Finite Group Databasebeta

← #29686830

Subgroups — #29686830

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455425C211yesmaximal, characteristic
2C2×#16839877C211yesmaximal, characteristic, complemented
3#21454513C211yesmaximal, characteristic, complemented
4C2×#16839717C211yesmaximal, characteristic, complemented
5#21446711C211yesmaximal, characteristic, complemented
6#21446569C211yesmaximal, characteristic, complemented
7#21446533C211yesmaximal, characteristic, complemented
8#21446529C211yesmaximal, characteristic, complemented
9#21446268C211yesmaximal, characteristic, complemented
10#21446265C211yesmaximal, characteristic, complemented
11#21445859C211yesmaximal, characteristic, complemented
12#21445699C211yesmaximal, characteristic, complemented
13#21445658C211yesmaximal, characteristic, complemented
14#21445654C211yesmaximal, characteristic, complemented
15C61×#2929C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18C22×#11758194C2211yescharacteristic
19#16840273C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840018C412yescomplemented
22#16840013C412yescomplemented
23#16840010C412yescomplemented
24#16840006C412yescomplemented
25#16839877—24no—
26#16839833—22no—
27C8×C22⋊D61—22no—
28C2×#14519712C2211yescharacteristic
29C2×#14519712C2211yescharacteristic
30#16839798C2211yescharacteristic
31#16839788C2211yescharacteristic
32C2×#14519704C2211yescharacteristic
33C2×#14519704C2211yescharacteristic
34C2×#14519703C2211yescharacteristic
35C2×#14519703C2211yescharacteristic
36#16839758C2211yescharacteristic
37#16839756C2211yescharacteristic
38#16839728C2211yescharacteristic
39#16839726C2211yescharacteristic
40#16839717—24no—
41#16839717C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45C2×C8×Dic61C2211yescharacteristic
46#16839526C2211yescharacteristic
47#16839526C2211yescharacteristic
48#16839523C2211yescharacteristic
49#16839523C2211yescharacteristic
50#16839521C2211yescharacteristic
51#16839521C2211yescharacteristic
52C2×#131×C61C2211yescharacteristic
53C23×C8×C61C2211yescharacteristic
54C61×#525C2211yescharacteristic
55C61×#435—22no—
56C8×D4×C61—22no—
57C61×#424C2211yescharacteristic
58C61×#408C2211yescharacteristic
59C61×#407C2211yescharacteristic
60C61×#405C2211yescharacteristic
61C2×#11758222C2311yescharacteristic
62#14519797—22no—
63#14519797—22no—
64C2×#11758202C2×C411yescharacteristic
65C2×#11758200C2×C411yescharacteristic
66C2×#11758201C2×C411yescharacteristic
67C2×#11758199C2×C411yescharacteristic
68#14519783—22no—
69#14519783—22no—
70C2×#11758194—22no—
71C2×#11758194—22no—
72C2×#11758194C2×C412yes—
73C2×#11758194C2×C412yes—
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758189—21no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C2×C8×D61—21no—
80#14519760—21no—
81#14519756C2×C412yes—
82#14519751C2×C412yes—
83#14519746—21no—
84C4×C22⋊D61—22no—
85#14519736—22no—
86#14519736—22no—
87#14519736C2311yescharacteristic
88C2×C4⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519712—22no—
93#14519712—22no—
94#14519710C2×C412yes—
95#14519708C2×C412yes—
96#14519706—22no—
97#14519704—22no—
98#14519704—22no—
99#14519703—22no—
100#14519703—22no—
101#14519701C2×C412yes—
102#14519700C2×C412yes—
103#14519698C2×C412yes—
104#14519697C2×C412yes—
105C22⋊#92418—22no—
106#14519689—22no—
107#14519687—21no—
108#14519687—22no—
109#14519687—22no—
110C8×Dic61—22no—
111C8×Dic61—21no—
112C8×Dic61D414yescomplemented
113C2×#41×C61C2311yescharacteristic
114C23×C4×C61C2311yescharacteristic
115#131×C61—22no—
116#131×C61—22no—
117C2×M16×C61—21no—
118C2×M16×C61C2311yescharacteristic
119C2×M16×C61C2311yescharacteristic
120C22×C8×C61—22no—
121C22×C8×C61—22no—
122C22×C8×C61—21no—
123C22×C8×C61D412yes—
124C22×C8×C61D412yes—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127#123×C61—21no—
128C22⋊Q8×C61C2×C412yes—
129#121×C61C2×C412yes—
130#120×C61—21no—
131C4×D4×C61—22no—
132#117×C61C2311yescharacteristic
133C4⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61—22no—
136#97×C61—21no—
137C4×C8×C61—21no—
138C24×C61#4111yescharacteristic
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4111yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160#11758222—21no—
161#11758222—21no—
162#11758222—21no—
163#11758222—21no—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×Dic122C22×C411yescharacteristic
167M16×C61—21no—
168M16×C61—21no—
169M16×C61—21no—
170M16×C61—21no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—24no—
175C2×C8×C61—24no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—42no—
179C2×C8×C61C2×D412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61#4112yes—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C42×C61—21no—
190C22⋊Dic61—24no—
191C22⋊Dic61—22no—
192C22⋊Dic61—21no—
193#11758202—22no—
194#11758201—22no—
195#11758200—22no—
196#11758199—22no—
197#11758198—22no—
198C4⋊Dic61—24no—
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61C2×D412yes—
205C4×Dic61#4112yes—
206#11758194—24no—
207#11758194—24no—
208#11758194—21no—
209#11758194—21no—
210#11758194—21no—
211#11758194—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C22×C412yes—
217C2×#92418C22×C411yescharacteristic
218C2×#92418C22×C411yescharacteristic
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221#11758189—42no—
222C8×D61—42no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—42no—
227C23×C61C4×D412yes—
228C23×C61C2×#4111yescharacteristic
229C22×D61—21no—
230Q8×C61—21no—
231Q8×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—44no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C4×D412yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—44no—
255C22⋊D61—41no—
256C22⋊D61—41no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263D244—21no—
264D244—21no—
265C4×D61—41no—
266C4×D61—41no—
267Dic122—21no—
268Dic122—21no—
269C8×C61—22no—
270C8×C61—41no—
271C8×C61—41no—
272C8×C61#12112yes—
273C8×C61#12112yes—
274#92418—22no—
275#92418—41no—
276#92418—41no—
277#92418C4×D412yes—
278#92418C4×D412yes—
279#21548—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—41no—
288C22×C61C2×C4×D411yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61#52911yescharacteristic
291C2×D61—42no—
292C2×D61—41no—
293C4×C61—22no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61C2×C4×D411yescharacteristic
297C4×C61C4×#4111yescharacteristic
298C4×C61C2×#12111yescharacteristic
299C4×C61#53211yescharacteristic
300Dic61—42no—
301Dic61—42no—
302Dic61—42no—
303Dic61—41no—
304#3054—611no—
305C2×#443—611no—
306#2929—611no—
307#2929D6111yescharacteristic, complemented
308C2×#405—611no—
309#1975—611no—
310#1946—611no—
311#1938—611no—
312#1938—611no—
313#1897—611no—
314#1897—611no—
315#1811—611no—
316#1772—611no—
317#1764—611no—
318#1764—611no—
319C2×C61—21no—
320C2×C61—21no—
321C2×C61—42no—
322C2×C61—81no—
323C2×C61C4×#12111yescharacteristic
324C2×C61#295512yes—
325D61—81no—
326C2×#135—611no—
327C2×#131—611no—
328C2×#131C2×D6111yescharacteristic
329C22×M16—611no—
330C23×C8C2×D6111yescharacteristic
331#525—611no—
332#525C2×D6111yescharacteristic
333C2×#117—611no—
334#470—612no—
335#469—612no—
336#468—612no—
337#468—612no—
338#443—1224no—
339#435—22no—
340#435—1222no—
341C8×D4—22no—
342C8×D4—1222no—
343C2×#108—611no—
344C2×#108—611no—
345#427—611no—
346#424—611no—
347#424C2×D6111yescharacteristic
348C2×#104—611no—
349C2×#104—611no—
350C2×#104—611no—
351C2×#104—611no—
352#416—611no—
353#416—611no—
354#408—611no—
355#408C2×D6111yescharacteristic
356#407—611no—
357#407C2×D6111yescharacteristic
358#405—611no—
359#405—1224no—
360#405C2×D6111yescharacteristic
361C2×#97—611no—
362C2×#97—611no—
363C2×C4×C8—611no—
364C2×C4×C8—611no—
365#341—611no—
366#341—611no—
367#339—611no—
368#339—611no—
369#339—611no—
370#339—611no—
371C61#2154811yescharacteristic, complemented, Sylow
372C2×#41—611no—
373C2×#41C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375#135—1222no—
376#135—1222no—
377C2×Q16—611no—
378C2×SD16—611no—
379C2×SD16—611no—
380C2×D8—611no—
381#131—22no—
382#131—22no—
383#131—1222no—
384#131—1222no—
385C2×M16—21no—
386C2×M16—612no—
387C2×M16—612no—
388C2×M16—611no—
389C2×M16—611no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1221no—
393C2×M16C22×D6111yescharacteristic
394C2×M16C22×D6111yescharacteristic
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—21no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—1221no—
401C22×C8C22⋊D6112yes—
402C22×C8C22⋊D6112yes—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405#123—21no—
406#123—1221no—
407C22⋊Q8—612no—
408C22⋊Q8C4×D6112yes—
409#121—612no—
410#121C4×D6112yes—
411#120—21no—
412#120—1221no—
413C4×D4—22no—
414C4×D4—1222no—
415#117—611no—
416#117—1222no—
417#117—1222no—
418#117C22×D6111yescharacteristic
419C2×C4⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423#108—1222no—
424#108—1222no—
425#107—612no—
426#106—612no—
427C4⋊C8—22no—
428C4⋊C8—1222no—
429#104—1222no—
430#104—1222no—
431#104—1222no—
432#104—1222no—
433#103—612no—
434#103—612no—
435#102—612no—
436#102—612no—
437C22⋊C8—22no—
438C22⋊C8—22no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441#97—21no—
442#97—1222no—
443#97—1222no—
444#97—1221no—
445C4×C8—21no—
446C4×C8—614no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24#1175822211yescharacteristic
450#41—21no—
451#41—21no—
452#41—21no—
453#41—21no—
454#41—1221no—
455#41—1221no—
456#41—1221no—
457#41—1221no—
458C2×Q8—611no—
459C2×Q8C2×C4×D6111yescharacteristic
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—611no—
463C2×D4—1221no—
464C2×D4—1222no—
465C2×D4C2×C4×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4C2×C4×D6112yes—
475C22×C4#1175822211yescharacteristic
476C22×C4C2×C22⋊D6112yes—
477C22×C4C23×D6111yescharacteristic
478Q16—1222no—
479SD16—1222no—
480SD16—1222no—
481D8—1222no—
482M16—21no—
483M16—21no—
484M16—21no—
485M16—21no—
486M16—42no—
487M16—1224no—
488M16—1224no—
489M16—1221no—
490M16—1221no—
491M16—1221no—
492M16—1221no—
493M16—2442no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—42no—
501C2×C8—611no—
502C2×C8—611no—
503C2×C8—612no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—2442no—
511C2×C8D4×D6112yes—
512C2×C8#1175822412yes—
513C2×C8C2×C22⋊D6112yes—
514C2×C8C2×C22⋊D6111yescharacteristic
515C2×C8C2×C22⋊D6111yescharacteristic
516C4⋊C4—22no—
517C4⋊C4—612no—
518C4⋊C4—1224no—
519C4⋊C4—1222no—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—1224no—
525C22⋊C4—1221no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—21no—
529C42—612no—
530C42—612no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—1221no—
539C23C4×C22⋊D6112yes—
540C23C2×#1175822211yescharacteristic
541Q8—21no—
542Q8—21no—
543Q8—1221no—
544Q8—1221no—
545D4—21no—
546D4—21no—
547D4—44no—
548D4—41no—
549D4—41no—
550D4—1221no—
551D4—1221no—
552D4—2444no—
553D4—2441no—
554D4—2441no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C4×C22⋊D6112yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C8—22no—
581C8—41no—
582C8—41no—
583C8—612no—
584C8—612no—
585C8—1222no—
586C8—2441no—
587C8—2441no—
588C8#1451974912yes—
589C8#1451974912yes—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—2442no—
599C22—2441no—
600C22C2×C4×C22⋊D6111yescharacteristic
601C22#1684027111yescharacteristic
602C22#1684029211yescharacteristic
603C4—22no—
604C4—42no—
605C4—41no—
606C4—2442no—
607C4—2442no—
608C4—2442no—
609C4—2441no—
610C4C4×D4×D6111yescharacteristic
611C4C4×#1175822411yescharacteristic
612C4C2×#1451974911yescharacteristic
613C4#1684030811yescharacteristic
614C2—21no—
615C2—21no—
616C2—42no—
617C2—81no—
618C2—4881no—
619C2C4×#1451974911yescharacteristic
620C2#2145472012yes—
subgroupG/H
centerC2×C8C2×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2929C2
socleC22×C61C2×C4×D4
prime radical#21455425C2
cosocleC22×C4×C61C24
hypercenter#2929D61
hypocenterC61#21548
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. C2×#16839877C2semidirect
3. #21454513C2semidirect
4. C2×#16839717C2semidirect
5. #21446711C2semidirect
6. #21446569C2semidirect
7. #21446533C2semidirect
8. #21446529C2semidirect
9. #21446268C2semidirect
10. #21446265C2semidirect
11. #21445859C2semidirect
12. #21445699C2semidirect
13. #21445658C2semidirect
14. #21445654C2semidirect
15. C61×#2929C2semidirect
21. #16840018C4semidirect
22. #16840013C4semidirect
23. #16840010C4semidirect
24. #16840006C4semidirect
112. C8×Dic61D4semidirect
307. #2929D61semidirect
371. C61#21548semidirect

Maximal subgroups

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

subgroupindex
#214554252
C2×#168398772
#214545132
C2×#168397172
#214467112
#214465692
#214465332
#214465292
#214462682
#214462652
#214458592
#214456992
#214456582
#214456542
C61×#29292
#2154861