Finite Group Databasebeta

← #29646099

Subgroups — #29646099

Order 15616.

1730 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519750C211yesmaximal, characteristic
2#407×D61C212yesmaximal, complemented
3#21444177C218yesmaximal, complemented
4#21444146C211yesmaximal, characteristic, complemented
5#21444145C212yesmaximal, complemented
6C61×#1462C211yesmaximal, characteristic, complemented
7C23×D244C411yescharacteristic
8C23×C22⋊D61C411yescharacteristic
9C23×C22⋊D61C411yescharacteristic
10C23×C4×D61C2211yescharacteristic
11C2×M16×D61—22no—
12C2×#14519750—22no—
13C2×#14519750—22no—
14C2×#14519750—22no—
15C2×#14519750—22no—
16C2×#14519750C414yes—
17C2×#14519750C2214yes—
18C22×#11758196C411yescharacteristic
19C22×#11758198C2211yescharacteristic
20#16839726C2212yes—
21#16839725—22no—
22#16839724—22no—
23#16839724C2214yescomplemented
24C22⋊C8×D61—22no—
25C22⋊C8×D61C2214yescomplemented
26#16839486C2214yescomplemented
27#16839485C2218yescomplemented
28C22×C22⋊C4×C61C2211yescharacteristic
29C61×#407C2212yes—
30C61×#323C2214yescomplemented
31C24×D61C2×C411yescharacteristic
32C24×D61C2×C411yescharacteristic
33C22×D244—22no—
34C22×D244—21no—
35C22×D244—21no—
36C22×D244—44no—
37C22×D244C2×C414yes—
38C22×C22⋊D61—22no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61—21no—
41C22×C22⋊D61—21no—
42C22×C22⋊D61—22no—
43C22×C22⋊D61—21no—
44C22×C22⋊D61—44no—
45C22×C22⋊D61—44no—
46C22×C22⋊D61C2×C414yes—
47C22×C22⋊D61C2×C414yes—
48C23×Dic61C2×C411yescharacteristic
49C23×Dic61C2×C411yescharacteristic
50C22×C4×D61—44no—
51C22×C4×D61C2×C412yes—
52C22×C4×D61D412yes—
53C22×C4×D61D412yes—
54C22×C4×D61D412yes—
55C22×C4×D61D412yes—
56C22×C4×D61C2312yes—
57C2×#11758194—22no—
58C2×#11758189—24no—
59M16×D61—48no—
60C2×C8×D61—24no—
61#14519750—24no—
62#14519750—24no—
63#14519750—24no—
64#14519750—24no—
65#14519750—24no—
66#14519750—24no—
67#14519750—24no—
68#14519750—24no—
69C2×#11758196—21no—
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×#11758196C2×C414yes—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198C2×C412yes—
76C2×#11758198C2312yes—
77C22⋊#92418—22no—
78C22⋊#92418C2314yes—
79#14519689—24no—
80#14519689—24no—
81C25×C61C2×C411yescharacteristic
82C23×C4×C61C2×C411yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C2×M16×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61C2×C412yes—
88C2×C22⋊C4×C61C2312yes—
89C22⋊C8×C61—22no—
90C22⋊C8×C61C2314yes—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61—42no—
94C24×C61—42no—
95C24×C61—42no—
96C24×C61C22×C411yescharacteristic
97C24×C61C22×C412yes—
98C23×D61—24no—
99C23×D61—21no—
100C23×D61—21no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61C22⋊C412yes—
107C23×D61C22⋊C411yescharacteristic
108C23×D61C22⋊C411yescharacteristic
109C23×D61C22×C412yes—
110C23×D61C22×C412yes—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—24no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—48no—
126C2×C22⋊D61—48no—
127C2×C22⋊D61—48no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—48no—
131C2×C22⋊D61—44no—
132C22×Dic61—24no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—42no—
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C22⋊C412yes—
138C22×Dic61C22⋊C411yescharacteristic
139C22×Dic61C22×C412yes—
140C22×Dic61C22×C412yes—
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—44no—
144C2×D244—48no—
145C2×D244—44no—
146C2×D244—48no—
147C2×C4×D61—28no—
148C2×C4×D61—44no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×C4×D61—44no—
152C2×C4×D61—44no—
153C2×C4×D61C22⋊C412yes—
154C2×C4×D61C22⋊C412yes—
155C2×C4×D61C22⋊C412yes—
156C2×C4×D61C22⋊C412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161M16×C61—44no—
162C2×C8×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165#11758198—24no—
166#11758198—24no—
167#11758196—24no—
168#11758196—24no—
169#11758194—44no—
170C2×#92418—24no—
171#11758189—48no—
172C8×D61—48no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—88no—
187C23×C61C2×C22⋊C411yescharacteristic
188C23×C61C2×C22⋊C411yescharacteristic
189C23×C61C23×C411yescharacteristic
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—22no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—22no—
196C22×D61—24no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—42no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—41no—
203C22×D61—416no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—88no—
213C22×D61#10014yes—
214C22×D61C2×C22⋊C412yes—
215C22×D61C2×C22⋊C412yes—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C22×D412yes—
226C22⋊D61—48no—
227C22⋊D61—48no—
228C22⋊D61—48no—
229C22⋊D61—48no—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61#9914yes—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240D244—48no—
241D244—48no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C8×C61—44no—
251#92418—44no—
252#17396—611nomaximal, Sylow
253C22×C61—28no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—84no—
264C22×C61—84no—
265C22×C61—84no—
266C22×C61C2×#9912yes—
267C22×C61C2×#10012yes—
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61#56812yes—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—44no—
276C2×D61—48no—
277C2×D61—48no—
278C2×D61—44no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—84no—
283C2×D61—84no—
284C2×D61—84no—
285C2×D61C2×#10014yes—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—42no—
289Dic61—22no—
290Dic61—42no—
291Dic61C2×#9914yes—
292C22×#121—611no—
293C2×#407—612no—
294#1471—618no—
295#1462—612no—
296#1462—611no—
297#1462D6111yescharacteristic, complemented
298C2×C61—22no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—82no—
302C2×C61#287911yescharacteristic
303C2×C61C22×#9911yescharacteristic
304C2×C61C22×#10011yescharacteristic
305D61—24no—
306D61—42no—
307D61—82no—
308C23×D4—611no—
309C23×D4—611no—
310C23×D4—611no—
311C24×C4—611no—
312C22×M16—1222no—
313C2×#121—614no—
314C2×#121—614no—
315C2×#121—1222no—
316C2×#121—1222no—
317C2×#121—1222no—
318C2×#121—1222no—
319C22×C4⋊C4—611no—
320C22×C22⋊C4—611no—
321C22×C22⋊C4C2×D6111yescharacteristic
322#407—612no—
323#407—614no—
324#407—1222no—
325#407—1222no—
326#407C2×D6112yes—
327C2×C22⋊C8—614no—
328C2×C22⋊C8—1222no—
329#323—614no—
330#323—618no—
331#323C2×D6114yescomplemented
332C61#1739611yescharacteristic, complemented, Sylow
333C25—611no—
334C25—611no—
335C25C4×D6111yescharacteristic
336C22×D4—614no—
337C22×D4—614no—
338C22×D4—614no—
339C22×D4—1221no—
340C22×D4—1222no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4—1222no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4—2444no—
349C22×D4—2444no—
350C22×D4—2444no—
351C23×C4—611no—
352C23×C4—612no—
353C23×C4—612no—
354C23×C4—612no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4—612no—
358C23×C4—611no—
359C23×C4—2444no—
360C23×C4C4×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C2×M16—22no—
363C2×M16—1224no—
364C2×M16—1222no—
365C2×M16—2448no—
366C22×C8—1224no—
367#121—1224no—
368#121—1224no—
369#121—1224no—
370#121—1224no—
371#121—1224no—
372#121—1224no—
373#121—1224no—
374#121—1224no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1221no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4C4×D6112yes—
386C2×C22⋊C4C22×D6112yes—
387C22⋊C8—22no—
388C22⋊C8—614no—
389C22⋊C8—1224no—
390C22⋊C8—1224no—
391C22⋊C8—1222no—
392C22⋊C8C22×D6114yes—
393C24—21no—
394C24—21no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—611no—
399C24—612no—
400C24—612no—
401C24—611no—
402C24—612no—
403C24—1224no—
404C24—1221no—
405C24—1221no—
406C24—2442no—
407C24—2442no—
408C24—2442no—
409C24—2442no—
410C24—2442no—
411C24C2×C4×D6111yescharacteristic
412C24C2×C4×D6112yes—
413C2×D4—1224no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1224no—
419C2×D4—2444no—
420C2×D4—2448no—
421C2×D4—2444no—
422C2×D4—2448no—
423C2×D4—2444no—
424C2×D4—2448no—
425C2×D4—2444no—
426C2×D4—2448no—
427C2×D4—2444no—
428C2×D4—2448no—
429C2×D4—2444no—
430C2×D4—2448no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—1228no—
449C22×C4—1224no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2442no—
458C22×C4C2×C4×D6112yes—
459C22×C4C2×C4×D6111yescharacteristic
460C22×C4D4×D6112yes—
461C22×C4D4×D6112yes—
462C22×C4C23×D6111yescharacteristic
463M16—44no—
464M16—2448no—
465M16—2444no—
466C2×C8—24no—
467C2×C8—1224no—
468C2×C8—2448no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—88no—
489C23—612no—
490C23—614no—
491C23—612no—
492C23—1224no—
493C23—1224no—
494C23—1224no—
495C23—1222no—
496C23—1224no—
497C23—1224no—
498C23—1222no—
499C23—4888no—
500C23—2442no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23—2442no—
505C23—2441no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—24416no—
513C23—2442no—
514C23—2442no—
515C23C22⋊C4×D6111yescharacteristic
516C23C22⋊C4×D6111yescharacteristic
517C23C22×C4×D6111yescharacteristic
518D4—2448no—
519D4—2448no—
520D4—2448no—
521D4—2448no—
522D4—2448no—
523D4—2448no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—612no—
533C2×C4—614no—
534C2×C4—612no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6112yes—
551C2×C4C2×D4×D6112yes—
552C8—44no—
553C8—2444no—
554C22—28no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—614no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—4884no—
574C22—4884no—
575C22—4884no—
576C22—2444no—
577C22—2448no—
578C22—2448no—
579C22—2444no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22#99×D6112yes—
584C22#100×D6112yes—
585C22C2×C22⋊C4×D6111yescharacteristic
586C22#1684052812yes—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—614no—
591C4—1222no—
592C4—2442no—
593C2—22no—
594C2—22no—
595C2—22no—
596C2—82no—
597C2—1224no—
598C2—4882no—
599C2—2442no—
600C2#2145426211yescharacteristic
601C2C2×#99×D6111yescharacteristic
602C2C2×#100×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1462C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#14519750C2
cosocleC22×C4×C61C24
hypercenter#1462D61
hypocenterC61#17396
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #407×D61C2semidirect
3. #21444177C2semidirect
4. #21444146C2semidirect
5. #21444145C2semidirect
6. C61×#1462C2semidirect
23. #16839724C22semidirect
25. C22⋊C8×D61C22semidirect
26. #16839486C22semidirect
27. #16839485C22semidirect
30. C61×#323C22semidirect
297. #1462D61semidirect
331. #323C2×D61semidirect
332. C61#17396semidirect

Maximal subgroups

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

subgroupindex
C22×#145197502
#407×D612
#214441772
#214441462
#214441452
C61×#14622
#1739661