Finite Group Databasebeta

← #29682722

Subgroups — #29682722

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839764C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×M16×Dic61C211yesmaximal, characteristic, complemented
4#21448307C211yesmaximal, characteristic
5C2×#16839661C211yesmaximal, characteristic, complemented
6#21446517C212yesmaximal, complemented
7#21446510C212yesmaximal, complemented
8#21446503C212yesmaximal, complemented
9#21446496C212yesmaximal, complemented
10#21445540C211yesmaximal, characteristic, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×C4⋊Dic61—21no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839764—24no—
18#16839761C418yescomplemented
19C2×#14519702C2212yes—
20C2×#14519700C2212yes—
21C2×#14519699C2212yes—
22C2×#14519697C2212yes—
23M16×Dic61—24no—
24C2×#14519687C2212yes—
25C2×C8×Dic61C2212yes—
26#16839667C2211yescharacteristic
27#16839667C2211yescharacteristic
28#16839663C2211yescharacteristic
29#16839663C2211yescharacteristic
30#16839661—21no—
31#16839661—21no—
32#16839661C2212yes—
33#16839661C2212yes—
34#16839639—21no—
35#16839639—21no—
36C4×#11758198—21no—
37C4×#11758198—21no—
38C2×#14519684—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C23×Dic61D411yescharacteristic
57C23×Dic61D411yescharacteristic
58C22×C4×D61—21no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194C2×C414yes—
62C22×#92418C2312yes—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—44no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80#14519702—24no—
81#14519700C2×C418yescomplemented
82#14519699—24no—
83#14519697C2×C418yescomplemented
84#14519687—24no—
85C8×Dic61—24no—
86#14519684—22no—
87#14519684—22no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C23×C4×C61—21no—
93C23×C4×C61C2311yescharacteristic
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61D414yes—
97C22×C8×C61C2312yes—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61C2×C414yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110C22×C4×C61—24no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—41no—
134C22×Dic61—41no—
135C22×Dic61—42no—
136C22×Dic61—42no—
137C22×Dic61—42no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142#11758222—28no—
143#11758222—28no—
144#11758222—44no—
145#11758222—44no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C412yes—
161M16×C61—28no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61C2×D414yes—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C414yes—
172C22⋊C4×C61—42no—
173C42×C61—42no—
174#11758198—42no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—44no—
181C4⋊Dic61—42no—
182C4⋊Dic61—44no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189#11758194—28no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yes—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61C4×D412yes—
198C23×C61#12112yes—
199C23×C61#12412yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C2×C4×C61—24no—
209C2×C4×C61—48no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61#11714yes—
219C2×C4×C61C4×D414yes—
220C2×C4×C61C4×D412yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12314yes—
223C2×C4×C61#12412yes—
224C2×C4×C61#12614yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C22⋊D61—48no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—44no—
244C2×Dic61—44no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—84no—
251C2×Dic61AGL(1,ℤ/8)12yes—
252C2×Dic61#13712yes—
253D244—22no—
254D244—22no—
255D244—41no—
256D244—41no—
257C4×D61—48no—
258Dic122—22no—
259Dic122—22no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—24no—
263#92418—24no—
264#21093—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61#38814yes—
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×#12411yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61C2×#13711yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57511yescharacteristic
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C4×C61—44no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#38814yes—
290C4×C61C2×#11711yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×#12311yescharacteristic
293C4×C61C2×#12611yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—41no—
298Dic61—42no—
299Dic61—82no—
300C2×#417—611no—
301C2×#417—611no—
302C2×C4×M16—611no—
303#2297—611no—
304C2×#389—611no—
305#1936—612no—
306#1935—612no—
307#1934—612no—
308#1933—612no—
309#1736—611no—
310#1736D6111yescharacteristic, complemented
311C2×C61—24no—
312C2×C61C2×#38811yescharacteristic
313C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C61C4×#13711yescharacteristic
315C2×C61#305511yescharacteristic
316C2×C61#305611yescharacteristic
317C2×C61#314111yescharacteristic
318C2×C61#314211yescharacteristic
319D61—82no—
320C22×#41—611no—
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323C2×#117C2×D6111yescharacteristic
324C22×C4⋊C4—611no—
325C22×C4⋊C4—1221no—
326C22×C4⋊C4C2×D6111yescharacteristic
327C22×C42—611no—
328#417—618no—
329#417—1224no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#102—612no—
333C2×#102—612no—
334C4×M16—1224no—
335C2×#97—612no—
336C2×C4×C8—612no—
337#391—611no—
338#391—611no—
339#390—611no—
340#390—611no—
341#389—612no—
342#389—612no—
343#389—1221no—
344#389—1221no—
345#386—1221no—
346#386—1221no—
347C4×C22⋊C4—1221no—
348C4×C22⋊C4—1221no—
349C2×#95—1221no—
350#340—612no—
351#340—612no—
352#340C2×D6112yes—
353#340C2×D6112yes—
354C61#2109311yescharacteristic, complemented, Sylow
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1222no—
361C22×Q8—611no—
362C22×D4—611no—
363C22×D4—1221no—
364C23×C4—21no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—614no—
373C2×M16—1222no—
374C2×M16—1222no—
375C2×M16D24414yes—
376C22×C8—612no—
377C22×C8C22×D6112yes—
378#117—22no—
379#117—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—2444no—
389C2×C4⋊C4C4×D6114yes—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—21no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—1222no—
404C2×C42—1222no—
405#103—618no—
406#103—1224no—
407#102—618no—
408#102—1224no—
409#97—1224no—
410C4×C8—1224no—
411#95—1222no—
412#95—1222no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417C24—1221no—
418C24D4×D6111yescharacteristic
419#41—1228no—
420#41—1228no—
421#41—2444no—
422#41—2444no—
423C2×Q8—612no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×D4—612no—
430C2×D4—1224no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—2442no—
437C2×D4—2442no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1224no—
454C22×C4—1222no—
455C22×C4—2441no—
456C22×C4—2441no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4C2×C4×D6112yes—
463C22×C4C2×D24412yes—
464C22×C4#1175822212yes—
465C22×C4D4×D6111yescharacteristic
466C22×C4#1175822412yes—
467C22×C4#1175822412yes—
468C22×C4#1175822612yes—
469C22×C4C23×D6111yescharacteristic
470M16—28no—
471M16—1228no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—614no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8C2×D24414yes—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—2444no—
487C4⋊C4—2444no—
488C4⋊C4—2442no—
489C4⋊C4C2×C4×D6114yes—
490C22⋊C4—42no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C42—42no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—1222no—
501C42—1222no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1222no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23#1451973712yes—
514C23#1451975212yes—
515C23#1451976312yes—
516C23C2×D4×D6111yescharacteristic
517Q8—1222no—
518Q8—1222no—
519Q8—2441no—
520Q8—2441no—
521D4—1222no—
522D4—1222no—
523D4—2448no—
524D4—2441no—
525D4—2441no—
526C2×C4—24no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—4884no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2448no—
558C2×C4#1451973514yes—
559C2×C4#1451973712yes—
560C2×C4C4×D24414yes—
561C2×C4#1451975212yes—
562C2×C4#1451976114yes—
563C2×C4#1451976312yes—
564C2×C4#1451977214yes—
565C2×C4C22×C4×D6111yescharacteristic
566C2×C4C22×D24411yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C2×C4C2×#1175822611yescharacteristic
571C8—24no—
572C8—1224no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22#1683965414yes—
587C22C2×#1451973711yescharacteristic
588C22C2×#1451975211yescharacteristic
589C22C2×#1451976311yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22#137×D6111yescharacteristic
592C22#1684055811yescharacteristic
593C22#1684056111yescharacteristic
594C4—44no—
595C4—81no—
596C4—81no—
597C4—1222no—
598C4—1222no—
599C4—4882no—
600C4—2442no—
601C4—2441no—
602C4—2441no—
603C4#1683965414yes—
604C4C2×#1451973511yescharacteristic
605C4C2×C4×D24411yescharacteristic
606C4C2×#1451976111yescharacteristic
607C4C2×#1451977211yescharacteristic
608C2—24no—
609C2—4882no—
610C2C2×#1683965411yescharacteristic
611C2#2145462311yescharacteristic
612C2#2145463411yescharacteristic
613C2#2145543911yescharacteristic
614C2#2145545011yescharacteristic
615C2#2145621011yescharacteristic
616C2#2145621711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#21093
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×M16×Dic61C2semidirect
5. C2×#16839661C2semidirect
6. #21446517C2semidirect
7. #21446510C2semidirect
8. #21446503C2semidirect
9. #21446496C2semidirect
10. #21445540C2semidirect
11. C61×#1736C2semidirect
18. #16839761C4semidirect
81. #14519700C2×C4semidirect
83. #14519697C2×C4semidirect
310. #1736D61semidirect
354. C61#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#168397642
C2×#168397612
C2×M16×Dic612
#214483072
C2×#168396612
#214465172
#214465102
#214465032
#214464962
#214455402
C61×#17362
#2109361