Finite Group Databasebeta

← #30596443

Subgroups — #30596443

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2#21462263C211yesmaximal, characteristic
3#21460667C214yesmaximal, complemented
4#21448198C211yesmaximal, characteristic
5C2×D61⋊M16—21no—
6C2×#14519856—21no—
7#16840734C2212yes—
8#16840733—21no—
9#16840733C412yescomplemented
10#16840732—21no—
11D61⋊(C22⋊C8)—21no—
12D61⋊(C22⋊C8)C412yescomplemented
13#16840705C2212yes—
14#16840704—21no—
15#16840704C412yescomplemented
16#16840703—21no—
17C4×C22⋊#6923—21no—
18C4×C22⋊#6923C412yescomplemented
19C23×C4×D61C2211yescharacteristic
20C2×#14519736C411yescharacteristic, complemented
21#16839623—21no—
22#16839619—21no—
23#16839619—21no—
24#16839617—21no—
25C4×#11758198C412yescomplemented
26C4×#11758198C2212yes—
27C2×#117×C61C411yescharacteristic, complemented
28C2×#11758213—22no—
29D61⋊M16—42no—
30D61⋊M16—42no—
31C2×#11758217—22no—
32C2×D61⋊C8—22no—
33#14519856—42no—
34#14519856—42no—
35C2×C4⋊#6923—22no—
36C2×C22⋊#6923—22no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923—22no—
39C2×C4×#6923—22no—
40#14519834—22no—
41#14519834C2×C414yes—
42C22⋊#92420—22no—
43C22⋊#92420—22no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4C2×C414yes—
46C24×D61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48C22×C4×D61—42no—
49C22×C4×D61—42no—
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61Q811yescharacteristic
60C22×C4×D61Q811yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519736—42no—
63C2×C4⋊Dic61—21no—
64C2×C22⋊Dic61—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61C2×C412yes—
68#14519684—22no—
69#14519684—22no—
70C23×C4×C61C2×C411yescharacteristic
71#117×C61—42no—
72C2×C4⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C42×C61C2×C412yes—
75C24×C61C22⋊C411yescharacteristic
76C23×D61—42no—
77C23×D61—42no—
78C23×D61—41no—
79C23×D61—41no—
80C23×D61C22⋊C411yescharacteristic
81C23×D61C22⋊C411yescharacteristic
82C23×D61C22⋊C411yescharacteristic
83C23×D61C4⋊C411yescharacteristic
84C23×D61C4⋊C411yescharacteristic
85C23×D61C2×D411yescharacteristic
86C22×#6923—24no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61C4211yescharacteristic
91C22×C4×C61C4211yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C4⋊C411yescharacteristic
94C22×C4×C61C4⋊C411yescharacteristic
95C22×C4×C61C22×C411yescharacteristic
96C22×Dic61—22no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22⋊C411yescharacteristic
101C22×Dic61C22⋊C411yescharacteristic
102C22×Dic61C4⋊C411yescharacteristic
103C22×Dic61C4⋊C411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C4×D61—41no—
121C2×C4×D61C4211yescharacteristic
122C2×C4×D61C4211yescharacteristic
123C2×C4×D61C22⋊C411yescharacteristic
124C2×C4×D61C22⋊C411yescharacteristic
125C2×C4×D61C22⋊C411yescharacteristic
126C2×C4×D61C22⋊C411yescharacteristic
127C2×C4×D61C4⋊C411yescharacteristic
128C2×C4×D61C4⋊C411yescharacteristic
129C2×C4×D61C4⋊C411yescharacteristic
130C2×C4×D61C4⋊C411yescharacteristic
131C2×C4×D61C4⋊C411yescharacteristic
132C2×C4×D61C4⋊C411yescharacteristic
133C2×C4×D61C22×C411yescharacteristic
134C2×C4×D61C22×C411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×Q811yescharacteristic
137C22⋊#6923—44no—
138C22⋊#6923—44no—
139C22⋊#6923—42no—
140C22⋊#6923—42no—
141#11758217—42no—
142#11758217—42no—
143C2×#92420—24no—
144C4⋊#6923—44no—
145C4×#6923—44no—
146#11758213—44no—
147D61⋊C8—44no—
148C4⋊C4×C61—42no—
149C22⋊C4×C61—42no—
150C42×C61—42no—
151C22⋊Dic61—42no—
152#11758198—44no—
153#11758198—44no—
154C4⋊Dic61—42no—
155C4×Dic61—42no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61#9511yescharacteristic
161C23×C61#9511yescharacteristic
162C23×C61C2×C22⋊C411yescharacteristic
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—48no—
184C22×D61—48no—
185C22×D61#9511yescharacteristic
186C22×D61#9511yescharacteristic
187C22×D61C2×C22⋊C411yescharacteristic
188C22×D61C2×C4⋊C411yescharacteristic
189C2×#6923—44no—
190C2×#6923—44no—
191C2×#6923—44no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61#9511yescharacteristic
202C2×C4×C61#9511yescharacteristic
203C2×C4×C61#9912yes—
204C2×C4×C61#9912yes—
205C2×C4×C61C2×C4211yescharacteristic
206C2×C4×C61C2×C4⋊C411yescharacteristic
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61#9511yescharacteristic
219C2×Dic61#9511yescharacteristic
220C2×Dic61C2×C22⋊C411yescharacteristic
221C2×Dic61C2×C4⋊C411yescharacteristic
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—22no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—41no—
237C4×D61—41no—
238#92420—44no—
239#7946—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61C2×#9511yescharacteristic
251C22×C61C2×#9911yescharacteristic
252C22×C61C2×#9911yescharacteristic
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—44no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—44no—
269C2×D61—44no—
270C2×D61—44no—
271C2×D61H(3,ℤ/4)12yes—
272C2×D61#42112yes—
273#6923—84no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—82no—
277C4×C61#34212yes—
278C4×C61#34212yes—
279Dic61—21no—
280Dic61—21no—
281Dic61—21no—
282Dic61—21no—
283Dic61—82no—
284C2×#407—611no—
285#2280—611no—
286#2280—611no—
287#1284—614no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—41no—
291C2×C61—41no—
292C2×C61C2×H(3,ℤ/4)11yescharacteristic
293C2×C61#173311yescharacteristic
294C2×C61C2×#34211yescharacteristic
295D61—22no—
296D61—22no—
297D61—41no—
298D61—41no—
299C24×C4—611no—
300C22×M16—1221no—
301C2×#117—611no—
302C2×#117—1221no—
303C2×#117#692311yescharacteristic, complemented
304#407—612no—
305#407—612no—
306#407—1221no—
307#407—1221no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—1221no—
310#381—612no—
311#381—1221no—
312#380—612no—
313#380—1221no—
314#380—1221no—
315#380—1221no—
316#379—1221no—
317#379—1221no—
318C4×C22⋊C4—612no—
319C4×C22⋊C4—612no—
320C4×C22⋊C4—612no—
321C4×C22⋊C4—1221no—
322C61#794611yescharacteristic, 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—2442no—
338C23×C4—2442no—
339C23×C4C2×#692311yescharacteristic
340C2×M16—1222no—
341C2×M16—1222no—
342C2×M16—2442no—
343C2×M16—2442no—
344C22×C8—1222no—
345#117—42no—
346#117—2442no—
347#117—2442no—
348#117—2442no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1221no—
359C2×C42—612no—
360C2×C42—1222no—
361C2×C42C2×#692312yes—
362C22⋊C8—614no—
363C22⋊C8—1222no—
364C22⋊C8—1222no—
365C22⋊C8—1222no—
366#95—614no—
367#95—1222no—
368#95—1222no—
369#95—1222no—
370C24—611no—
371C24—611no—
372C24—611no—
373C24—611no—
374C24—611no—
375C24—611no—
376C24—2441no—
377C24—2441no—
378C24—2442no—
379C24—2442no—
380C24C22⋊#692311yescharacteristic
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
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—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1224no—
411C22×C4—1222no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4C4×#692311yescharacteristic
427C22×C4C4×#692311yescharacteristic
428C22×C4C4⋊#692311yescharacteristic
429C22×C4C4⋊#692311yescharacteristic
430C22×C4C22⋊#692311yescharacteristic
431C22×C4C22×#692311yescharacteristic
432M16—2444no—
433M16—2442no—
434M16—2442no—
435C2×C8—1224no—
436C2×C8—2444no—
437C4⋊C4—42no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C22⋊C4—42no—
441C22⋊C4—2444no—
442C22⋊C4—2444no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2444no—
447C22⋊C4—2444no—
448C42—42no—
449C42—2442no—
450C42—2444no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—611no—
456C23—611no—
457C23—611no—
458C23—611no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—1221no—
466C23—1221no—
467C23—1221no—
468C23—1221no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2448no—
480C23—2448no—
481C23(C2×#6923)⋊C411yescharacteristic
482C23(C2×#6923)⋊C411yescharacteristic
483C23C2×C22⋊#692311yescharacteristic
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—611no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2444no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4—2442no—
527C2×C4(C2×#6923)⋊C411yescharacteristic
528C2×C4(C2×#6923)⋊C411yescharacteristic
529C2×C4#1451983512yes—
530C2×C4#1451983612yes—
531C2×C4C2×C4×#692311yescharacteristic
532C2×C4C2×C4⋊#692311yescharacteristic
533C8—2444no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—612no—
545C22—612no—
546C22—1221no—
547C22—1221no—
548C22—1221no—
549C22—1221no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—1221no—
554C22—1221no—
555C22—1221no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22—2442no—
560C22—2444no—
561C22—2444no—
562C22—2444no—
563C22—2444no—
564C22C2×(C2×#6923)⋊C411yescharacteristic
565C22C2×#1451983511yescharacteristic
566C22C2×#1451983611yescharacteristic
567C4—21no—
568C4—21no—
569C4—82no—
570C4—1221no—
571C4—1221no—
572C4—1221no—
573C4—1221no—
574C4—4882no—
575C4—4884no—
576C4#1684064012yes—
577C4#1684064012yes—
578C2—21no—
579C2—21no—
580C2—41no—
581C2—41no—
582C2—1222no—
583C2—1222no—
584C2—2441no—
585C2—2441no—
586C2#2146148911yescharacteristic
587C2#2146150011yescharacteristic
588C2C2×#1684064011yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×#117#6923
hypocenterC61#7946
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21460667C2semidirect
9. #16840733C4semidirect
12. D61⋊(C22⋊C8)C4semidirect
15. #16840704C4semidirect
18. C4×C22⋊#6923C4semidirect
20. C2×#14519736C4semidirect
25. C4×#11758198C4semidirect
27. C2×#117×C61C4semidirect
303. C2×#117#6923semidirect
322. C61#7946semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
#214622632
#214606672
#214481982
#794661