Finite Group Databasebeta

← #29732204

Subgroups — #29732204

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839996C212yesmaximal, complemented
2#21454150C211yesmaximal, characteristic, complemented
3#21447144C211yesmaximal, characteristic, complemented
4#21447050C211yesmaximal, characteristic, complemented
5#21446952C211yesmaximal, characteristic, complemented
6#21446890C211yesmaximal, characteristic, complemented
7#21446248C212yesmaximal, complemented
8#21446167C212yesmaximal, complemented
9#21446066C212yesmaximal, complemented
10C2×#16839517C211yesmaximal, characteristic, complemented
11C2×C61×#532C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C23×#92418C2212yes—
14C2×#14519760—21no—
15C2×#14519751C2211yescharacteristic
16#16839996—28no—
17C2×#14519710C2212yes—
18C2×#14519698C2212yes—
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519697C2212yes—
22#16839664C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839660C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839653—21no—
27#16839653—21no—
28#16839641—21no—
29#16839622C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2211yescharacteristic
34#16839517C2211yescharacteristic
35C22×#41×C61—21no—
36C61×#532—28no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#127×C61C2211yescharacteristic
39C2×#124×C61—21no—
40C2×#121×C61—21no—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2212yes—
44C22×C42×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C2×#11758199—24no—
48C2×#11758199—24no—
49C2×#11758199—42no—
50C22×#92418—24no—
51C22×#92418—24no—
52C22×#92418—42no—
53C22×#92418C2312yes—
54C22×#92418C2312yes—
55#14519760—44no—
56#14519751—24no—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68#14519710—24no—
69#14519698—24no—
70#14519698—24no—
71#14519698—42no—
72#14519697—24no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76C2×#41×C61—28no—
77C2×#41×C61—44no—
78C2×#41×C61—44no—
79C22×Q8×C61—21no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61C2312yes—
84C23×C4×C61—22no—
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C4⋊Q8×C61—24no—
89#127×C61—24no—
90#124×C61—28no—
91#121×C61—28no—
92#121×C61—24no—
93#121×C61D418yes—
94C4×D4×C61—28no—
95C4×D4×C61—24no—
96C4×D4×C61—24no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C42×C61—24no—
102C2×C42×C61D411yescharacteristic
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C24×C61—22no—
107C24×C61C2×D411yescharacteristic
108C23×D61—21no—
109#41×C61—432no—
110C2×Q8×C61—24no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—28no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—44no—
124C2×D4×C61—44no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61C2×D414yes—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—28no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—41no—
138C22×C4×C61—44no—
139C22×C4×C61—44no—
140C22×C4×C61D814yescomplemented
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—42no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—21no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D414yes—
161C22⋊C4×C61—28no—
162C22⋊C4×C61—28no—
163C42×C61—22no—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—22no—
167C22⋊Dic61—42no—
168#11758199—48no—
169#11758198—42no—
170#11758198—42no—
171#11758198—42no—
172#11758198—42no—
173#11758198—42no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178#11758196—42no—
179#11758196—42no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—48no—
186C2×#92418#4114yescomplemented
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—24no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61#12012yes—
199C23×C61C2×D812yes—
200C23×C61AGL(1,ℤ/8)12yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209Q8×C61—48no—
210D4×C61—24no—
211D4×C61—24no—
212D4×C61—48no—
213D4×C61—48no—
214D4×C61—416no—
215D4×C61—42no—
216D4×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—28no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—21no—
229C2×C4×C61—24no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—44no—
233C2×C4×C61—416no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61#12012yes—
237C2×C4×C61C2×D812yes—
238C2×C4×C61C2×D812yes—
239C2×C4×C61#13512yes—
240C2×C4×C61C22×D412yes—
241C2×C4×C61C2×#4112yes—
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C22⋊D61—84no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254D244—42no—
255D244—42no—
256#92418—24no—
257#92418—42no—
258#25496—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—44no—
266C22×C61—44no—
267C22×C61—44no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61#44714yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61#53211yescharacteristic
275C22×C61#53711yescharacteristic
276C22×C61C22×D811yescharacteristic
277C22×C61C2×#13511yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—44no—
292C4×C61—41no—
293C4×C61#54612yes—
294C4×C61#54712yes—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298C2×#532D6111yescharacteristic, complemented
299C2×#466—612no—
300#2854—611no—
301#2057—611no—
302#2041—611no—
303#2023—611no—
304#2011—611no—
305#1893—612no—
306#1879—612no—
307#1862—612no—
308C2×#336—611no—
309C2×C61—24no—
310C2×C61—84no—
311C2×C61#243011yescharacteristic
312C2×C61C2×#44711yescharacteristic
313C2×C61#300211yescharacteristic
314C2×C61#316911yescharacteristic
315C2×C61#318011yescharacteristic
316C2×C61#324511yescharacteristic
317C2×C61#324711yescharacteristic
318D61—161no—
319C22×#41—21no—
320C22×D8—612no—
321C23×C8—612no—
322#532—28no—
323C2×C4⋊Q8C2×D6111yescharacteristic
324C2×#127C2×D6111yescharacteristic
325C2×#124—21no—
326C2×#123—1221no—
327C2×#121—21no—
328C2×#121—611no—
329C2×#121C2×D6112yes—
330C2×C4×D4—21no—
331C2×C4×D4C2×D6112yes—
332C22×C42C2×D6111yescharacteristic
333#466—1228no—
334C2×#107—612no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339#390—611no—
340#390—611no—
341#389—611no—
342#389—611no—
343#388—1221no—
344#388—1221no—
345#386—1221no—
346#380—611no—
347#340—612no—
348#340—612no—
349#336—611no—
350#336—612no—
351#336—611no—
352C61#2549611yescharacteristic, complemented, Sylow
353C2×#41—28no—
354C2×#41—44no—
355C2×#41—44no—
356C22×Q8—21no—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—611no—
361C22×D4—1221no—
362C22×D4C22×D6112yes—
363C23×C4—22no—
364C23×C4C22⋊D6111yescharacteristic
365C23×C4C22⋊D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C2×D8—1224no—
368C2×D8—1224no—
369C2×D8—2442no—
370C22×C8—612no—
371C22×C8—612no—
372C22×C8—1224no—
373C22×C8—1224no—
374C22×C8—2442no—
375C4⋊Q8—24no—
376#127—24no—
377#124—28no—
378#123—2444no—
379#121—28no—
380#121—24no—
381#121—1224no—
382#121C22⋊D6118yes—
383C4×D4—28no—
384C4×D4—24no—
385C4×D4—24no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—24no—
402C2×C42C22⋊D6111yescharacteristic
403C2×C42C22⋊D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406#107—1224no—
407#102—1224no—
408#102—1224no—
409#102—1224no—
410#102—2442no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414C24—22no—
415C24—1221no—
416C24C2×C22⋊D6111yescharacteristic
417#41—432no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—28no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—2442no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4C2×C22⋊D6114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4#1175819914yescomplemented
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6114yes—
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466D8—2448no—
467C2×C8—614no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—2448no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2442no—
481C4⋊C4—2442no—
482C4⋊C4C2×C22⋊D6114yes—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C22⋊C4—2442no—
488C22⋊C4—2442no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23#1451974612yes—
514C23C2×#1175819912yes—
515C23#1451980312yes—
516C23C22×C22⋊D6111yescharacteristic
517Q8—48no—
518D4—24no—
519D4—24no—
520D4—48no—
521D4—48no—
522D4—416no—
523D4—42no—
524D4—42no—
525D4—4884no—
526D4—2442no—
527D4—2442no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451974612yes—
557C2×C4C2×#1175819912yes—
558C2×C4C2×#1175819912yes—
559C2×C4#1451979712yes—
560C2×C4C22×C22⋊D6112yes—
561C2×C4#41×D6112yes—
562C2×C4#1451982311yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—1224no—
565C8—2442no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22#1683989614yes—
587C22C2×#1451974611yescharacteristic
588C22#1684030811yescharacteristic
589C22#1684034011yescharacteristic
590C22C22×#1175819911yescharacteristic
591C22C2×#1451979711yescharacteristic
592C22C2×#1451980311yescharacteristic
593C22#1684056011yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—44no—
599C4—41no—
600C4—4881no—
601C4—4881no—
602C4—4881no—
603C4#1684040012yes—
604C4#1684041112yes—
605C2—24no—
606C2—84no—
607C2—9761no—
608C2#2144942311yescharacteristic
609C2C2×#1683989611yescharacteristic
610C2#2145503711yescharacteristic
611C2#2145639411yescharacteristic
612C2#2145651111yescharacteristic
613C2#2145712611yescharacteristic
614C2#2145713911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#532C2
socleC23×C61C22×D4
hypercenterC2×#532D61
hypocenterC61#25496
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16839996C2semidirect
2. #21454150C2semidirect
3. #21447144C2semidirect
4. #21447050C2semidirect
5. #21446952C2semidirect
6. #21446890C2semidirect
7. #21446248C2semidirect
8. #21446167C2semidirect
9. #21446066C2semidirect
10. C2×#16839517C2semidirect
11. C2×C61×#532C2semidirect
140. C22×C4×C61D8semidirect
186. C2×#92418#41semidirect
298. C2×#532D61semidirect
352. C61#25496semidirect
458. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168399962
#214541502
#214471442
#214470502
#214469522
#214468902
#214462482
#214461672
#214460662
C2×#168395172
C2×C61×#5322
#2549661