Finite Group Databasebeta

← #29707925

Subgroups — #29707925

Order 15616.

1358 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840562C211yesmaximal, characteristic
2C2×#16839810C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5C2×#16839744C211yesmaximal, characteristic, complemented
6#21454263C211yesmaximal, characteristic, complemented
7#21447121C214yesmaximal, complemented
8#21447114C214yesmaximal, complemented
9C2×C61×#343C211yesmaximal, 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
20C22×#11758194C2211yescharacteristic
21#16839810—24no—
22C2×#14519712C2214yescomplemented
23#16839761—22no—
24#16839761—22no—
25#16839761—22no—
26#16839761C2214yes—
27C2×#14519700—21no—
28C2×#14519700C2212yescomplemented
29C2×#14519697—21no—
30C2×#14519697C2212yescomplemented
31#16839744—24no—
32C2×#14519695C2212yescomplemented
33C2×#14519693C2212yescomplemented
34#16839730C2214yes—
35#16839728—22no—
36#16839725—22no—
37C22×M16×C61C2211yescharacteristic
38C22×M16×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#343C2214yescomplemented
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—22no—
77C2×#11758194—22no—
78C2×#11758194—21no—
79C2×#11758194—22no—
80C2×#11758194—22no—
81C2×#11758194D412yes—
82C2×#11758194D412yes—
83C2×#11758194C2312yes—
84C22×#92418—21no—
85C22×#92418C2312yes—
86#14519712—28no—
87#14519700—24no—
88#14519700—24no—
89#14519697—24no—
90#14519697—24no—
91#14519695—24no—
92#14519693—24no—
93#14519689—28no—
94C23×C4×C61—21no—
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—21no—
105C22×C8×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2312yes—
112C24×C61#4111yescharacteristic
113C23×D61—21no—
114C23×D61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4112yes—
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—22no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—41no—
133C2×C22⋊D61—41no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139#11758222—24no—
140#11758222—24no—
141#11758222—24no—
142#11758222—24no—
143#11758222—24no—
144#11758222—24no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—22no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152#11758222—42no—
153#11758222—42no—
154#11758222—42no—
155#11758222—42no—
156#11758222—42no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244C2×D412yes—
166C2×D244C2×D412yes—
167C2×C4×D61—22no—
168C2×C4×D61—24no—
169C2×C4×D61—22no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—41no—
173C2×C4×D61—41no—
174C2×C4×D61—42no—
175C2×Dic122—22no—
176C2×Dic122—22no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122C2×D412yes—
184C2×Dic122C2×D412yes—
185M16×C61—24no—
186M16×C61—22no—
187M16×C61—22no—
188M16×C61—22no—
189M16×C61—42no—
190M16×C61C2×D414yes—
191C2×C8×C61—24no—
192C2×C8×C61—22no—
193C2×C8×C61—22no—
194C2×C8×C61—42no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—41no—
198C4⋊C4×C61—41no—
199#11758194—24no—
200#11758194—24no—
201#11758194—22no—
202#11758194—22no—
203#11758194—42no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—42no—
208C2×#92418C2×D414yes—
209Dic244—28no—
210D488—28no—
211#11758190—28no—
212#11758190—28no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61#12111yescharacteristic
216C23×C61#12111yescharacteristic
217C23×C61#12312yes—
218C23×C61#12412yes—
219C23×C61C2×#4111yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—42no—
225C22×D61—81no—
226C22×D61—81no—
227C22×D61—81no—
228C22×D61—81no—
229C2×C4×C61—42no—
230C2×C4×C61—82no—
231C2×C4×C61—82no—
232C2×C4×C61—82no—
233C2×C4×C61#12012yes—
234C2×C4×C61#12012yes—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12112yes—
238C2×C4×C61#12112yes—
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12312yes—
242C2×C4×C61#12312yes—
243C2×C4×C61#12312yes—
244C2×C4×C61#12412yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C22⋊D61—42no—
252C22⋊D61—42no—
253C22⋊D61—44no—
254C22⋊D61—44no—
255C22⋊D61—44no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261D244—22no—
262D244—22no—
263D244—22no—
264D244—22no—
265D244—41no—
266D244—41no—
267D244—42no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—44no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—22no—
277Dic122—41no—
278Dic122—41no—
279Dic122—42no—
280C8×C61—24no—
281C8×C61—42no—
282#92418—24no—
283#92418—42no—
284#23533—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61#39412yes—
294C22×C61#39412yes—
295C22×C61C2×#12111yescharacteristic
296C22×C61C2×#12311yescharacteristic
297C22×C61C2×#12411yescharacteristic
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—84no—
303C2×D61—82no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—82no—
307C4×C61—82no—
308C4×C61#39412yes—
309C4×C61#39412yes—
310C4×C61C2×#12011yescharacteristic
311C4×C61C2×#12111yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12311yescharacteristic
314Dic61—42no—
315Dic61—42no—
316C2×#575—611no—
317C2×#430—611no—
318C2×#417—611no—
319C2×#417—611no—
320C2×#413—611no—
321#2879—611no—
322#2052—614no—
323#2051—614no—
324C2×#343D6111yescharacteristic, complemented
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—22no—
328C2×C61—22no—
329C2×C61C2×#39411yescharacteristic
330C2×C61#306811yescharacteristic
331C2×C61#306911yescharacteristic
332D61—82no—
333D61—82no—
334C22×#41—611no—
335C22×#41—611no—
336#575—1224no—
337#575—1224no—
338C2×#137—612no—
339C2×#137—1221no—
340C2×AGL(1,ℤ/8)—612no—
341C2×AGL(1,ℤ/8)—1221no—
342C2×#135—1222no—
343C22×M16—611no—
344C22×M16—611no—
345C22×M16C2×D6111yescharacteristic
346C22×M16C2×D6111yescharacteristic
347C22×C4⋊C4C2×D6111yescharacteristic
348#430—1224no—
349C2×#108—614no—
350#417—614no—
351#417—1222no—
352#417—1222no—
353#417—1222no—
354C2×#103—612no—
355C2×#103—1221no—
356C2×#102—612no—
357C2×#102—1221no—
358#413—1224no—
359C2×#101—612no—
360C2×#100—612no—
361#408—614no—
362#408—1222no—
363#407—1222no—
364#343C2×D6114yescomplemented
365C61#2353311yescharacteristic, complemented, Sylow
366C2×#41—612no—
367C2×#41—614no—
368C2×#41—612no—
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41—1222no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1221no—
377C2×#41—1221no—
378C2×#41—1222no—
379C2×#41—1221no—
380C2×#41—1221no—
381C2×#41—1222no—
382C22×Q8—611no—
383C22×Q8—611no—
384C22×D4—611no—
385C22×D4—611no—
386C22×D4—1221no—
387C22×D4—1221no—
388C23×C4—21no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392#137—1224no—
393#137—1228no—
394#137—1224no—
395AGL(1,ℤ/8)—1224no—
396AGL(1,ℤ/8)—1228no—
397AGL(1,ℤ/8)—1224no—
398#135—12216no—
399C2×Q16—1222no—
400C2×SD16—1222no—
401C2×SD16—1222no—
402C2×D8—1222no—
403C2×M16—21no—
404C2×M16—22no—
405C2×M16—21no—
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—612no—
409C2×M16—612no—
410C2×M16—612no—
411C2×M16—1222no—
412C2×M16—1222no—
413C2×M16—1221no—
414C2×M16—1222no—
415C2×M16—1222no—
416C2×M16D24414yes—
417C2×M16C22×D6112yes—
418C2×M16C22×D6112yes—
419C22×C8—21no—
420C22×C8—21no—
421C22×C8—612no—
422C22×C8—1221no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4C22×D6112yes—
429#108—1228no—
430#103—1224no—
431#103—1224no—
432#102—1224no—
433#102—1224no—
434#101—1224no—
435#100—1224no—
436C22⋊C8—1228no—
437C24—1221no—
438C24—1221no—
439C24#1175822211yescharacteristic
440#41—1222no—
441#41—1222no—
442#41—1222no—
443#41—1222no—
444#41—1224no—
445#41—1224no—
446#41—1224no—
447#41—1224no—
448#41—1224no—
449#41—1224no—
450#41—2442no—
451#41—2442no—
452#41—2442no—
453#41—2442no—
454#41—2442no—
455#41—2442no—
456#41—2442no—
457#41—2442no—
458C2×Q8—612no—
459C2×Q8—612no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×D4—612no—
469C2×D4—612no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1224no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1222no—
481C2×D4—2441no—
482C2×D4—2441no—
483C2×D4—2442no—
484C2×D4—2442no—
485C2×D4—2442no—
486C22×C4—22no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—82no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1224no—
495C22×C4—2442no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4C2×D24412yes—
501C22×C4#1175822212yes—
502C22×C4#1175822211yescharacteristic
503C22×C4D4×D6112yes—
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4#1175822612yes—
509C22×C4C23×D6111yescharacteristic
510Q16—1228no—
511SD16—1228no—
512SD16—1228no—
513D8—1228no—
514M16—22no—
515M16—24no—
516M16—22no—
517M16—22no—
518M16—42no—
519M16—1224no—
520M16—1224no—
521M16—1222no—
522M16—1222no—
523M16—2442no—
524M16C2×D24414yes—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—42no—
529C2×C8—614no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—2442no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C23—41no—
539C23—41no—
540C23—1222no—
541C23—1222no—
542C23—4881no—
543C23—4881no—
544C23—4881no—
545C23—4881no—
546C23—2442no—
547C23—2441no—
548C23—2441no—
549C23#1451975011yescharacteristic
550C23#1451975011yescharacteristic
551C23#1451976212yes—
552C23#1451976512yes—
553C23C2×#1175822211yescharacteristic
554Q8—1222no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558Q8—2442no—
559Q8—2441no—
560Q8—2441no—
561D4—1222no—
562D4—1222no—
563D4—1222no—
564D4—1222no—
565D4—2444no—
566D4—2444no—
567D4—2444no—
568D4—2442no—
569D4—2442no—
570D4—2442no—
571D4—2441no—
572D4—2441no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—2442no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4#1451974712yes—
588C2×C4#1451974712yes—
589C2×C4#1451975011yescharacteristic
590C2×C4#1451975011yescharacteristic
591C2×C4#1451975012yes—
592C2×C4#1451975012yes—
593C2×C4#1451975212yes—
594C2×C4#1451975212yes—
595C2×C4#1451976212yes—
596C2×C4#1451976212yes—
597C2×C4#1451976212yes—
598C2×C4#1451976512yes—
599C2×C4C2×D4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C22×D24411yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4C2×#1175822611yescharacteristic
605C8—24no—
606C8—42no—
607C8—1224no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—4882no—
618C22—4884no—
619C22—4882no—
620C22—4881no—
621C22—4881no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22#1683968012yes—
627C22#1683968012yes—
628C22C2×#1451975011yescharacteristic
629C22C2×#1451976211yescharacteristic
630C22C2×#1451976511yescharacteristic
631C4—82no—
632C4—2442no—
633C4—2442no—
634C4#1683968012yes—
635C4#1683968012yes—
636C4C2×#1451974711yescharacteristic
637C4C2×#1451975011yescharacteristic
638C4C2×#1451975211yescharacteristic
639C4C2×#1451976211yescharacteristic
640C2—22no—
641C2—21no—
642C2—21no—
643C2—22no—
644C2—4882no—
645C2—4882no—
646C2C2×#1683968011yescharacteristic
647C2#2145554611yescharacteristic
648C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C2×#121
prime radicalC2×#16840562C2
hypercenterC2×#343D61
hypocenterC61#23533
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16839810C2semidirect
3. C2×#16839761C2semidirect
4. C2×#16839761C2semidirect
5. C2×#16839744C2semidirect
6. #21454263C2semidirect
7. #21447121C2semidirect
8. #21447114C2semidirect
9. C2×C61×#343C2semidirect
22. C2×#14519712C22semidirect
28. C2×#14519700C22semidirect
30. C2×#14519697C22semidirect
32. C2×#14519695C22semidirect
33. C2×#14519693C22semidirect
40. C61×#343C22semidirect
324. C2×#343D61semidirect
364. #343C2×D61semidirect
365. C61#23533semidirect

Maximal subgroups

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

subgroupindex
C2×#168405622
C2×#168398102
C2×#168397612
C2×#168397442
#214542632
#214471212
#214471142
C2×C61×#3432
#2353361