Finite Group Databasebeta

← #29707699

Subgroups — #29707699

Order 15616.

1522 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C2×#16839759C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5#21448301C211yesmaximal, characteristic, complemented
6#21447063C212yesmaximal, complemented
7#21447058C212yesmaximal, complemented
8#21447056C212yesmaximal, complemented
9#21447051C212yesmaximal, complemented
10C2×#16839516C211yesmaximal, characteristic, complemented
11C2×C61×#416C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840308—24no—
15#16840308—24no—
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×#14519764C2211yescharacteristic
20C2×#14519749—22no—
21C2×C4×C22⋊D61—22no—
22C2×#14519736C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839759—24no—
25#16839758—24no—
26#16839756—24no—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33#16839658C2212yes—
34#16839658C2212yes—
35#16839634—22no—
36C4×#11758196—22no—
37#16839516C2212yes—
38#16839516C2212yes—
39C22×#41×C61C2211yescharacteristic
40C23×C8×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#416—24no—
43C2×#103×C61C2212yes—
44C2×#102×C61C2212yes—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222D414yes—
51C22×Dic122C2311yescharacteristic
52C22×D244C2311yescharacteristic
53C22×C22⋊D61—21no—
54C22×C22⋊D61—22no—
55C23×Dic61D412yes—
56C22×C4×D61—21no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418—22no—
60C22×#92418C2312yes—
61#14519775—22no—
62#14519775—22no—
63#14519773—22no—
64#14519773—22no—
65#14519766—22no—
66#14519766—22no—
67#14519764—22no—
68#14519764—22no—
69#14519749—216no—
70C4×C22⋊D61—216no—
71#14519736—22no—
72#14519736D414yescomplemented
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D412yes—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519702—24no—
87#14519701—24no—
88#14519700—24no—
89#14519699—24no—
90#14519698—24no—
91#14519697—24no—
92#14519684—22no—
93#14519684—22no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61—22no—
99C2×#41×C61D414yes—
100C22×Q8×C61C2311yescharacteristic
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61—22no—
106C22×C8×C61—22no—
107C22×C8×C61—22no—
108C22×C8×C61C2312yes—
109#117×C61—22no—
110#117×C61—22no—
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—24no—
115#102×C61—24no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119#41×C61—28no—
120#41×C61—28no—
121#41×C61—44no—
122#41×C61—44no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61C2×D412yes—
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61C2×D412yes—
138C22×C4×C61—21no—
139C22×C4×C61—24no—
140C22×C4×C61—22no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4112yes—
148C22×C4×C61#4112yes—
149C22×C4×C61#4112yes—
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—28no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C2×C22⋊D61—44no—
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—24no—
161C22×Dic61—42no—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164#11758222—28no—
165#11758222—28no—
166#11758222—44no—
167#11758222—44no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244C2×D412yes—
174C2×C4×D61—24no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122C2×D412yes—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61—44no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—42no—
190C42×C61—42no—
191C22⋊Dic61—28no—
192C22⋊Dic61—24no—
193#11758198—28no—
194C4⋊Dic61—22no—
195C4⋊Dic61C2×D414yes—
196#11758196—28no—
197#11758196—44no—
198#11758196—44no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—44no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61#12012yes—
217C23×C61#12112yes—
218C23×C61#12412yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—42no—
224C22×D61—42no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—48no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—48no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61#12012yes—
242C2×C4×C61#12114yes—
243C2×C4×C61#12114yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12314yes—
246C2×C4×C61#12314yes—
247C2×C4×C61#12412yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—416no—
255C22⋊D61—48no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—48no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61#13514yes—
269D244—22no—
270D244—22no—
271D244—41no—
272D244—41no—
273C4×D61—48no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—42no—
279#92418—42no—
280#23504—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—24no—
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—82no—
290C22×C61#39414yes—
291C22×C61C2×#12011yescharacteristic
292C22×C61C2×#12111yescharacteristic
293C22×C61C2×#12411yescharacteristic
294C22×C61C2×#13512yes—
295C22×C61#57512yes—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—82no—
301C4×C61—42no—
302C4×C61—82no—
303C4×C61#39414yes—
304C4×C61C2×#12111yescharacteristic
305C4×C61C2×#12111yescharacteristic
306C4×C61C2×#12311yescharacteristic
307C4×C61C2×#12311yescharacteristic
308Dic61—24no—
309Dic61—42no—
310Dic61—42no—
311Dic61—44no—
312C2×#532—611no—
313C2×#416—611no—
314C2×#416—611no—
315C2×#416—611no—
316C2×#416D6111yescharacteristic, complemented
317#2296—611no—
318#2043—612no—
319#2042—612no—
320#2042—612no—
321#2041—612no—
322C2×#336—611no—
323C2×C61—24no—
324C2×C61—82no—
325C2×C61C2×#39411yescharacteristic
326C2×C61#300212yes—
327C2×C61#303512yes—
328C2×C61#313111yescharacteristic
329C2×C61#313511yescharacteristic
330D61—82no—
331C22×#41—611no—
332C22×#41C2×D6111yescharacteristic
333C23×C8—611no—
334C23×C8C2×D6111yescharacteristic
335#532—1224no—
336#532—1224no—
337C2×C4⋊Q8—611no—
338C2×#127—611no—
339C2×#124—611no—
340C2×#124—611no—
341C2×#121—1222no—
342C2×C4×D4—1222no—
343C2×#117—611no—
344C2×#117C2×D6111yescharacteristic
345C22×C42—611no—
346#416—24no—
347#416—1224no—
348#416—1224no—
349#416—1224no—
350C2×#103—612no—
351C2×#103—612no—
352C2×#103—612no—
353C2×#103C2×D6112yes—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102C2×D6112yes—
358#389—612no—
359#389—612no—
360#385—1222no—
361C4×C4⋊C4—1222no—
362#336—612no—
363#336—612no—
364C61#2350411yescharacteristic, complemented, Sylow
365C2×#41—22no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41—614no—
371C2×#41—1222no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41C22⋊D6114yes—
377C22×Q8—611no—
378C22×Q8C22×D6111yescharacteristic
379C22×D4—21no—
380C22×D4—611no—
381C22×D4—1222no—
382C22×D4—1221no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—21no—
385C23×C4—612no—
386C23×C4—1221no—
387C23×C4C22×D6111yescharacteristic
388C22×C8—22no—
389C22×C8—22no—
390C22×C8—22no—
391C22×C8—612no—
392C22×C8—1222no—
393C22×C8—1222no—
394C22×C8—1222no—
395C22×C8C22×D6112yes—
396C4⋊Q8—1222no—
397C4⋊Q8—1222no—
398#127—1222no—
399#127—1222no—
400#124—1222no—
401#124—1222no—
402#124—1222no—
403#124—1222no—
404#121—12216no—
405C4×D4—12216no—
406#117—22no—
407#117—22no—
408#117—614no—
409#117—1222no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4C22×D6112yes—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1221no—
419C2×C42—21no—
420C2×C42—612no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—1222no—
424C2×C42—1221no—
425C2×C42—1222no—
426#103—24no—
427#103—1224no—
428#103—1224no—
429#103—1224no—
430#102—24no—
431#102—1224no—
432#102—1224no—
433#102—1224no—
434#95—1222no—
435#95—1222no—
436C24—21no—
437C24—1221no—
438C24D4×D6111yescharacteristic
439#41—28no—
440#41—28no—
441#41—44no—
442#41—44no—
443#41—1228no—
444#41—1228no—
445#41—2444no—
446#41—2444no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—612no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8C2×C22⋊D6112yes—
459C2×D4—24no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—612no—
468C2×D4—1228no—
469C2×D4—1224no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—2444no—
476C2×D4—2444no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4C2×C22⋊D6112yes—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—1224no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1224no—
491C22×C4—1224no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4#1175822212yes—
496C22×C4#1175822212yes—
497C22×C4D4×D6112yes—
498C22×C4D4×D6112yes—
499C22×C4D4×D6111yescharacteristic
500C22×C4#1175822412yes—
501C22×C4C2×C22⋊D6112yes—
502C22×C4C23×D6111yescharacteristic
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—44no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2444no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—614no—
514C4⋊C4—1228no—
515C4⋊C4—1222no—
516C4⋊C4—2444no—
517C4⋊C4—2444no—
518C22⋊C4—42no—
519C22⋊C4—1228no—
520C22⋊C4—1228no—
521C22⋊C4—1224no—
522C42—42no—
523C42—1221no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—1224no—
528C42—1224no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—1222no—
537C23—2441no—
538C23—2441no—
539C23—2442no—
540C23—2442no—
541C23#1451974512yes—
542C23#1451975012yes—
543C23#1451976312yes—
544C23C2×D4×D6111yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—1222no—
550Q8—1222no—
551Q8—2441no—
552Q8—2441no—
553D4—22no—
554D4—22no—
555D4—48no—
556D4—41no—
557D4—41no—
558D4—1222no—
559D4—1222no—
560D4—24416no—
561D4—2448no—
562D4—2441no—
563D4—2441no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—614no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1224no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—2448no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2448no—
585C2×C4#1451974512yes—
586C2×C4#1451974914yes—
587C2×C4#1451975014yes—
588C2×C4#1451975012yes—
589C2×C4#1451975914yes—
590C2×C4#1451976014yes—
591C2×C4#1451976312yes—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C2×#1175822411yescharacteristic
596C2×C4C2×#1175822211yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C8—42no—
599C8—2442no—
600C22—22no—
601C22—22no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—44no—
606C22—81no—
607C22—81no—
608C22—82no—
609C22—4881no—
610C22—4881no—
611C22—4882no—
612C22—2442no—
613C22—2444no—
614C22#1683967714yes—
615C22C2×#1451974511yescharacteristic
616C22C2×#1451975011yescharacteristic
617C22C2×#1451976311yescharacteristic
618C22#135×D6112yes—
619C22#1684055912yes—
620C4—42no—
621C4—82no—
622C4—1224no—
623C4—2442no—
624C4—2444no—
625C4—2442no—
626C4#1683967714yes—
627C4C2×#1451974911yescharacteristic
628C4C2×#1451975011yescharacteristic
629C4C2×#1451975911yescharacteristic
630C4C2×#1451976011yescharacteristic
631C2—24no—
632C2—82no—
633C2—4882no—
634C2C2×#1683967711yescharacteristic
635C2#2145503312yes—
636C2#2145527212yes—
637C2#2145613911yescharacteristic
638C2#2145616211yescharacteristic
subgroupG/H
centerC22×C4D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
hypercenterC2×#416D61
hypocenterC61#23504
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C2×#16839759C2semidirect
3. C2×#16839758C2semidirect
4. C2×#16839756C2semidirect
5. #21448301C2semidirect
6. #21447063C2semidirect
7. #21447058C2semidirect
8. #21447056C2semidirect
9. #21447051C2semidirect
10. C2×#16839516C2semidirect
11. C2×C61×#416C2semidirect
72. #14519736D4semidirect
316. C2×#416D61semidirect
364. C61#23504semidirect

Maximal subgroups

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

subgroupindex
C2×#168403082
C2×#168397592
C2×#168397582
C2×#168397562
#214483012
#214470632
#214470582
#214470562
#214470512
C2×#168395162
C2×C61×#4162
#2350461