Finite Group Databasebeta

← #29736571

Subgroups — #29736571

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840421C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic, complemented
3C2×#16840054C211yesmaximal, characteristic, complemented
4#21447022C212yesmaximal, complemented
5#21447001C212yesmaximal, complemented
6#21446751C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21445757C212yesmaximal, complemented
9#21445753C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2547C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14#16840421—24no—
15#16840421—24no—
16C2×#14519765C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519751C2211yescharacteristic
21C2×#14519749—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519737—21no—
24#16840124—24no—
25#16840054—24no—
26C2×#14519710C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×C22⋊#92418C2212yes—
35C2×C22⋊#92418C2212yes—
36C2×#14519688C2211yescharacteristic
37#16839524C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#124×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#397—21no—
42C61×#397—21no—
43C61×#391—21no—
44C61×#380C2212yes—
45C61×#380C2212yes—
46C4×C4⋊C4×C61C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—24no—
50C2×#11758222—42no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122—22no—
54C22×Dic122—24no—
55C22×Dic122—42no—
56C22×Dic122C2311yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61—21no—
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65#14519765—22no—
66#14519765D414yes—
67#14519763—28no—
68#14519756—22no—
69#14519756—22no—
70#14519755—28no—
71#14519751—22no—
72#14519751—22no—
73#14519749—28no—
74C4×Dic122—22no—
75C4×Dic122D414yes—
76#14519737—28no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85#14519710—24no—
86C4⋊#92418—22no—
87#14519700—24no—
88#14519700—24no—
89#14519700—24no—
90#14519700—42no—
91#14519697—24no—
92#14519697—42no—
93C22⋊#92418—42no—
94C22⋊#92418—42no—
95#14519688—22no—
96C22×Q8×C61—21no—
97C22×D4×C61D411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61—22no—
102#124×C61—44no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#95×C61—21no—
116#95×C61—22no—
117#95×C61—22no—
118C24×C61C2×D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C23×D61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—42no—
123C2×Q8×C61—42no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—41no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—44no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—21no—
148C22×Dic61—24no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—44no—
152#11758222—48no—
153#11758222—44no—
154#11758222—44no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244C2×D412yes—
159C2×C4×D61—24no—
160C2×C4×D61—44no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—24no—
166C2×Dic122—21no—
167C2×Dic122—48no—
168C2×Dic122—44no—
169C2×Dic122—44no—
170C2×Dic122—44no—
171C2×Dic122C2×D412yes—
172C2×Dic122C2×D412yes—
173C2×Dic122#4114yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—42no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—44no—
183C22⋊C4×C61—44no—
184C42×C61—42no—
185C42×C61C2×D412yes—
186C42×C61C2×D412yes—
187C22⋊Dic61—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191C4⋊Dic61—21no—
192C4⋊Dic61C2×D412yes—
193#11758196—24no—
194#11758196—24no—
195C4×Dic61—24no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61AGL(1,ℤ/8)12yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—42no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—42no—
218Q8×C61—84no—
219D4×C61—44no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61#12012yes—
240C2×C4×C61#12012yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C2×#4112yes—
244C2×C4×C61H(5,𝔽2)12yes—
245C22⋊D61—48no—
246C22⋊D61—48no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—48no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255D244—24no—
256D244—41no—
257D244—41no—
258C4×D61—48no—
259Dic122—22no—
260Dic122—24no—
261Dic122—24no—
262Dic122—48no—
263Dic122—41no—
264Dic122—41no—
265Dic122#12414yescomplemented
266#92418—42no—
267#92418—42no—
268#25896—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—42no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61C2×#12011yescharacteristic
280C22×C61#54612yes—
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61#57511yescharacteristic
283C22×C61#57812yes—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—82no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—82no—
294C4×C61—82no—
295C4×C61C2×#12411yescharacteristic
296C4×C61#53511yescharacteristic
297C4×C61#54612yes—
298Dic61—24no—
299Dic61—42no—
300Dic61—42no—
301C2×#548—611no—
302C2×#489—611no—
303C2×#477—611no—
304#2547D6111yescharacteristic, complemented
305#2036—612no—
306#2033—612no—
307#1983—611no—
308#1955—611no—
309#1788—612no—
310#1787—612no—
311#1763—611no—
312C2×C61—42no—
313C2×C61—81no—
314C2×C61#243511yescharacteristic
315C2×C61#300611yescharacteristic
316C2×C61#301111yescharacteristic
317C2×C61#330812yes—
318C2×C61#331312yes—
319D61—82no—
320C22×#41—611no—
321C23×Q8—611no—
322#548—1224no—
323#548—1224no—
324C2×C4⋊Q8C2×D6111yescharacteristic
325C2×#124—21no—
326C2×#124—611no—
327C2×#124—1221no—
328C2×C22⋊Q8—611no—
329C2×C22⋊Q8—1221no—
330C2×#121—611no—
331C2×#121—1221no—
332C2×C4×Q8—611no—
333C2×C4×D4—1221no—
334C2×C4×D4C2×D6111yescharacteristic
335#489—1224no—
336#477—1224no—
337C2×#107—612no—
338C2×C4⋊C8—611no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103—612no—
343C2×#102—612no—
344C2×#102—612no—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—612no—
347C2×#97—611no—
348#397—21no—
349#397—21no—
350#391—21no—
351#380C2×D6112yes—
352#380C2×D6112yes—
353C4×C4⋊C4C2×D6111yescharacteristic
354#340—612no—
355C61#2589611yescharacteristic, complemented, Sylow
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1224no—
359C2×#41—2442no—
360C22×Q8—21no—
361C22×Q8—611no—
362C22×Q8—611no—
363C22×Q8—1222no—
364C22×Q8—1222no—
365C22×Q8—1224no—
366C22×Q8—1221no—
367C22×Q8—2442no—
368C22×D4—611no—
369C22×D4—1221no—
370C22×D4—1221no—
371C22×D4C22⋊D6111yescharacteristic
372C23×C4—1221no—
373C23×C4—1221no—
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C22×C8—612no—
377C22×C8—612no—
378C4⋊Q8—22no—
379C4⋊Q8—22no—
380#124—44no—
381#124—614no—
382#124—1228no—
383#124—1222no—
384C22⋊Q8—1228no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387#121—1222no—
388#121—1222no—
389#121—1228no—
390C4×Q8—614no—
391C4×Q8—1222no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—1228no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4C22⋊D6111yescharacteristic
400C2×C4⋊C4C22×D6112yes—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6111yescharacteristic
410C2×C42—21no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—1224no—
414C4⋊C8—1222no—
415#103—1224no—
416#103—1224no—
417#103—1224no—
418#103—2442no—
419#102—1224no—
420#102—2442no—
421C22⋊C8—2442no—
422C22⋊C8—2442no—
423#97—1222no—
424#95—21no—
425#95—22no—
426#95—22no—
427C24—1221no—
428C24C2×C22⋊D6111yescharacteristic
429C24C2×C22⋊D6111yescharacteristic
430#41—2448no—
431#41—2444no—
432#41—2444no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—612no—
437C2×Q8—612no—
438C2×Q8—614no—
439C2×Q8—1224no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—2444no—
446C2×Q8—2444no—
447C2×Q8—2448no—
448C2×Q8—2444no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—612no—
453C2×D4—1224no—
454C2×D4—1224no—
455C2×D4—1221no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—2444no—
459C2×D4—2442no—
460C2×D4—2442no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—1224no—
472C22×C4—1221no—
473C22×C4—1224no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—2444no—
477C22×C4—2444no—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C23×D6111yescharacteristic
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—612no—
496C4⋊C4—1224no—
497C4⋊C4—1221no—
498C4⋊C4—1224no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C42—42no—
508C42—1224no—
509C42D4×D6112yes—
510C42D4×D6112yes—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—22no—
520C23—1222no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2442no—
525C23#1451980312yes—
526C23C22×C22⋊D6111yescharacteristic
527Q8—84no—
528Q8—614no—
529Q8—1222no—
530Q8—1224no—
531Q8—1224no—
532Q8—2448no—
533Q8—2441no—
534Q8—2441no—
535D4—44no—
536D4—1224no—
537D4—2448no—
538D4—2448no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—1222no—
561C2×C4—1224no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2448no—
565C2×C4—2448no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2442no—
569C2×C4#1451974512yes—
570C2×C4#1451974512yes—
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4#41×D6112yes—
574C2×C4#1451982312yes—
575C8—2442no—
576C8—2442no—
577C22—24no—
578C22—22no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—41no—
583C22—42no—
584C22—42no—
585C22—41no—
586C22—41no—
587C22—4882no—
588C22—4882no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22C2×#1451974511yescharacteristic
593C22#1684040012yes—
594C22C2×#1451980311yescharacteristic
595C22#1684056011yescharacteristic
596C22#1684057712yes—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—82no—
602C4—82no—
603C4—1224no—
604C4—2442no—
605C4—2442no—
606C4#124×D6111yescharacteristic
607C4#1684033011yescharacteristic
608C4#1684040512yes—
609C2—42no—
610C2—81no—
611C2—4882no—
612C2#2144946711yescharacteristic
613C2#2145506911yescharacteristic
614C2#2145511211yescharacteristic
615C2#2145780412yes—
616C2#2145787912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2547C2
socleC23×C61C22×D4
prime radicalC2×#16840421C2
hypercenter#2547D61
hypocenterC61#25896
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C2×#16840124C2semidirect
3. C2×#16840054C2semidirect
4. #21447022C2semidirect
5. #21447001C2semidirect
6. #21446751C2semidirect
7. #21446618C2semidirect
8. #21445757C2semidirect
9. #21445753C2semidirect
10. #21445652C2semidirect
11. C61×#2547C2semidirect
265. Dic122#124semidirect
304. #2547D61semidirect
355. C61#25896semidirect

Maximal subgroups

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

subgroupindex
C2×#168404212
C2×#168401242
C2×#168400542
#214470222
#214470012
#214467512
#214466182
#214457572
#214457532
#214456522
C61×#25472
#2589661