Finite Group Databasebeta

← #29707899

Subgroups — #29707899

Order 15616.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic, complemented
3C2×#16839759C211yesmaximal, characteristic, complemented
4C2×#16839744C211yesmaximal, characteristic, complemented
5C2×#16839728C211yesmaximal, characteristic, complemented
6#21447120C214yesmaximal, complemented
7#21447113C214yesmaximal, complemented
8#21445557C211yesmaximal, characteristic, complemented
9C22×#108×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758222C2211yescharacteristic
12#16840562—24no—
13#16840562—24no—
14C2×#14519810—21no—
15C2×#14519810C2212yes—
16C2×#14519805—21no—
17C2×#14519805C2212yes—
18C2×#14519799—22no—
19C22×#11758194C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839772—21no—
22#16839768—21no—
23#16839764C2214yes—
24#16839759—22no—
25#16839759—22no—
26#16839759—22no—
27C2×#14519702C2212yescomplemented
28C2×#14519699C2212yescomplemented
29#16839744—24no—
30C2×#14519695C2212yescomplemented
31C2×#14519693C2212yescomplemented
32#16839728—22no—
33#16839728C2214yes—
34C2×#14519689—22no—
35#16839530C2214yescomplemented
36C22×M16×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C2×#108×C61—24no—
40C2×#108×C61C2214yescomplemented
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—22no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—21no—
47C2×#11758222—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222—22no—
54C2×#11758222D414yes—
55C2×#11758222D412yes—
56C2×#11758222C2312yes—
57C22×Dic122D411yescharacteristic
58C22×Dic122C2311yescharacteristic
59C22×D244D411yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65#14519810—28no—
66#14519810—24no—
67#14519810—24no—
68#14519805—28no—
69#14519805—24no—
70#14519805—24no—
71#14519799—216no—
72C2×Dic244—22no—
73C2×#11758190—22no—
74C2×#11758190—22no—
75C2×D488—22no—
76C2×#11758194—21no—
77C2×#11758194—22no—
78C2×#11758194—22no—
79C2×#11758194C2312yes—
80C22×#92418—21no—
81#14519736—21no—
82#14519736—21no—
83#14519736—21no—
84#14519736—21no—
85C2×C4⋊Dic61C2312yes—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88#14519702—24no—
89#14519702—24no—
90#14519699—24no—
91#14519699—24no—
92#14519695—24no—
93#14519693—24no—
94#14519689—28no—
95C23×C4×C61C2311yescharacteristic
96C2×M16×C61—21no—
97C2×M16×C61—22no—
98C2×M16×C61—21no—
99C2×M16×C61—22no—
100C2×M16×C61—22no—
101C2×M16×C61D414yes—
102C2×M16×C61C2312yes—
103C2×M16×C61C2312yes—
104C22×C8×C61—22no—
105C22×C8×C61—21no—
106C22×C8×C61—22no—
107C22×C8×C61—21no—
108C22×C8×C61D412yes—
109C22×C8×C61D412yes—
110C22×C8×C61C2312yes—
111#108×C61—28no—
112C24×C61#4111yescharacteristic
113C23×D61—21no—
114C23×D61—21no—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4112yes—
122C22×C4×C61#4111yescharacteristic
123C22×C4×C61#4112yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—22no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—41no—
129C2×C22⋊D61—41no—
130C2×C22⋊D61—42no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136#11758222—24no—
137#11758222—24no—
138#11758222—24no—
139#11758222—24no—
140#11758222—24no—
141#11758222—24no—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—42no—
147#11758222—42no—
148#11758222—42no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152#11758222—42no—
153#11758222—42no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C2×D412yes—
163C2×D244C2×D412yes—
164C2×C4×D61—22no—
165C2×C4×D61—24no—
166C2×C4×D61—22no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171C2×C4×D61—42no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C2×D412yes—
181C2×Dic122C2×D412yes—
182M16×C61—24no—
183M16×C61—22no—
184M16×C61—22no—
185M16×C61—22no—
186M16×C61—42no—
187M16×C61C2×D414yes—
188C2×C8×C61—24no—
189C2×C8×C61—24no—
190C2×C8×C61—24no—
191C2×C8×C61—22no—
192C2×C8×C61—22no—
193C2×C8×C61—22no—
194C2×C8×C61—22no—
195C2×C8×C61—42no—
196C2×C8×C61C2×D414yes—
197C22⋊Dic61—41no—
198C22⋊Dic61—41no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208Dic244—28no—
209D488—28no—
210#11758190—28no—
211#11758190—28no—
212C23×C61—42no—
213C23×C61#12111yescharacteristic
214C23×C61#12111yescharacteristic
215C23×C61#12312yes—
216C23×C61#12412yes—
217C23×C61C2×#4111yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—44no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—42no—
224C22×D61—82no—
225C2×C4×C61#12012yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12312yes—
234C2×C4×C61#12312yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12412yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C22⋊D61—42no—
244C22⋊D61—42no—
245C22⋊D61—44no—
246C22⋊D61—44no—
247C22⋊D61—44no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—82no—
255D244—22no—
256D244—22no—
257D244—22no—
258D244—22no—
259D244—41no—
260D244—41no—
261D244—42no—
262C4×D61—42no—
263C4×D61—42no—
264C4×D61—44no—
265C4×D61—44no—
266C4×D61—44no—
267Dic122—22no—
268Dic122—22no—
269Dic122—22no—
270Dic122—22no—
271Dic122—41no—
272Dic122—41no—
273Dic122—42no—
274C8×C61—24no—
275C8×C61—24no—
276C8×C61—42no—
277#92418—42no—
278#23530—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61#39412yes—
285C22×C61#39412yes—
286C22×C61C2×#12111yescharacteristic
287C22×C61C2×#12311yescharacteristic
288C22×C61C2×#12411yescharacteristic
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—44no—
292C2×D61—84no—
293C2×D61—82no—
294C2×D61—82no—
295C2×D61—82no—
296C4×C61#39412yes—
297C4×C61#39412yes—
298C4×C61C2×#12011yescharacteristic
299C4×C61C2×#12111yescharacteristic
300C4×C61C2×#12111yescharacteristic
301C4×C61C2×#12311yescharacteristic
302Dic61—42no—
303Dic61—42no—
304Dic61—82no—
305C2×#575—611no—
306C22×#108D6111yescharacteristic, complemented
307#2896—611no—
308C2×#416—611no—
309C2×#413—611no—
310C2×#408—611no—
311#2052—614no—
312#2051—614no—
313#1742—611no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—22no—
317C2×C61—22no—
318C2×C61C2×#39411yescharacteristic
319C2×C61#306712yes—
320D61—82no—
321D61—82no—
322C22×#41—611no—
323C22×#41—611no—
324#575—1224no—
325#575—1224no—
326C2×#137—612no—
327C2×#137—1221no—
328C2×AGL(1,ℤ/8)—612no—
329C2×AGL(1,ℤ/8)—1221no—
330C2×#135—1222no—
331C22×M16—611no—
332C22×M16C2×D6111yescharacteristic
333C22×M16C2×D6111yescharacteristic
334C23×C8C2×D6111yescharacteristic
335C2×#117—611no—
336C2×#108—24no—
337C2×#108C2×D6114yescomplemented
338#419—1221no—
339#418—1221no—
340#417—614no—
341#416—1222no—
342#416—1222no—
343#416—1222no—
344C2×#103—612no—
345C2×#102—612no—
346#413—1224no—
347C2×#101—612no—
348C2×#100—612no—
349#408—614no—
350#408—1222no—
351C2×C22⋊C8—1222no—
352#343—614no—
353C61#2353011yescharacteristic, complemented, Sylow
354C2×#41—612no—
355C2×#41—614no—
356C2×#41—612no—
357C2×#41—1221no—
358C2×#41—1222no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1222no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1222no—
366C2×#41—1222no—
367C2×#41—1222no—
368C2×#41—1222no—
369C2×#41—1222no—
370C22×Q8—611no—
371C22×Q8—611no—
372C22×D4—611no—
373C22×D4—611no—
374C22×D4—1221no—
375C22×D4—1221no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4C22×D6111yescharacteristic
379#137—1224no—
380#137—1228no—
381#137—1224no—
382AGL(1,ℤ/8)—1224no—
383AGL(1,ℤ/8)—1228no—
384AGL(1,ℤ/8)—1224no—
385#135—12216no—
386C2×Q16—1222no—
387C2×SD16—1222no—
388C2×SD16—1222no—
389C2×D8—1222no—
390C2×M16—21no—
391C2×M16—22no—
392C2×M16—21no—
393C2×M16—22no—
394C2×M16—22no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1222no—
398C2×M16—1222no—
399C2×M16C22⋊D6114yes—
400C2×M16C22×D6112yes—
401C2×M16C22×D6112yes—
402C22×C8—22no—
403C22×C8—21no—
404C22×C8—22no—
405C22×C8—21no—
406C22×C8—1221no—
407C22×C8D24412yes—
408C22×C8D24412yes—
409C22×C8C22×D6112yes—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414C2×C4⋊C4—612no—
415C2×C22⋊C4—1221no—
416C2×C42—1221no—
417#108—28no—
418#103—1224no—
419#103—1224no—
420#102—1224no—
421#102—1224no—
422#101—1224no—
423#100—1224no—
424C22⋊C8—1228no—
425C24—1221no—
426C24—1221no—
427C24#1175822611yescharacteristic
428#41—1222no—
429#41—1222no—
430#41—1222no—
431#41—1222no—
432#41—1224no—
433#41—1224no—
434#41—1224no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—2442no—
439#41—2442no—
440#41—2442no—
441#41—2442no—
442#41—2442no—
443#41—2442no—
444#41—2442no—
445#41—2442no—
446C2×Q8—612no—
447C2×Q8—612no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×D4—612no—
457C2×D4—612no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1224no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1222no—
467C2×D4—1221no—
468C2×D4—1222no—
469C2×D4—2441no—
470C2×D4—2441no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4—2442no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—2442no—
481C22×C4—2441no—
482C22×C4—2442no—
483C22×C4—2441no—
484C22×C4—2442no—
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4#1175822212yes—
488C22×C4D4×D6112yes—
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4#1175822612yes—
492C22×C4#1175822611yescharacteristic
493C22×C4C2×C22⋊D6112yes—
494C22×C4C23×D6111yescharacteristic
495Q16—1228no—
496SD16—1228no—
497SD16—1228no—
498D8—1228no—
499M16—22no—
500M16—24no—
501M16—22no—
502M16—22no—
503M16—42no—
504M16—1222no—
505M16—1222no—
506M16—2442no—
507M16C2×C22⋊D6114yes—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8—1222no—
517C2×C8—2442no—
518C2×C8C2×D24414yes—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C22⋊C4—2441no—
522C22⋊C4—2441no—
523C42—2441no—
524C42—2441no—
525C23—42no—
526C23—1222no—
527C23—1222no—
528C23—4882no—
529C23—2444no—
530C23—2441no—
531C23—2441no—
532C23—2442no—
533C23#1451975211yescharacteristic
534C23#1451975211yescharacteristic
535C23#1451976212yes—
536C23#1451976612yes—
537C23C2×#1175822611yescharacteristic
538Q8—1222no—
539Q8—1222no—
540Q8—1222no—
541Q8—1222no—
542Q8—2441no—
543Q8—2441no—
544Q8—2442no—
545D4—1222no—
546D4—1222no—
547D4—1222no—
548D4—1222no—
549D4—2442no—
550D4—2442no—
551D4—2444no—
552D4—2444no—
553D4—2444no—
554D4—2441no—
555D4—2441no—
556D4—2442no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—4882no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4#1451974512yes—
570C2×C4#1451974512yes—
571C2×C4#1451975112yes—
572C2×C4#1451975112yes—
573C2×C4#1451975212yes—
574C2×C4#1451975212yes—
575C2×C4#1451975211yescharacteristic
576C2×C4#1451975211yescharacteristic
577C2×C4#1451976212yes—
578C2×C4#1451976212yes—
579C2×C4#1451976212yes—
580C2×C4#1451976612yes—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C22×D24411yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822611yescharacteristic
587C8—24no—
588C8—24no—
589C8—42no—
590C8—2442no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—4882no—
597C22—4882no—
598C22—4884no—
599C22—4882no—
600C22—2442no—
601C22—2442no—
602C22—2444no—
603C22#1683967912yes—
604C22#1683967912yes—
605C22C2×#1451975211yescharacteristic
606C22C2×#1451976211yescharacteristic
607C22C2×#1451976611yescharacteristic
608C4—4882no—
609C4—2442no—
610C4—2442no—
611C4#1683967912yes—
612C4#1683967912yes—
613C4C2×#1451974511yescharacteristic
614C4C2×#1451975111yescharacteristic
615C4C2×#1451975211yescharacteristic
616C4C2×#1451976211yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—4882no—
622C2—4882no—
623C2C2×#1683967911yescharacteristic
624C2#2145554012yes—
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#108×C61C2
socleC22×C61C2×#121
prime radicalC2×#16840562C2
hypercenterC22×#108D61
hypocenterC61#23530
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
2. #21454339C2semidirect
3. C2×#16839759C2semidirect
4. C2×#16839744C2semidirect
5. C2×#16839728C2semidirect
6. #21447120C2semidirect
7. #21447113C2semidirect
8. #21445557C2semidirect
9. C22×#108×C61C2semidirect
27. C2×#14519702C22semidirect
28. C2×#14519699C22semidirect
30. C2×#14519695C22semidirect
31. C2×#14519693C22semidirect
35. #16839530C22semidirect
40. C2×#108×C61C22semidirect
306. C22×#108D61semidirect
337. C2×#108C2×D61semidirect
353. C61#23530semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
#214543392
C2×#168397592
C2×#168397442
C2×#168397282
#214471202
#214471132
#214455572
C22×#108×C612
#2353061