Finite Group Databasebeta

← #29645889

Subgroups — #29645889

Order 15616.

1738 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic
2#407×D61C212yesmaximal, complemented
3#21444156C214yesmaximal, complemented
4#21444149C211yesmaximal, characteristic, complemented
5#21444145C212yesmaximal, complemented
6#21444136C214yesmaximal, complemented
7C61×#1463C211yesmaximal, characteristic, complemented
8C23×D244C411yescharacteristic
9C23×C4×D61C411yescharacteristic
10C23×C4×D61C2211yescharacteristic
11C2×M16×D61—22no—
12C2×#14519739—22no—
13C2×#14519739—22no—
14C2×#14519739—22no—
15C2×#14519739—22no—
16C2×#14519739C412yes—
17C2×#14519739C412yes—
18C2×#14519739C2212yes—
19C2×#14519739C2212yes—
20C22×#11758198C411yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×C4×Dic61C411yescharacteristic
23#16839726C2212yes—
24#16839725—22no—
25#16839724—22no—
26#16839724C2214yescomplemented
27C22⋊C8×D61—22no—
28C22⋊C8×D61C2214yescomplemented
29#16839488C2214yescomplemented
30#16839485C2218yescomplemented
31C22×C4⋊C4×C61C2211yescharacteristic
32C61×#407C2212yes—
33C61×#324C2214yescomplemented
34C24×D61C2×C411yescharacteristic
35C24×D61C2×C411yescharacteristic
36C22×D244—22no—
37C22×D244—22no—
38C22×D244—28no—
39C22×D244C2×C412yes—
40C22×D244C2×C412yes—
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C22×C4×D61—21no—
44C22×C4×D61—21no—
45C22×C4×D61—21no—
46C22×C4×D61—21no—
47C22×C4×D61—44no—
48C22×C4×D61—44no—
49C22×C4×D61C2×C412yes—
50C22×C4×D61C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D412yes—
54C22×C4×D61D412yes—
55C22×C4×D61D412yes—
56C22×C4×D61D412yes—
57C22×C4×D61C2311yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758194—22no—
60C2×#11758189—24no—
61M16×D61—48no—
62C2×C8×D61—24no—
63#14519739—24no—
64#14519739—24no—
65#14519739—24no—
66#14519739—24no—
67#14519739—24no—
68#14519739—24no—
69#14519739—24no—
70#14519739—24no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198C2×C412yes—
76C2×#11758198C2×C412yes—
77C2×#11758198C2×C412yes—
78C2×#11758198C2312yes—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2×C412yes—
84C2×C4×Dic61C2×C412yes—
85C22⋊#92418—22no—
86C22⋊#92418C2314yes—
87#14519689—24no—
88#14519689—24no—
89C23×C4×C61C2×C411yescharacteristic
90C23×C4×C61C2×C411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61C2×C412yes—
96C2×C4⋊C4×C61C2312yes—
97C22⋊C8×C61—22no—
98C22⋊C8×C61C2314yes—
99C24×C61C22×C411yescharacteristic
100C23×D61—21no—
101C23×D61—21no—
102C23×D61—24no—
103C23×D61—24no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61C22⋊C412yes—
109C23×D61C22⋊C412yes—
110C23×D61C22×C411yescharacteristic
111C23×D61C22×C412yes—
112C23×D61C22×C411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—24no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—24no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—42no—
130C22×Dic61C22⋊C411yescharacteristic
131C22×Dic61C22⋊C411yescharacteristic
132C22×Dic61C22⋊C411yescharacteristic
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C412yes—
137C2×D244—216no—
138C2×D244—216no—
139C2×D244—24no—
140C2×D244—24no—
141C2×D244—24no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244M1618yescomplemented
145C2×C4×D61—28no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—44no—
153C2×C4×D61—44no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—44no—
157C2×C4×D61—44no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61C22⋊C412yes—
164C2×C4×D61C22⋊C412yes—
165C2×C4×D61C22⋊C412yes—
166C2×C4×D61C22⋊C412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D412yes—
171M16×C61—44no—
172C2×C8×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175#11758198—24no—
176#11758198—24no—
177#11758198—24no—
178#11758198—24no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183#11758194—44no—
184C2×#92418—24no—
185#11758189—48no—
186C8×D61—48no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61C2×C22⋊C411yescharacteristic
191C23×C61C2×C22⋊C411yescharacteristic
192C23×C61C23×C411yescharacteristic
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—42no—
209C22×D61—416no—
210C22×D61—416no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61C2×C22⋊C412yes—
217C22×D61C2×C22⋊C412yes—
218C22×D61C2×M1614yes—
219C22×D61C2×M1614yescomplemented
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×M1614yescomplemented
234C2×C4×C61C22×D412yes—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61#9914yes—
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249D244—216no—
250D244—216no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—42no—
259C4×D61—44no—
260C4×D61—44no—
261C4×D61—44no—
262C4×D61—44no—
263C8×C61—44no—
264#92418—44no—
265#17362—611nomaximal, Sylow
266C22×C61—28no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61C2×#9912yes—
274C22×C61#41312yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C22×M1612yes—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—24no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—44no—
284C2×D61—48no—
285C2×D61—48no—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—48no—
289C2×D61—48no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—44no—
293C2×D61#40714yes—
294C4×C61—24no—
295C4×C61—22no—
296C4×C61—22no—
297C4×C61—42no—
298Dic61—21no—
299Dic61—21no—
300Dic61—42no—
301Dic61C2×#9912yes—
302Dic61C2×#9912yes—
303C22×C4×D4—611no—
304C2×#407—612no—
305#1465—614no—
306#1463—611no—
307#1463D6111yescharacteristic, complemented
308#1462—612no—
309#1459—614no—
310C2×C61—22no—
311C2×C61—22no—
312C2×C61—22no—
313C2×C61C2×#40711yescharacteristic
314C2×C61C22×#9911yescharacteristic
315C2×C61C2×#41311yescharacteristic
316D61—24no—
317D61—42no—
318D61—44no—
319C23×D4—611no—
320C24×C4—611no—
321C24×C4—611no—
322C22×M16—1222no—
323C2×C4×D4—612no—
324C2×C4×D4—612no—
325C2×C4×D4—612no—
326C2×C4×D4—612no—
327C2×C4×D4—1222no—
328C2×C4×D4—1222no—
329C2×C4×D4—1222no—
330C2×C4×D4—1222no—
331C22×C4⋊C4C2×D6111yescharacteristic
332C22×C22⋊C4—611no—
333C22×C22⋊C4—611no—
334C22×C42—611no—
335#407—612no—
336#407—614no—
337#407—1222no—
338#407—1222no—
339#407C2×D6112yes—
340C2×C22⋊C8—614no—
341C2×C22⋊C8—1222no—
342#324—614no—
343#324C2×D6114yescomplemented
344#323—618no—
345C61#1736211yescharacteristic, complemented, Sylow
346C25—611no—
347C25—611no—
348C22×D4—612no—
349C22×D4—612no—
350C22×D4—1228no—
351C22×D4—1222no—
352C22×D4—1222no—
353C23×C4—611no—
354C23×C4—612no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—612no—
359C23×C4—612no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—612no—
364C23×C4—612no—
365C23×C4—1221no—
366C23×C4—1221no—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4—2444no—
370C23×C4—2444no—
371C23×C4C4×D6111yescharacteristic
372C23×C4C4×D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C2×M16—22no—
375C2×M16—1224no—
376C2×M16—1222no—
377C2×M16—2448no—
378C22×C8—1224no—
379C4×D4—1224no—
380C4×D4—1224no—
381C4×D4—1224no—
382C4×D4—1224no—
383C4×D4—1224no—
384C4×D4—1224no—
385C4×D4—1224no—
386C4×D4—1224no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4C4×D6112yes—
390C2×C4⋊C4C22×D6112yes—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—612no—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—612no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C42—612no—
400C2×C42—612no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C22⋊C8—22no—
406C22⋊C8—614no—
407C22⋊C8—1224no—
408C22⋊C8—1224no—
409C22⋊C8—1222no—
410C22⋊C8C22×D6114yes—
411C24—611no—
412C24—612no—
413C24—612no—
414C24—611no—
415C24—612no—
416C24—1224no—
417C24—1224no—
418C24—1221no—
419C24—1221no—
420C24—2442no—
421C24—2442no—
422C24—2442no—
423C24—2442no—
424C24C2×C4×D6111yescharacteristic
425C2×D4—618no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—12216no—
431C2×D4—12216no—
432C2×D4—1224no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1228no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—2444no—
464C22×C4—2444no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4—2444no—
468C22×C4—2444no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4—2444no—
474C22×C4—2444no—
475C22×C4—2444no—
476C22×C4—2442no—
477C22×C4C2×C4×D6112yes—
478C22×C4C2×C4×D6112yes—
479C22×C4C2×C4×D6111yescharacteristic
480C22×C4D4×D6112yes—
481C22×C4D4×D6112yes—
482C22×C4C23×D6111yescharacteristic
483M16—44no—
484M16—2448no—
485M16—2444no—
486C2×C8—24no—
487C2×C8—1224no—
488C2×C8—2448no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C42—1222no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—612no—
503C23—614no—
504C23—612no—
505C23—614no—
506C23—1224no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—1222no—
511C23—1224no—
512C23—1224no—
513C23—1222no—
514C23—1222no—
515C23—2441no—
516C23—2441no—
517C23—2442no—
518C23—2441no—
519C23—2441no—
520C23—2442no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—24416no—
525C23—24416no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23C22⋊C4×D6111yescharacteristic
530C23C22⋊C4×D6111yescharacteristic
531C23C22×C4×D6111yescharacteristic
532D4—12216no—
533D4—12216no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—614no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2444no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4C22⋊C4×D6112yes—
573C2×C4M16×D6114yescomplemented
574C2×C4C2×D4×D6112yes—
575C8—44no—
576C8—2444no—
577C22—28no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—614no—
585C22—1222no—
586C22—1222no—
587C22—1222no—
588C22—1224no—
589C22—1222no—
590C22—1222no—
591C22—2444no—
592C22—2448no—
593C22—2448no—
594C22—2444no—
595C22—2448no—
596C22—2448no—
597C22—2444no—
598C22—2444no—
599C22—2442no—
600C22—2442no—
601C22#99×D6112yes—
602C22#1683974512yes—
603C22C2×C22⋊C4×D6111yescharacteristic
604C22C2×M16×D6112yes—
605C4—24no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—612no—
610C4—612no—
611C4—1221no—
612C4—1221no—
613C4—2442no—
614C2—22no—
615C2—22no—
616C2—22no—
617C2—1224no—
618C2—2442no—
619C2—2444no—
620C2#407×D6111yescharacteristic
621C2C2×#99×D6111yescharacteristic
622C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1463C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D244C4
cosocleC22×C4×C61C24
hypercenter#1463D61
hypocenterC61#17362
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #407×D61C2semidirect
3. #21444156C2semidirect
4. #21444149C2semidirect
5. #21444145C2semidirect
6. #21444136C2semidirect
7. C61×#1463C2semidirect
26. #16839724C22semidirect
28. C22⋊C8×D61C22semidirect
29. #16839488C22semidirect
30. #16839485C22semidirect
33. C61×#324C22semidirect
144. C2×D244M16semidirect
219. C22×D61C2×M16semidirect
233. C2×C4×C61C2×M16semidirect
307. #1463D61semidirect
343. #324C2×D61semidirect
345. C61#17362semidirect
573. C2×C4M16×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
#407×D612
#214441562
#214441492
#214441452
#214441362
C61×#14632
#1736261