Finite Group Databasebeta

← #29684052

Subgroups — #29684052

Order 15616.

1562 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2C2×#16839779C211yesmaximal, characteristic, complemented
3C2×#16839778C211yesmaximal, characteristic, complemented
4C22×#14519705C211yesmaximal, characteristic, complemented
5#21454237C211yesmaximal, characteristic, complemented
6#21448302C211yesmaximal, characteristic, complemented
7#21446736C218yesmaximal, complemented
8C22×#104×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11#16840340—22no—
12#16840336C412yescomplemented
13#16840333C412yescomplemented
14#16840329C412yescomplemented
15#16840326C412yescomplemented
16#16840321—22no—
17#16840308C2214yes—
18#16840273—22no—
19#16840269—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839779—24no—
23#16839778—24no—
24C2×#14519705—24no—
25C2×#14519705C2214yescomplemented
26C2×#14519704C2214yescomplemented
27C2×#14519703C2214yescomplemented
28#16839720—22no—
29#16839717—22no—
30M16×Dic61C2214yescomplemented
31#16839660—22no—
32#16839635—22no—
33C4×#11758196C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×C42×C61C2211yescharacteristic
37C2×#104×C61—24no—
38C2×#104×C61C2214yescomplemented
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C412yes—
46C2×#11758222C2312yes—
47C22×Dic122C2×C411yescharacteristic
48C22×D244C2×C411yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C2×#11758194D414yescomplemented
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519775C2×C414yescomplemented
57#14519773C2×C414yescomplemented
58#14519766C2×C414yescomplemented
59#14519764C2×C414yescomplemented
60#14519760—24no—
61#14519758—24no—
62#14519756—24no—
63#14519753—24no—
64#14519751—24no—
65#14519749—28no—
66#14519748—24no—
67#14519746—24no—
68#14519745—24no—
69C4×C22⋊D61—28no—
70#14519736—22no—
71#14519736—22no—
72#14519736C2×C412yes—
73#14519736C2×C412yes—
74#14519736D412yes—
75#14519736D412yes—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D412yes—
78C2×#11758196—24no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61D412yes—
81C2×C4×Dic61C2312yes—
82C2×C4×Dic61C2312yes—
83#14519705—28no—
84#14519704—28no—
85#14519703—28no—
86#14519687—24no—
87C8×Dic61—24no—
88#14519684—24no—
89C2×#41×C61—21no—
90C2×#41×C61—21no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61C2×C412yes—
96C2×#41×C61C2312yes—
97C22×Q8×C61C2×C411yescharacteristic
98C22×D4×C61—21no—
99C22×D4×C61C2×C411yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×M16×C61—21no—
104C2×M16×C61—22no—
105C2×M16×C61D414yescomplemented
106C2×M16×C61C2312yes—
107C22×C8×C61—21no—
108C2×C42×C61—22no—
109C2×C42×C61—22no—
110C2×C42×C61—21no—
111C2×C42×C61C2312yes—
112#104×C61—28no—
113C24×C61—21no—
114C24×C61C2×D411yescharacteristic
115C23×D61—21no—
116#41×C61—24no—
117#41×C61—24no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C22×C412yes—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—44no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61#4112yes—
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—28no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—42no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61C2×D414yes—
164#11758222—24no—
165#11758222—24no—
166#11758222—22no—
167#11758222—22no—
168#11758222—22no—
169#11758222—22no—
170#11758222—42no—
171#11758222—42no—
172#11758222—42no—
173#11758222—42no—
174C2×D244—22no—
175C2×D244—22no—
176C2×D244—21no—
177C2×D244C22×C412yes—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122C22×C412yes—
186M16×C61—24no—
187M16×C61—22no—
188M16×C61C2×D414yes—
189C2×C8×C61—24no—
190C2×C8×C61—22no—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—42no—
194C22⋊Dic61—28no—
195C22⋊Dic61—42no—
196C22⋊Dic61C22⋊C414yes—
197C22⋊Dic61C22⋊C414yes—
198#11758198—28no—
199C4⋊Dic61—42no—
200C4⋊Dic61C22⋊C414yes—
201C4⋊Dic61C22⋊C414yes—
202#11758196—28no—
203#11758196—48no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61C22×C414yes—
207C4×Dic61C2×D414yes—
208#11758194—24no—
209#11758194—22no—
210#11758194C2×D414yes—
211C2×#92418—24no—
212C2×#92418—22no—
213C23×C61—22no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—82no—
217C23×C61C2×C22⋊C411yescharacteristic
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61#12412yes—
220C23×C61#12712yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—42no—
224C22×D61—82no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—42no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—44no—
233D4×C61—42no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61—48no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—44no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C4×D412yes—
246C2×C4×C61C4×D412yes—
247C2×C4×C61C4×D412yes—
248C2×C4×C61C4×D412yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61#12412yes—
254C2×C4×C61#12712yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C22⋊D61—42no—
262C22⋊D61—42no—
263C22⋊D61—44no—
264C22⋊D61—88no—
265C2×Dic61—22no—
266C2×Dic61—44no—
267C2×Dic61—42no—
268C2×Dic61—44no—
269C2×Dic61—42no—
270C2×Dic61C2×C22⋊C418yes—
271D244—22no—
272D244—22no—
273D244—42no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—44no—
277Dic122—22no—
278Dic122—22no—
279Dic122—42no—
280C8×C61—24no—
281#92418—24no—
282#21253—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—44no—
289C22×C61—42no—
290C22×C61—84no—
291C22×C61#39012yes—
292C22×C61#39012yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61C2×#12411yescharacteristic
295C22×C61C2×#12711yescharacteristic
296C2×D61—44no—
297C2×D61—42no—
298C2×D61—84no—
299C4×C61—44no—
300C4×C61—42no—
301C4×C61#39012yes—
302C4×C61#39012yes—
303C4×C61C2×C4×D411yescharacteristic
304C4×C61C2×C4×D411yescharacteristic
305C4×C61C2×#12111yescharacteristic
306C4×C61C2×#12111yescharacteristic
307Dic61—28no—
308Dic61—42no—
309Dic61—82no—
310#3449—611no—
311C2×#421—611no—
312C2×#421—611no—
313C22×#104—611no—
314C22×#104D6111yescharacteristic, complemented
315#2873—611no—
316#2296—611no—
317#1980—618no—
318C2×C61—21no—
319C2×C61—22no—
320C2×C61—22no—
321C2×C61—21no—
322C2×C61—82no—
323C2×C61C2×#39011yescharacteristic
324C2×C61#290112yes—
325D61—82no—
326C22×#41—611no—
327C22×#41C2×D6111yescharacteristic
328C22×M16—611no—
329C22×M16C2×D6111yescharacteristic
330#537—1222no—
331#536—612no—
332#536—612no—
333#535—612no—
334#535—612no—
335#534—1222no—
336#532—614no—
337#525—1222no—
338#525—1222no—
339C2×#117—611no—
340C2×#117—611no—
341C22×C42C2×D6111yescharacteristic
342#421—1224no—
343#421—1224no—
344C2×#104—24no—
345C2×#104—614no—
346C2×#104—614no—
347C2×#104—614no—
348C2×#104—1224no—
349C2×#104C2×D6114yescomplemented
350#405—1222no—
351#405—1222no—
352C4×M16—614no—
353#389—1222no—
354#385—1222no—
355C4×C4⋊C4—614no—
356C61#2125311yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—612no—
364C2×#41—612no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1222no—
368C2×#41—1222no—
369C2×#41—1222no—
370C2×#41—1222no—
371C2×#41C4×D6112yes—
372C2×#41C22×D6112yes—
373C22×Q8—611no—
374C22×Q8C4×D6111yescharacteristic
375C22×D4—21no—
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4C4×D6111yescharacteristic
379C23×C4—21no—
380C23×C4—21no—
381C23×C4—1221no—
382C23×C4C22×D6111yescharacteristic
383C2×M16—21no—
384C2×M16—22no—
385C2×M16—614no—
386C2×M16—612no—
387C2×M16—1222no—
388C2×M16—1221no—
389C2×M16C22⋊D6114yescomplemented
390C2×M16C22×D6112yes—
391C22×C8—21no—
392C22×C8—1221no—
393C4⋊Q8—614no—
394#127—614no—
395#124—614no—
396#124—614no—
397#123—1224no—
398#123—1224no—
399C22⋊Q8—1224no—
400C22⋊Q8—1224no—
401#121—1224no—
402#121—1224no—
403#121—1228no—
404#120—1224no—
405#120—1224no—
406C4×D4—1228no—
407#117—612no—
408#117—612no—
409#117—612no—
410#117—612no—
411#117—1222no—
412#117—1222no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—1221no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—21no—
421C2×C42—612no—
422C2×C42—612no—
423C2×C42C22×D6112yes—
424#104—28no—
425#104—1228no—
426#104—1228no—
427#104—1228no—
428#97—1224no—
429C4×C8—1224no—
430#95—1224no—
431C24—21no—
432C24—1221no—
433C24D4×D6111yescharacteristic
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—24no—
439#41—24no—
440#41—42no—
441#41—42no—
442#41—42no—
443#41—42no—
444#41—1222no—
445#41—1222no—
446#41—1222no—
447#41—1222no—
448#41—1224no—
449#41—1224no—
450#41—2442no—
451#41—2442no—
452#41—2442no—
453#41—2442no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—612no—
458C2×Q8—1221no—
459C2×Q8—1222no—
460C2×Q8—1222no—
461C2×Q8C2×C4×D6112yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—612no—
470C2×D4—1228no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1221no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4C2×C4×D6112yes—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—614no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4#1175822212yes—
497C22×C4D4×D6112yes—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4#1175822412yes—
502C22×C4C2×C22⋊D6112yes—
503C22×C4C23×D6111yescharacteristic
504M16—22no—
505M16—24no—
506M16—614no—
507M16—1222no—
508M16—1224no—
509M16C2×C22⋊D6114yes—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—1222no—
513C2×C8—1224no—
514C4⋊C4—614no—
515C4⋊C4—614no—
516C4⋊C4—1228no—
517C4⋊C4—2448no—
518C4⋊C4—2442no—
519C22⋊C4—614no—
520C22⋊C4—614no—
521C22⋊C4—1228no—
522C22⋊C4—1228no—
523C22⋊C4—2442no—
524C42—22no—
525C42—22no—
526C42—42no—
527C42—614no—
528C42—614no—
529C42—1222no—
530C42—1222no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—82no—
535C23—1222no—
536C23—4882no—
537C23—2442no—
538C23C22⋊C4×D6111yescharacteristic
539C23C22⋊C4×D6111yescharacteristic
540C23#1451976312yes—
541C23#1451977312yes—
542C23C2×D4×D6111yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—42no—
546Q8—1222no—
547Q8—1222no—
548Q8—2442no—
549D4—22no—
550D4—22no—
551D4—42no—
552D4—42no—
553D4—44no—
554D4—42no—
555D4—1222no—
556D4—1222no—
557D4—4888no—
558D4—2442no—
559D4—2442no—
560D4—2444no—
561D4—2442no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—618no—
572C2×C4—1222no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2444no—
580C2×C4C22⋊C4×D6111yescharacteristic
581C2×C4C22⋊C4×D6111yescharacteristic
582C2×C4#1451973712yes—
583C2×C4#1451973712yes—
584C2×C4C4×C22⋊D6112yes—
585C2×C4C4×C22⋊D6112yes—
586C2×C4#1451974912yes—
587C2×C4#1451974912yes—
588C2×C4#1451975012yes—
589C2×C4#1451975012yes—
590C2×C4#1451976312yes—
591C2×C4#1451977312yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C8—24no—
599C8—1224no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—84no—
608C22—4884no—
609C22—2444no—
610C22—2442no—
611C22#1683966212yes—
612C22#1683966212yes—
613C22C2×C22⋊C4×D6111yescharacteristic
614C22C2×#1451976311yescharacteristic
615C22C2×#1451977311yescharacteristic
616C4—44no—
617C4—42no—
618C4—1228no—
619C4—4882no—
620C4—2442no—
621C4#1683966212yes—
622C4#1683966212yes—
623C4C2×#1451973711yescharacteristic
624C4C2×C4×C22⋊D6111yescharacteristic
625C4C2×#1451974911yescharacteristic
626C4C2×#1451975011yescharacteristic
627C2—21no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2—82no—
632C2—4882no—
633C2C2×#1683966211yescharacteristic
634C2#2145436312yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#104×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459148C2
cosocleC22×C4×C61C24
hypercenterC22×#104D61
hypocenterC61#21253
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16839779C2semidirect
3. C2×#16839778C2semidirect
4. C22×#14519705C2semidirect
5. #21454237C2semidirect
6. #21448302C2semidirect
7. #21446736C2semidirect
8. C22×#104×C61C2semidirect
12. #16840336C4semidirect
13. #16840333C4semidirect
14. #16840329C4semidirect
15. #16840326C4semidirect
25. C2×#14519705C22semidirect
26. C2×#14519704C22semidirect
27. C2×#14519703C22semidirect
30. M16×Dic61C22semidirect
33. C4×#11758196C22semidirect
38. C2×#104×C61C22semidirect
53. C2×#11758194D4semidirect
56. #14519775C2×C4semidirect
57. #14519773C2×C4semidirect
58. #14519766C2×C4semidirect
59. #14519764C2×C4semidirect
105. C2×M16×C61D4semidirect
314. C22×#104D61semidirect
349. C2×#104C2×D61semidirect
356. C61#21253semidirect
389. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214591482
C2×#168397792
C2×#168397782
C22×#145197052
#214542372
#214483022
#214467362
C22×#104×C612
#2125361