Finite Group Databasebeta

← #30602547

Subgroups — #30602547

Order 15616.

1230 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
1C2×#16840753C212yesmaximal
2C2×#16840641C212yesmaximal, complemented
3#21461511C211yesmaximal, characteristic, complemented
4C2×#16840636C211yesmaximal, characteristic, complemented
5C2×#131×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C22×C4⋊#6923C2212yes—
8#16840753—24no—
9#16840753C418yescomplemented
10C2×#14519840—22no—
11C2×#14519839—22no—
12#16840641—24no—
13#16840638—21no—
14#16840638—21no—
15#16840636—21no—
16#16840636—21no—
17C2×#41×D61C2211yescharacteristic
18C2×#14519785—21no—
19C2×#14519785C411yescharacteristic, complemented
20C2×#14519784C411yescharacteristic, complemented
21C2×#14519783—21no—
22C2×#14519783C411yescharacteristic, complemented
23#131×D61—24no—
24#131×D61—24no—
25C2×M16×D61—21no—
26C2×M16×D61C2211yescharacteristic
27C22×C8×D61—21no—
28C22×C8×D61C2211yescharacteristic
29C2×#131×C61C411yescharacteristic, complemented
30C23×#6923—22no—
31C2×#11758213—22no—
32D61⋊M16—22no—
33D61⋊M16—22no—
34D61⋊M16—22no—
35D61⋊M16C2×C414yes—
36C2×#11758217—22no—
37C2×D61⋊C8—22no—
38C2×C4⋊#6923—22no—
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—22no—
42C2×C4⋊#6923—22no—
43C2×C4⋊#6923C2×C414yes—
44#14519840—28no—
45#14519839—28no—
46C2×#11758226Q811yescharacteristic
47C2×#11758222—21no—
48C2×#11758222C2×C411yescharacteristic
49C2×#11758224—21no—
50C2×#11758224Q811yescharacteristic
51#41×D61—22no—
52#41×D61—22no—
53#41×D61C2×C414yes—
54#41×D61D414yes—
55C2×Q8×D61D411yescharacteristic
56C2×D4×D61—21no—
57C2×D4×D61D411yescharacteristic
58C22×C4×D61—21no—
59C22×C4×D61C2311yescharacteristic
60#14519785—28no—
61#14519785—22no—
62#14519785—22no—
63#14519784—22no—
64#14519784—22no—
65#14519783—28no—
66#14519783—22no—
67#14519783—22no—
68C2×#11758194—21no—
69C2×#11758194C2×C411yescharacteristic
70C2×#11758189—21no—
71C2×#11758189—21no—
72C2×#11758189—21no—
73M16×D61—28no—
74M16×D61—22no—
75M16×D61D414yes—
76C22×#92418—21no—
77C22×#92418C2×C411yescharacteristic
78C2×C8×D61—28no—
79C2×C8×D61—21no—
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84C2×C8×D61D411yescharacteristic
85C2×C8×D61D411yescharacteristic
86C2×C8×D61D411yescharacteristic
87C2×C8×D61D411yescharacteristic
88C2×#41×C61C2×C411yescharacteristic
89#131×C61—22no—
90#131×C61—22no—
91C2×M16×C61—21no—
92C2×M16×C61C2×C411yescharacteristic
93C22×C8×C61—21no—
94C22×C8×C61C2×C411yescharacteristic
95C23×D61—21no—
96C23×D61#4111yescharacteristic
97C22×#6923—24no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—82no—
101C22×#6923—82no—
102#41×C61—21no—
103#41×C61—21no—
104#41×C61C4212yes—
105#41×C61C22⋊C412yes—
106C2×Q8×C61C4⋊C411yescharacteristic
107C2×D4×C61—21no—
108C2×D4×C61C4⋊C411yescharacteristic
109C22×C4×C61—21no—
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61—21no—
112C2×C22⋊D61—21no—
113C2×C22⋊D61—21no—
114C22×Dic61—21no—
115C22×Dic61#4111yescharacteristic
116#11758226—22no—
117#11758226C4⋊C414yes—
118Q8×D61—22no—
119Q8×D61C22⋊C414yes—
120#11758224—24no—
121#11758224—24no—
122#11758224—22no—
123#11758224C4⋊C414yes—
124D4×D61—24no—
125D4×D61—24no—
126D4×D61—22no—
127D4×D61C22⋊C414yes—
128#11758222—21no—
129#11758222—21no—
130#11758222—24no—
131#11758222—24no—
132#11758222C4212yes—
133#11758222C22⋊C412yes—
134C2×D244—21no—
135C2×D244C4⋊C411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—24no—
140C2×C4×D61—24no—
141C2×C4×D61C22×C412yes—
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D411yescharacteristic
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×Q811yescharacteristic
148C2×C4×D61#4112yes—
149C2×Dic122—21no—
150C2×Dic122C4⋊C411yescharacteristic
151#11758217—24no—
152#11758217—42no—
153C2×#92420—42no—
154C4⋊#6923—24no—
155C4⋊#6923—24no—
156C4⋊#6923—42no—
157C4⋊#6923—42no—
158#11758213—24no—
159#11758213—42no—
160D61⋊C8—24no—
161D61⋊C8—42no—
162M16×C61—24no—
163M16×C61—21no—
164M16×C61C22⋊C412yes—
165C2×C8×C61—24no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C22⋊C411yescharacteristic
170C2×C8×C61C22⋊C411yescharacteristic
171#11758194—24no—
172#11758194—21no—
173#11758194C22⋊C412yes—
174C2×#92418—24no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418C22⋊C411yescharacteristic
179C2×#92418C22⋊C411yescharacteristic
180#11758189—24no—
181#11758189—24no—
182#11758189—24no—
183C8×D61—24no—
184C8×D61—24no—
185C8×D61—22no—
186C8×D61—24no—
187C8×D61—22no—
188C8×D61—22no—
189C8×D61—22no—
190C23×C61—21no—
191C23×C61#11711yescharacteristic
192C22×D61—22no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—24no—
197C22×D61—42no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61C4×D412yes—
201C22×D61#12111yescharacteristic
202C22×D61C22⋊Q811yescharacteristic
203C22×D61#12312yes—
204C2×#6923—44no—
205C2×#6923—84no—
206C2×#6923—82no—
207C2×#6923—82no—
208C2×#6923—84no—
209Q8×C61—21no—
210Q8×C61#9512yes—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—21no—
214D4×C61#9512yes—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61C2×C4211yescharacteristic
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C4⋊C411yescharacteristic
223C2×C4×C61#11711yescharacteristic
224C22⋊D61—22no—
225C22⋊D61—22no—
226C22⋊D61—24no—
227C22⋊D61—24no—
228C2×Dic61—21no—
229C2×Dic61—24no—
230C2×Dic61C4×D412yes—
231C2×Dic61#12111yescharacteristic
232C2×Dic61C22⋊Q811yescharacteristic
233C2×Dic61#12312yes—
234D244—24no—
235D244—21no—
236D244#9512yes—
237C4×D61—24no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—24no—
241C4×D61C2×C22⋊C412yes—
242C4×D61C2×C4⋊C412yes—
243C4×D61C4×D412yes—
244C4×D61C4×D412yes—
245C4×D61#12012yes—
246C4×D61#12012yes—
247C4×D61#12112yes—
248C4×D61C22⋊Q812yes—
249Dic122—21no—
250Dic122—24no—
251Dic122#9512yes—
252#92420—44no—
253C8×C61—21no—
254C8×C61—22no—
255C8×C61—21no—
256#92418—21no—
257#92418—22no—
258#92418—21no—
259#10774—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61C4×C22⋊C411yescharacteristic
266C22×C61#38011yescharacteristic
267C22×C61#38111yescharacteristic
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—44no—
278C2×D61—44no—
279C2×D61#38512yes—
280C2×D61#38612yes—
281C2×D61#47112yes—
282C2×D61#47211yescharacteristic
283C2×D61#47311yescharacteristic
284#6923—84no—
285C4×C61—22no—
286C4×C61C2×#9511yescharacteristic
287C4×C61C4×C22⋊C411yescharacteristic
288C4×C61#37911yescharacteristic
289C4×C61#38011yescharacteristic
290Dic61—22no—
291Dic61#38512yes—
292Dic61#38612yes—
293C22×#131—611no—
294C2×#417—612no—
295C2×#343—612no—
296C2×#341—611no—
297C2×#340—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—42no—
301C2×C61C22⋊#9511yescharacteristic
302C2×C61#181511yescharacteristic
303C2×C61#181611yescharacteristic
304D61—22no—
305D61—42no—
306D61#190712yes—
307D61#190812yes—
308C22×#41—611no—
309C2×#131—611no—
310C2×#131—611no—
311C2×#131—611no—
312C2×#131—1221no—
313C2×#131—1221no—
314C2×#131—1224no—
315C2×#131—1224no—
316C2×#131#692311yescharacteristic, complemented
317C22×M16—611no—
318C22×M16—612no—
319C22×M16—1221no—
320C23×C8—611no—
321C23×C8—1221no—
322C22×C4⋊C4—612no—
323#417—618no—
324#417—1224no—
325C2×#103—1222no—
326C2×#102—1222no—
327#343—1224no—
328#341—1221no—
329#341—1221no—
330#340—1221no—
331#340—1221no—
332C61#1077411yescharacteristic, complemented, Sylow
333C2×#41—614no—
334C2×#41—614no—
335C2×#41—611no—
336C2×#41—611no—
337C2×#41—611no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1222no—
341C2×#41—1222no—
342C2×#41C2×#692311yescharacteristic
343C22×Q8—611no—
344C22×D4—611no—
345C22×D4—1221no—
346C23×C4—611no—
347C23×C4—1221no—
348C23×C4—1222no—
349#131—22no—
350#131—22no—
351#131—1222no—
352#131—1222no—
353#131—1228no—
354#131—1222no—
355#131—1228no—
356#131—1222no—
357#131—1222no—
358#131—1222no—
359C2×M16—21no—
360C2×M16—611no—
361C2×M16—614no—
362C2×M16—614no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1222no—
367C2×M16—1228no—
368C2×M16—1222no—
369C2×M16—1221no—
370C2×M16—1222no—
371C2×M16—1222no—
372C2×M16—1222no—
373C2×M16—1222no—
374C2×M16C2×#692311yescharacteristic
375C22×C8—21no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—1221no—
382C22×C8—1222no—
383C22×C8—1222no—
384C22×C8—1221no—
385C22×C8—1228no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—1222no—
389C22×C8C2×#692311yescharacteristic
390C2×C4⋊C4—614no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396#103—1228no—
397#102—1228no—
398C24—611no—
399C24—1221no—
400#41—21no—
401#41—21no—
402#41—614no—
403#41—614no—
404#41—612no—
405#41—612no—
406#41—1224no—
407#41—1224no—
408#41—1224no—
409#41—1224no—
410#41—1222no—
411#41—1222no—
412#41—1221no—
413#41—1221no—
414#41C4×#692312yes—
415#41C22⋊#692312yes—
416C2×Q8—614no—
417C2×Q8—611no—
418C2×Q8—1221no—
419C2×Q8—1222no—
420C2×Q8C4⋊#692311yescharacteristic
421C2×D4—21no—
422C2×D4—614no—
423C2×D4—611no—
424C2×D4—1221no—
425C2×D4—1222no—
426C2×D4—1221no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4C4⋊#692311yescharacteristic
432C22×C4—21no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—612no—
442C22×C4—1224no—
443C22×C4—1221no—
444C22×C4—1224no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1224no—
449C22×C4—4882no—
450C22×C4—4882no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4C22×#692311yescharacteristic
454M16—24no—
455M16—21no—
456M16—612no—
457M16—1224no—
458M16—1224no—
459M16—1224no—
460M16—1224no—
461M16—1221no—
462M16—1224no—
463M16—1224no—
464M16—2442no—
465M16—2442no—
466M16C22⋊#692312yes—
467C2×C8—24no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1224no—
478C2×C8—1224no—
479C2×C8—1224no—
480C2×C8—1224no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1224no—
485C2×C8—2442no—
486C2×C8—2442no—
487C2×C8C22⋊#692311yescharacteristic
488C2×C8C22⋊#692311yescharacteristic
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C23—21no—
494C23—611no—
495C23—611no—
496C23—612no—
497C23—612no—
498C23—1221no—
499C23—1222no—
500C23—1221no—
501C23—1224no—
502C23—1221no—
503C23—2442no—
504C23—2441no—
505C23—2441no—
506C23#1451985611yescharacteristic
507Q8—21no—
508Q8—612no—
509Q8—1224no—
510Q8—1221no—
511Q8(C2×#6923)⋊C412yes—
512D4—22no—
513D4—22no—
514D4—21no—
515D4—612no—
516D4—1224no—
517D4—1224no—
518D4—1224no—
519D4—1222no—
520D4—1222no—
521D4—1221no—
522D4(C2×#6923)⋊C412yes—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—4884no—
547C2×C4—4884no—
548C2×C4—2444no—
549C2×C4C2×C4×#692311yescharacteristic
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C2×C22⋊#692311yescharacteristic
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C2×C4⋊#692311yescharacteristic
554C2×C4#1451985611yescharacteristic
555C8—21no—
556C8—22no—
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8—1222no—
561C8—2444no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—612no—
568C22—612no—
569C22—611no—
570C22—611no—
571C22—612no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1222no—
576C22—1221no—
577C22—1221no—
578C22—1222no—
579C22—2444no—
580C22—2444no—
581C22—2441no—
582C22—2441no—
583C22C4×C22⋊#692311yescharacteristic
584C22#1684070411yescharacteristic
585C22#1684070511yescharacteristic
586C4—22no—
587C4—612no—
588C4—612no—
589C4—1222no—
590C4—4884no—
591C4C2×(C2×#6923)⋊C411yescharacteristic
592C4C4×C22⋊#692311yescharacteristic
593C4#1684070311yescharacteristic
594C4#1684070411yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2—612no—
599C2—612no—
600C2—1222no—
601C2—2442no—
602C2#2146099311yescharacteristic
603C2#2146170711yescharacteristic
604C2#2146170811yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#131×C61C4
socleC22×C61#380
prime radicalC2×#41×D61C22
cosocleC22×C4×D61C23
hypercenterC2×#131#6923
hypocenterC61#10774
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. C2×#16840641C2semidirect
3. #21461511C2semidirect
4. C2×#16840636C2semidirect
9. #16840753C4semidirect
19. C2×#14519785C4semidirect
20. C2×#14519784C4semidirect
22. C2×#14519783C4semidirect
29. C2×#131×C61C4semidirect
316. C2×#131#6923semidirect
332. C61#10774semidirect

Maximal subgroups

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

subgroupindex
C2×#168407532
C2×#168406412
#214615112
C2×#168406362
C2×#131×D612
#1077461