Finite Group Databasebeta

← #29681825

Subgroups — #29681825

Order 15616.

1718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840338C211yesmaximal, characteristic
2C2×#16840239C211yesmaximal, characteristic, complemented
3C2×#16839735C211yesmaximal, characteristic
4C2×#16839735C211yesmaximal, characteristic
5#21448403C211yesmaximal, characteristic, complemented
6#21446276C218yesmaximal, complemented
7#21445552C211yesmaximal, characteristic, complemented
8C61×#1740C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11#16840338—22no—
12#16840338—22no—
13#16840338—22no—
14#16840338—22no—
15C2×#14519777C412yescomplemented
16C2×#14519765C412yescomplemented
17C2×#14519761C2214yes—
18C2×#14519757C414yescomplemented
19#16840239—24no—
20#16840239—24no—
21C2×C4×Dic122—22no—
22C2×#14519742—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×#14519733C2214yes—
26#16839735—24no—
27#16839735—24no—
28C2×#14519691C2214yes—
29C2×#14519691C2214yes—
30#16839668—22no—
31#16839667—22no—
32#16839623C2214yes—
33#16839529C2214yes—
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#342C2214yes—
38C2×#11758222—22no—
39C2×#11758222—22no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—22no—
50C22×Dic122C2×C411yescharacteristic
51C22×Dic122C2×C412yes—
52C22×Dic122C2×C412yes—
53C22×Dic122D412yes—
54C22×Dic122D412yes—
55C22×Dic122D412yes—
56C22×Dic122D412yes—
57C22×D244C2×C411yescharacteristic
58C22×C22⋊D61C2312yes—
59C23×Dic61C2312yes—
60C22×C4×D61C2×C412yes—
61#14519777—24no—
62#14519777—24no—
63#14519765—24no—
64#14519765—24no—
65#14519761—28no—
66#14519761—28no—
67#14519757—28no—
68#14519757—28no—
69C4×Dic122—216no—
70#14519742—216no—
71#14519736—22no—
72#14519736—22no—
73#14519736—22no—
74#14519736D414yes—
75#14519733—28no—
76#14519733—28no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2×C418yes—
80C2×#11758196—24no—
81C2×C22⋊Dic61C2312yes—
82C2×C22⋊Dic61C2312yes—
83C2×#11758198C2×C418yes—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—24no—
86C2×C4×Dic61—21no—
87#14519691—28no—
88#14519691—28no—
89#14519684—28no—
90C23×C4×C61C2311yescharacteristic
91#117×C61—22no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—22no—
97#117×C61—22no—
98#117×C61D414yes—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2×C412yes—
102C2×C22⋊C4×C61C2312yes—
103C2×C22⋊C4×C61C2312yes—
104C2×C22⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2×C412yes—
108C24×C61C2411yescharacteristic
109C23×D61C22×C412yes—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61C22×C414yes—
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61C22×C412yes—
136C22×Dic61C22×C414yes—
137C22×Dic61C2×D414yes—
138C22×Dic61C2×D414yes—
139#11758222—42no—
140#11758222—42no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144#11758222—42no—
145#11758222—42no—
146#11758222—42no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—22no—
150C2×D244—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—24no—
154C2×C4×D61—24no—
155C2×Dic122—24no—
156C2×Dic122—24no—
157C2×Dic122—24no—
158C2×Dic122—24no—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C2×Dic122—24no—
162C2×Dic122—24no—
163C2×Dic122—24no—
164C2×Dic122—24no—
165C2×Dic122—24no—
166C2×Dic122—24no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122C22⋊C414yes—
172C2×Dic122C22⋊C414yes—
173C2×Dic122C22⋊C414yes—
174C2×Dic122C22⋊C414yes—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—42no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61C2×D414yes—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—24no—
188C42×C61—42no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61C2×D414yes—
193#11758198—216no—
194C4⋊Dic61—24no—
195C4⋊Dic61—216no—
196C4⋊Dic61—42no—
197#11758196—216no—
198C4×Dic61—216no—
199C4×Dic61—24no—
200C4×Dic61—42no—
201C23×C61—42no—
202C23×C61C23×C411yescharacteristic
203C23×C61C22×D411yescharacteristic
204C23×C61C22×D411yescharacteristic
205C23×C61C22×D411yescharacteristic
206C23×C61C22×D411yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4111yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—44no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—44no—
216C2×C4×C61—44no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C4×D412yes—
224C2×C4×C61C4×D412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61#12012yes—
228C2×C4×C61#12012yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12312yes—
234C2×C4×C61#12312yes—
235C22⋊D61—42no—
236C22⋊D61—42no—
237C22⋊D61—42no—
238C22⋊D61—42no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—44no—
250C2×Dic61—44no—
251C2×Dic61—42no—
252C2×Dic61C2×C22⋊C418yes—
253D244—42no—
254D244—42no—
255C4×D61—44no—
256C4×D61—44no—
257Dic122—28no—
258Dic122—28no—
259Dic122—28no—
260Dic122—28no—
261Dic122—42no—
262Dic122—42no—
263#20986—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×C4×D411yescharacteristic
274C22×C61C2×#12011yescharacteristic
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×#12311yescharacteristic
278C2×D61—44no—
279C2×D61—42no—
280C2×D61—82no—
281C2×D61—82no—
282C4×C61—44no—
283C4×C61—44no—
284C4×C61—82no—
285C4×C61#38612yes—
286C4×C61#38612yes—
287C4×C61#38612yes—
288C4×C61#38612yes—
289Dic61—28no—
290Dic61—44no—
291Dic61—42no—
292Dic61—82no—
293C2×#536—611no—
294C2×#519—611no—
295C2×#410—611no—
296C2×#410—611no—
297#2310—611no—
298#1899—618no—
299#1740—611no—
300#1740D6111yescharacteristic, complemented
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61C2×#38611yescharacteristic
308C2×C61#288412yes—
309D61—82no—
310C22×#41—611no—
311C23×Q8—611no—
312#536—1222no—
313#536—1222no—
314#536—1222no—
315#536—1222no—
316C2×C4⋊Q8—612no—
317C2×#124—612no—
318C2×#123—614no—
319C2×C22⋊Q8—614no—
320#519—1224no—
321#519—1224no—
322C2×C4×Q8—1222no—
323C2×C4×Q8—1222no—
324C2×#117—611no—
325C2×#117—611no—
326C2×#117—614no—
327C2×#117C2×D6111yescharacteristic
328C2×#117C2×D6111yescharacteristic
329C2×#117C2×D6111yescharacteristic
330#410—1224no—
331#410—1224no—
332C2×#99—614no—
333C2×#99—614no—
334#391—1222no—
335#391—1222no—
336#381—614no—
337#342—614no—
338#342C2×D6114yes—
339C61#2098611yescharacteristic, complemented, Sylow
340C2×#41—1222no—
341C2×#41—1222no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—612no—
349C22×Q8—612no—
350C22×Q8—611no—
351C22×Q8—612no—
352C22×Q8—612no—
353C22×Q8—612no—
354C22×Q8—612no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×Q8—1222no—
358C22×Q8—1222no—
359C22×D4—612no—
360C22×D4—611no—
361C23×C4—612no—
362C23×C4—612no—
363C23×C4C22×D6111yescharacteristic
364C4⋊Q8—1224no—
365C4⋊Q8—1224no—
366#124—1224no—
367#124—1224no—
368#123—1228no—
369#123—1228no—
370C22⋊Q8—1228no—
371C22⋊Q8—1228no—
372C4×Q8—12216no—
373C4×Q8—12216no—
374#117—22no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—22no—
380#117—22no—
381#117—614no—
382#117—1228no—
383#117—1222no—
384#117—1228no—
385#117—1222no—
386#117—1222no—
387#117D24414yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—618no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C4×D6112yes—
395C2×C22⋊C4—612no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—612no—
398C2×C22⋊C4C22×D6112yes—
399C2×C22⋊C4C22×D6112yes—
400C2×C22⋊C4C22×D6112yes—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—1224no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42C4×D6112yes—
407#99—1228no—
408#99—1228no—
409#95—1228no—
410C24—612no—
411C24C23×D6111yescharacteristic
412#41—2442no—
413#41—2442no—
414#41—2442no—
415#41—2442no—
416#41—2442no—
417#41—2442no—
418#41—2442no—
419#41—2442no—
420C2×Q8—614no—
421C2×Q8—614no—
422C2×Q8—614no—
423C2×Q8—614no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1224no—
427C2×Q8—1224no—
428C2×Q8—1222no—
429C2×Q8—1224no—
430C2×Q8—1224no—
431C2×Q8—1222no—
432C2×Q8—1224no—
433C2×Q8—1224no—
434C2×Q8—1224no—
435C2×Q8—1224no—
436C2×Q8—1224no—
437C2×Q8—1224no—
438C2×Q8—1224no—
439C2×Q8—1224no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1222no—
449C2×D4—1222no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—614no—
453C22×C4—614no—
454C22×C4—614no—
455C22×C4—612no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4C2×C4×D6114yes—
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4C2×D24412yes—
470C22×C4#1175822212yes—
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6112yes—
476C22×C4#1175822612yes—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—12216no—
482C4⋊C4—12216no—
483C4⋊C4—1224no—
484C4⋊C4—2442no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—614no—
491C22⋊C4—12216no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4C2×D24414yes—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—42no—
500C42—12216no—
501C42—1224no—
502C42—2442no—
503C23—42no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—2444no—
508C23C22×C4×D6111yescharacteristic
509C23C2×D4×D6111yescharacteristic
510C23C2×D4×D6111yescharacteristic
511C23C2×D4×D6111yescharacteristic
512C23C22×D24411yescharacteristic
513C23C2×#1175822211yescharacteristic
514C23C2×#1175822611yescharacteristic
515Q8—1228no—
516Q8—1228no—
517Q8—1228no—
518Q8—1228no—
519Q8—2442no—
520Q8—2442no—
521D4—2442no—
522D4—2442no—
523D4—2442no—
524D4—2442no—
525D4—2442no—
526D4—2442no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—618no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1224no—
544C2×C4—2444no—
545C2×C4—2442no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22⋊C4×D6111yescharacteristic
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4#1451973912yes—
554C2×C4#1451973912yes—
555C2×C4C4×D24412yes—
556C2×C4C4×D24412yes—
557C2×C4#1451974712yes—
558C2×C4#1451974712yes—
559C2×C4#1451975012yes—
560C2×C4#1451975012yes—
561C2×C4#1451975212yes—
562C2×C4#1451975212yes—
563C2×C4#1451976212yes—
564C2×C4#1451976212yes—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—4882no—
573C22—4882no—
574C22—2442no—
575C22—2444no—
576C22C2×C22⋊C4×D6111yescharacteristic
577C22C2×#1451973911yescharacteristic
578C22C2×C4×D24411yescharacteristic
579C22C2×#1451974711yescharacteristic
580C22C2×#1451975011yescharacteristic
581C22C2×#1451975211yescharacteristic
582C22C2×#1451976211yescharacteristic
583C4—44no—
584C4—44no—
585C4—82no—
586C4—1228no—
587C4—4882no—
588C4—2444no—
589C4—2442no—
590C4#1683964012yes—
591C4#1683964012yes—
592C4#1683964012yes—
593C4#1683964012yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—4882no—
601C2C2×#1683964011yescharacteristic
602C2#2145428112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1740C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C22
cosocleC24×C61C24
hypercenter#1740D61
hypocenterC61#20986
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C2×#16840239C2semidirect
5. #21448403C2semidirect
6. #21446276C2semidirect
7. #21445552C2semidirect
8. C61×#1740C2semidirect
15. C2×#14519777C4semidirect
16. C2×#14519765C4semidirect
18. C2×#14519757C4semidirect
300. #1740D61semidirect
339. C61#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
C2×#168402392
C2×#168397352
#214484032
#214462762
#214455522
C61×#17402
#2098661