Finite Group Databasebeta

← #29707631

Subgroups — #29707631

Order 15616.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840336C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic
4#21454336C211yesmaximal, characteristic
5C22×#14519696C211yesmaximal, characteristic, complemented
6#21447036C218yesmaximal, complemented
7#21445529C211yesmaximal, characteristic, complemented
8C61×#2310C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840336—22no—
13#16840336—22no—
14#16840336—22no—
15#16840336C2214yes—
16C2×#14519775—21no—
17C2×#14519766—21no—
18C2×#14519760—22no—
19C2×#14519753—22no—
20C2×#14519736C2211yescharacteristic
21#16839805—22no—
22#16839802C2214yescomplemented
23#16839798—22no—
24#16839780—24no—
25C2×#14519705C2214yes—
26#16839770—21no—
27#16839766—21no—
28#16839761C2214yes—
29#16839756—22no—
30C2×#14519696—24no—
31C2×#14519696C2214yescomplemented
32#16839522C2214yescomplemented
33C23×Q8×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#391—22no—
37C61×#391C2214yescomplemented
38C61×#381—22no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222D412yes—
46C2×#11758222C2312yes—
47C22×Dic122D411yescharacteristic
48C22×D244D411yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C2×#11758194—21no—
54C2×#11758194—22no—
55C2×#11758194—22no—
56C2×#11758194D414yes—
57C2×#11758194C2312yes—
58C2×#11758194C2312yes—
59C22×#92418—21no—
60C22×#92418—21no—
61#14519775—24no—
62#14519775—24no—
63#14519766—24no—
64#14519766—24no—
65#14519760—28no—
66#14519760—28no—
67#14519753—28no—
68#14519753—28no—
69#14519736—21no—
70#14519736—21no—
71#14519736D412yes—
72#14519736C2312yes—
73C2×C4⋊Dic61D412yes—
74C2×#11758196—24no—
75C2×C22⋊Dic61—21no—
76C2×#11758198—24no—
77C2×C4×Dic61—21no—
78#14519710—24no—
79#14519708—24no—
80#14519705—28no—
81#14519700—24no—
82#14519697—24no—
83#14519696—28no—
84C22×Q8×C61—22no—
85C22×Q8×C61—21no—
86C22×Q8×C61—21no—
87C22×Q8×C61—21no—
88C22×Q8×C61—21no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61D412yes—
93C22×Q8×C61C2312yes—
94C23×C4×C61D412yes—
95C23×C4×C61C2311yescharacteristic
96#117×C61—22no—
97#117×C61—22no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2312yes—
108#95×C61—28no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—24no—
114C2×Q8×C61—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—42no—
122C2×Q8×C61—42no—
123C2×Q8×C61—42no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×Q8×C61—42no—
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61C2×D414yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152#11758222—24no—
153#11758222—24no—
154#11758222—22no—
155#11758222—22no—
156#11758222—22no—
157#11758222—22no—
158#11758222—42no—
159#11758222—42no—
160#11758222—42no—
161#11758222—42no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—21no—
165C2×D244C2×D412yes—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122C2×D412yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—42no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61—41no—
183C42×C61—41no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186#11758198—216no—
187C4⋊Dic61—22no—
188C4⋊Dic61C2×D414yes—
189#11758196—216no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192#11758194—24no—
193#11758194—22no—
194#11758194—22no—
195#11758194—22no—
196#11758194—42no—
197#11758194C2×D414yes—
198C2×#92418—24no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418—42no—
202C23×C61—42no—
203C23×C61#12011yescharacteristic
204C23×C61#12011yescharacteristic
205C23×C61#12112yes—
206C23×C61#12312yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—22no—
209C22×D61—44no—
210C22×D61—42no—
211Q8×C61—42no—
212Q8×C61—42no—
213Q8×C61—42no—
214Q8×C61—42no—
215Q8×C61—44no—
216Q8×C61—44no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—44no—
221C2×C4×C61—44no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12312yes—
234C2×C4×C61#12312yes—
235C2×C4×C61#12312yes—
236C2×C4×C61#12412yes—
237C2×C4×C61#12412yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—44no—
247C2×Dic61—24no—
248C2×Dic61—22no—
249C2×Dic61—44no—
250C2×Dic61—42no—
251D244—22no—
252D244—22no—
253D244—42no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—44no—
257Dic122—22no—
258Dic122—22no—
259Dic122—42no—
260#92418—24no—
261#92418—42no—
262#23494—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61#39412yes—
269C22×C61#39412yes—
270C22×C61C2×#12011yescharacteristic
271C22×C61C2×#12111yescharacteristic
272C22×C61C2×#12311yescharacteristic
273C2×D61—42no—
274C2×D61—44no—
275C2×D61—84no—
276C4×C61—44no—
277C4×C61—82no—
278C4×C61—82no—
279C4×C61#39412yes—
280C4×C61#39412yes—
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12311yescharacteristic
284C4×C61C2×#12411yescharacteristic
285Dic61—42no—
286Dic61—44no—
287C2×#536—611no—
288#2911—611no—
289C2×#421—611no—
290#2896—611no—
291C22×#101—611no—
292#2310D6111yescharacteristic, complemented
293#2038—618no—
294#1733—611no—
295C2×C61—21no—
296C2×C61—22no—
297C2×C61—21no—
298C2×C61—22no—
299C2×C61C2×#39411yescharacteristic
300C2×C61#302012yes—
301D61—82no—
302C22×#41—611no—
303C23×Q8C2×D6111yescharacteristic
304C22×M16—611no—
305C22×M16—611no—
306#536—614no—
307#536—1222no—
308#536—1222no—
309#536—1222no—
310C2×C4⋊Q8—1221no—
311C2×#124—1221no—
312C2×#123—1222no—
313C2×C22⋊Q8—1222no—
314C2×#117—611no—
315C2×#117C2×D6111yescharacteristic
316C2×#117C2×D6111yescharacteristic
317#428—614no—
318#428—1222no—
319#427—1222no—
320#421—1224no—
321C2×#104—614no—
322#419—1221no—
323#418—1221no—
324#417—614no—
325#416—1222no—
326C2×#101—614no—
327C2×#101—1224no—
328#391—22no—
329#391C2×D6114yescomplemented
330#381—22no—
331#339—614no—
332C61#2349411yescharacteristic, complemented, Sylow
333C2×#41—612no—
334C2×#41—612no—
335C2×#41—1221no—
336C2×#41—1221no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—21no—
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—22no—
349C22×Q8—611no—
350C22×Q8C22⋊D6112yes—
351C22×Q8C22×D6112yes—
352C22×D4—611no—
353C22×D4—1221no—
354C23×C4—1221no—
355C23×C4C22⋊D6112yes—
356C23×C4C22×D6111yescharacteristic
357C2×M16—614no—
358C2×M16—612no—
359C2×M16—612no—
360C2×M16—1222no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1222no—
364C2×M16—1222no—
365C22×C8—1221no—
366C22×C8—1221no—
367C4⋊Q8—1224no—
368C4⋊Q8—1224no—
369#124—1224no—
370#124—1224no—
371#123—1228no—
372#123—1228no—
373C22⋊Q8—1228no—
374C22⋊Q8—1228no—
375#117—22no—
376#117—22no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117—612no—
382#117—612no—
383#117—1221no—
384#117—1221no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4C22×D6112yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—1221no—
395C2×C42C22×D6112yes—
396#107—1224no—
397#106—1224no—
398#104—1228no—
399#103—1224no—
400#102—1224no—
401#101—1228no—
402#95—28no—
403C24—1221no—
404C24C2×C22⋊D6111yescharacteristic
405#41—1222no—
406#41—1222no—
407#41—1222no—
408#41—1222no—
409#41—1224no—
410#41—1224no—
411#41—2442no—
412#41—2442no—
413#41—2442no—
414#41—2442no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—612no—
434C2×Q8—1221no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×D4—612no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1221no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—2442no—
444C2×D4—2442no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4#1175822212yes—
458C22×C4#1175822212yes—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6112yes—
464C22×C4#1175822412yes—
465C22×C4C2×C22⋊D6114yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468M16—614no—
469M16—1222no—
470M16—1224no—
471M16—1222no—
472M16—1222no—
473M16—2442no—
474C2×C8—1222no—
475C2×C8—1224no—
476C2×C8—1222no—
477C2×C8—2442no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—614no—
482C4⋊C4—12216no—
483C4⋊C4—1222no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—42no—
487C22⋊C4—12216no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—1222no—
495C42—1222no—
496C23—42no—
497C23—1222no—
498C23—2444no—
499C23—2442no—
500C23#1451974511yescharacteristic
501C23#1451974511yescharacteristic
502C23#1451974912yes—
503C23#1451976012yes—
504C23C22×C22⋊D6111yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—44no—
510Q8—44no—
511Q8—1222no—
512Q8—1222no—
513Q8—2442no—
514D4—1222no—
515D4—1222no—
516D4—2442no—
517D4—2442no—
518D4—2444no—
519D4—2442no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—1224no—
530C2×C4—1222no—
531C2×C4—2444no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4#1451974511yescharacteristic
537C2×C4#1451974511yescharacteristic
538C2×C4#1451974912yes—
539C2×C4#1451975012yes—
540C2×C4#1451975012yes—
541C2×C4#1451975012yes—
542C2×C4#1451975012yes—
543C2×C4#1451975912yes—
544C2×C4#1451975912yes—
545C2×C4#1451976012yes—
546C2×C4#1451976312yes—
547C2×C4#1451976312yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C8—1224no—
555C8—2442no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—4884no—
562C22—2442no—
563C22—2444no—
564C22#1683967712yes—
565C22#1683967712yes—
566C22C2×#1451974511yescharacteristic
567C22C2×#1451974911yescharacteristic
568C22C2×#1451976011yescharacteristic
569C4—44no—
570C4—82no—
571C4—82no—
572C4—2442no—
573C4—2444no—
574C4#1683967712yes—
575C4#1683967712yes—
576C4C2×#1451975011yescharacteristic
577C4C2×#1451975011yescharacteristic
578C4C2×#1451975911yescharacteristic
579C4C2×#1451976311yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—4882no—
585C2C2×#1683967711yescharacteristic
586C2#2145518212yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
hypercenter#2310D61
hypocenterC61#23494
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
5. C22×#14519696C2semidirect
6. #21447036C2semidirect
7. #21445529C2semidirect
8. C61×#2310C2semidirect
22. #16839802C22semidirect
31. C2×#14519696C22semidirect
32. #16839522C22semidirect
37. C61×#391C22semidirect
292. #2310D61semidirect
329. #391C2×D61semidirect
332. C61#23494semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
#214544082
C2×#168397802
#214543362
C22×#145196962
#214470362
#214455292
C61×#23102
#2349461