Finite Group Databasebeta

← #12025893

Subgroups — #12025893

Order 1280.

2086 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4024C211yesmaximal, characteristic
2C22×#3963C211yesmaximal, characteristic
3C2×#65231C211yesmaximal, characteristic
4C5×C22⋊#95C211yesmaximal, characteristic
5#10602920C211yesmaximal, characteristic
6#10602337C211yesmaximal, characteristic
7#10602337C211yesmaximal, characteristic
8#10600217C212yesmaximal, complemented
9#10600214C212yesmaximal, complemented
10#10600204C212yesmaximal, complemented
11#10600202C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×Dic5C2211yescharacteristic
14C24×Dic5C2211yescharacteristic
15C23×Dic10C2211yescharacteristic
16C22×C5×C22⋊C4C2211yescharacteristic
17C22×C5×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic5C2211yescharacteristic
19C22×C22⋊Dic5C2211yescharacteristic
20C22×C4⋊Dic5C2211yescharacteristic
21C22×#683C2211yescharacteristic
22C22×#683C2211yescharacteristic
23C22×#683C2212yes—
24C2×#4024—24no—
25C2×#4024—24no—
26C2×#4024C418yescomplemented
27C2×C5×#95C2212yes—
28C2×C5×#95C2212yes—
29C2×#3963—24no—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38C2×#3963C2212yes—
39#65231—24no—
40#14281—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—42no—
45C23×C4×C5—42no—
46C23×C4×C5Q814yescomplemented
47C23×C4×C5C2312yes—
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2312yes—
51C23×Dic5—22no—
52C23×Dic5—22no—
53C23×Dic5—42no—
54C23×Dic5—42no—
55C23×Dic5—42no—
56C23×Dic5—42no—
57C23×Dic5C2×C414yes—
58C23×Dic5D414yes—
59C23×Dic5D414yes—
60C23×Dic5Q814yescomplemented
61C23×Dic5C2312yes—
62C23×Dic5C2311yescharacteristic
63C23×Dic5C2311yescharacteristic
64C23×Dic5C2312yes—
65C23×Dic5C2311yescharacteristic
66C23×Dic5C2311yescharacteristic
67C22×Dic10—22no—
68C22×Dic10—22no—
69C22×Dic10—42no—
70C22×Dic10—42no—
71C22×Dic10C2×C414yescomplemented
72C2×C5×C22⋊C4—22no—
73C2×C5×C22⋊C4—22no—
74C2×C5×C22⋊C4—22no—
75C2×C5×C22⋊C4—22no—
76C2×C5×C22⋊C4—22no—
77C2×C5×C22⋊C4C2×C414yes—
78C2×C22⋊Dic5—22no—
79C2×C22⋊Dic5—22no—
80C2×C22⋊Dic5—22no—
81C2×C22⋊Dic5—22no—
82C2×C22⋊Dic5—22no—
83C2×C22⋊Dic5C2×C414yes—
84C2×C4⋊Dic5—22no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5C2×C414yescomplemented
87C2×#683—22no—
88C2×#683—22no—
89C2×#683—22no—
90C2×#683—22no—
91C2×#683—24no—
92C2×#683—42no—
93C2×#683—42no—
94C2×#683C2×C414yescomplemented
95C2×#683C2×C414yescomplemented
96#4024—232no—
97C5×#95—42no—
98C5×#95—42no—
99#3963—24no—
100#3963—24no—
101#3963—24no—
102#3963—42no—
103#3963—42no—
104#3963—42no—
105#3963—42no—
106#3963—42no—
107#3963—42no—
108C22×C22⋊Q8—51no—
109C2×#380—51no—
110C22×#95—51no—
111#1446—52no—
112#1445—52no—
113#1441—52no—
114#1441—52no—
115C22⋊#95—51no—
116C22⋊#95—51no—
117C22⋊#95—51no—
118C22⋊#95D511yescharacteristic
119C24×C5—22no—
120C24×C5—22no—
121C24×C5—22no—
122C24×C5—22no—
123C24×C5C22×C412yes—
124C24×C5C2×D412yes—
125C24×C5C2×D412yes—
126C24×C5C2×Q812yes—
127C24×C5C2×Q812yes—
128C24×C5#4112yes—
129C24×C5#4112yes—
130C24×C5C2411yescharacteristic
131C22×C4×C5—28no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
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—22no—
143C22×C4×C5—22no—
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—22no—
152C22×C4×C5—22no—
153C22×C4×C5—48no—
154C22×C4×C5—44no—
155C22×C4×C5—44no—
156C22×C4×C5—48no—
157C22×C4×C5C22×C412yes—
158C22×C4×C5C22×C412yes—
159C22×C4×C5C2×Q814yes—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—28no—
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—21no—
168C22×Dic5—21no—
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—22no—
179C22×Dic5—22no—
180C22×Dic5—22no—
181C22×Dic5—22no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5—22no—
185C22×Dic5—22no—
186C22×Dic5—48no—
187C22×Dic5—48no—
188C22×Dic5—44no—
189C22×Dic5—44no—
190C22×Dic5—48no—
191C22×Dic5—44no—
192C22×Dic5—44no—
193C22×Dic5—48no—
194C22×Dic5C22⋊C4116yes—
195C22×Dic5C22×C412yes—
196C22×Dic5C22×C412yes—
197C22×Dic5C22×C412yes—
198C22×Dic5C22×C412yes—
199C22×Dic5C2×D412yes—
200C22×Dic5C2×D412yes—
201C22×Dic5C2×D412yes—
202C22×Dic5C2×D412yes—
203C22×Dic5C2×Q814yes—
204C2×Dic10—28no—
205C2×Dic10—48no—
206C2×Dic10—44no—
207C2×Dic10—48no—
208C2×Dic10—44no—
209C5×C22⋊C4—28no—
210C5×C22⋊C4—44no—
211C22⋊Dic5—28no—
212C22⋊Dic5—44no—
213C4⋊Dic5—28no—
214#683—28no—
215#683—28no—
216#683—48no—
217C23×Q8—51no—
218C24×C4—51no—
219C24×C4—51no—
220C24×C4C2×D511yescharacteristic
221C2×C22⋊Q8—58no—
222C2×C22⋊Q8—104no—
223C2×C22⋊Q8—104no—
224C22×C4⋊C4—51no—
225C22×C4⋊C4—51no—
226C22×C4⋊C4—51no—
227C22×C4⋊C4—52no—
228C22×C22⋊C4—51no—
229C22×C22⋊C4—51no—
230C22×C22⋊C4C2×D511yescharacteristic
231C22×C22⋊C4C2×D511yescharacteristic
232#380—104no—
233C2×#95—52no—
234C2×#95—52no—
235C2×#95—52no—
236C2×#95—52no—
237C2×#95—52no—
238C2×#95—52no—
239C2×#95—52no—
240C2×#95—52no—
241C2×#95—52no—
242C2×#95—104no—
243C2×#95C2×D512yes—
244C2×#95C2×D512yes—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—24no—
250C23×C5—22no—
251C23×C5—22no—
252C23×C5—24no—
253C23×C5—24no—
254C23×C5—24no—
255C23×C5—24no—
256C23×C5—24no—
257C23×C5—24no—
258C23×C5—24no—
259C23×C5—24no—
260C23×C5—24no—
261C23×C5C2×C22⋊C414yes—
262C23×C5C4×Q814yes—
263C23×C5C4×Q814yes—
264C23×C5C22⋊Q814yes—
265C23×C5C22⋊Q814yes—
266C23×C5#12414yes—
267C23×C5C4⋊Q814yes—
268C23×C5C23×C411yescharacteristic
269C23×C5C22×D411yescharacteristic
270C23×C5C22×D411yescharacteristic
271C23×C5C22×Q811yescharacteristic
272C23×C5C22×Q811yescharacteristic
273C23×C5C2×#4111yescharacteristic
274C23×C5C2×#4111yescharacteristic
275C23×C5H(5,𝔽2)12yes—
276C23×C5H(5,𝔽2)12yes—
277C23×C5H(5,𝔽2)12yes—
278C23×C52−1+412yes—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—24no—
282C2×C4×C5—24no—
283C2×C4×C5—24no—
284C2×C4×C5—24no—
285C2×C4×C5—24no—
286C2×C4×C5—416no—
287C2×C4×C5—41no—
288C2×C4×C5—42no—
289C2×C4×C5—41no—
290C2×C4×C5—42no—
291C2×C4×C5—41no—
292C2×C4×C5—42no—
293C2×C4×C5—41no—
294C2×C4×C5—42no—
295C2×C4×C5—42no—
296C2×C4×C5—42no—
297C2×C4×C5—42no—
298C2×C4×C5—42no—
299C2×C4×C5—42no—
300C2×C4×C5—42no—
301C2×C4×C5—42no—
302C2×C4×C5—42no—
303C2×C4×C5—42no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—24no—
307C2×Dic5—22no—
308C2×Dic5—24no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—22no—
314C2×Dic5—24no—
315C2×Dic5—24no—
316C2×Dic5—24no—
317C2×Dic5—24no—
318C2×Dic5—24no—
319C2×Dic5—416no—
320C2×Dic5—416no—
321C2×Dic5—41no—
322C2×Dic5—42no—
323C2×Dic5—41no—
324C2×Dic5—42no—
325C2×Dic5—41no—
326C2×Dic5—42no—
327C2×Dic5—41no—
328C2×Dic5—42no—
329C2×Dic5—42no—
330C2×Dic5—42no—
331C2×Dic5—42no—
332C2×Dic5—42no—
333C2×Dic5—42no—
334C2×Dic5—42no—
335C2×Dic5—42no—
336C2×Dic5—42no—
337C2×Dic5—42no—
338C2×Dic5C2×C22⋊C414yes—
339C2×Dic5C2×C22⋊C414yes—
340Dic10—416no—
341C25C22×D511yescharacteristic
342C22×Q8—54no—
343C22×Q8—102no—
344C22×Q8—102no—
345C22×Q8—202no—
346C22×Q8—202no—
347C23×C4—22no—
348C23×C4—22no—
349C23×C4—42no—
350C23×C4—42no—
351C23×C4—51no—
352C23×C4—51no—
353C23×C4—54no—
354C23×C4—54no—
355C23×C4—54no—
356C23×C4—51no—
357C23×C4—51no—
358C23×C4—52no—
359C23×C4—54no—
360C23×C4—52no—
361C23×C4—102no—
362C23×C4—102no—
363C23×C4—202no—
364C23×C4—202no—
365C23×C4—202no—
366C23×C4—202no—
367C23×C4Dic1014yescomplemented
368C23×C4C22×D512yes—
369C23×C4C22×D512yes—
370C23×C4C22×D511yescharacteristic
371C23×C4C22×D511yescharacteristic
372C22⋊Q8—1032no—
373C2×C4⋊C4—54no—
374C2×C4⋊C4—54no—
375C2×C4⋊C4—54no—
376C2×C4⋊C4—102no—
377C2×C4⋊C4—102no—
378C2×C4⋊C4—102no—
379C2×C4⋊C4—102no—
380C2×C4⋊C4—102no—
381C2×C4⋊C4—102no—
382C2×C4⋊C4—104no—
383C2×C4⋊C4—202no—
384C2×C4⋊C4—202no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—54no—
391C2×C22⋊C4—102no—
392C2×C22⋊C4—102no—
393C2×C22⋊C4—102no—
394C2×C22⋊C4—102no—
395C2×C22⋊C4—102no—
396C2×C22⋊C4C4×D514yes—
397#95—42no—
398#95—42no—
399#95—104no—
400#95—104no—
401#95—104no—
402#95—202no—
403#95—202no—
404#95—202no—
405#95—202no—
406#95—202no—
407#95—202no—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5—28no—
411C22×C5—28no—
412C22×C5—24no—
413C22×C5—24no—
414C22×C5—24no—
415C22×C5—24no—
416C22×C5—28no—
417C22×C5—28no—
418C22×C5#39118yes—
419C22×C5C22×C22⋊C411yescharacteristic
420C22×C5C2×C4×Q811yescharacteristic
421C22×C5C2×C4×Q811yescharacteristic
422C22×C5#51812yes—
423C22×C5#52012yes—
424C22×C5C2×C22⋊Q811yescharacteristic
425C22×C5C2×C22⋊Q811yescharacteristic
426C22×C5C2×#12411yescharacteristic
427C22×C5C2×C4⋊Q811yescharacteristic
428C22×C5#53411yescharacteristic
429C22×C5#53412yes—
430C22×C5#53411yescharacteristic
431C22×C5#53511yescharacteristic
432C22×C5#53611yescharacteristic
433C22×C5#53712yes—
434C22×C5#53812yes—
435C22×C5#53812yes—
436C22×C5#53912yes—
437C22×C5#53912yes—
438C22×C5#54312yes—
439C22×C5#54312yes—
440C22×C5#54412yes—
441C22×C5#54412yes—
442C4×C5—42no—
443C4×C5—42no—
444C4×C5—44no—
445C4×C5—82no—
446Dic5—24no—
447Dic5—24no—
448Dic5—42no—
449Dic5—42no—
450Dic5—44no—
451Dic5—82no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24C2×Dic1012yes—
457C24C2×C4×D512yes—
458C24#70912yes—
459C24D4×D512yes—
460C24D4×D512yes—
461C24#71112yes—
462C24Q8×D512yes—
463C24C23×D511yescharacteristic
464C2×Q8—108no—
465C2×Q8—208no—
466C2×Q8—204no—
467C2×Q8—208no—
468C2×Q8—204no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—48no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—52no—
496C22×C4—52no—
497C22×C4—52no—
498C22×C4—52no—
499C22×C4—52no—
500C22×C4—52no—
501C22×C4—516no—
502C22×C4—52no—
503C22×C4—52no—
504C22×C4—54no—
505C22×C4—102no—
506C22×C4—102no—
507C22×C4—102no—
508C22×C4—102no—
509C22×C4—102no—
510C22×C4—102no—
511C22×C4—101no—
512C22×C4—101no—
513C22×C4—101no—
514C22×C4—101no—
515C22×C4—102no—
516C22×C4—102no—
517C22×C4—102no—
518C22×C4—102no—
519C22×C4—102no—
520C22×C4—102no—
521C22×C4—102no—
522C22×C4—102no—
523C22×C4—108no—
524C22×C4—102no—
525C22×C4—102no—
526C22×C4—102no—
527C22×C4—102no—
528C22×C4—102no—
529C22×C4—102no—
530C22×C4—102no—
531C22×C4—208no—
532C22×C4—204no—
533C22×C4—208no—
534C22×C4—204no—
535C22×C4—208no—
536C22×C4—208no—
537C22×C4—204no—
538C22×C4—204no—
539C22×C4C2×Dic1014yes—
540C22×C4C2×C4×D512yes—
541C22×C4C2×C4×D512yes—
542C4⋊C4—108no—
543C4⋊C4—108no—
544C4⋊C4—108no—
545C4⋊C4—208no—
546C22⋊C4—28no—
547C22⋊C4—44no—
548C22⋊C4—108no—
549C22⋊C4—204no—
550C2×C5—28no—
551C2×C5C2×#39111yescharacteristic
552C2×C5#230811yescharacteristic
553C2×C5#231111yescharacteristic
554C2×C5#232912yes—
555C2×C5#232912yes—
556C2×C5#262611yescharacteristic
557C2×C5#262611yescharacteristic
558C2×C5#262711yescharacteristic
559C2×C5#262711yescharacteristic
560C2×C5#263912yes—
561C2×C5#265912yes—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23C4×Dic1014yes—
579C23#402414yes—
580C23D5×C22⋊C414yes—
581C23#403114yes—
582C23#403314yes—
583C23#403414yes—
584C23#404214yes—
585C23C22×Dic1011yescharacteristic
586C23C22×C4×D511yescharacteristic
587C23C2×#70911yescharacteristic
588C23C2×D4×D511yescharacteristic
589C23C2×D4×D511yescharacteristic
590C23C2×#71111yescharacteristic
591C23#414412yes—
592C23#414412yes—
593C23C2×Q8×D511yescharacteristic
594C23#414912yes—
595C23#415012yes—
596Q8—2016no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—54no—
623C2×C4—54no—
624C2×C4—104no—
625C2×C4—104no—
626C2×C4—102no—
627C2×C4—102no—
628C2×C4—104no—
629C2×C4—104no—
630C2×C4—102no—
631C2×C4—102no—
632C2×C4—102no—
633C2×C4—102no—
634C2×C4—104no—
635C2×C4—104no—
636C2×C4—104no—
637C2×C4—104no—
638C2×C4—104no—
639C2×C4—2016no—
640C2×C4—2016no—
641C2×C4—202no—
642C2×C4—202no—
643C2×C4—201no—
644C2×C4—202no—
645C2×C4—201no—
646C2×C4—202no—
647C2×C4—202no—
648C2×C4—201no—
649C2×C4—202no—
650C2×C4—201no—
651C2×C4—202no—
652C2×C4—202no—
653C2×C4—202no—
654C2×C4—202no—
655C2×C4—202no—
656C2×C4—202no—
657C2×C4—202no—
658C5#1428111yescharacteristic, complemented, Sylow
659C22—24no—
660C22—24no—
661C22—28no—
662C22—24no—
663C22—28no—
664C22—28no—
665C22—24no—
666C22—28no—
667C22—24no—
668C22—24no—
669C22#6493518yes—
670C22C2×C4×Dic1011yescharacteristic
671C22C2×#402411yescharacteristic
672C22#6581712yes—
673C22C2×D5×C22⋊C411yescharacteristic
674C22C2×#403111yescharacteristic
675C22C2×#403311yescharacteristic
676C22C2×#403411yescharacteristic
677C22C2×#404211yescharacteristic
678C22#6585312yes—
679C22#6587912yes—
680C22#6588012yes—
681C22#6589412yes—
682C22#6589812yes—
683C22#6592311yescharacteristic
684C22#6592611yescharacteristic
685C22#6592712yes—
686C22#6593112yes—
687C22#6593912yes—
688C22#6595712yes—
689C22#6596812yes—
690C22#6598211yescharacteristic
691C22#6598711yescharacteristic
692C22#6599912yes—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—82no—
697C4—104no—
698C4—104no—
699C4—202no—
700C4—204no—
701C4—202no—
702C4—402no—
703C2—28no—
704C2C2×#6493511yescharacteristic
705C2#1060517812yes—
706C2#1060556711yescharacteristic
707C2#1060568812yes—
708C2#1060582111yescharacteristic
709C2#1060611812yes—
710C2#1060623311yescharacteristic
711C2#1060635211yescharacteristic
712C2#1060639311yescharacteristic
713C2#1060651612yes—
714C2#1060660511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×C22⋊#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22⋊#95D5
hypocenterC5#14281
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
8. #10600217C2semidirect
9. #10600214C2semidirect
10. #10600204C2semidirect
11. #10600202C2semidirect
26. C2×#4024C4semidirect
46. C23×C4×C5Q8semidirect
60. C23×Dic5Q8semidirect
71. C22×Dic10C2×C4semidirect
86. C2×C4⋊Dic5C2×C4semidirect
94. C2×#683C2×C4semidirect
95. C2×#683C2×C4semidirect
367. C23×C4Dic10semidirect
658. C5#14281semidirect

Maximal subgroups

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

subgroupindex
C22×#40242
C22×#39632
C2×#652312
C5×C22⋊#952
#106029202
#106023372
#106002172
#106002142
#106002042
#106002022
#142815