Finite Group Databasebeta

← #30609973

Subgroups — #30609973

Order 15616.

1738 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C212yesmaximal, complemented
2#21461074C214yesmaximal, complemented
3#21461069C212yesmaximal, complemented
4#21461064C212yesmaximal, complemented
5#21461054C214yesmaximal, complemented
6C22×C4×C22⋊D61C211yesmaximal, characteristic
7C2×D61⋊M16—22no—
8#16840734—22no—
9#16840733—22no—
10#16840733C2214yescomplemented
11#16840732C2214yes—
12D61⋊(C22⋊C8)—22no—
13D61⋊(C22⋊C8)C2214yescomplemented
14#16840608C2218yescomplemented
15#16840605C2218yescomplemented
16C23×C22⋊D61C411yescharacteristic
17C23×C4×D61C2211yescharacteristic
18C2×C4×C22⋊D61—22no—
19C2×C4×C22⋊D61—22no—
20C2×C4×C22⋊D61—22no—
21C2×C4×C22⋊D61—22no—
22C2×C4×C22⋊D61C412yes—
23C2×C4×C22⋊D61C412yes—
24C2×C4×C22⋊D61C2212yes—
25C2×C4×C22⋊D61C2212yes—
26C22×#11758196C2211yescharacteristic
27C22×C22⋊Dic61C2211yescharacteristic
28C22×#11758198C411yescharacteristic
29C22×C4×Dic61C411yescharacteristic
30C24×C4×C61C411yescharacteristic
31C2×#11758213—24no—
32D61⋊M16—48no—
33C2×#11758217—24no—
34C2×D61⋊C8—24no—
35#14519834—24no—
36#14519834—24no—
37C22⋊#92420—24no—
38C22⋊#92420C2318yes—
39C24×D61C2×C411yescharacteristic
40C22×C22⋊D61—22no—
41C22×C22⋊D61—22no—
42C22×C22⋊D61—28no—
43C22×C22⋊D61C2×C412yes—
44C22×C22⋊D61C2×C412yes—
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2311yescharacteristic
48C22×C4×D61—44no—
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61D412yes—
52C22×C4×D61D412yes—
53C22×C4×D61D412yes—
54C22×C4×D61D412yes—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C4×C22⋊D61—24no—
58C4×C22⋊D61—24no—
59C4×C22⋊D61—24no—
60C4×C22⋊D61—24no—
61C4×C22⋊D61—24no—
62C4×C22⋊D61—24no—
63C4×C22⋊D61—24no—
64C4×C22⋊D61—24no—
65C2×#11758196—22no—
66C2×#11758196—22no—
67C2×#11758196C2×C412yes—
68C2×#11758196C2312yes—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61C2×C412yes—
72C2×C22⋊Dic61C2312yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C412yes—
82C2×C4×Dic61C2×C412yes—
83C25×C61C2×C411yescharacteristic
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C23×C4×C61—44no—
89C23×C4×C61C2×C411yescharacteristic
90C23×C4×C61C2×C412yes—
91C23×C4×C61C2×C412yes—
92C23×C4×C61C2×C411yescharacteristic
93C24×C61—21no—
94C24×C61—21no—
95C24×C61—24no—
96C24×C61—42no—
97C24×C61—42no—
98C24×C61C22×C411yescharacteristic
99C24×C61C22×C412yes—
100C23×D61—24no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61C22⋊C412yes—
104C23×D61C22⋊C412yes—
105C23×D61C22×C411yescharacteristic
106C23×D61C22×C411yescharacteristic
107C22×C4×C61—24no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—44no—
115C22×C4×C61—44no—
116C22×C4×C61—44no—
117C22×C4×C61—44no—
118C22×C4×C61—44no—
119C22×C4×C61—44no—
120C22×C4×C61—42no—
121C22×C4×C61C22⋊C411yescharacteristic
122C22×C4×C61C22⋊C411yescharacteristic
123C22×C4×C61C22⋊C411yescharacteristic
124C22×C4×C61C22⋊C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C412yes—
128C2×C22⋊D61—216no—
129C2×C22⋊D61—216no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61M1618yescomplemented
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—24no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C412yes—
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148C22×Dic61C2411yescharacteristic
149C2×C4×D61—28no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—44no—
153C2×C4×D61—44no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—44no—
157C2×C4×D61C22⋊C412yes—
158C2×C4×D61C22⋊C412yes—
159C2×C4×D61C22⋊C412yes—
160C2×C4×D61C22⋊C412yes—
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165#11758217—48no—
166C2×#92420—28no—
167#11758213—48no—
168D61⋊C8—48no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171#11758198—24no—
172#11758198—24no—
173#11758196—24no—
174#11758196—24no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—42no—
187C23×C61—416no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61C2×C22⋊C411yescharacteristic
194C23×C61C2×C22⋊C411yescharacteristic
195C23×C61C2×M1614yescomplemented
196C23×C61C23×C411yescharacteristic
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—416no—
209C22×D61C2×C22⋊C412yes—
210C22×D61C2×C22⋊C412yes—
211C22×D61C2×M1614yes—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—42no—
225C2×C4×C61#9914yes—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C22⋊D61—216no—
231C22⋊D61—216no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×M1614yescomplemented
246C2×Dic61C22×D412yes—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—24no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255#92420—48no—
256#17362—611nomaximal, Sylow
257C22×C61—28no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—24no—
265C22×C61—44no—
266C22×C61—44no—
267C22×C61—48no—
268C22×C61—48no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61C2×#9912yes—
272C22×C61#41312yes—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61C22×M1612yes—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—44no—
281C2×D61—48no—
282C2×D61—48no—
283C2×D61—44no—
284C2×D61#40714yes—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—42no—
288C4×C61C2×#9912yes—
289C4×C61C2×#9912yes—
290Dic61—24no—
291Dic61—22no—
292Dic61—22no—
293Dic61—42no—
294C22×C4×D4—611no—
295C2×#407—612no—
296#1465—614no—
297#1463—612no—
298#1462—612no—
299#1459—614no—
300C2×C61—22no—
301C2×C61—22no—
302C2×C61—22no—
303C2×C61—44no—
304C2×C61C2×#40711yescharacteristic
305C2×C61C22×#9911yescharacteristic
306C2×C61C2×#41311yescharacteristic
307D61—24no—
308D61—42no—
309C23×D4—611no—
310C24×C4—611no—
311C24×C4#692311yescharacteristic
312C22×M16—1222no—
313C2×C4×D4—612no—
314C2×C4×D4—612no—
315C2×C4×D4—612no—
316C2×C4×D4—612no—
317C2×C4×D4—1222no—
318C2×C4×D4—1222no—
319C2×C4×D4—1222no—
320C2×C4×D4—1222no—
321C22×C4⋊C4—611no—
322C22×C22⋊C4—611no—
323C22×C22⋊C4—611no—
324C22×C42—611no—
325#407—614no—
326#407—614no—
327#407—1222no—
328#407—1222no—
329C2×C22⋊C8—614no—
330C2×C22⋊C8—1222no—
331#324—618no—
332#323—618no—
333C61#1736211yescharacteristic, complemented, Sylow
334C25—611no—
335C25C2×#692311yescharacteristic
336C22×D4—612no—
337C22×D4—612no—
338C22×D4—1228no—
339C22×D4—1222no—
340C22×D4—1222no—
341C23×C4—21no—
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—44no—
346C23×C4—611no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—2444no—
358C23×C4C2×#692311yescharacteristic
359C23×C4C2×#692312yes—
360C23×C4C2×#692312yes—
361C23×C4C2×#692311yescharacteristic
362C2×M16—1224no—
363C2×M16—1224no—
364C2×M16—2448no—
365C22×C8—1224no—
366C4×D4—1224no—
367C4×D4—1224no—
368C4×D4—1224no—
369C4×D4—1224no—
370C4×D4—1224no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1224no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4—1222no—
386C2×C42—612no—
387C2×C42—612no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C22⋊C8—618no—
393C22⋊C8—1224no—
394C22⋊C8—1224no—
395C22⋊C8—1224no—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—611no—
402C24—612no—
403C24—612no—
404C24—611no—
405C24—1224no—
406C24—2442no—
407C24—2442no—
408C24C22×#692311yescharacteristic
409C24C22×#692312yes—
410C2×D4—618no—
411C2×D4—1224no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—1224no—
415C2×D4—12216no—
416C2×D4—12216no—
417C2×D4—1224no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1228no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2444no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4C22⋊#692311yescharacteristic
462C22×C4C22⋊#692311yescharacteristic
463C22×C4C22⋊#692311yescharacteristic
464C22×C4C22⋊#692311yescharacteristic
465C22×C4C22×#692311yescharacteristic
466C22×C4C22×#692311yescharacteristic
467C22×C4C22×#692312yes—
468M16—2448no—
469M16—2448no—
470C2×C8—1228no—
471C2×C8—2448no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C42—1222no—
479C42—1222no—
480C42—1222no—
481C42—1222no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—416no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—612no—
497C23—614no—
498C23—612no—
499C23—1224no—
500C23—1224no—
501C23—1224no—
502C23—1224no—
503C23—1222no—
504C23—2442no—
505C23—2441no—
506C23—2441no—
507C23—2442no—
508C23—2441no—
509C23—2441no—
510C23—24416no—
511C23C2×C22⋊#692311yescharacteristic
512C23C2×C22⋊#692311yescharacteristic
513C23D61⋊M1614yescomplemented
514C23C23×#692311yescharacteristic
515D4—12216no—
516D4—12216no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—614no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1222no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4#1451983514yes—
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C2×C22⋊#692311yescharacteristic
558C8—2448no—
559C22—28no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—24no—
567C22—44no—
568C22—48no—
569C22—48no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—614no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—2444no—
580C22—2448no—
581C22—2448no—
582C22—2444no—
583C22C2×#1451983512yes—
584C22#1684074812yes—
585C22C22×C22⋊#692311yescharacteristic
586C22C2×D61⋊M1612yes—
587C4—21no—
588C4—21no—
589C4—42no—
590C4—1224no—
591C4—1222no—
592C4—1222no—
593C4—2442no—
594C4C2×#1451983512yes—
595C4C2×#1451983512yes—
596C2—22no—
597C2—22no—
598C2—22no—
599C2—44no—
600C2—1224no—
601C2—2442no—
602C2D61⋊#40711yescharacteristic
603C2C22×#1451983511yescharacteristic
604C2C2×#1684074811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC24×C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×Dic61C24
hypercenterC24×C4#6923
hypocenterC61#17362
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. D61⋊#407C2semidirect
2. #21461074C2semidirect
3. #21461069C2semidirect
4. #21461064C2semidirect
5. #21461054C2semidirect
10. #16840733C22semidirect
13. D61⋊(C22⋊C8)C22semidirect
14. #16840608C22semidirect
15. #16840605C22semidirect
135. C2×C22⋊D61M16semidirect
195. C23×C61C2×M16semidirect
245. C2×Dic61C2×M16semidirect
333. C61#17362semidirect
513. C23D61⋊M16semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
#214610742
#214610692
#214610642
#214610542
C22×C4×C22⋊D612
#1736261