Finite Group Databasebeta

← #29707633

Subgroups — #29707633

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×#16840338C211yesmaximal, characteristic
2C2×#16839778C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C22×#14519695C211yesmaximal, characteristic, complemented
5#21448403C211yesmaximal, characteristic, complemented
6#21447038C218yesmaximal, complemented
7#21445528C211yesmaximal, characteristic, complemented
8C61×#2911C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×Dic122C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840338—22no—
13#16840338—22no—
14#16840338—22no—
15#16840338C2214yes—
16C2×#14519777—21no—
17C2×#14519765—21no—
18C2×#14519761—22no—
19C2×#14519757—22no—
20C2×#14519736C2211yescharacteristic
21#16839778—24no—
22C2×#14519703C2214yes—
23#16839770—21no—
24#16839766—21no—
25#16839761C2214yes—
26#16839756—22no—
27C2×#14519695—24no—
28C2×#14519695C2214yescomplemented
29#16839668—22no—
30#16839667C2214yescomplemented
31#16839623—22no—
32#16839521C2214yescomplemented
33C22×M16×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#428—22no—
37C61×#428C2214yescomplemented
38C61×#427—22no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222D412yes—
46C2×#11758222C2312yes—
47C22×Dic122—22no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122—22no—
54C22×Dic122—22no—
55C22×Dic122D412yes—
56C22×Dic122D411yescharacteristic
57C22×Dic122C2312yes—
58C22×D244D411yescharacteristic
59C22×C22⋊D61—21no—
60C23×Dic61D412yes—
61C22×C4×D61—21no—
62C2×#11758194—21no—
63C2×#11758194—22no—
64C2×#11758194—22no—
65C2×#11758194C2312yes—
66C22×#92418—21no—
67#14519777—24no—
68#14519777—24no—
69#14519765—24no—
70#14519765—24no—
71#14519761—28no—
72#14519761—28no—
73#14519757—28no—
74#14519757—28no—
75#14519736—22no—
76#14519736—22no—
77C2×C4⋊Dic61—24no—
78C2×C4⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—24no—
81C2×C4×Dic61C2312yes—
82#14519703—28no—
83#14519700—24no—
84#14519697—24no—
85#14519695—28no—
86#14519684—28no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—21no—
89C2×M16×C61—22no—
90C2×M16×C61D414yes—
91C2×M16×C61C2312yes—
92C22×C8×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98#117×C61—21no—
99#117×C61D412yes—
100#117×C61C2312yes—
101C2×C4⋊C4×C61D412yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107#107×C61—24no—
108#106×C61—24no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61#4112yes—
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—24no—
131C22×Dic61—24no—
132C22×Dic61—22no—
133C22×Dic61C2×D414yes—
134#11758222—24no—
135#11758222—24no—
136#11758222—22no—
137#11758222—22no—
138#11758222—22no—
139#11758222—22no—
140#11758222—42no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—21no—
147C2×D244C2×D412yes—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—21no—
157C2×Dic122—24no—
158C2×Dic122—24no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—42no—
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122C2×D412yes—
174M16×C61—24no—
175M16×C61—22no—
176M16×C61C2×D414yes—
177C2×C8×C61—24no—
178C2×C8×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C2×D414yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—41no—
190C42×C61—41no—
191C22⋊Dic61—42no—
192#11758198—216no—
193C4⋊Dic61—216no—
194C4⋊Dic61—42no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—42no—
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—
211C2×C4×C61—24no—
212C2×C4×C61—44no—
213C2×C4×C61—44no—
214C2×C4×C61—42no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#12112yes—
218C2×C4×C61#12112yes—
219C2×C4×C61#12112yes—
220C2×C4×C61#12112yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12312yes—
223C2×C4×C61#12312yes—
224C2×C4×C61#12312yes—
225C2×C4×C61#12412yes—
226C2×C4×C61#12412yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—42no—
234C22⋊D61—42no—
235C22⋊D61—44no—
236C2×Dic61—22no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—42no—
241C2×Dic61—44no—
242C2×Dic61—44no—
243C2×Dic61—44no—
244C2×Dic61—42no—
245D244—22no—
246D244—22no—
247D244—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—44no—
251Dic122—22no—
252Dic122—22no—
253Dic122—42no—
254Dic122—42no—
255Dic122—42no—
256Dic122—42no—
257Dic122—44no—
258Dic122—44no—
259Dic122—42no—
260C8×C61—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#39412yes—
279C4×C61#39412yes—
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12311yescharacteristic
283C4×C61C2×#12411yescharacteristic
284Dic61—42no—
285Dic61—44no—
286Dic61—82no—
287C2×#536—611no—
288#2911D6111yescharacteristic, complemented
289C2×#421—611no—
290#2896—611no—
291C22×#101—611no—
292#2310—611no—
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×Q8—611no—
304C22×M16—611no—
305C22×M16C2×D6111yescharacteristic
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—22no—
318#428C2×D6114yescomplemented
319#427—22no—
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—614no—
329#391—1222no—
330#381—1222no—
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—612no—
342C22×Q8—612no—
343C22×Q8—611no—
344C22×Q8—1222no—
345C22×Q8—1222no—
346C22×Q8—1221no—
347C22×Q8—1221no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8—1221no—
351C22×Q8—1221no—
352C22×D4—611no—
353C22×D4—1221no—
354C23×C4—612no—
355C23×C4—1221no—
356C23×C4C22×D6111yescharacteristic
357C2×M16—22no—
358C2×M16—21no—
359C2×M16—612no—
360C2×M16—1221no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16D24414yes—
364C2×M16C22×D6112yes—
365C22×C8—21no—
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—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117—1222no—
382#117—1222no—
383#117C22⋊D6112yes—
384#117C22×D6112yes—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22⋊D6112yes—
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—21no—
395C2×C42—612no—
396#107—24no—
397#106—24no—
398#104—1228no—
399#103—1224no—
400#102—1224no—
401#101—1228no—
402#95—1228no—
403C24—1221no—
404C24D4×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—612no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—1221no—
425C2×Q8—1224no—
426C2×Q8—1224no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×Q8—2442no—
432C2×Q8—2442no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×Q8—2442no—
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—614no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4C2×D24412yes—
459C22×C4#1175822212yes—
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4#1175822612yes—
464C22×C4#1175822612yes—
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468M16—22no—
469M16—24no—
470M16—1222no—
471M16—1222no—
472M16—2442no—
473M16C2×D24414yes—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—1222no—
477C2×C8—2442no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—12216no—
482C4⋊C4—2442no—
483C4⋊C4C2×C22⋊D6114yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—12216no—
489C22⋊C4—2442no—
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#1451975212yes—
503C23#1451976212yes—
504C23C2×D4×D6111yescharacteristic
505Q8—1222no—
506Q8—1222no—
507Q8—2442no—
508Q8—2442no—
509Q8—2442no—
510Q8—2442no—
511Q8—2444no—
512Q8—2444no—
513Q8—2442no—
514D4—1222no—
515D4—1222no—
516D4—2442no—
517D4—2442no—
518D4—2444no—
519D4—2442no—
520C2×C4—24no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—1224no—
525C2×C4—1222no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—2444no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4#1451974511yescharacteristic
537C2×C4#1451974511yescharacteristic
538C2×C4#1451975112yes—
539C2×C4#1451975112yes—
540C2×C4#1451975212yes—
541C2×C4#1451975212yes—
542C2×C4#1451975212yes—
543C2×C4#1451976212yes—
544C2×C4#1451976212yes—
545C2×C4#1451976212yes—
546C2×C4#1451976612yes—
547C2×C4#1451976612yes—
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C22×C22⋊D6111yescharacteristic
550C2×C4C22×D24411yescharacteristic
551C2×C4C2×#1175822211yescharacteristic
552C2×C4C2×#1175822611yescharacteristic
553C2×C4C2×#1175822611yescharacteristic
554C8—24no—
555C8—2442no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—4884no—
562C22—2442no—
563C22—2444no—
564C22#1683967912yes—
565C22#1683967912yes—
566C22C2×#1451974511yescharacteristic
567C22C2×#1451975211yescharacteristic
568C22C2×#1451976211yescharacteristic
569C4—44no—
570C4—82no—
571C4—4882no—
572C4—2444no—
573C4—2442no—
574C4#1683967912yes—
575C4#1683967912yes—
576C4C2×#1451975111yescharacteristic
577C4C2×#1451975211yescharacteristic
578C4C2×#1451976211yescharacteristic
579C4C2×#1451976611yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—4882no—
585C2C2×#1683967911yescharacteristic
586C2#2145518112yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2911C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
hypercenter#2911D61
hypocenterC61#23494
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
4. C22×#14519695C2semidirect
5. #21448403C2semidirect
6. #21447038C2semidirect
7. #21445528C2semidirect
8. C61×#2911C2semidirect
28. C2×#14519695C22semidirect
30. #16839667C22semidirect
32. #16839521C22semidirect
37. C61×#428C22semidirect
288. #2911D61semidirect
318. #428C2×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×#168403382
C2×#168397782
#214543362
C22×#145196952
#214484032
#214470382
#214455282
C61×#29112
#2349461