Finite Group Databasebeta

← #12022974

Subgroups — #12022974

Order 1280.

1862 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×#3963C211yesmaximal, characteristic
2C2×#65235C211yesmaximal, characteristic
3C2×C22⋊C4×Dic5C211yesmaximal, characteristic
4C5×C22⋊#95C211yesmaximal, characteristic
5#10602920C211yesmaximal, characteristic
6#10602337C211yesmaximal, characteristic
7#10602337C211yesmaximal, characteristic
8#10600214C212yesmaximal, complemented
9#10600211C212yesmaximal, complemented
10#10600198C212yesmaximal, complemented
11#10600194C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×Dic5C2211yescharacteristic
14C24×Dic5C2211yescharacteristic
15C22×C5×C22⋊C4C2211yescharacteristic
16C22×C5×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2211yescharacteristic
19C22×#683C2212yes—
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C2×C5×#95C2212yes—
23C2×C5×#95C2212yes—
24C2×#3963—24no—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2211yescharacteristic
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36#65235C418yescomplemented
37C22⋊C4×Dic5—24no—
38#14040—51nomaximal, Sylow
39C25×C5C2311yescharacteristic
40C23×C4×C5—22no—
41C23×C4×C5—24no—
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—42no—
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2312yes—
49C23×Dic5—22no—
50C23×Dic5—22no—
51C23×Dic5—22no—
52C23×Dic5—22no—
53C23×Dic5—24no—
54C23×Dic5—42no—
55C23×Dic5—42no—
56C23×Dic5—42no—
57C23×Dic5C2×C414yes—
58C23×Dic5D414yes—
59C23×Dic5C2312yes—
60C23×Dic5C2312yes—
61C23×Dic5C2311yescharacteristic
62C23×Dic5C2311yescharacteristic
63C23×Dic5C2311yescharacteristic
64C23×Dic5C2311yescharacteristic
65C2×C5×C22⋊C4—22no—
66C2×C5×C22⋊C4—22no—
67C2×C5×C22⋊C4—22no—
68C2×C5×C22⋊C4—22no—
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4C2×C414yes—
71C2×C22⋊Dic5—22no—
72C2×C22⋊Dic5—22no—
73C2×C22⋊Dic5—22no—
74C2×C22⋊Dic5—22no—
75C2×C22⋊Dic5C2×C414yes—
76C2×C22⋊Dic5Q814yes—
77C2×#683—42no—
78C2×#683—42no—
79C2×#683C2×C418yescomplemented
80C2×C4×Dic5—22no—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—42no—
84C5×#95—24no—
85C5×#95—42no—
86#3963—24no—
87#3963—22no—
88#3963—22no—
89#3963—24no—
90#3963—24no—
91#3963—24no—
92#3963—42no—
93#3963—42no—
94#3963—42no—
95#3963—42no—
96#3963C2×C418yescomplemented
97C2×#385—51no—
98C2×C4×C22⋊C4—51no—
99C22×#95—51no—
100#1445—52no—
101#1443—52no—
102#1436—52no—
103#1434—52no—
104C22⋊#95—51no—
105C22⋊#95—51no—
106C22⋊#95—51no—
107C22⋊#95D511yescharacteristic
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—22no—
112C24×C5C22×C412yes—
113C24×C5C2×D412yes—
114C24×C5C2×Q812yes—
115C24×C5#4112yes—
116C24×C5#4112yes—
117C24×C5#4112yes—
118C24×C5#4112yes—
119C24×C5C2411yescharacteristic
120C22×C4×C5—28no—
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C22×C4×C5—21no—
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×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—48no—
143C22×C4×C5—48no—
144C22×C4×C5—44no—
145C22×C4×C5—44no—
146C22×C4×C5C22×C412yes—
147C22×C4×C5C22×C412yes—
148C22×C4×C5#4114yes—
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5—22no—
152C22×Dic5—28no—
153C22×Dic5—28no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—22no—
163C22×Dic5—21no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—28no—
179C22×Dic5—48no—
180C22×Dic5—44no—
181C22×Dic5—48no—
182C22×Dic5—48no—
183C22×Dic5—44no—
184C22×Dic5—44no—
185C22×Dic5—48no—
186C22×Dic5C22×C414yes—
187C22×Dic5C22×C414yes—
188C22×Dic5C2×D412yes—
189C22×Dic5C2×D412yes—
190C22×Dic5C2×Q812yes—
191C22×Dic5C2×Q812yes—
192C22×Dic5#4114yes—
193C5×C22⋊C4—28no—
194C5×C22⋊C4—44no—
195C22⋊Dic5—28no—
196C22⋊Dic5—28no—
197#683—48no—
198C4×Dic5—44no—
199C4×Dic5—44no—
200C24×C4—51no—
201C24×C4—51no—
202C24×C4C2×D511yescharacteristic
203C22×C4⋊C4—52no—
204C22×C22⋊C4—51no—
205C22×C22⋊C4—51no—
206C22×C22⋊C4C2×D511yescharacteristic
207C22×C22⋊C4C2×D511yescharacteristic
208C22×C42—51no—
209C22×C42—51no—
210#385—58no—
211C4×C22⋊C4—104no—
212C2×#95—52no—
213C2×#95—52no—
214C2×#95—52no—
215C2×#95—52no—
216C2×#95—51no—
217C2×#95—52no—
218C2×#95—51no—
219C2×#95—52no—
220C2×#95—52no—
221C2×#95—52no—
222C2×#95—52no—
223C2×#95—104no—
224C2×#95C2×D512yes—
225C2×#95C2×D512yes—
226C23×C5—24no—
227C23×C5—24no—
228C23×C5—22no—
229C23×C5—22no—
230C23×C5—24no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—24no—
234C23×C5—24no—
235C23×C5—24no—
236C23×C5—24no—
237C23×C5—24no—
238C23×C5—24no—
239C23×C5—24no—
240C23×C5—24no—
241C23×C5—24no—
242C23×C5#11714yes—
243C23×C5C4×D414yes—
244C23×C5C4×Q814yes—
245C23×C5C22⋊Q814yes—
246C23×C5#12314yes—
247C23×C5#12514yes—
248C23×C5#12614yes—
249C23×C5C23×C411yescharacteristic
250C23×C5C22×D411yescharacteristic
251C23×C5C22×Q811yescharacteristic
252C23×C5C2×#4111yescharacteristic
253C23×C5C2×#4111yescharacteristic
254C23×C5C2×#4111yescharacteristic
255C23×C5C2×#4111yescharacteristic
256C23×C5C2×#4112yes—
257C23×C5C2×#4112yes—
258C23×C5H(5,𝔽2)12yes—
259C23×C5H(5,𝔽2)12yes—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—24no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—416no—
268C2×C4×C5—41no—
269C2×C4×C5—42no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—24no—
288C2×Dic5—24no—
289C2×Dic5—24no—
290C2×Dic5—24no—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—24no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—416no—
303C2×Dic5—416no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—42no—
308C2×Dic5—41no—
309C2×Dic5—42no—
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—42no—
320C2×Dic5#12314yes—
321C25C22×D511yescharacteristic
322C23×C4—22no—
323C23×C4—24no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—42no—
327C23×C4—51no—
328C23×C4—52no—
329C23×C4—51no—
330C23×C4—51no—
331C23×C4—52no—
332C23×C4—51no—
333C23×C4—54no—
334C23×C4—54no—
335C23×C4—104no—
336C23×C4—102no—
337C23×C4—102no—
338C23×C4—102no—
339C23×C4—102no—
340C23×C4—202no—
341C23×C4—202no—
342C23×C4—202no—
343C23×C4C22×D512yes—
344C23×C4C22×D511yescharacteristic
345C23×C4C22×D511yescharacteristic
346C23×C4C22×D512yes—
347C2×C4⋊C4—58no—
348C2×C4⋊C4—202no—
349C2×C4⋊C4—202no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—54no—
356C2×C22⋊C4—54no—
357C2×C22⋊C4—102no—
358C2×C22⋊C4—102no—
359C2×C22⋊C4—102no—
360C2×C22⋊C4—102no—
361C2×C22⋊C4C4×D514yes—
362C2×C42—102no—
363C2×C42—102no—
364C2×C42—104no—
365C2×C42—202no—
366#95—24no—
367#95—42no—
368#95—58no—
369#95—104no—
370#95—104no—
371#95—102no—
372#95—102no—
373#95—104no—
374#95—104no—
375#95—202no—
376#95—202no—
377#95—202no—
378#95—202no—
379C22×C5—24no—
380C22×C5—24no—
381C22×C5—28no—
382C22×C5—28no—
383C22×C5—24no—
384C22×C5—24no—
385C22×C5—24no—
386C22×C5—28no—
387C22×C5—28no—
388C22×C5—24no—
389C22×C5#38718yes—
390C22×C5C2×#11711yescharacteristic
391C22×C5C2×C4×D411yescharacteristic
392C22×C5C2×C4×Q811yescharacteristic
393C22×C5C4×#4112yes—
394C22×C5#51812yes—
395C22×C5C2×C22⋊Q811yescharacteristic
396C22×C5C2×#12311yescharacteristic
397C22×C5C2×#12311yescharacteristic
398C22×C5#52511yescharacteristic
399C22×C5C2×#12511yescharacteristic
400C22×C5C2×#12611yescharacteristic
401C22×C5#52912yes—
402C22×C5#53411yescharacteristic
403C22×C5#53511yescharacteristic
404C22×C5#53812yes—
405C22×C5#55112yes—
406C22×C5#55112yes—
407C22×C5#55112yes—
408C22×C5#55112yes—
409C22×C5#55312yes—
410C22×C5#55312yes—
411C22×C5#55412yes—
412C22×C5#55412yes—
413C4×C5—42no—
414C4×C5—44no—
415C4×C5—42no—
416C4×C5—82no—
417Dic5—24no—
418Dic5—44no—
419Dic5—42no—
420Dic5—44no—
421Dic5—42no—
422Dic5—42no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24C2×C4×D512yes—
428C24#70912yes—
429C24#70912yes—
430C24D4×D512yes—
431C24#71112yes—
432C24#71112yes—
433C24Q8×D512yes—
434C24C23×D511yescharacteristic
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—52no—
462C22×C4—54no—
463C22×C4—54no—
464C22×C4—54no—
465C22×C4—52no—
466C22×C4—52no—
467C22×C4—52no—
468C22×C4—108no—
469C22×C4—101no—
470C22×C4—101no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—101no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—102no—
478C22×C4—101no—
479C22×C4—101no—
480C22×C4—102no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—101no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—102no—
492C22×C4—102no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—108no—
497C22×C4—108no—
498C22×C4—204no—
499C22×C4—208no—
500C22×C4—208no—
501C22×C4—208no—
502C22×C4—204no—
503C22×C4—208no—
504C22×C4—204no—
505C22×C4C2×C4×D512yes—
506C22×C4C2×C4×D512yes—
507C22×C4#71114yes—
508C4⋊C4—208no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—108no—
512C22⋊C4—108no—
513C42—204no—
514C42—204no—
515C2×C5—28no—
516C2×C5C2×#38711yescharacteristic
517C2×C5C4×#12311yescharacteristic
518C2×C5#232111yescharacteristic
519C2×C5#235312yes—
520C2×C5#236812yes—
521C2×C5#248211yescharacteristic
522C2×C5#248311yescharacteristic
523C2×C5#248711yescharacteristic
524C2×C5#250512yes—
525C2×C5#251012yes—
526C2×C5#257911yescharacteristic
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23#401814yes—
544C23#402514yes—
545C23#402714yes—
546C23#402914yes—
547C23#403314yes—
548C23#403514yes—
549C23#404214yes—
550C23C22×C4×D511yescharacteristic
551C23C2×#70911yescharacteristic
552C23C2×#70911yescharacteristic
553C23C2×D4×D511yescharacteristic
554C23C2×#71111yescharacteristic
555C23C2×#71111yescharacteristic
556C23C2×#71112yes—
557C23#414412yes—
558C23C2×Q8×D511yescharacteristic
559C23D5×#4112yes—
560C23#414912yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—416no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—54no—
587C2×C4—102no—
588C2×C4—104no—
589C2×C4—102no—
590C2×C4—104no—
591C2×C4—104no—
592C2×C4—104no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—104no—
600C2×C4—104no—
601C2×C4—104no—
602C2×C4—102no—
603C2×C4—102no—
604C2×C4—2016no—
605C2×C4—2016no—
606C2×C4—202no—
607C2×C4—202no—
608C2×C4—202no—
609C2×C4—202no—
610C2×C4—202no—
611C2×C4—202no—
612C2×C4—202no—
613C2×C4—201no—
614C2×C4—202no—
615C2×C4—201no—
616C2×C4—202no—
617C2×C4—202no—
618C2×C4—201no—
619C2×C4—201no—
620C2×C4—202no—
621C2×C4—202no—
622C5#1404011yescharacteristic, complemented, Sylow
623C22—24no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—28no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—24no—
633C22#6494118yes—
634C22C2×#401811yescharacteristic
635C22C2×#402511yescharacteristic
636C22C2×#402711yescharacteristic
637C22C2×#402911yescharacteristic
638C22#6582912yes—
639C22C2×#403311yescharacteristic
640C22C2×#403511yescharacteristic
641C22C2×#404211yescharacteristic
642C22#6586112yes—
643C22C4×#71112yes—
644C22#6587412yes—
645C22#6587912yes—
646C22#6589912yes—
647C22#6592011yescharacteristic
648C22#6592112yes—
649C22#6592611yescharacteristic
650C22#6593412yes—
651C22#6595112yes—
652C22#6596912yes—
653C22#6597012yes—
654C22D5×#12311yescharacteristic
655C22#6598711yescharacteristic
656C22#6599212yes—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—82no—
661C4—104no—
662C4—204no—
663C4—202no—
664C4—202no—
665C4—202no—
666C4—204no—
667C2—28no—
668C2C2×#6494111yescharacteristic
669C2#1060519312yes—
670C2#1060556311yescharacteristic
671C2#1060565812yes—
672C2#1060586111yescharacteristic
673C2#1060613012yes—
674C2#1060618811yescharacteristic
675C2#1060621611yescharacteristic
676C2#1060639211yescharacteristic
677C2#1060647412yes—
678C2#1060660611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×C22⋊#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22⋊#95D5
hypocenterC5#14040
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
8. #10600214C2semidirect
9. #10600211C2semidirect
10. #10600198C2semidirect
11. #10600194C2semidirect
36. #65235C4semidirect
79. C2×#683C2×C4semidirect
96. #3963C2×C4semidirect
622. C5#14040semidirect

Maximal subgroups

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

subgroupindex
C22×#39632
C2×#652352
C2×C22⋊C4×Dic52
C5×C22⋊#952
#106029202
#106023372
#106002142
#106002112
#106001982
#106001942
#140405