Finite Group Databasebeta

← #29699881

Subgroups — #29699881

Order 15616.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839864C211yesmaximal, characteristic
2#21448794C211yesmaximal, characteristic
3#21448778C211yesmaximal, characteristic, complemented
4#21446698C211yesmaximal, characteristic
5#21446671C211yesmaximal, characteristic
6#21446459C211yesmaximal, characteristic, complemented
7#21446452C211yesmaximal, characteristic, complemented
8#21446445C211yesmaximal, characteristic
9#21446438C211yesmaximal, characteristic
10#21445831C211yesmaximal, characteristic, complemented
11#21445816C211yesmaximal, characteristic
12#21445719C211yesmaximal, characteristic, complemented
13#107×Dic61C211yesmaximal, characteristic, complemented
14#21445691C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16839864C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839667C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839654—21no—
35#16839654—21no—
36#16839643—21no—
37#16839643—21no—
38#16839627—21no—
39#16839624—21no—
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839609—21no—
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190C2×C414yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C4×Dic122—22no—
69C4×Dic122C2×C414yes—
70C4×D244—22no—
71C4×D244C2×C414yes—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711C2×C414yes—
90C4⋊#92418—22no—
91#14519702C2×C414yes—
92#14519700—41no—
93#14519700—41no—
94#14519699C2×C414yes—
95#14519697—41no—
96#14519697—41no—
97#14519687—22no—
98C8×Dic61—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#107×C61—41no—
120#106×C61—22no—
121#106×C61—41no—
122#97×C61C2×C414yes—
123#95×C61—21no—
124#95×C61—21no—
125C23×D61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—24no—
147C2×C4×D61—81no—
148C2×C4×D61—81no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122C22×C412yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61C22×C412yes—
158C2×C8×C61C22×C412yes—
159C2×C8×C61#4112yes—
160C2×C8×C61#4112yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—81no—
171C42×C61C22×C412yes—
172C42×C61#4112yes—
173#11758198—24no—
174#11758198—42no—
175#11758198—42no—
176#11758198—81no—
177#11758198—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—22no—
181C4⋊Dic61—41no—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C22×C412yes—
184#11758196—24no—
185C4×Dic61—24no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—81no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200#11758190—28no—
201C23×C61C22×D411yescharacteristic
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61#13712yes—
254D244—22no—
255D244—22no—
256D244—41no—
257D244—41no—
258C4×D61—84no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—22no—
264C8×C61#11714yes—
265#92418—41no—
266#92418—41no—
267#22819—611nomaximal, Sylow
268C22×C61C2×C4×D411yescharacteristic
269C22×C61#54711yescharacteristic
270C22×C61#55011yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57611yescharacteristic
274C22×C61#57711yescharacteristic
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61C2×#11711yescharacteristic
289C4×C61#52011yescharacteristic
290C4×C61#55511yescharacteristic
291C4×C61#55611yescharacteristic
292Dic61—22no—
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Dic61—42no—
297Dic61—42no—
298Dic61—42no—
299C2×#440—611no—
300#2357—611no—
301#2355—611no—
302#1973—611no—
303#1968—611no—
304#1928—611no—
305#1927—611no—
306#1926—611no—
307#1925—611no—
308#1847D6111yescharacteristic, complemented
309#1805—611no—
310#1802—611no—
311#1776—611no—
312C4×#107—611no—
313#1771—611no—
314C2×C61#235411yescharacteristic
315C2×C61C4×#13711yescharacteristic
316C2×C61#294611yescharacteristic
317C2×C61#334611yescharacteristic
318C2×C61#335011yescharacteristic
319C2×C61#340011yescharacteristic
320C2×C61#340311yescharacteristic
321D61—82no—
322C22×SD16—611no—
323C2×C4×Q8—611no—
324C2×C4×D4—611no—
325#440—618no—
326C2×#107—611no—
327C2×#107C2×D6111yescharacteristic
328C2×#107C2×D6111yescharacteristic
329C2×#106C2×D6111yescharacteristic
330C2×#106C2×D6111yescharacteristic
331C2×C4⋊C8—611no—
332C2×#103—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#102—611no—
338C2×#97—611no—
339C2×#97C2×D6111yescharacteristic
340C2×C4×C8—611no—
341#391—611no—
342#391—611no—
343#390—611no—
344#390—611no—
345#388—1221no—
346#388—1221no—
347#387—1221no—
348#387—1221no—
349#384C2×D6111yescharacteristic
350#382—1221no—
351#382C2×D6111yescharacteristic
352#381—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#336—611no—
365#336—611no—
366C61#2281911yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×D4—611no—
369C23×C4—1221no—
370C2×SD16—614no—
371C2×SD16—1222no—
372C2×SD16—1222no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4×Q8—614no—
378C4×Q8—1222no—
379C4×D4—614no—
380C4×D4—1222no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405#107—22no—
406#107—41no—
407#107—614no—
408#106—22no—
409#106—41no—
410C4⋊C8—1222no—
411#103—614no—
412#103—2441no—
413#103—2441no—
414#102—614no—
415#102—2441no—
416#102—2441no—
417#97—1222no—
418#97C4×D6114yes—
419C4×C8—1222no—
420#95—21no—
421#95—21no—
422#95—611no—
423#95—611no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432C24—1221no—
433C2×Q8—612no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—612no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1224no—
457C22×C4—4881no—
458C22×C4—4881no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462SD16—1228no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×C4×D6112yes—
472C2×C8C2×C4×D6112yes—
473C2×C8#1175822412yes—
474C2×C8#1175822612yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1224no—
491C4⋊C4—2441no—
492C22⋊C4—1224no—
493C22⋊C4—4881no—
494C22⋊C4—4881no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—1224no—
502C42—4881no—
503C42—4881no—
504C42—4881no—
505C42—4881no—
506C42C2×C4×D6112yes—
507C42#1175822612yes—
508C23—1222no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23C2×D4×D6111yescharacteristic
516Q8—1222no—
517Q8—1222no—
518Q8—2441no—
519Q8—2441no—
520D4—1222no—
521D4—1222no—
522D4—2441no—
523D4—2441no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—612no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—4884no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4#1451973912yes—
563C2×C4#1451973912yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4C2×#1175822611yescharacteristic
567C2×C4C2×#1175822611yescharacteristic
568C2×C4#1451982211yescharacteristic
569C2×C4#1451982511yescharacteristic
570C8—22no—
571C8—2441no—
572C8—2441no—
573C8#1451973514yes—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×#1451973911yescharacteristic
582C22#1684041511yescharacteristic
583C22#1684043211yescharacteristic
584C22#137×D6111yescharacteristic
585C22#1684055811yescharacteristic
586C22#1684056611yescharacteristic
587C22#1684057111yescharacteristic
588C4—22no—
589C4—22no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—1222no—
595C4—2442no—
596C4—2442no—
597C4—2442no—
598C4—2441no—
599C4—2441no—
600C4—2441no—
601C4C2×#1451973511yescharacteristic
602C4#1684024311yescharacteristic
603C4#1684047811yescharacteristic
604C4#1684048611yescharacteristic
605C2—4882no—
606C2#2144877011yescharacteristic
607C2#2145463711yescharacteristic
608C2#2145464911yescharacteristic
609C2#2145830311yescharacteristic
610C2#2145836311yescharacteristic
611C2#2145886911yescharacteristic
612C2#2145890411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#22819
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
3. #21448778C2semidirect
6. #21446459C2semidirect
7. #21446452C2semidirect
10. #21445831C2semidirect
12. #21445719C2semidirect
13. #107×Dic61C2semidirect
14. #21445691C2semidirect
15. C61×#1847C2semidirect
19. #16839864C4semidirect
308. #1847D61semidirect
366. C61#22819semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214487942
#214487782
#214466982
#214466712
#214464592
#214464522
#214464452
#214464382
#214458312
#214458162
#214457192
#107×Dic612
#214456912
C61×#18472
#2281961