Finite Group Databasebeta

← #29679233

Subgroups — #29679233

Order 15616.

4038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758226C211yesmaximal, characteristic
2C2×#16839762C214yesmaximal, complemented
3C2×#16839724C211yesmaximal, characteristic, complemented
4#380×D61C211yesmaximal, characteristic, complemented
5#21445758C214yesmaximal, complemented
6#21445754C211yesmaximal, characteristic, complemented
7#21445751C211yesmaximal, characteristic, complemented
8#21445749C211yesmaximal, characteristic, complemented
9C61×#1787C211yesmaximal, characteristic, complemented
10C22×#11758226—216no—
11C22×#11758226—24no—
12C22×#11758226—21no—
13C22×#11758226—21no—
14C22×#11758226C418yescomplemented
15C22×#11758226C2212yes—
16C22×#11758226C2212yes—
17C23×D244—21no—
18C23×D244C2211yescharacteristic
19C23×C4×D61—21no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758189C2212yescomplemented
22C2×C4⋊C4×D61C2212yescomplemented
23C2×C22⋊C4×D61—21no—
24#16839762—216no—
25C2×#14519701C2214yescomplemented
26C2×#14519699C2214yescomplemented
27C2×#14519697C2214yescomplemented
28#16839724—24no—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yescomplemented
31#16839622C2211yescharacteristic
32#16839621—21no—
33#16839620—21no—
34#16839619C2212yescomplemented
35#95×D61—21no—
36C23×Q8×C61C2211yescharacteristic
37C2×#103×C61C2214yescomplemented
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#380C2211yescharacteristic
40C24×D61—21no—
41C24×D61D411yescharacteristic
42C2×#11758226—216no—
43C2×#11758226—264no—
44C2×#11758226—216no—
45C2×#11758226—28no—
46C2×#11758226—264no—
47C2×#11758226—28no—
48C2×#11758226—28no—
49C2×#11758226—28no—
50C2×#11758226—24no—
51C2×#11758226—24no—
52C2×#11758226—28no—
53C2×#11758226C2×C418yes—
54C2×#11758226C2×C418yescomplemented
55C2×#11758226D4116yes—
56C22×D244—216no—
57C22×D244—28no—
58C22×D244—22no—
59C22×D244—22no—
60C22×D244—21no—
61C22×D244—22no—
62C22×D244—28no—
63C22×D244C2×C414yescomplemented
64C22×D244C2312yes—
65C23×Dic61D411yescharacteristic
66C22×C4×D61—216no—
67C22×C4×D61—28no—
68C22×C4×D61—22no—
69C22×C4×D61—22no—
70C22×C4×D61—28no—
71C22×C4×D61—22no—
72C22×C4×D61C2×C414yes—
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61D414yes—
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2311yescharacteristic
78C2×#11758189—24no—
79C2×#11758189—28no—
80C22×#92418C2312yes—
81C2×C4⋊Dic61C2312yes—
82C2×#11758196—22no—
83C4⋊C4×D61—28no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—22no—
86C22⋊C4×D61—44no—
87#14519701—28no—
88#14519699—28no—
89#14519697—28no—
90C22⋊#92418—22no—
91#14519689—24no—
92#14519684—21no—
93#14519684—22no—
94C22×Q8×C61—28no—
95C22×Q8×C61—22no—
96C22×Q8×C61—21no—
97C22×Q8×C61C2×C414yescomplemented
98C22×Q8×C61C2312yes—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104#103×C61—28no—
105C22⋊C8×C61—22no—
106#95×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—28no—
109C23×D61—24no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—24no—
114C23×D61—42no—
115C23×D61—44no—
116C23×D61—42no—
117C23×D61—42no—
118C23×D61C22⋊C414yes—
119C23×D61C2×D411yescharacteristic
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—216no—
122C2×Q8×C61—28no—
123C2×Q8×C61—216no—
124C2×Q8×C61—24no—
125C2×Q8×C61—22no—
126C2×Q8×C61—24no—
127C2×Q8×C61C22×C414yes—
128C2×Q8×C61C2×D418yes—
129C22×C4×C61—28no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—24no—
139C22×Dic61—22no—
140C22×Dic61C22⋊C414yes—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143#11758226—264no—
144#11758226—232no—
145#11758226—264no—
146#11758226—216no—
147#11758226—216no—
148#11758226—216no—
149#11758226—232no—
150#11758226C22⋊C4132yescomplemented
151C2×D244—216no—
152C2×D244—28no—
153C2×D244—264no—
154C2×D244—216no—
155C2×D244—28no—
156C2×D244—28no—
157C2×D244—216no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—22no—
161C2×D244—216no—
162C2×D244C22×C414yes—
163C2×D244C2×D418yes—
164C2×C4×D61—216no—
165C2×C4×D61—264no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61—28no—
169C2×C4×D61—28no—
170C2×C4×D61—216no—
171C2×C4×D61—24no—
172C2×C4×D61—216no—
173C2×C4×D61—216no—
174C2×C4×D61—44no—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61C22⋊C414yes—
178C2×C4×D61C22⋊C414yes—
179C2×C4×D61C22×C412yes—
180C2×C4×D61C22×C412yes—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D418yes—
183C2×C4×D61C2×D412yes—
184C2×C8×C61—22no—
185C2×C8×C61—24no—
186C4⋊C4×C61—24no—
187C22⋊C4×C61—42no—
188C22⋊Dic61—42no—
189#11758198—44no—
190C4⋊Dic61—24no—
191#11758196—44no—
192C2×#92418—22no—
193C2×#92418—24no—
194#11758189—48no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61C2×C22⋊C412yes—
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61#13712yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—22no—
204C22×D61—216no—
205C22×D61—24no—
206C22×D61—22no—
207C22×D61—28no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—28no—
212C22×D61—28no—
213C22×D61—44no—
214C22×D61—48no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61—416no—
218C22×D61—416no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—44no—
222C22×D61—48no—
223C22×D61C2×C22⋊C412yes—
224C22×D61C2×C22⋊C412yes—
225C22×D61#12014yes—
226Q8×C61—216no—
227Q8×C61—24no—
228Q8×C61—28no—
229Q8×C61C2×C22⋊C418yescomplemented
230C2×C4×C61—216no—
231C2×C4×C61—24no—
232C2×C4×C61—28no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D414yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—28no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61#12014yes—
251D244—232no—
252D244—216no—
253D244—232no—
254D244—24no—
255D244—28no—
256D244C2×C22⋊C418yescomplemented
257C4×D61—232no—
258C4×D61—28no—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—232no—
262C4×D61—28no—
263C4×D61—88no—
264C4×D61C2×C22⋊C418yes—
265C4×D61#12018yes—
266C8×C61—42no—
267#92418—42no—
268#20601—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—22no—
273C22×C61#41714yescomplemented
274C22×C61C22×C22⋊C411yescharacteristic
275C22×C61C2×#12012yes—
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57512yes—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—24no—
282C2×D61—24no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—28no—
287C2×D61—416no—
288C2×D61—432no—
289C2×D61—44no—
290C2×D61—44no—
291C2×D61—42no—
292C2×D61—48no—
293C2×D61—42no—
294C2×D61—48no—
295C2×D61#37914yes—
296C4×C61—28no—
297C4×C61—22no—
298C4×C61—82no—
299C4×C61C22×C22⋊C412yes—
300C4×C61C2×#12012yes—
301Dic61—22no—
302Dic61—82no—
303Dic61#37914yes—
304C23×#41—611no—
305C2×#417—614no—
306C2×#407—611no—
307C2×#380—611no—
308#1788—614no—
309#1787—611no—
310#1787D6111yescharacteristic, complemented
311#1786—611no—
312#1786—611no—
313C2×C61—24no—
314C2×C61C2×#37911yescharacteristic
315C2×C61C2×#41711yescharacteristic
316C2×C61#289611yescharacteristic
317C2×C61#300612yes—
318C2×C61#300712yes—
319D61—24no—
320D61—42no—
321D61—48no—
322C22×#41—618no—
323C22×#41—612no—
324C22×#41—612no—
325C22×#41—1221no—
326C22×#41—1221no—
327C22×#41—1224no—
328C22×#41—12216no—
329C23×Q8C2×D6111yescharacteristic
330C23×D4—611no—
331C23×D4—1221no—
332C24×C4—611no—
333C24×C4—1221no—
334C22×M16—612no—
335C22×C4⋊C4—612no—
336C22×C22⋊C4—1221no—
337#417—12216no—
338C2×#103—614no—
339C2×#103C2×D6114yescomplemented
340C2×#102—614no—
341C2×#102—614no—
342#407—1224no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—612no—
345C2×C22⋊C8C2×D6111yescharacteristic
346#380—611no—
347#380—612no—
348#380—1221no—
349#380—1221no—
350#380C2×D6111yescharacteristic
351C2×#95—1221no—
352C61#2060111yescharacteristic, complemented, Sylow
353C25—611no—
354C25—1221no—
355C2×#41—618no—
356C2×#41—618no—
357C2×#41—6116no—
358C2×#41—1228no—
359C2×#41—1224no—
360C2×#41—1224no—
361C2×#41—1228no—
362C2×#41—1228no—
363C2×#41—1228no—
364C2×#41—12264no—
365C2×#41—12216no—
366C2×#41—12264no—
367C2×#41—12216no—
368C2×#41—1228no—
369C22×Q8—28no—
370C22×Q8—21no—
371C22×Q8—22no—
372C22×Q8C4×D6114yescomplemented
373C22×Q8C22×D6112yes—
374C22×D4—614no—
375C22×D4—612no—
376C22×D4—1228no—
377C22×D4—1222no—
378C22×D4—1222no—
379C22×D4—1228no—
380C22×D4—12216no—
381C22×D4—1222no—
382C22×D4—1221no—
383C23×C4—21no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—614no—
390C23×C4—611no—
391C23×C4—1228no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4—1228no—
395C23×C4—12216no—
396C23×C4—1222no—
397C23×C4C22×D6111yescharacteristic
398C2×M16—1224no—
399C2×M16—1228no—
400C22×C8—612no—
401C22×C8C22×D6112yes—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1228no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4C22×D6112yes—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—2444no—
410#103—28no—
411#103—1228no—
412#102—1228no—
413#102—1228no—
414C22⋊C8—22no—
415C22⋊C8—1224no—
416C22⋊C8—1222no—
417#95—21no—
418#95—1222no—
419#95—1221no—
420C24—611no—
421C24—611no—
422C24—614no—
423C24—1224no—
424C24—1228no—
425C24—1222no—
426C24—1222no—
427C24—1224no—
428C24—1222no—
429C24—2442no—
430C24—2444no—
431C24—2442no—
432C24—2442no—
433C24D4×D6111yescharacteristic
434#41—6132no—
435#41—12216no—
436#41—12232no—
437#41—12216no—
438#41—12264no—
439#41—12264no—
440#41—12232no—
441#41—12216no—
442C2×Q8—216no—
443C2×Q8—28no—
444C2×Q8—216no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8C2×C4×D6114yes—
449C2×Q8D4×D6118yes—
450C2×D4—614no—
451C2×D4—618no—
452C2×D4—12216no—
453C2×D4—1228no—
454C2×D4—1228no—
455C2×D4—12216no—
456C2×D4—12264no—
457C2×D4—12216no—
458C2×D4—1228no—
459C2×D4—12216no—
460C2×D4—1224no—
461C2×D4—1222no—
462C2×D4—1224no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—611no—
469C22×C4—612no—
470C22×C4—618no—
471C22×C4—612no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—1224no—
479C22×C4—12216no—
480C22×C4—12216no—
481C22×C4—1228no—
482C22×C4—1228no—
483C22×C4—12216no—
484C22×C4—1224no—
485C22×C4—12264no—
486C22×C4—12216no—
487C22×C4—1224no—
488C22×C4—1224no—
489C22×C4—1222no—
490C22×C4—2444no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4C2×C4×D6112yes—
494C22×C4D4×D6112yes—
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497M16—2448no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—1222no—
501C2×C8—1224no—
502C4⋊C4—24no—
503C4⋊C4—1224no—
504C4⋊C4—2444no—
505C22⋊C4—42no—
506C22⋊C4—2444no—
507C22⋊C4—2442no—
508C23—22no—
509C23—22no—
510C23—612no—
511C23—614no—
512C23—612no—
513C23—1228no—
514C23—12216no—
515C23—1221no—
516C23—1221no—
517C23—1222no—
518C23—1224no—
519C23—1228no—
520C23—1222no—
521C23—1228no—
522C23—1221no—
523C23—1221no—
524C23—1222no—
525C23—2444no—
526C23—2448no—
527C23—2448no—
528C23—2448no—
529C23—24416no—
530C23—24416no—
531C23—2448no—
532C23—2442no—
533C23—2442no—
534C23—2444no—
535C23C22⋊C4×D6112yes—
536C23#1451980412yes—
537C23#1451980912yes—
538C23C2×D4×D6111yescharacteristic
539Q8—216no—
540Q8—24no—
541Q8—28no—
542Q8C22⋊C4×D6118yescomplemented
543D4—618no—
544D4—12232no—
545D4—12232no—
546D4—12216no—
547D4—1224no—
548D4—1228no—
549C2×C4—216no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—612no—
559C2×C4—614no—
560C2×C4—618no—
561C2×C4—618no—
562C2×C4—612no—
563C2×C4—1228no—
564C2×C4—12232no—
565C2×C4—1228no—
566C2×C4—12232no—
567C2×C4—1228no—
568C2×C4—1222no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—4888no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4C22⋊C4×D6112yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6114yes—
578C2×C4C2×D4×D6111yescharacteristic
579C8—42no—
580C8—2442no—
581C22—22no—
582C22—24no—
583C22—24no—
584C22—22no—
585C22—614no—
586C22—1222no—
587C22—1224no—
588C22—1224no—
589C22—1222no—
590C22—1222no—
591C22—1228no—
592C22—1222no—
593C22—1222no—
594C22—24416no—
595C22—24432no—
596C22—2444no—
597C22—2442no—
598C22—2448no—
599C22—2442no—
600C22—2448no—
601C22—2444no—
602C22#1683976214yescomplemented
603C22C2×C22⋊C4×D6111yescharacteristic
604C22#120×D6112yes—
605C22C2×#1451980411yescharacteristic
606C22C2×#1451980911yescharacteristic
607C22#1684056112yes—
608C4—28no—
609C4—22no—
610C4—82no—
611C4—614no—
612C4—1222no—
613C4—4882no—
614C4C2×C22⋊C4×D6112yes—
615C4#120×D6112yes—
616C2—24no—
617C2—1224no—
618C2—2442no—
619C2—2448no—
620C2#379×D6111yescharacteristic
621C2C2×#1683976211yescharacteristic
622C2#2145433711yescharacteristic
623C2#2145507012yes—
624C2#2145507812yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC23×#11758226C2
cosocleC22×C4×C61C24
hypercenter#1787D61
hypocenterC61#20601
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C2×#16839762C2semidirect
3. C2×#16839724C2semidirect
4. #380×D61C2semidirect
5. #21445758C2semidirect
6. #21445754C2semidirect
7. #21445751C2semidirect
8. #21445749C2semidirect
9. C61×#1787C2semidirect
14. C22×#11758226C4semidirect
21. C22×#11758189C22semidirect
22. C2×C4⋊C4×D61C22semidirect
25. C2×#14519701C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519697C22semidirect
30. C2×#14519689C22semidirect
34. #16839619C22semidirect
37. C2×#103×C61C22semidirect
54. C2×#11758226C2×C4semidirect
63. C22×D244C2×C4semidirect
97. C22×Q8×C61C2×C4semidirect
150. #11758226C22⋊C4semidirect
229. Q8×C61C2×C22⋊C4semidirect
256. D244C2×C22⋊C4semidirect
273. C22×C61#417semidirect
310. #1787D61semidirect
339. C2×#103C2×D61semidirect
352. C61#20601semidirect
372. C22×Q8C4×D61semidirect
542. Q8C22⋊C4×D61semidirect
602. C22#16839762semidirect

Maximal subgroups

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

subgroupindex
C23×#117582262
C2×#168397622
C2×#168397242
#380×D612
#214457582
#214457542
#214457512
#214457492
C61×#17872
#2060161