—
323C2×Dic5—42no—
324C2×Dic5—42no—
325C2×Dic5—42no—
326C2×Dic5—42no—
327C2×Dic5—42no—
328C2×Dic5—42no—
329C2×Dic5—42no—
330C2×Dic5—42no—
331C2×Dic5C4×D418yes—
332C25C22×D511yescharacteristic
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—24no—
336C23×C4—42no—
337C23×C4—52no—
338C23×C4—54no—
339C23×C4—52no—
340C23×C4—52no—
341C23×C4—52no—
342C23×C4—104no—
343C23×C4—104no—
344C23×C4—104no—
345C23×C4—102no—
346C23×C4—102no—
347C23×C4—102no—
348C23×C4—102no—
349C23×C4—102no—
350C23×C4—202no—
351C23×C4C22⋊D514yes—
352C23×C4C22×D511yescharacteristic
353C23×C4C22×D511yescharacteristic
354C23×C4C22×D511yescharacteristic
355C23×C4C22×D511yescharacteristic
356C23×C4C22×D511yescharacteristic
357C23×C4C22×D511yescharacteristic
358C2×C4⋊C4—58no—
359C2×C4⋊C4—58no—
360C2×C4⋊C4—104no—
361C2×C4⋊C4—104no—
362C2×C4⋊C4—104no—
363C2×C4⋊C4—104no—
364C2×C4⋊C4—1032no—
365C2×C4⋊C4—104no—
366C2×C4⋊C4—104no—
367C2×C4⋊C4—104no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—54no—
373C2×C22⋊C4—102no—
374C2×C22⋊C4—102no—
375C2×C22⋊C4—102no—
376C2×C22⋊C4—102no—
377C2×C22⋊C4—102no—
378C2×C22⋊C4Dic1014yes—
379C2×C22⋊C4C4×D514yes—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—42no—
384#95—22no—
385#95—22no—
386#95—58no—
387#95—104no—
388#95—104no—
389#95—104no—
390#95—104no—
391#95—104no—
392#95—202no—
393#95—202no—
394#95—202no—
395C22×C5—24no—
396C22×C5—24no—
397C22×C5—28no—
398C22×C5—28no—
399C22×C5—24no—
400C22×C5—24no—
401C22×C5—28no—
402C22×C5—28no—
403C22×C5—24no—
404C22×C5—24no—
405C22×C5#38718yescomplemented
406C22×C5C2×#11711yescharacteristic
407C22×C5C2×C4×D411yescharacteristic
408C22×C5C2×C4×D412yes—
409C22×C5C2×C4×Q811yescharacteristic
410C22×C5#52012yes—
411C22×C5C2×#12011yescharacteristic
412C22×C5C2×C22⋊Q811yescharacteristic
413C22×C5C2×#12311yescharacteristic
414C22×C5#52512yes—
415C22×C5#52511yescharacteristic
416C22×C5C2×#12511yescharacteristic
417C22×C5C2×#12611yescharacteristic
418C22×C5#53411yescharacteristic
419C22×C5#53611yescharacteristic
420C22×C5#53912yes—
421C22×C5#54612yes—
422C22×C5#54612yes—
423C22×C5#54812yes—
424C22×C5D4×Q812yes—
425C22×C5#55212yes—
426C22×C5#55212yes—
427C22×C5#55312yes—
428C22×C5#55412yes—
429C4×C5—24no—
430C4×C5—42no—
431C4×C5—42no—
432C4×C5—42no—
433C4×C5—42no—
434C4×C5—42no—
435Dic5—28no—
436Dic5—44no—
437Dic5—44no—
438Dic5—82no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24C2×Dic1012yes—
444C24C2×C4×D512yes—
445C24#70912yes—
446C24#70912yes—
447C24#70912yes—
448C24#70912yes—
449C24C2×C22⋊D512yes—
450C24C23×D511yescharacteristic
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—54no—
480C22×C4—54no—
481C22×C4—54no—
482C22×C4—54no—
483C22×C4—54no—
484C22×C4—1016no—
485C22×C4—108no—
486C22×C4—1016no—
487C22×C4—108no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—108no—
493C22×C4—108no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—102no—
498C22×C4—102no—
499C22×C4—102no—
500C22×C4—102no—
501C22×C4—102no—
502C22×C4—102no—
503C22×C4—102no—
504C22×C4—102no—
505C22×C4—102no—
506C22×C4—102no—
507C22×C4—102no—
508C22×C4—102no—
509C22×C4—102no—
510C22×C4—102no—
511C22×C4—102no—
512C22×C4—102no—
513C22×C4—208no—
514C22×C4—208no—
515C22×C4—204no—
516C22×C4—204no—
517C22×C4C2×Dic1012yes—
518C22×C4C2×Dic1012yes—
519C22×C4C2×C4×D512yes—
520C22×C4C2×C4×D512yes—
521C22×C4D4×D514yes—
522C22×C4C2×C22⋊D512yes—
523C22×C4C2×C22⋊D512yes—
524C4⋊C4—1016no—
525C4⋊C4—1016no—
526C4⋊C4—1016no—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—108no—
530C22⋊C4—204no—
531C42—44no—
532C42—44no—
533C2×C5—28no—
534C2×C5C2×#38711yescharacteristic
535C2×C5C4×#12011yescharacteristic
536C2×C5#232211yescharacteristic
537C2×C5#235212yes—
538C2×C5#236012yes—
539C2×C5#240511yescharacteristic
540C2×C5#240911yescharacteristic
541C2×C5#247312yes—
542C2×C5#248112yes—
543C2×C5#248411yescharacteristic
544C2×C5#257911yescharacteristic
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—22no—
561C23C4×Dic1014yes—
562C23#401614yes—
563C23#401814yes—
564C23#402214yes—
565C23#407014yes—
566C23C4×C22⋊D514yes—
567C23#407514yes—
568C23C22×Dic1011yescharacteristic
569C23C22×C4×D511yescharacteristic
570C23C2×#70911yescharacteristic
571C23C2×#70911yescharacteristic
572C23C2×#70911yescharacteristic
573C23C2×#70911yescharacteristic
574C23C2×D4×D512yes—
575C23#414412yes—
576C23D5×#4112yes—
577C23#415012yes—
578C23C22×C22⋊D511yescharacteristic
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—58no—
610C2×C4—1032no—
611C2×C4—104no—
612C2×C4—104no—
613C2×C4—104no—
614C2×C4—104no—
615C2×C4—104no—
616C2×C4—104no—
617C2×C4—104no—
618C2×C4—104no—
619C2×C4—104no—
620C2×C4—104no—
621C2×C4—104no—
622C2×C4—104no—
623C2×C4—2016no—
624C2×C4—202no—
625C2×C4—202no—
626C2×C4—202no—
627C2×C4—202no—
628C2×C4—202no—
629C2×C4—202no—
630C2×C4—202no—
631C2×C4—202no—
632C2×C4—202no—
633C2×C4—202no—
634C2×C4—202no—
635C2×C4—202no—
636C2×C4—202no—
637C2×C4—202no—
638C2×C4—202no—
639C2×C4#408314yes—
640C5#1403911yescharacteristic, complemented, Sylow
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—24no—
651C22#6522218yescomplemented
652C22C2×C4×Dic1011yescharacteristic
653C22C2×#401611yescharacteristic
654C22C2×#401811yescharacteristic
655C22C2×#402211yescharacteristic
656C22#6582412yes—
657C22#6582412yes—
658C22#6586512yes—
659C22#6586512yes—
660C22D4×Dic1012yes—
661C22#6587312yes—
662C22#6587412yes—
663C22C4×D4×D512yes—
664C22#6588012yes—
665C22#6588112yes—
666C22#6588612yes—
667C22#6589812yes—
668C22C2×#407011yescharacteristic
669C22C2×C4×C22⋊D511yescharacteristic
670C22C2×#407511yescharacteristic
671C22C2×#408311yescharacteristic
672C22#6613811yescharacteristic
673C22#6615911yescharacteristic
674C22#6616211yescharacteristic
675C4—24no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—108no—
682C4—204no—
683C4—204no—
684C4—402no—
685C2—28no—
686C2#1060519212yes—
687C2#1060519712yes—
688C2#1060523212yes—
689C2#1060523412yes—
690C2C2×#6522211yescharacteristic
691C2#1061385711yescharacteristic
692C2#1061398511yescharacteristic
693C2#1061405711yescharacteristic
694C2#1061408011yescharacteristic
695C2#1061411811yescharacteristic
696C2C4×#408311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C22⋊C4C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5#14039
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#65222C2semidirect
9. #10602329C2semidirect
10. #10600207C2semidirect
11. #10600202C2semidirect
38. #65235C4semidirect
91. C2×#683C2×C4semidirect
103. #3963C2×C4semidirect
405. C22×C5#387semidirect
640. C5#14039semidirect
651. C22#65222semidirect

Maximal subgroups

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

subgroupindex
C23×#6832
C2×C4×C5×C22⋊C42
C2×#652352
C2×#652222
#106029202
#106023372
#106023292
#106002072
#106002022
#140395