Finite Group Databasebeta

← #29654148

Subgroups — #29654148

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456621C211yesmaximal, characteristic, complemented
2#21456616C211yesmaximal, characteristic
3#21451299C211yesmaximal, characteristic, complemented
4#21444850C214yesmaximal, complemented
5#21444748C214yesmaximal, complemented
6#21444339C211yesmaximal, characteristic, complemented
7#21444335C211yesmaximal, characteristic, complemented
8#21444149C211yesmaximal, characteristic, complemented
9C2×C61×#539C211yesmaximal, characteristic, complemented
10C2×#14519751C2211yescharacteristic
11#16840095C2212yescomplemented
12#16840080—21no—
13#16840078C2212yes—
14#16840075—21no—
15#16840013—22no—
16#16840010—22no—
17#16839912C2212yescomplemented
18#16839901—21no—
19#16839896—21no—
20#16839895C2212yes—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839701C2212yescomplemented
24#16839679C2212yescomplemented
25#16839652—22no—
26#16839641—21no—
27#16839641—21no—
28C4×C22⋊Dic61C2211yescharacteristic
29#16839499C2214yescomplemented
30#16839496C2214yes—
31#16839488C2214yescomplemented
32C61×#539—22no—
33C61×#539—22no—
34C61×#539—22no—
35C61×#539C2214yes—
36C2×#126×C61—21no—
37C2×#125×C61—21no—
38C2×#123×C61—21no—
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×C4×D4×C61—21no—
44C22×C4⋊C4×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C2×#11758201—22no—
48C2×#11758199—22no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51#14519751—21no—
52#14519751—21no—
53#14519751—21no—
54#14519751—21no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—21no—
61#14519710—22no—
62#14519708—22no—
63#14519701—22no—
64#14519698—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418C2312yes—
70C22⋊#92418C2312yes—
71#14519684—22no—
72#14519684—22no—
73#14519684C2312yes—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61—21no—
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C23×C4×C61D411yescharacteristic
79C23×C4×C61D411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81#126×C61—24no—
82#126×C61—24no—
83#125×C61—24no—
84#125×C61—24no—
85#123×C61—24no—
86#123×C61—24no—
87#123×C61—24no—
88#123×C61—24no—
89C22⋊Q8×C61—24no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61D412yes—
94C22⋊Q8×C61C2312yes—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61D412yes—
98#121×C61C2312yes—
99C4×D4×C61—24no—
100C4×D4×C61—24no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61D412yes—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—81no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—44no—
129C2×D4×C61—81no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—21no—
139C22×C4×C61—42no—
140C22×C4×C61—81no—
141C22×C4×C61—81no—
142C22×C4×C61—82no—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—44no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—81no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61—24no—
180C22⋊Dic61—42no—
181#11758201—82no—
182#11758199—82no—
183#11758198—44no—
184#11758198—41no—
185#11758198—41no—
186#11758198—82no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4×Dic61—81no—
190#11758194—81no—
191#11758194—81no—
192C2×#92418—22no—
193C2×#92418—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61#12011yescharacteristic
202C23×C61#12011yescharacteristic
203C23×C61C22×D411yescharacteristic
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—42no—
208Q8×C61—42no—
209Q8×C61—81no—
210D4×C61—42no—
211D4×C61—44no—
212D4×C61—44no—
213D4×C61—81no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—81no—
235C2×C4×C61—84no—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—82no—
241C22⋊D61—82no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247D244—42no—
248D244—42no—
249#92418—42no—
250#92418—42no—
251#18320—611nomaximal, Sylow
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—44no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—84no—
264C22×C61#45412yes—
265C22×C61C2×#12011yescharacteristic
266C22×C61#57512yes—
267C2×D61—84no—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—42no—
280Dic61—82no—
281Dic61—81no—
282C2×#539D6111yescharacteristic, complemented
283#3193—611no—
284#3192—611no—
285#2591—611no—
286#1615—614no—
287#1597—614no—
288#1508—611no—
289#1507—611no—
290#1463—611no—
291C2×C61—22no—
292C2×C61—22no—
293C2×C61—41no—
294C2×C61—82no—
295C2×C61#300511yescharacteristic
296C2×C61C2×#45411yescharacteristic
297C2×C61#302611yescharacteristic
298D61—161no—
299#539—22no—
300#539—22no—
301#539—22no—
302#539C2×D6114yes—
303C2×#126—21no—
304C2×#125—21no—
305C2×#123—21no—
306C2×#123—21no—
307C2×C22⋊Q8—21no—
308C2×C22⋊Q8C2×D6111yescharacteristic
309C2×#121—611no—
310C2×#121C2×D6111yescharacteristic
311C2×C4×D4—21no—
312C22×C4⋊C4C2×D6111yescharacteristic
313#484—612no—
314#481—612no—
315#481—1221no—
316#480—1221no—
317#469—1222no—
318#468—1222no—
319#450—612no—
320#448—1221no—
321#447—612no—
322#447—1221no—
323#407—611no—
324#407—611no—
325#400—612no—
326#394—612no—
327#388—1222no—
328#386—1221no—
329#386—1221no—
330C4×C22⋊C4—611no—
331#328—614no—
332#327—614no—
333#324—614no—
334C61#1832011yescharacteristic, complemented, Sylow
335C22×Q8C22⋊D6111yescharacteristic
336C22×D4—21no—
337C22×D4—611no—
338C22×D4—1221no—
339C22×D4C22⋊D6111yescharacteristic
340C23×C4—21no—
341C23×C4C22⋊D6111yescharacteristic
342C23×C4C22⋊D6111yescharacteristic
343C23×C4C22×D6111yescharacteristic
344C2×SD16—1222no—
345C2×D8—1222no—
346C2×M16—1221no—
347C2×M16—1221no—
348#126—24no—
349#126—24no—
350#125—24no—
351#125—24no—
352#123—24no—
353#123—24no—
354#123—24no—
355#123—24no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8C22⋊D6112yes—
361C22⋊Q8C22×D6112yes—
362#121—21no—
363#121—21no—
364#121—1221no—
365#121—1221no—
366#121—1221no—
367#121—1221no—
368#121C22⋊D6112yes—
369#121C22×D6112yes—
370C4×D4—24no—
371C4×D4—24no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4C22⋊D6112yes—
382C2×C4⋊C4C22⋊D6111yescharacteristic
383C2×C4⋊C4C22⋊D6111yescharacteristic
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1222no—
394C2×C42—21no—
395C2×C42—1221no—
396#107—1222no—
397#106—1222no—
398#103—1222no—
399#102—1222no—
400#102—1222no—
401#102—1222no—
402C22⋊C8—612no—
403C22⋊C8—612no—
404C22⋊C8—1221no—
405C22⋊C8—1221no—
406#95—612no—
407#95—1222no—
408#95—1222no—
409C24—21no—
410C24—1221no—
411C24C2×C22⋊D6111yescharacteristic
412C2×Q8—21no—
413C2×Q8—81no—
414C2×Q8C2×C22⋊D6112yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—44no—
419C2×D4—81no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—4881no—
423C2×D4—2444no—
424C2×D4—2441no—
425C2×D4—2441no—
426C2×D4C2×C22⋊D6112yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—82no—
439C22×C4—1222no—
440C22×C4—1221no—
441C22×C4C2×C22⋊D6112yes—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449SD16—4882no—
450D8—4882no—
451M16—4881no—
452M16—4881no—
453C2×C8—1222no—
454C2×C8—1222no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4C2×C22⋊D6112yes—
469C4⋊C4C2×C22⋊D6112yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—4882no—
478C22⋊C4—2444no—
479C22⋊C4—2442no—
480C22⋊C4—2441no—
481C22⋊C4—2441no—
482C42—24no—
483C42—4881no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—2442no—
492C23—2442no—
493C23—2441no—
494C23—2442no—
495C23#1451974611yescharacteristic
496C23#1451974611yescharacteristic
497C23C22×C22⋊D6111yescharacteristic
498Q8—42no—
499Q8—81no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—81no—
504D4—4882no—
505D4—4882no—
506D4—2442no—
507D4—2442no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—84no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4#1451974611yescharacteristic
536C2×C4#1451974611yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C8—2442no—
540C8—2442no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—84no—
553C22—4884no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22#1683993512yes—
558C22C2×#1451974611yescharacteristic
559C22#1684056012yes—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—4882no—
570C4—4881no—
571C2—22no—
572C2—22no—
573C2—41no—
574C2—82no—
575C2—9761no—
576C2#2145506111yescharacteristic
577C2C2×#1683993511yescharacteristic
578C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#539C2
socleC22×C61C2×#120
prime radical#21456616C2
hypercenterC2×#539D61
hypocenterC61#18320
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. #21456621C2semidirect
3. #21451299C2semidirect
4. #21444850C2semidirect
5. #21444748C2semidirect
6. #21444339C2semidirect
7. #21444335C2semidirect
8. #21444149C2semidirect
9. C2×C61×#539C2semidirect
11. #16840095C22semidirect
17. #16839912C22semidirect
23. #16839701C22semidirect
24. #16839679C22semidirect
29. #16839499C22semidirect
31. #16839488C22semidirect
282. C2×#539D61semidirect
334. C61#18320semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214566162
#214512992
#214448502
#214447482
#214443392
#214443352
#214441492
C2×C61×#5392
#1832061