Finite Group Databasebeta

← #473×#6923 #30617097

Subgroups — #473×#6923 #30617097

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×#6923C212yesmaximal
2C2×Q16×#6923C211yesmaximal, characteristic, complemented
3#131×#6923C211yesmaximal, characteristic
4#21461981C211yesmaximal, characteristic, complemented
5#21461937C211yesmaximal, characteristic, complemented
6#21461904C212yesmaximal, complemented
7#21461859C212yesmaximal
8#21461708C211yesmaximal, characteristic
9#108×#6923C211yesmaximal, characteristic, complemented
10#101×#6923C212yesmaximal, complemented
11#473×D61C211yesmaximal, characteristic
12D61⋊#131C2211yescharacteristic
13#41×#6923C2211yescharacteristic
14C2×Q8×#6923C2212yes—
15#16840802—22no—
16#16840801—22no—
17Q16×#6923—22no—
18Q16×#6923—21no—
19Q16×#6923—21no—
20Q16×#6923—21no—
21Q16×#6923—21no—
22SD16×#6923—22no—
23D61⋊#108C2211yescharacteristic
24#16840773C2212yes—
25C2×#14519846C2211yescharacteristic
26#16840755C2212yes—
27#16840753C2212yes—
28C2×#14519840C2212yes—
29D61⋊#101C2212yes—
30#16840728—21no—
31#16840728C2211yescharacteristic
32#16840725—21no—
33#16840725C2211yescharacteristic
34M16×#6923—21no—
35M16×#6923C2212yes—
36M16×#6923C2211yescharacteristic
37C2×C8×#6923—21no—
38C2×C8×#6923C2211yescharacteristic
39#16840641C2212yes—
40#16840641C2212yes—
41#16840637C2212yes—
42#137×D61C2212yes—
43C2×Q16×D61C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840038C411yescharacteristic, complemented
46#16840037C411yescharacteristic, complemented
47#16840036C412yescomplemented
48#16840035C412yescomplemented
49#16840034C411yescharacteristic, complemented
50#108×D61C2211yescharacteristic
51#101×D61C2212yes—
52C61×#473C411yescharacteristic, complemented
53#14519865—21no—
54#14519864—21no—
55#14519864—22no—
56D61⋊M16—21no—
57D61⋊M16C2312yes—
58D61⋊M16C2311yescharacteristic
59C2×D61⋊C8—21no—
60C2×D61⋊C8C2311yescharacteristic
61Q8×#6923—22no—
62Q8×#6923—22no—
63Q8×#6923—22no—
64Q8×#6923—22no—
65Q8×#6923D411yescharacteristic
66Q8×#6923D411yescharacteristic
67D4×#6923—21no—
68D4×#6923—21no—
69D4×#6923D411yescharacteristic
70D4×#6923D411yescharacteristic
71#14519856—21no—
72#14519856C2311yescharacteristic
73C2×C4⋊#6923—22no—
74C2×C4⋊#6923C2312yes—
75C2×C4×#6923—21no—
76C2×C4×#6923—22no—
77C2×C4×#6923C2311yescharacteristic
78#14519848—21no—
79#14519847—21no—
80#14519846—22no—
81#14519846—21no—
82#14519846—21no—
83#14519845—22no—
84#14519840—22no—
85#14519840—22no—
86#14519840—22no—
87#14519840—22no—
88#14519840—22no—
89#14519839—22no—
90#14519838—24no—
91#14519831—21no—
92#14519831—22no—
93#14519831—21no—
94#14519831—21no—
95C8×#6923—21no—
96C8×#6923—22no—
97C8×#6923—21no—
98C8×#6923—21no—
99C8×#6923D411yescharacteristic
100C8×#6923D411yescharacteristic
101C8×#6923D411yescharacteristic
102C8×#6923D411yescharacteristic
103#41×D61C2311yescharacteristic
104C2×Q8×D61C2312yes—
105#14519810C2×C412yes—
106#14519809—22no—
107#14519808C2×C412yes—
108#14519807C2×C412yes—
109#14519806—22no—
110C2×Dic244C2×C411yescharacteristic, complemented
111C2×#11758202C2×C412yescomplemented
112Q16×D61—22no—
113Q16×D61—21no—
114Q16×D61—21no—
115Q16×D61—21no—
116Q16×D61—21no—
117SD16×D61—22no—
118#14519785—21no—
119#14519785C2×C411yescharacteristic
120#14519784C2×C411yescharacteristic
121#14519783—21no—
122#14519783C2×C411yescharacteristic
123M16×D61—21no—
124M16×D61C2311yescharacteristic
125M16×D61C2312yes—
126C2×C8×D61—21no—
127C2×C8×D61C2311yescharacteristic
128#14519713C2×C411yescharacteristic, complemented
129#14519712C2×C412yescomplemented
130#14519696C2×C412yescomplemented
131#14519695C2×C412yescomplemented
132#14519695C2×C412yescomplemented
133#137×C61C2×C412yes—
134C2×Q16×C61C2×C411yescharacteristic, complemented
135#131×C61C2×C411yescharacteristic
136#108×C61C2×C411yescharacteristic, complemented
137#101×C61C2×C412yescomplemented
138C22×#6923—21no—
139C22×#6923—21no—
140C22×#6923—22no—
141C22×#6923#4111yescharacteristic
142C22×#6923#4111yescharacteristic
143#41×C61C22×C411yescharacteristic
144C2×Q8×C61C22×C412yes—
145#11758226#4111yescharacteristic
146Q8×D61—22no—
147Q8×D61—22no—
148Q8×D61—22no—
149Q8×D61—22no—
150Q8×D61C2×D411yescharacteristic
151#11758224—21no—
152#11758224#4111yescharacteristic
153D4×D61—21no—
154D4×D61C2×D411yescharacteristic
155#11758222—21no—
156#11758222C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×C4×D61C2411yescharacteristic
160C2×Dic122—22no—
161C2×Dic122C22×C412yes—
162C22⋊#6923—21no—
163C22⋊#6923—21no—
164C22⋊#6923—21no—
165#11758217—22no—
166#11758217—22no—
167#11758217#4112yes—
168C2×#92420—22no—
169C2×#92420—21no—
170C4⋊#6923—21no—
171C4⋊#6923—21no—
172C4⋊#6923—22no—
173C4⋊#6923—22no—
174C4⋊#6923—42no—
175C4⋊#6923—42no—
176C4⋊#6923C2×D411yescharacteristic
177C4⋊#6923C2×D411yescharacteristic
178C4×#6923—21no—
179C4×#6923—21no—
180C4×#6923—21no—
181C4×#6923—42no—
182C4×#6923—42no—
183C4×#6923C2×D411yescharacteristic
184C4×#6923C2×D411yescharacteristic
185C4×#6923C2×D411yescharacteristic
186C4×#6923C2×D411yescharacteristic
187#11758213—22no—
188#11758213—21no—
189#11758213—22no—
190D61⋊C8—22no—
191D61⋊C8—21no—
192D61⋊C8—21no—
193D61⋊C8—22no—
194D61⋊C8C2×D411yescharacteristic
195D61⋊C8C2×D411yescharacteristic
196Q16×C61—22no—
197Q16×C61—21no—
198Q16×C61—21no—
199SD16×C61—22no—
200M16×C61—21no—
201M16×C61C22×C412yes—
202M16×C61C22×C411yescharacteristic
203C2×C8×C61—21no—
204C2×C8×C61C22×C411yescharacteristic
205#11758202—22no—
206#11758202—22no—
207#11758202—22no—
208#11758202—22no—
209#11758201—22no—
210#11758200—22no—
211#11758194—21no—
212#11758194C22×C412yes—
213#11758194C22×C411yescharacteristic
214C2×#92418—21no—
215C2×#92418C22×C411yescharacteristic
216Dic244—22no—
217Dic244—21no—
218Dic244—21no—
219#11758190—22no—
220#11758189—22no—
221#11758189—21no—
222#11758189—21no—
223#11758189—21no—
224C8×D61—21no—
225C8×D61—22no—
226C8×D61—21no—
227C8×D61—21no—
228C8×D61C2×D411yescharacteristic
229C8×D61C2×D411yescharacteristic
230C8×D61#4111yescharacteristic
231C8×D61#4111yescharacteristic
232C22×D61—21no—
233C22×D61C2×#4111yescharacteristic
234C2×#6923—22no—
235C2×#6923—21no—
236C2×#6923—21no—
237C2×#6923—41no—
238C2×#6923—42no—
239C2×#6923—42no—
240C2×#6923—42no—
241C2×#6923—82no—
242C2×#6923#12111yescharacteristic
243C2×#6923#12111yescharacteristic
244C2×#6923#12111yescharacteristic
245C2×#6923#12111yescharacteristic
246Q8×C61—22no—
247Q8×C61—22no—
248Q8×C61C4×D411yescharacteristic
249D4×C61—21no—
250D4×C61C4×D411yescharacteristic
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61C23×C411yescharacteristic
254C22⋊D61—21no—
255C22⋊D61—21no—
256C22⋊D61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61C2×#4111yescharacteristic
260D244—21no—
261D244C4×D411yescharacteristic
262C4×D61—21no—
263C4×D61—21no—
264C4×D61—21no—
265C4×D61—42no—
266C4×D61—42no—
267C4×D61C22×D411yescharacteristic
268C4×D61C22×D411yescharacteristic
269C4×D61C2×#4111yescharacteristic
270C4×D61C2×#4111yescharacteristic
271Dic122—21no—
272Dic122—22no—
273Dic122—22no—
274Dic122—42no—
275Dic122—42no—
276Dic122C4×D411yescharacteristic
277#92420—21no—
278#92420—22no—
279#92420—21no—
280#92420—42no—
281C8×C61—21no—
282C8×C61—22no—
283C8×C61C4×D411yescharacteristic
284C8×C61C4×D411yescharacteristic
285#92418—21no—
286#92418—22no—
287#92418C4×D411yescharacteristic
288#92418C4×D411yescharacteristic
289C4×#473—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61C4×#4111yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61C2×#12111yescharacteristic
298C2×D61#52911yescharacteristic
299#6923—21no—
300#6923—21no—
301#6923—21no—
302#6923—41no—
303#6923—41no—
304#6923—82no—
305#6923#47312yesdirect product factor
306C4×C61—21no—
307C4×C61—42no—
308C4×C61C2×C4×D411yescharacteristic
309C4×C61C2×C4×D411yescharacteristic
310Dic61—21no—
311Dic61—42no—
312Dic61#52511yescharacteristic
313Dic61#53211yescharacteristic
314C2×#473—611no—
315C4×#137—612no—
316C2×C4×Q16—611no—
317C4×#131—611no—
318#1979—611no—
319#1952—611no—
320#1930—612no—
321#1908—612no—
322#1816—611no—
323C4×#108—611no—
324C4×#101—612no—
325C2×C61—21no—
326C2×C61—41no—
327C2×C61C4×#12111yescharacteristic
328D61—21no—
329D61—41no—
330D61C2×#47311yescharacteristic
331D61#306711yescharacteristic
332C2×#137—612no—
333C22×Q16—611no—
334C2×#131—611no—
335C2×#131—611no—
336C4×#41—611no—
337C2×C4×Q8—612no—
338#473—612no—
339#473—611no—
340#473—611no—
341#473—611no—
342#473—612no—
343#473#692311yescharacteristic, direct product factor
344#441—1222no—
345#440—1222no—
346C4×Q16—1221no—
347C4×Q16—1221no—
348C4×Q16—1221no—
349C4×Q16—1221no—
350C4×Q16—1222no—
351C4×SD16—1222no—
352C2×#108—611no—
353C2×#108—611no—
354#428—612no—
355C2×#107—611no—
356#419—612no—
357#417—612no—
358C2×#103—612no—
359C2×#101—612no—
360C2×#101—612no—
361#405—611no—
362#405—611no—
363#405—1221no—
364#405—1221no—
365C4×M16—611no—
366C4×M16—612no—
367C4×M16—1221no—
368C2×C4×C8—611no—
369C2×C4×C8—1221no—
370#343—612no—
371#343—612no—
372#341—612no—
373C61C4×#47311yescharacteristic, complemented, Sylow
374C2×#41—611no—
375C22×Q8—612no—
376#137—612no—
377#137—612no—
378#137—612no—
379#137—1222no—
380#137—1222no—
381#137C2×#692312yes—
382C2×Q16—612no—
383C2×Q16—611no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×Q16—1221no—
387C2×Q16—1221no—
388C2×Q16—1222no—
389C2×Q16C2×#692311yescharacteristic, complemented
390C2×SD16—1222no—
391#131—611no—
392#131—611no—
393#131—611no—
394#131—1221no—
395#131—1221no—
396#131—1222no—
397#131—1221no—
398#131—1221no—
399#131C2×#692311yescharacteristic
400C2×M16—611no—
401C2×M16—611no—
402C2×M16—612no—
403C2×M16—612no—
404C2×M16—1221no—
405C2×M16—1221no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—1221no—
409C22×C8—1221no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×D4—611no—
417C4×D4—611no—
418C4×D4—1221no—
419C4×D4—1221no—
420#117—611no—
421#117—1221no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—1222no—
424C2×C42—611no—
425C2×C42—1221no—
426C2×C42—1222no—
427#108—612no—
428#108—611no—
429#108—1221no—
430#108—1221no—
431#108C2×#692311yescharacteristic, complemented
432#107—1222no—
433#107—1221no—
434#107—1221no—
435#106—1222no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#102—1222no—
442#101—612no—
443#101—612no—
444#101—612no—
445#101—1224no—
446#101C2×#692312yescomplemented
447#97—1221no—
448#97—1222no—
449#97—1221no—
450#97—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—1221no—
456C4×C8—1222no—
457C4×C8—1221no—
458C4×C8—1221no—
459#41—611no—
460#41—611no—
461#41—611no—
462#41—1221no—
463#41—1221no—
464#41C22×#692311yescharacteristic
465C2×Q8—611no—
466C2×Q8—612no—
467C2×Q8—1222no—
468C2×Q8—1222no—
469C2×Q8—1222no—
470C2×Q8—1222no—
471C2×Q8—1222no—
472C2×Q8C22×#692312yes—
473C2×D4—611no—
474C2×D4—1221no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1222no—
483Q16—22no—
484Q16—21no—
485Q16—21no—
486Q16—1222no—
487Q16—1222no—
488Q16—1222no—
489Q16—1222no—
490Q16—1222no—
491Q16—1221no—
492Q16—1221no—
493SD16—22no—
494SD16—1222no—
495SD16—1222no—
496SD16—1222no—
497M16—21no—
498M16—612no—
499M16—611no—
500M16—612no—
501M16—1221no—
502M16—1221no—
503M16—1222no—
504M16—1221no—
505M16—1221no—
506M16—1222no—
507M16—1222no—
508M16—1221no—
509M16—1222no—
510M16—1222no—
511M16C22×#692311yescharacteristic
512M16C22×#692312yes—
513C2×C8—21no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—611no—
519C2×C8—611no—
520C2×C8—611no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1222no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1222no—
529C2×C8—1222no—
530C2×C8—1221no—
531C2×C8—1222no—
532C2×C8C22×#692311yescharacteristic
533C4⋊C4—611no—
534C4⋊C4—611no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C22⋊C4—1221no—
542C22⋊C4—1221no—
543C22⋊C4—1221no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—2442no—
552C42—2442no—
553C23—611no—
554C23—1221no—
555Q8—22no—
556Q8—22no—
557Q8—611no—
558Q8—1221no—
559Q8—1222no—
560Q8—1222no—
561Q8—2442no—
562Q8—2442no—
563Q8D4×#692311yescharacteristic
564D4—21no—
565D4—611no—
566D4—1221no—
567D4—1221no—
568D4—1221no—
569D4—1221no—
570D4D4×#692311yescharacteristic
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1222no—
590C2×C4—4882no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2441no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4C23×#692311yescharacteristic
598C8—21no—
599C8—22no—
600C8—611no—
601C8—611no—
602C8—1221no—
603C8—1221no—
604C8—1222no—
605C8—1221no—
606C8—1222no—
607C8—2442no—
608C8D4×#692311yescharacteristic
609C8D4×#692311yescharacteristic
610C22—21no—
611C22—611no—
612C22—611no—
613C22—1221no—
614C22—1221no—
615C22—1221no—
616C22—2441no—
617C22—2441no—
618C22#41×#692311yescharacteristic
619C4—21no—
620C4—42no—
621C4—612no—
622C4—611no—
623C4—611no—
624C4—1221no—
625C4—1221no—
626C4—1221no—
627C4—1221no—
628C4—4882no—
629C4—2441no—
630C4—2441no—
631C4—2442no—
632C4C2×D4×#692311yescharacteristic
633C4C2×D4×#692311yescharacteristic
634C2—21no—
635C2—41no—
636C2—611no—
637C2—611no—
638C2—1221no—
639C2—2441no—
640C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#473C4
socleC2×C61C4×#121
prime radical#41×D61C23
cosocleC2×C4×D61C24
hypercenter#473#6923
hypocenterC61C4×#473
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
2. C2×Q16×#6923C2semidirect
4. #21461981C2semidirect
5. #21461937C2semidirect
6. #21461904C2semidirect
9. #108×#6923C2semidirect
10. #101×#6923C2semidirect
45. #16840038C4semidirect
46. #16840037C4semidirect
47. #16840036C4semidirect
48. #16840035C4semidirect
49. #16840034C4semidirect
52. C61×#473C4semidirect
110. C2×Dic244C2×C4semidirect
111. C2×#11758202C2×C4semidirect
128. #14519713C2×C4semidirect
129. #14519712C2×C4semidirect
130. #14519696C2×C4semidirect
131. #14519695C2×C4semidirect
132. #14519695C2×C4semidirect
134. C2×Q16×C61C2×C4semidirect
136. #108×C61C2×C4semidirect
137. #101×C61C2×C4semidirect
305. #6923#473direct (N × H)
343. #473#6923direct (N × H)
373. C61C4×#473semidirect
389. C2×Q16C2×#6923semidirect
431. #108C2×#6923semidirect
446. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#137×#69232
C2×Q16×#69232
#131×#69232
#214619812
#214619372
#214619042
#214618592
#214617082
#108×#69232
#101×#69232
#473×D612
C4×#47361