Finite Group Databasebeta

← #30602415

Subgroups — #30602415

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic
3C2×#16840637C214yesmaximal, complemented
4C22×M16×D61C211yesmaximal, characteristic
5C2×D61⋊M16C2212yes—
6C22×D61⋊C8C2212yes—
7#16840734—21no—
8#16840733C412yes—
9#16840733C412yes—
10#16840732—21no—
11C2×#14519834—21no—
12D61⋊(C22⋊C8)—21no—
13D61⋊(C22⋊C8)—21no—
14D61⋊(C22⋊C8)—21no—
15D61⋊(C22⋊C8)—21no—
16D61⋊(C22⋊C8)C412yes—
17D61⋊(C22⋊C8)C412yes—
18C2×C22⋊#92420—21no—
19#16840637—28no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758194C411yescharacteristic
22C22×#11758189—21no—
23C2×M16×D61—28no—
24C2×M16×D61—21no—
25C2×M16×D61—21no—
26C2×M16×D61C412yes—
27C2×M16×D61C2212yes—
28C22×C8×D61—21no—
29C22×M16×C61C411yescharacteristic
30C2×#11758213C2×C414yes—
31D61⋊M16—24no—
32D61⋊M16—24no—
33C2×#11758217—22no—
34C2×D61⋊C8—24no—
35C2×D61⋊C8—24no—
36C2×D61⋊C8—22no—
37C2×D61⋊C8—22no—
38C2×D61⋊C8C2×C414yes—
39C22×#92420—22no—
40#14519834—22no—
41#14519834—22no—
42#14519834—22no—
43#14519834—22no—
44C22⋊#92420—22no—
45C22⋊#92420—22no—
46C22⋊#92420—22no—
47C22⋊#92420—22no—
48C24×D61Q811yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—24no—
51C22×C4×D61—24no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61Q811yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758194—21no—
65C2×#11758194—24no—
66C2×#11758194C2×C412yes—
67C2×#11758189—28no—
68C2×#11758189—22no—
69C2×#11758189—22no—
70M16×D61—28no—
71M16×D61—28no—
72M16×D61—24no—
73M16×D61—24no—
74M16×D61—24no—
75M16×D61D418yes—
76C22×#92418—21no—
77C2×C8×D61—28no—
78C2×C8×D61—22no—
79C2×C8×D61—22no—
80C23×C4×C61C2×C411yescharacteristic
81C2×M16×C61—21no—
82C2×M16×C61—24no—
83C2×M16×C61C2×C412yes—
84C22×C8×C61—21no—
85C24×C61C4⋊C411yescharacteristic
86C23×D61—24no—
87C23×D61—24no—
88C23×D61—42no—
89C23×D61C4⋊C411yescharacteristic
90C23×D61C4⋊C411yescharacteristic
91C23×D61C4⋊C411yescharacteristic
92C23×D61C4⋊C411yescharacteristic
93C23×D61C4⋊C411yescharacteristic
94C23×D61C2×Q811yescharacteristic
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61C4211yescharacteristic
98C22×C4×C61C4211yescharacteristic
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C4⋊C411yescharacteristic
102C22×C4×C61C22×C411yescharacteristic
103C22×Dic61—24no—
104C22×Dic61C22⋊C411yescharacteristic
105C22×Dic61C22⋊C411yescharacteristic
106C22×Dic61C22⋊C411yescharacteristic
107C22×Dic61C22⋊C411yescharacteristic
108C22×Dic61C4⋊C411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110C2×C4×D61—24no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—24no—
114C2×C4×D61—24no—
115C2×C4×D61—24no—
116C2×C4×D61—24no—
117C2×C4×D61—24no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61C4211yescharacteristic
127C2×C4×D61C4211yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C22⋊C411yescharacteristic
133C2×C4×D61C22⋊C411yescharacteristic
134C2×C4×D61C4⋊C411yescharacteristic
135C2×C4×D61C4⋊C411yescharacteristic
136C2×C4×D61C4⋊C411yescharacteristic
137C2×C4×D61C4⋊C411yescharacteristic
138C2×C4×D61C22×C411yescharacteristic
139C2×C4×D61C22×C411yescharacteristic
140C2×C4×D61C2×D414yes—
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61#4114yes—
144#11758217—44no—
145C2×#92420—24no—
146C2×#92420—24no—
147C2×#92420—44no—
148#11758213—28no—
149D61⋊C8—28no—
150D61⋊C8—42no—
151D61⋊C8—42no—
152D61⋊C8—42no—
153D61⋊C8—42no—
154M16×C61—24no—
155M16×C61—22no—
156M16×C61C22⋊C414yes—
157C2×C8×C61—24no—
158C2×C8×C61—22no—
159#11758194—24no—
160#11758194—22no—
161#11758194C22⋊C414yes—
162C2×#92418—24no—
163C2×#92418—22no—
164#11758189—28no—
165#11758189—28no—
166C8×D61—28no—
167C8×D61—28no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—41no—
171C23×C61#9511yescharacteristic
172C23×C61#9511yescharacteristic
173C23×C61C2×C4⋊C411yescharacteristic
174C22×D61—22no—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—416no—
191C22×D61#9511yescharacteristic
192C22×D61#9511yescharacteristic
193C22×D61#10814yes—
194C22×D61C2×C4⋊C411yescharacteristic
195C22×D61C2×C4⋊C411yescharacteristic
196C22×D61C22⋊Q814yes—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61#9511yescharacteristic
205C2×C4×C61#9511yescharacteristic
206C2×C4×C61C2×C4211yescharacteristic
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61#11712yes—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61#9511yescharacteristic
216C2×Dic61#9511yescharacteristic
217C2×Dic61C2×C22⋊C411yescharacteristic
218C2×Dic61C2×C22⋊C411yescharacteristic
219C2×Dic61#12114yes—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—21no—
227C4×D61—21no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61C4×D414yes—
235C4×D61C4×D414yes—
236C4×D61#12012yes—
237C4×D61#12012yes—
238C4×D61#12312yes—
239C4×D61#12312yes—
240#92420—44no—
241#92420—44no—
242C8×C61—24no—
243#92418—24no—
244#10734—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—44no—
253C22×C61#34112yescomplemented
254C22×C61C2×#9511yescharacteristic
255C22×C61#38012yes—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—28no—
261C2×D61—28no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61—44no—
269C2×D61#34112yescomplemented
270C2×D61#38512yes—
271C2×D61#38512yes—
272C2×D61C2×#10812yes—
273C2×D61#43012yes—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61C4×C22⋊C412yes—
277C4×C61#37911yescharacteristic
278C4×C61#38111yescharacteristic
279Dic61—21no—
280Dic61—21no—
281Dic61#38612yes—
282Dic61#38612yes—
283C23×M16—611no—
284C2×#407—611no—
285C22×C22⋊C8—611no—
286C2×#341—614no—
287C2×C61—22no—
288C2×C61—22no—
289C2×C61—41no—
290C2×C61C22⋊#9511yescharacteristic
291C2×C61C2×#34111yescharacteristic
292C2×C61#173811yescharacteristic
293D61—22no—
294D61—22no—
295D61—41no—
296D61#185514yes—
297C24×C4—611no—
298C22×M16—612no—
299C22×M16—612no—
300C22×M16—611no—
301C22×M16—612no—
302C22×M16—1221no—
303C22×M16—1228no—
304C22×M16—1221no—
305C22×M16—1221no—
306C22×M16#692311yescharacteristic
307C23×C8—612no—
308C23×C8—1221no—
309#407—612no—
310#407—612no—
311#407—1221no—
312#407—1221no—
313C2×C22⋊C8—612no—
314C2×C22⋊C8—612no—
315C2×C22⋊C8—1221no—
316C2×C22⋊C8—1221no—
317C2×C22⋊C8—1221no—
318C2×C22⋊C8—1221no—
319C2×C22⋊C8—1221no—
320C2×C22⋊C8—1221no—
321#341—1228no—
322C61#1073411yescharacteristic, complemented, Sylow
323C25—611no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—1224no—
338C23×C4—1224no—
339C23×C4C2×#692311yescharacteristic
340C2×M16—21no—
341C2×M16—24no—
342C2×M16—612no—
343C2×M16—618no—
344C2×M16—614no—
345C2×M16—1222no—
346C2×M16—1222no—
347C2×M16—1228no—
348C2×M16—1224no—
349C2×M16—1224no—
350C2×M16—1228no—
351C2×M16—1228no—
352C2×M16—1224no—
353C2×M16—1221no—
354C2×M16—1224no—
355C2×M16—1222no—
356C2×M16—1224no—
357C2×M16—1224no—
358C2×M16C2×#692312yes—
359C22×C8—21no—
360C22×C8—614no—
361C22×C8—1222no—
362C22×C8—1222no—
363C22×C8—1228no—
364C22×C8—1221no—
365C22×C8—1222no—
366C22×C8—1224no—
367C22×C8—1224no—
368C22×C8—1222no—
369C22×C8—1222no—
370C22⋊C8—1222no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374C22⋊C8—1222no—
375C22⋊C8—1222no—
376C22⋊C8—1222no—
377C22⋊C8—1222no—
378C24—611no—
379C24—611no—
380C24—611no—
381C24—611no—
382C24—611no—
383C24—611no—
384C24—1224no—
385C24—1224no—
386C24—2442no—
387C24C4⋊#692311yescharacteristic
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—614no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—614no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1224no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1224no—
424C22×C4—1224no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1224no—
428C22×C4—1224no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4C4×#692311yescharacteristic
432C22×C4C4×#692311yescharacteristic
433C22×C4C4⋊#692311yescharacteristic
434C22×C4C22⋊#692311yescharacteristic
435C22×C4C22⋊#692311yescharacteristic
436C22×C4C22×#692311yescharacteristic
437M16—24no—
438M16—22no—
439M16—614no—
440M16—1228no—
441M16—1228no—
442M16—1224no—
443M16—1222no—
444M16—1228no—
445M16—2444no—
446M16C22⋊#692314yes—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—1228no—
450C2×C8—1228no—
451C2×C8—1224no—
452C2×C8—1222no—
453C2×C8—1224no—
454C2×C8—1224no—
455C2×C8—1228no—
456C2×C8—2442no—
457C2×C8—2442no—
458C2×C8—2444no—
459C2×C8—2442no—
460C2×C8—2442no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—611no—
465C23—614no—
466C23—614no—
467C23—611no—
468C23—611no—
469C23—611no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—24416no—
487C23(C2×#6923)⋊C411yescharacteristic
488C23(C2×#6923)⋊C411yescharacteristic
489C23C2×C4⋊#692311yescharacteristic
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—614no—
498C2×C4—611no—
499C2×C4—611no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—614no—
503C2×C4—614no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4(C2×#6923)⋊C411yescharacteristic
528C2×C4(C2×#6923)⋊C411yescharacteristic
529C2×C4C2×C4×#692311yescharacteristic
530C2×C4C2×C22⋊#692312yes—
531C2×C4C2×C22⋊#692311yescharacteristic
532C2×C4#1451985612yes—
533C8—24no—
534C8—1224no—
535C8—2444no—
536C8—2444no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—612no—
546C22—612no—
547C22—612no—
548C22—612no—
549C22—612no—
550C22—1228no—
551C22—1228no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—1221no—
557C22—1221no—
558C22—1221no—
559C22—2444no—
560C22—2444no—
561C22—2444no—
562C22—2444no—
563C22#1684063712yescomplemented
564C22C2×(C2×#6923)⋊C411yescharacteristic
565C22#1684070412yes—
566C4—21no—
567C4—21no—
568C4—612no—
569C4—612no—
570C4—1221no—
571C4—1221no—
572C4C4×C22⋊#692312yes—
573C4#1684070311yescharacteristic
574C4#1684070511yescharacteristic
575C2—22no—
576C2—22no—
577C2—41no—
578C2—614no—
579C2—1222no—
580C2—1222no—
581C2—2441no—
582C2#2146099311yescharacteristic
583C2C2×#1684063711yescharacteristic
584C2#2146151211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61Q8
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#10734
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16840637C2semidirect
253. C22×C61#341semidirect
269. C2×D61#341semidirect
322. C61#10734semidirect
563. C22#16840637semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
C2×D61⋊(C22⋊C8)2
C2×#168406372
C22×M16×D612
#1073461