Finite Group Databasebeta

← D61×#2071 #29708741

Subgroups — D61×#2071 #29708741

Order 15616.

1366 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#102×D61C212yesmaximal
2#390×D61C211yesmaximal, characteristic
3#384×D61C211yesmaximal, characteristic, complemented
4#21447226C212yesmaximal, complemented
5#21447225C212yesmaximal, complemented
6#21447224C211yesmaximal, characteristic, complemented
7#21447223C211yesmaximal, characteristic, complemented
8#21447222C211yesmaximal, characteristic, complemented
9#340×D61C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C22×C8×D61C2212yes—
14C2×C4⋊C4×D61—21no—
15C2×C4⋊C4×D61—21no—
16C2×C4⋊C4×D61C2212yes—
17C2×C4⋊C4×D61C2211yescharacteristic
18C2×C22⋊C4×D61—21no—
19C2×C22⋊C4×D61—21no—
20C2×C42×D61C2211yescharacteristic
21C2×#14519699C2212yes—
22C2×#14519698C2212yes—
23C2×#14519697C2212yes—
24#102×D61—28no—
25#16839665C2211yescharacteristic
26#16839664C2211yescharacteristic
27#16839663C2211yescharacteristic
28#16839662—21no—
29#16839662—21no—
30#16839632C2211yescharacteristic, complemented
31#16839631—21no—
32#16839631—21no—
33#16839631C2212yescomplemented
34#16839525C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2211yescharacteristic, complemented
38#16839516C2212yescomplemented
39C2×#102×C61C2212yes—
40C61×#390C2211yescharacteristic
41C61×#384C2211yescharacteristic, complemented
42C61×#340C2212yescomplemented
43C61×#336C2211yescharacteristic, complemented
44C24×D61—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C2×D4×D61—24no—
48C2×D4×D61—24no—
49C2×D4×D61—22no—
50C2×D4×D61D414yes—
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×#92418C2312yes—
57C2×C8×D61—24no—
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196—21no—
68C4⋊C4×D61—24no—
69C4⋊C4×D61—24no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—44no—
72C4⋊C4×D61—44no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C22⋊C4×D61—44no—
80C22⋊C4×D61—44no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—24no—
85#14519699—24no—
86#14519698—24no—
87#14519697—24no—
88C22×D4×C61C2311yescharacteristic
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#102×C61—24no—
98C24×C61—21no—
99C23×D61—21no—
100C23×D61—24no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61C2×D411yescharacteristic
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61C2×D412yes—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—24no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125D4×D61—216no—
126D4×D61—48no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—21no—
130C2×D244C2×D412yes—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—44no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—44no—
139C2×C4×D61—42no—
140C2×C4×D61—44no—
141C2×C4×D61D814yes—
142C2×C4×D61SD1614yes—
143C2×C4×D61C2×D412yes—
144C2×C4×D61#4114yes—
145C2×C4×D61#4114yes—
146C2×C4×D61#4112yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—42no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160#11758198—42no—
161#11758198—42no—
162#11758198—42no—
163#11758198—42no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168C4⋊Dic61—42no—
169#11758196—44no—
170#11758196—44no—
171#11758196—42no—
172#11758196—44no—
173C4×Dic61—22no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C8×D61—48no—
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—48no—
196C22×D61—48no—
197C22×D61—48no—
198C22×D61—84no—
199C22×D61—84no—
200C22×D61—84no—
201C22×D61—84no—
202C22×D61#12112yes—
203C22×D61#12312yes—
204C22×D61#12412yes—
205C22×D61C2×D812yes—
206C22×D61C2×SD1612yes—
207C22×D61AGL(1,ℤ/8)12yes—
208C22×D61#13712yes—
209D4×C61—24no—
210D4×C61—42no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61C2×D812yes—
222C2×C4×C61C2×SD1612yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4112yes—
225C2×C4×C61C2×#4112yes—
226C2×C4×C61C2×#4111yescharacteristic
227C22⋊D61—48no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61#12112yes—
239C2×Dic61#12312yes—
240C2×Dic61#12412yes—
241C2×Dic61C2×D812yes—
242C2×Dic61C2×SD1612yes—
243D244—24no—
244D244—42no—
245C4×D61—28no—
246C4×D61—28no—
247C4×D61—88no—
248C4×D61—84no—
249C4×D61#12318yes—
250C4×D61#12618yes—
251C8×C61—42no—
252#92418—42no—
253C2×#2071—611nomaximal, Sylow
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×#12311yescharacteristic
263C22×C61C2×#12411yescharacteristic
264C22×C61C22×D811yescharacteristic
265C22×C61C22×SD1611yescharacteristic
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—816no—
276C2×D61#39714yes—
277C2×D61#45914yes—
278C2×D61#46114yes—
279C2×D61#48014yes—
280C2×D61#48114yes—
281C2×D61#48614yes—
282C2×D61#48914yes—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—81no—
286C4×C61—82no—
287C4×C61C2×#12312yes—
288C4×C61C2×#12612yes—
289Dic61—22no—
290Dic61—22no—
291Dic61—81no—
292Dic61—82no—
293Dic61#39714yes—
294C22×#102—612no—
295C2×#390—611no—
296C2×#384—611no—
297#2071—612no—
298#2071—611no—
299#2071—611no—
300#2071—612no—
301#2071—611no—
302#2071D6111yescharacteristic, direct product factor
303C2×#340—612no—
304C2×#336—611no—
305C2×C61—82no—
306C2×C61C2×#39711yescharacteristic
307C2×C61C2×#45911yescharacteristic
308C2×C61C2×#46111yescharacteristic
309C2×C61C2×#48011yescharacteristic
310C2×C61C2×#48111yescharacteristic
311C2×C61C2×#48611yescharacteristic
312C2×C61C2×#48911yescharacteristic
313D61—82no—
314D61#207118yesdirect product factor
315C23×D4—611no—
316C23×C8—612no—
317C22×C4⋊C4—612no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—1221no—
320C22×C4⋊C4—1221no—
321C22×C22⋊C4—1221no—
322C22×C22⋊C4—1221no—
323C22×C42—611no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102—612no—
327C2×#102—1228no—
328C2×#102C2×D6112yes—
329#390—611no—
330#390—611no—
331#390—611no—
332#390—1221no—
333#390—1221no—
334#390C2×D6111yescharacteristic
335#384—612no—
336#384—611no—
337#384—1221no—
338#384—1221no—
339#384C2×D6111yescharacteristic, complemented
340#340—612no—
341#340—612no—
342#340—612no—
343#340C2×D6112yescomplemented
344#336—612no—
345#336—611no—
346#336C2×D6111yescharacteristic, complemented
347C61C2×#207111yescharacteristic, complemented, Sylow
348C25—1221no—
349C22×D4—614no—
350C22×D4—611no—
351C22×D4—1224no—
352C22×D4—1224no—
353C22×D4—1222no—
354C22×D4—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4—1222no—
361C22×C8—612no—
362C22×C8—1224no—
363C22×C8—1224no—
364C22×C8—1224no—
365C22×C8C22×D6112yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1224no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—2444no—
380C2×C4⋊C4—2444no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6112yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—2444no—
392C2×C22⋊C4—2444no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1224no—
397C2×C42C22×D6111yescharacteristic
398#102—24no—
399#102—1224no—
400#102—1224no—
401#102—1224no—
402C24—21no—
403C24—611no—
404C24—1224no—
405C24—1221no—
406C24—2442no—
407C24—2442no—
408C24—2442no—
409C24—2442no—
410C24—2442no—
411C24—2442no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—612no—
416C2×D4—12216no—
417C2×D4—1224no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1221no—
421C2×D4—2448no—
422C2×D4—2442no—
423C2×D4—2442no—
424C2×D4D4×D6112yes—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—614no—
430C22×C4—612no—
431C22×C4—614no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1221no—
441C22×C4—1222no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2444no—
445C22×C4—2442no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4D4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4C23×D6111yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—2448no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—2442no—
467C4⋊C4—2444no—
468C4⋊C4—2442no—
469C4⋊C4—2444no—
470C4⋊C4—2444no—
471C4⋊C4—2442no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C22⋊C4—2442no—
477C22⋊C4—2442no—
478C22⋊C4—2442no—
479C22⋊C4—2442no—
480C42—22no—
481C42—1222no—
482C42—1222no—
483C42—1222no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—612no—
492C23—612no—
493C23—612no—
494C23—612no—
495C23—612no—
496C23—612no—
497C23—612no—
498C23—1222no—
499C23—4884no—
500C23—4884no—
501C23—4884no—
502C23—4884no—
503C23—2448no—
504C23—2448no—
505C23—2448no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23C2×D4×D6111yescharacteristic
513D4—24no—
514D4—42no—
515D4—1224no—
516D4—2448no—
517D4—2442no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—618no—
532C2×C4—618no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—1228no—
536C2×C4—1228no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—4888no—
542C2×C4—4884no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4D8×D6112yes—
550C2×C4SD16×D6112yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4#41×D6112yes—
553C2×C4#41×D6112yes—
554C2×C4#41×D6111yescharacteristic
555C8—42no—
556C8—2442no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—614no—
565C22—614no—
566C22—614no—
567C22—614no—
568C22—614no—
569C22—614no—
570C22—614no—
571C22—48816no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22#121×D6111yescharacteristic
580C22#123×D6111yescharacteristic
581C22#124×D6111yescharacteristic
582C22C2×D8×D6111yescharacteristic
583C22C2×SD16×D6111yescharacteristic
584C22AGL(1,ℤ/8)×D6111yescharacteristic
585C22#137×D6111yescharacteristic
586C4—22no—
587C4—22no—
588C4—82no—
589C4—81no—
590C4—614no—
591C4—1222no—
592C4—1222no—
593C4—4882no—
594C4—4881no—
595C4#123×D6112yes—
596C4#126×D6112yes—
597C2—82no—
598C2—618no—
599C2—4882no—
600C2#397×D6111yescharacteristic
601C2#459×D6111yescharacteristic
602C2#461×D6111yescharacteristic
603C2#480×D6111yescharacteristic
604C2#481×D6111yescharacteristic
605C2#486×D6111yescharacteristic
606C2#489×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
hypercenter#2071D61
hypocenterC61C2×#2071
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #384×D61C2semidirect
4. #21447226C2semidirect
5. #21447225C2semidirect
6. #21447224C2semidirect
7. #21447223C2semidirect
8. #21447222C2semidirect
9. #340×D61C2semidirect
10. #336×D61C2semidirect
11. C61×#2071C2semidirect
30. #16839632C22semidirect
33. #16839631C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
38. #16839516C22semidirect
41. C61×#384C22semidirect
42. C61×#340C22semidirect
43. C61×#336C22semidirect
302. #2071D61direct (N × H)
314. D61#2071direct (N × H)
339. #384C2×D61semidirect
343. #340C2×D61semidirect
346. #336C2×D61semidirect
347. C61C2×#2071semidirect

Maximal subgroups

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

subgroupindex
C2×#102×D612
#390×D612
#384×D612
#214472262
#214472252
#214472242
#214472232
#214472222
#340×D612
#336×D612
C61×#20712
C2×#207161