Finite Group Databasebeta

← #29828833

Subgroups — #29828833

Order 15616.

1868 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic
2C2×#16839966C212yesmaximal, complemented
3C2×#16839915C211yesmaximal, characteristic
4C2×#16839912C211yesmaximal, characteristic
5#21454167C211yesmaximal, characteristic, complemented
6#21446154C212yesmaximal, complemented
7#21446150C212yesmaximal, complemented
8#21446126C211yesmaximal, characteristic, complemented
9#21445818C212yesmaximal, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C61×#1872C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×#11758190C2212yes—
15C22×#11758201C2212yes—
16C2×#14519775C2211yescharacteristic, complemented
17C2×#14519756C2211yescharacteristic
18C2×#14519739—28no—
19C2×#14519739—21no—
20C2×#14519739—21no—
21C2×#14519739C2212yes—
22C2×#14519739C2212yes—
23C22×#11758198—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839966—28no—
26#16839915—24no—
27#16839912—24no—
28C2×#14519706C2212yescomplemented
29C2×#14519700C2212yescomplemented
30C2×#14519699C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33#16839669C2211yescharacteristic, complemented
34#16839667C2211yescharacteristic, complemented
35#16839657—21no—
36#16839657—21no—
37#16839631C2211yescharacteristic, complemented
38#16839620—21no—
39#16839620—21no—
40#16839524C2212yescomplemented
41#16839516C2212yescomplemented
42C2×C22⋊Q8×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#103×C61C2212yescomplemented
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#340C2212yescomplemented
47C24×D61—21no—
48C22×Dic122C2311yescharacteristic
49C22×D244—21no—
50C22×D244—22no—
51C22×D244—24no—
52C22×D244—24no—
53C22×D244D414yes—
54C22×D244C2312yes—
55C23×Dic61D411yescharacteristic
56C23×Dic61D411yescharacteristic
57C22×C4×D61—28no—
58C22×C4×D61—22no—
59C22×C4×D61—22no—
60C22×C4×D61—44no—
61C22×C4×D61—44no—
62C2×#11758190—24no—
63C2×#11758190—24no—
64C2×#11758190—24no—
65C2×#11758201—24no—
66C2×#11758201—24no—
67C2×#11758201—24no—
68C22×#92418C2312yes—
69#14519775—24no—
70#14519756—22no—
71#14519756—22no—
72#14519739—28no—
73#14519739—28no—
74#14519739—24no—
75#14519739—24no—
76#14519739—24no—
77#14519739D418yes—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—28no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88C2×C4×Dic61—24no—
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519706—24no—
94#14519700—24no—
95#14519699—24no—
96#14519697—24no—
97C22⋊#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—24no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114#103×C61—24no—
115C22⋊C8×C61—22no—
116C24×C61C2×D411yescharacteristic
117C23×D61—24no—
118C23×D61—24no—
119C23×D61—24no—
120C23×D61—22no—
121C23×D61—42no—
122C23×D61—42no—
123C23×D61—42no—
124C23×D61—42no—
125C2×Q8×C61—22no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—41no—
148C22×Dic61SD1614yescomplemented
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—24no—
154C2×D244—216no—
155C2×D244—24no—
156C2×D244—24no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—216no—
161C2×D244C2×D414yes—
162C2×D244C2×D414yes—
163C2×D244#4114yes—
164C2×C4×D61—28no—
165C2×C4×D61—28no—
166C2×C4×D61—44no—
167C2×C4×D61—432no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C2×D414yes—
184C22⋊C4×C61—42no—
185C22⋊Dic61—42no—
186#11758201—48no—
187#11758198—28no—
188#11758198—28no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61C2×D414yes—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203#11758190—48no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61#12112yes—
209C23×C61C2×SD1612yes—
210C23×C61#13712yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—216no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—48no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—48no—
226Q8×C61—42no—
227Q8×C61—42no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—48no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61#12112yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D412yes—
245C2×C4×C61C2×#4112yes—
246C2×C4×C61C2×#4111yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—28no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—44no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#12012yes—
263C2×Dic61#12012yes—
264C2×Dic61C2×SD1612yes—
265C2×Dic61C2×SD1612yes—
266C2×Dic61AGL(1,ℤ/8)12yes—
267D244—216no—
268D244—24no—
269D244—24no—
270D244—24no—
271D244#12118yes—
272C4×D61—416no—
273C4×D61—416no—
274Dic122—42no—
275Dic122—42no—
276C8×C61—42no—
277#92418—42no—
278#32905—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61#46114yescomplemented
286C22×C61C2×#12011yescharacteristic
287C22×C61C2×#12111yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61C22×SD1611yescharacteristic
290C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57511yescharacteristic
293C2×D61—28no—
294C2×D61—432no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C4×C61—22no—
302C4×C61—44no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C2×#12112yes—
306C4×C61#52512yes—
307Dic61—22no—
308Dic61—22no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312Dic61#45014yescomplemented
313C22×C4×D4—611no—
314C2×#461—612no—
315C2×#450—611no—
316C2×#450—611no—
317#2857—611no—
318#1877—612no—
319#1877—612no—
320#1872—611no—
321#1872D6111yescharacteristic, complemented
322#1802—612no—
323#1797—611no—
324C2×C61—24no—
325C2×C61#284711yescharacteristic
326C2×C61C2×#45011yescharacteristic
327C2×C61#300711yescharacteristic
328C2×C61C2×#46111yescharacteristic
329C2×C61#303611yescharacteristic
330C2×C61#310111yescharacteristic
331C2×C61#311011yescharacteristic
332D61—48no—
333C23×D4—611no—
334C24×C4—1221no—
335C22×SD16—612no—
336C22×SD16—612no—
337C2×C4⋊Q8—611no—
338C2×C22⋊Q8—611no—
339C2×C22⋊Q8C2×D6111yescharacteristic
340C2×C4×D4—612no—
341C2×C4×D4—612no—
342C2×C4×D4—1221no—
343C2×C4×D4—1221no—
344C2×C4×D4—1228no—
345C22×C4⋊C4C2×D6111yescharacteristic
346C22×C22⋊C4—1221no—
347C22×C42—611no—
348#461—1228no—
349#450—1224no—
350#450—1224no—
351C2×C4⋊C8—612no—
352C2×#103—612no—
353C2×#103C2×D6112yescomplemented
354C2×#102—612no—
355C2×#102—612no—
356C2×C22⋊C8—611no—
357C2×C22⋊C8C2×D6111yescharacteristic
358#391—611no—
359#391—611no—
360#389—1221no—
361#389—1221no—
362#384—611no—
363#380—1221no—
364#380—1221no—
365#340—612no—
366#340C2×D6112yescomplemented
367#336—612no—
368C61#3290511yescharacteristic, complemented, Sylow
369C25—1221no—
370C22×Q8—611no—
371C22×Q8C22×D6111yescharacteristic
372C22×D4—614no—
373C22×D4—612no—
374C22×D4—1224no—
375C22×D4—1224no—
376C22×D4—1222no—
377C22×D4—1221no—
378C23×C4—21no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—1228no—
384C23×C4—2444no—
385C23×C4—2444no—
386C23×C4C22×D6111yescharacteristic
387C2×SD16—1224no—
388C2×SD16—1224no—
389C2×SD16—1224no—
390C2×SD16—1224no—
391C2×SD16—1224no—
392C2×SD16—1224no—
393C22×C8—612no—
394C22×C8C22×D6112yes—
395C4⋊Q8—1224no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C4×D4—618no—
401C4×D4—1224no—
402C4×D4—1224no—
403C4×D4—1228no—
404C4×D4—1228no—
405C4×D4—1224no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—24no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6112yes—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1224no—
427C4⋊C8—1224no—
428#103—24no—
429#103—1224no—
430#102—1224no—
431#102—1224no—
432C22⋊C8—22no—
433C22⋊C8—1222no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—1224no—
439C24—1224no—
440C24—1222no—
441C24—1224no—
442C24—2442no—
443C24—2442no—
444C24—2442no—
445C24—2442no—
446C24D4×D6111yescharacteristic
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—1222no—
452C2×Q8—1221no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×D4—614no—
456C2×D4—614no—
457C2×D4—614no—
458C2×D4—12216no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—12216no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1224no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—614no—
478C22×C4—611no—
479C22×C4—1228no—
480C22×C4—1228no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—24432no—
491C22×C4—2444no—
492C22×C4—2441no—
493C22×C4#1175822212yes—
494C22×C4D4×D6112yes—
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497SD16—2448no—
498SD16—2448no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4D4×D6114yes—
518C22⋊C4—42no—
519C22⋊C4—1228no—
520C22⋊C4—1228no—
521C22⋊C4—2442no—
522C42—614no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—12216no—
531C23—1224no—
532C23—1224no—
533C23—1224no—
534C23—2448no—
535C23—2448no—
536C23—2448no—
537C23—2448no—
538C23—2448no—
539C23—2448no—
540C23—2442no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23#1451975012yes—
545C23SD16×D6112yes—
546C23#1451980912yes—
547C23C2×D4×D6111yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—2442no—
551Q8—2442no—
552D4—618no—
553D4—12216no—
554D4—1224no—
555D4—1224no—
556D4—1224no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—1228no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—24416no—
585C2×C4—24416no—
586C2×C4—2444no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451975012yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×D4×D6112yes—
596C2×C4#41×D6112yes—
597C2×C4C2×#1175822211yescharacteristic
598C8—42no—
599C8—2442no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—1228no—
607C22—24432no—
608C22—2444no—
609C22—2444no—
610C22—2444no—
611C22—2444no—
612C22—2444no—
613C22—2444no—
614C22#1683996614yescomplemented
615C22#120×D6111yescharacteristic
616C22C2×#1451975011yescharacteristic
617C22#1684027011yescharacteristic
618C22C2×SD16×D6111yescharacteristic
619C22AGL(1,ℤ/8)×D6111yescharacteristic
620C22C2×#1451980911yescharacteristic
621C22#1684056111yescharacteristic
622C4—22no—
623C4—44no—
624C4—81no—
625C4—81no—
626C4—614no—
627C4—1222no—
628C4—1222no—
629C4—4881no—
630C4—4881no—
631C4—2441no—
632C4#121×D6112yes—
633C4#1684027112yes—
634C2—24no—
635C2—2448no—
636C2#2145410011yescharacteristic
637C2#450×D6111yescharacteristic
638C2#2145507811yescharacteristic
639C2C2×#1683996611yescharacteristic
640C2#2145528211yescharacteristic
641C2#2145582411yescharacteristic
642C2#2145591511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1872D61
hypocenterC61#32905
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16839966C2semidirect
5. #21454167C2semidirect
6. #21446154C2semidirect
7. #21446150C2semidirect
8. #21446126C2semidirect
9. #21445818C2semidirect
10. #21445797C2semidirect
11. C61×#1872C2semidirect
16. C2×#14519775C22semidirect
28. C2×#14519706C22semidirect
29. C2×#14519700C22semidirect
33. #16839669C22semidirect
34. #16839667C22semidirect
37. #16839631C22semidirect
40. #16839524C22semidirect
41. #16839516C22semidirect
44. C2×#103×C61C22semidirect
46. C61×#340C22semidirect
148. C22×Dic61SD16semidirect
285. C22×C61#461semidirect
312. Dic61#450semidirect
321. #1872D61semidirect
353. C2×#103C2×D61semidirect
366. #340C2×D61semidirect
368. C61#32905semidirect
614. C22#16839966semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
C2×#168399662
C2×#168399152
C2×#168399122
#214541672
#214461542
#214461502
#214461262
#214458182
#214457972
C61×#18722
#3290561