Finite Group Databasebeta

← #30602523

Subgroups — #30602523

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840748C211yesmaximal, characteristic
2C2×#16840741C211yesmaximal, characteristic
3#21461506C212yesmaximal, complemented
4#21461498C212yesmaximal, complemented
5C2×#16840243C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840748—22no—
9#16840748C414yescomplemented
10C2×#14519838—21no—
11C2×#14519837—21no—
12#16840741—22no—
13#16840741C414yescomplemented
14C2×#14519836—21no—
15C2×#14519835—21no—
16#16840635—24no—
17#16840632—24no—
18C22×#11758222C2211yescharacteristic
19#16840243—24no—
20#16840243—24no—
21C2×#14519742—21no—
22C2×#14519739—21no—
23C2×C4×C22⋊D61—21no—
24C2×C4×C22⋊D61—21no—
25C2×#14519736C411yescharacteristic, complemented
26C2×#14519735C411yescharacteristic, complemented
27C2×#14519735C2211yescharacteristic
28C2×C4⋊C4×D61C411yescharacteristic, complemented
29C2×C4⋊C4×D61C2211yescharacteristic
30C22×C4⋊C4×C61C411yescharacteristic, complemented
31C2×#11758213—22no—
32D61⋊M16—22no—
33D61⋊M16—22no—
34D61⋊M16—22no—
35D61⋊M16C2×C414yescomplemented
36C2×#11758217—22no—
37C2×D61⋊C8—22no—
38#14519856—22no—
39#14519856—22no—
40#14519856—22no—
41#14519856C2×C414yescomplemented
42C2×C4⋊#6923—22no—
43C2×C22⋊#6923—22no—
44C2×C4×#6923—22no—
45#14519838—24no—
46#14519837—24no—
47#14519836—24no—
48#14519835—24no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222C2×C412yes—
54C2×#11758222C2×C412yes—
55C2×#11758222D412yes—
56C2×#11758222Q812yes—
57C22×Dic122D411yescharacteristic
58C22×D244D411yescharacteristic
59C22×C22⋊D61D411yescharacteristic
60C22×C22⋊D61D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2×C411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519742—28no—
66#14519739—28no—
67C4×C22⋊D61—28no—
68C4×C22⋊D61—28no—
69#14519736—22no—
70#14519736—22no—
71#14519735—22no—
72#14519735—22no—
73#14519735—22no—
74#14519735D414yes—
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61D414yes—
83C2×C22⋊Dic61—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C23×C4×C61—21no—
90C23×C4×C61C2×C411yescharacteristic
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61C2×C411yescharacteristic
95C2×C4⋊C4×C61C2×C411yescharacteristic
96C24×C61C22⋊C411yescharacteristic
97C23×D61C22⋊C411yescharacteristic
98C23×D61C2×D411yescharacteristic
99C22×#6923—42no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—24no—
107C22×C4×C61—82no—
108C22×C4×C61C4212yes—
109C22×C4×C61C22⋊C411yescharacteristic
110C22×C4×C61C4⋊C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—22no—
116C2×C22⋊D61C22⋊C412yes—
117C2×C22⋊D61C22⋊C412yes—
118C2×C22⋊D61C4⋊C412yes—
119C2×C22⋊D61C4⋊C412yes—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124#11758222—24no—
125#11758222—24no—
126#11758222—24no—
127#11758222—24no—
128#11758222—24no—
129#11758222—24no—
130#11758222—24no—
131#11758222—24no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244C22⋊C412yes—
135C2×D244C4⋊C412yes—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×C4×D61C4212yes—
145C2×C4×D61C4⋊C412yes—
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×Q811yescharacteristic
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122C22⋊C412yes—
157C2×Dic122C4⋊C412yes—
158C22⋊#6923—24no—
159C22⋊#6923—42no—
160#11758217—24no—
161#11758217—42no—
162C2×#92420—42no—
163C4⋊#6923—24no—
164C4⋊#6923—42no—
165C4×#6923—24no—
166C4×#6923—42no—
167#11758213—24no—
168#11758213—42no—
169D61⋊C8—24no—
170D61⋊C8—42no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22⋊C412yes—
175C4⋊C4×C61C22⋊C412yes—
176C22⋊Dic61—24no—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22⋊C412yes—
182C4⋊Dic61C22⋊C412yes—
183#11758196—24no—
184#11758196—24no—
185C4×Dic61—24no—
186C4×Dic61—24no—
187C4×Dic61—24no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61#9512yes—
193C23×C61C2×C22⋊C411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61#9512yes—
201C22×D61C2×C22⋊C411yescharacteristic
202C22×D61C2×C4⋊C411yescharacteristic
203C22×D61#12012yes—
204C22×D61#12312yes—
205C2×#6923—44no—
206C2×#6923—82no—
207C2×#6923—82no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—84no—
219C2×C4×C61#9512yes—
220C2×C4×C61C2×C4211yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C4⋊C411yescharacteristic
224C2×C4×C61#11711yescharacteristic
225C2×C4×C61#11711yescharacteristic
226C22⋊D61—24no—
227C22⋊D61—24no—
228C22⋊D61—24no—
229C22⋊D61—24no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61#9512yes—
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61C2×C4⋊C411yescharacteristic
239C2×Dic61#12012yes—
240C2×Dic61#12312yes—
241D244—24no—
242D244—24no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—84no—
248C4×D61—84no—
249C4×D61C4×D412yes—
250C4×D61C4×D412yes—
251C4×D61C4×D412yes—
252C4×D61C4×D412yes—
253C4×D61#12112yes—
254C4×D61#12112yes—
255C4×D61C22⋊Q812yes—
256C4×D61C22⋊Q812yes—
257Dic122—24no—
258Dic122—24no—
259#92420—44no—
260#10765—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×#9511yescharacteristic
269C22×C61#37911yescharacteristic
270C22×C61#38111yescharacteristic
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—42no—
281C2×D61#38512yes—
282C2×D61#38612yes—
283#6923—84no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61C4×C22⋊C411yescharacteristic
288C4×C61C4×C22⋊C411yescharacteristic
289C4×C61#38011yescharacteristic
290C4×C61#38011yescharacteristic
291Dic61—22no—
292Dic61—42no—
293Dic61—42no—
294Dic61#38512yes—
295Dic61#38612yes—
296C2×#520—611no—
297C2×#413—611no—
298C2×#410—611no—
299#1736—612no—
300#1733—612no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—42no—
304C2×C61C22⋊#9511yescharacteristic
305C2×C61#179511yescharacteristic
306C2×C61#179611yescharacteristic
307D61—22no—
308D61—22no—
309D61—42no—
310C22×#41—611no—
311C22×M16—612no—
312#520—1224no—
313#520—1224no—
314C2×C4×Q8—1221no—
315C2×C4×D4—1221no—
316C2×C4×D4—1221no—
317C2×C4×D4—1221no—
318C2×#117—611no—
319C2×#117—611no—
320C2×#117—611no—
321C2×#117—612no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4#692311yescharacteristic, complemented
325#413—614no—
326#413—1222no—
327C2×#101—1221no—
328C2×#100—1221no—
329#410—614no—
330#410—1222no—
331C2×#99—1221no—
332C2×#99—1221no—
333#340—1224no—
334#339—1224no—
335C61#1076511yescharacteristic, complemented, Sylow
336C2×#41—612no—
337C2×#41—612no—
338C2×#41—612no—
339C2×#41—612no—
340C2×#41—1222no—
341C2×#41—1222no—
342C2×#41—1222no—
343C2×#41—1222no—
344C22×Q8—611no—
345C22×D4—611no—
346C22×D4—611no—
347C22×D4—611no—
348C23×C4—21no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1221no—
352C23×C4—1221no—
353C23×C4C2×#692311yescharacteristic
354C2×M16—614no—
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C2×M16—1222no—
360C22×C8—1222no—
361C4×Q8—1228no—
362C4×D4—1228no—
363C4×D4—1228no—
364C4×D4—1228no—
365#117—614no—
366#117—614no—
367#117—1222no—
368#117—1222no—
369#117—1222no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4C2×#692311yescharacteristic
388C2×C4⋊C4C2×#692311yescharacteristic
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1222no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1222no—
397#101—1224no—
398#100—1224no—
399#99—1224no—
400#99—1224no—
401C24—611no—
402C24—611no—
403C24C22⋊#692311yescharacteristic
404#41—1224no—
405#41—1224no—
406#41—1224no—
407#41—1224no—
408#41—1224no—
409#41—1224no—
410#41—1224no—
411#41—1224no—
412C2×Q8—612no—
413C2×Q8—612no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×D4—612no—
417C2×D4—612no—
418C2×D4—612no—
419C2×D4—612no—
420C2×D4—612no—
421C2×D4—612no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—82no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—4882no—
457C22×C4—4882no—
458C22×C4—2442no—
459C22×C4C4×#692312yes—
460C22×C4C4⋊#692312yes—
461C22×C4C22⋊#692311yescharacteristic
462C22×C4C22×#692311yescharacteristic
463M16—1224no—
464M16—1224no—
465M16—2442no—
466M16—2442no—
467C2×C8—1224no—
468C2×C8—2442no—
469C2×C8—2442no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—1224no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2442no—
481C4⋊C4C22⋊#692312yes—
482C4⋊C4C22⋊#692312yes—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—2442no—
488C42—1224no—
489C42—1224no—
490C42—1224no—
491C42—1224no—
492C42—2442no—
493C23—21no—
494C23—21no—
495C23—41no—
496C23—41no—
497C23—612no—
498C23—612no—
499C23—611no—
500C23—611no—
501C23—612no—
502C23—1221no—
503C23—1221no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23(C2×#6923)⋊C412yes—
509C23C2×C22⋊#692311yescharacteristic
510Q8—1224no—
511Q8—1224no—
512D4—1224no—
513D4—1224no—
514D4—1224no—
515D4—1224no—
516D4—1224no—
517D4—1224no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—84no—
529C2×C4—612no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—4884no—
551C2×C4—4884no—
552C2×C4—4882no—
553C2×C4—4882no—
554C2×C4—2444no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4(C2×#6923)⋊C412yes—
558C2×C4C2×C4×#692311yescharacteristic
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C2×C4⋊#692311yescharacteristic
562C2×C4#1451985611yescharacteristic
563C2×C4#1451985611yescharacteristic
564C8—2444no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—612no—
573C22—612no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—2442no—
578C22—2441no—
579C22—2441no—
580C22—2442no—
581C22—2442no—
582C22—2441no—
583C22—2441no—
584C22C2×(C2×#6923)⋊C411yescharacteristic
585C22#1684070311yescharacteristic
586C22#1684070511yescharacteristic
587C4—22no—
588C4—42no—
589C4—42no—
590C4—612no—
591C4—612no—
592C4—1222no—
593C4—4884no—
594C4—2442no—
595C4—2442no—
596C4C4×C22⋊#692311yescharacteristic
597C4C4×C22⋊#692311yescharacteristic
598C4#1684070411yescharacteristic
599C4#1684070411yescharacteristic
600C2—21no—
601C2—21no—
602C2—42no—
603C2—1222no—
604C2—1222no—
605C2—2442no—
606C2#2146099311yescharacteristic
607C2#2146166911yescharacteristic
608C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61#379
prime radicalC22×#11758222C22
cosocleC22×C4×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#10765
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21461506C2semidirect
4. #21461498C2semidirect
9. #16840748C4semidirect
13. #16840741C4semidirect
25. C2×#14519736C4semidirect
26. C2×#14519735C4semidirect
28. C2×C4⋊C4×D61C4semidirect
30. C22×C4⋊C4×C61C4semidirect
35. D61⋊M16C2×C4semidirect
41. #14519856C2×C4semidirect
324. C22×C4⋊C4#6923semidirect
335. C61#10765semidirect

Maximal subgroups

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

subgroupindex
C2×#168407482
C2×#168407412
#214615062
#214614982
C2×#168402432
#1076561