Finite Group Databasebeta

← #30615641

Subgroups — #30615641

Order 15616.

1690 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×#16840830C211yesmaximal, characteristic
2C2×#16840755C211yesmaximal, characteristic
3C2×#16840755C211yesmaximal, characteristic
4#21462293C211yesmaximal, characteristic
5#21461813C218yesmaximal, complemented
6#21461505C211yesmaximal, characteristic
7#21461484C211yesmaximal, characteristic
8C22×Q16×D61C211yesmaximal, characteristic
9C2×D61⋊M16C2211yescharacteristic
10#16840830—24no—
11#16840830—24no—
12C2×Q8×#6923—22no—
13C2×Q8×#6923C2214yes—
14C2×#14519856—22no—
15C2×#14519856C2211yescharacteristic
16C2×#14519856C2211yescharacteristic
17C2×#14519856C2211yescharacteristic
18#16840755—24no—
19#16840755—24no—
20C2×#14519840C2214yes—
21C2×#14519840C2214yes—
22(C2×#6923)⋊Q8—22no—
23(C2×#6923)⋊Q8C2214yes—
24#16840705—22no—
25#16840636C2214yes—
26(C2×#6923)⋊C8C2214yes—
27C22×Q8×D61C2211yescharacteristic
28C22×Q8×D61C2211yescharacteristic
29C22×Dic244C411yescharacteristic, complemented
30C22×#11758202C411yescharacteristic, complemented
31C22×#11758202C411yescharacteristic, complemented
32C2×Q16×D61—24no—
33C2×Q16×D61—24no—
34C2×Q16×D61C414yescomplemented
35C2×Q16×D61C2214yes—
36C22×C8×D61C2211yescharacteristic
37C22×Q16×C61C411yescharacteristic, complemented
38C2×#11758213—21no—
39D61⋊M16—22no—
40D61⋊M16—22no—
41C2×#11758217—21no—
42C2×D61⋊C8C2312yes—
43Q8×#6923—216no—
44Q8×#6923—28no—
45Q8×#6923—28no—
46#14519856—216no—
47#14519856—22no—
48#14519856—22no—
49#14519856—22no—
50#14519856—22no—
51#14519856—22no—
52#14519856D414yes—
53C2×C4⋊#6923—24no—
54C2×C4⋊#6923D412yes—
55C2×C4⋊#6923C2312yes—
56C2×C4⋊#6923C2312yes—
57C2×C22⋊#6923—21no—
58C2×C22⋊#6923—21no—
59C2×C22⋊#6923D412yes—
60C2×C4×#6923—24no—
61C2×C4×#6923—21no—
62C2×C4×#6923—21no—
63C2×C4×#6923C2312yes—
64#14519840—28no—
65#14519840—28no—
66(C2×#6923)⋊C4—28no—
67C22×Dic122—21no—
68C22×Dic122—21no—
69C22×Dic122C2×C411yescharacteristic
70C22×Dic122C2×C411yescharacteristic
71C2×Q8×D61—21no—
72C2×Q8×D61—22no—
73C2×Q8×D61—21no—
74C2×Q8×D61—22no—
75C2×Q8×D61—21no—
76C2×Q8×D61—21no—
77C2×Q8×D61—22no—
78C2×Q8×D61—22no—
79C2×Q8×D61—22no—
80C2×Q8×D61—22no—
81C2×Q8×D61C2×C412yes—
82C2×Q8×D61C2×C412yes—
83C2×Q8×D61D414yes—
84C2×Q8×D61D414yes—
85C2×Q8×D61C2312yes—
86C2×Q8×D61C2312yes—
87C22×C4×D61—21no—
88C22×C4×D61—21no—
89C22×C4×D61C2311yescharacteristic
90C2×Dic244—24no—
91C2×Dic244C2×C414yes—
92C2×#11758202—24no—
93C2×#11758202—24no—
94C2×#11758202C2×C414yes—
95C2×#11758202C2×C414yes—
96Q16×D61—216no—
97Q16×D61—28no—
98Q16×D61—28no—
99C22×#92418C2×C411yescharacteristic
100C2×C8×D61—22no—
101C2×C8×D61—22no—
102C2×C8×D61C2×C412yes—
103C2×C8×D61C2312yes—
104C22×Q8×C61C2×C411yescharacteristic
105C22×Q8×C61C2×C411yescharacteristic
106C2×Q16×C61—24no—
107C2×Q16×C61C2×C414yes—
108C22×C8×C61C2×C411yescharacteristic
109C23×D61C2×D411yescharacteristic
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923C2×D414yes—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61C22⋊C414yes—
119C2×Q8×C61C22×C412yes—
120C2×Q8×C61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C22×C411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61#4111yescharacteristic
127Q8×D61—24no—
128Q8×D61—24no—
129Q8×D61—24no—
130Q8×D61—24no—
131Q8×D61—24no—
132Q8×D61—22no—
133Q8×D61—22no—
134Q8×D61—24no—
135Q8×D61—24no—
136Q8×D61—22no—
137Q8×D61—22no—
138Q8×D61—22no—
139Q8×D61—22no—
140Q8×D61—42no—
141Q8×D61—42no—
142Q8×D61—42no—
143Q8×D61—42no—
144Q8×D61C22⋊C418yes—
145Q8×D61C2×D414yes—
146Q8×D61C2×D414yes—
147C2×C4×D61—22no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4112yes—
161C2×C4×D61#4111yescharacteristic
162C2×C4×D61C2411yescharacteristic
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122—24no—
167C2×Dic122—24no—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122C22⋊C414yes—
177C2×Dic122C22×C412yes—
178C2×Dic122C22×C412yes—
179C22⋊#6923—24no—
180C22⋊#6923—24no—
181C22⋊#6923—42no—
182#11758217—42no—
183C2×#92420—22no—
184C4⋊#6923—216no—
185C4⋊#6923—24no—
186C4⋊#6923—22no—
187C4⋊#6923—22no—
188C4⋊#6923—22no—
189C4⋊#6923C2×D414yes—
190C4×#6923—216no—
191C4×#6923—24no—
192C4×#6923—22no—
193C4×#6923—22no—
194C4×#6923—42no—
195#11758213—42no—
196D61⋊C8—22no—
197D61⋊C8—22no—
198Q16×C61—28no—
199C2×C8×C61—22no—
200C2×C8×C61C22×C412yes—
201#11758202—28no—
202#11758202—28no—
203C2×#92418—22no—
204C2×#92418C22×C412yes—
205Dic244—28no—
206C8×D61—24no—
207C8×D61—22no—
208C8×D61—22no—
209C23×C61C4×D411yescharacteristic
210C22×D61—24no—
211C22×D61C4×D411yescharacteristic
212C22×D61#12012yes—
213C22×D61#12112yes—
214C22×D61C22×D411yescharacteristic
215C2×#6923—24no—
216C2×#6923—24no—
217C2×#6923—28no—
218C2×#6923—44no—
219C2×#6923—42no—
220C2×#6923—44no—
221Q8×C61—22no—
222Q8×C61—24no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—42no—
226Q8×C61C2×C22⋊C414yes—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—42no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61#12112yes—
241C2×Dic61#12312yes—
242C2×Dic61C2×#4111yescharacteristic
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—28no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—44no—
249C4×D61C2×C22⋊C412yes—
250C4×D61C4×D412yes—
251C4×D61#12012yes—
252C4×D61#12112yes—
253C4×D61#12112yes—
254C4×D61#12312yes—
255C4×D61C22×D411yescharacteristic
256C4×D61C22×D411yescharacteristic
257C4×D61C22×D411yescharacteristic
258C4×D61C2×#4111yescharacteristic
259Dic122—22no—
260Dic122—24no—
261Dic122—24no—
262Dic122—24no—
263Dic122—28no—
264Dic122—22no—
265Dic122—22no—
266Dic122—42no—
267Dic122—42no—
268Dic122—42no—
269Dic122—44no—
270Dic122C2×C22⋊C414yes—
271#92420—42no—
272C8×C61—22no—
273#92418—22no—
274#20935—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61#38612yes—
277C22×C61C2×C4×D411yescharacteristic
278C2×D61—22no—
279C2×D61—24no—
280C2×D61—24no—
281C2×D61#38612yes—
282C2×D61C2×#12011yescharacteristic
283C2×D61C2×#12111yescharacteristic
284C2×D61#57814yes—
285#6923—44no—
286#6923—44no—
287#6923—82no—
288C4×C61—24no—
289C4×C61—42no—
290C4×C61#38612yes—
291C4×C61C22×C22⋊C411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293Dic61—24no—
294Dic61—42no—
295Dic61#38612yes—
296Dic61C2×#12111yescharacteristic
297Dic61C2×#12311yescharacteristic
298C23×Q16—611no—
299C2×#519—611no—
300C2×#419—611no—
301C2×#419—611no—
302#2310—611no—
303#1878—618no—
304#1735—611no—
305#1729—611no—
306C2×C61—22no—
307C2×C61C2×#38611yescharacteristic
308C2×C61#295012yes—
309D61—22no—
310D61#301112yes—
311D61#304412yes—
312C23×Q8—611no—
313C23×Q8—611no—
314C22×Q16—611no—
315C22×Q16—611no—
316C22×Q16—614no—
317C22×Q16—614no—
318C22×Q16—611no—
319C22×Q16—1224no—
320C22×Q16—1224no—
321C22×Q16#692311yescharacteristic, complemented
322C22×M16—611no—
323C23×C8—611no—
324#519—1224no—
325#519—1224no—
326C2×C4×Q8—614no—
327C2×C4×Q8—1222no—
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—611no—
331C2×#117—1222no—
332#419—1224no—
333#419—1224no—
334C2×#103—614no—
335C2×#103—614no—
336#391—614no—
337#391—1222no—
338#381—1222no—
339#340—614no—
340#336—614no—
341C61#2093511yescharacteristic, complemented, Sylow
342C22×Q8—614no—
343C22×Q8—614no—
344C22×Q8—612no—
345C22×Q8—612no—
346C22×Q8—612no—
347C22×Q8—612no—
348C22×Q8—611no—
349C22×Q8—611no—
350C22×Q8—1221no—
351C22×Q8—1221no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8—1221no—
356C22×Q8—1221no—
357C22×Q8—1221no—
358C22×Q8—1222no—
359C22×Q8—1222no—
360C22×Q8—1222no—
361C22×Q8—1222no—
362C22×Q8C2×#692311yescharacteristic
363C22×Q8C2×#692311yescharacteristic
364C23×C4—611no—
365C23×C4—1221no—
366C23×C4—1221no—
367C2×Q16—24no—
368C2×Q16—614no—
369C2×Q16—614no—
370C2×Q16—614no—
371C2×Q16—1224no—
372C2×Q16—1228no—
373C2×Q16—1228no—
374C2×Q16—12216no—
375C2×Q16—1224no—
376C2×Q16—1224no—
377C2×Q16C2×#692314yes—
378C2×M16—1221no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1221no—
382C22×C8—611no—
383C22×C8—612no—
384C22×C8—612no—
385C22×C8—612no—
386C22×C8—1222no—
387C22×C8—1222no—
388C22×C8C2×#692311yescharacteristic
389C4×Q8—1228no—
390C4×Q8—1228no—
391C4×Q8—12216no—
392#117—614no—
393#117—1222no—
394#117—1222no—
395#117—1222no—
396#117—12216no—
397#117—1222no—
398#117—1222no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1224no—
403C2×C22⋊C4—612no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—612no—
407C2×C42—1221no—
408C2×C42—1224no—
409C2×C42—1221no—
410#103—1228no—
411#103—1228no—
412#95—1228no—
413C24—611no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—614no—
420C2×Q8—614no—
421C2×Q8—618no—
422C2×Q8—614no—
423C2×Q8—612no—
424C2×Q8—612no—
425C2×Q8—1224no—
426C2×Q8—1224no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1224no—
432C2×Q8—1222no—
433C2×Q8—1224no—
434C2×Q8—1224no—
435C2×Q8—1224no—
436C2×Q8—1224no—
437C2×Q8—1222no—
438C2×Q8—1222no—
439C2×Q8—1224no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×Q8—1224no—
443C2×Q8—1222no—
444C2×Q8—1222no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—2442no—
450C2×Q8—2442no—
451C2×Q8—2442no—
452C2×Q8—2442no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×Q8C22⋊#692314yes—
456C2×Q8C22×#692312yes—
457C2×Q8C22×#692312yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—611no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—614no—
470C22×C4—1222no—
471C22×C4—1224no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4C22×#692311yescharacteristic
483Q16—28no—
484Q16—1228no—
485Q16—1228no—
486Q16—1228no—
487M16—2442no—
488M16—2442no—
489C2×C8—22no—
490C2×C8—612no—
491C2×C8—1224no—
492C2×C8—1222no—
493C2×C8—1222no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8C22×#692312yes—
499C4⋊C4—614no—
500C4⋊C4—1224no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—12216no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—2442no—
508C42—1222no—
509C42—1222no—
510C42—1224no—
511C42—12216no—
512C42—2442no—
513C23—611no—
514C23—612no—
515C23—612no—
516C23—611no—
517C23—1224no—
518C23D4×#692311yescharacteristic
519Q8—24no—
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—614no—
525Q8—1224no—
526Q8—1224no—
527Q8—1228no—
528Q8—1224no—
529Q8—1222no—
530Q8—1222no—
531Q8—1222no—
532Q8—2442no—
533Q8—2442no—
534Q8—2444no—
535Q8—2442no—
536Q8C2×C22⋊#692314yes—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—612no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—1224no—
556C2×C4—1224no—
557C2×C4—1228no—
558C2×C4—1224no—
559C2×C4—1222no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1228no—
563C2×C4—1222no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4—2442no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4C2×C22⋊#692312yes—
572C2×C4D4×#692311yescharacteristic
573C2×C4D4×#692312yes—
574C2×C4C23×#692311yescharacteristic
575C8—22no—
576C8—1222no—
577C8—2442no—
578C22—22no—
579C22—614no—
580C22—611no—
581C22—611no—
582C22—612no—
583C22—1224no—
584C22—1224no—
585C22—1222no—
586C22#1684071012yes—
587C22C2×D4×#692311yescharacteristic
588C4—24no—
589C4—42no—
590C4—611no—
591C4—611no—
592C4—612no—
593C4—1224no—
594C4—4882no—
595C4—2444no—
596C4—2444no—
597C4—2442no—
598C4#1684071012yes—
599C4C22×C22⋊#692311yescharacteristic
600C4C2×D4×#692311yescharacteristic
601C2—22no—
602C2—612no—
603C2—612no—
604C2—1222no—
605C2C2×#1684071011yescharacteristic
606C2#2146258012yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×Q16×C61C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC22×Q16#6923
hypocenterC61#20935
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
5. #21461813C2semidirect
29. C22×Dic244C4semidirect
30. C22×#11758202C4semidirect
31. C22×#11758202C4semidirect
34. C2×Q16×D61C4semidirect
37. C22×Q16×C61C4semidirect
321. C22×Q16#6923semidirect
341. C61#20935semidirect

Maximal subgroups

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

subgroupindex
C2×#168408302
C2×#168407552
#214622932
#214618132
#214615052
#214614842
C22×Q16×D612
#2093561