Finite Group Databasebeta

← #29645885

Subgroups — #29645885

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×#14519737C211yesmaximal, characteristic
2#407×D61C212yesmaximal, complemented
3#21444154C214yesmaximal, complemented
4#21444148C212yesmaximal, complemented
5#21444146C211yesmaximal, characteristic, complemented
6#21444134C214yesmaximal, complemented
7C61×#1462C211yesmaximal, characteristic, complemented
8C23×C22⋊D61C411yescharacteristic
9C24×Dic61C411yescharacteristic
10C23×C4×D61C2211yescharacteristic
11C2×M16×D61—22no—
12C2×#14519737—22no—
13C2×#14519737—22no—
14C2×#14519737—22no—
15C2×#14519737—22no—
16C2×#14519737C412yes—
17C2×#14519737C412yes—
18C2×#14519737C2212yes—
19C2×#14519737C2212yes—
20C22×#11758196C2211yescharacteristic
21C22×#11758198C411yescharacteristic
22C22×C4×Dic61C411yescharacteristic
23#16839726C2212yes—
24#16839725—22no—
25#16839724—22no—
26#16839724C2214yescomplemented
27C22⋊C8×D61—22no—
28C22⋊C8×D61C2214yescomplemented
29#16839487C2218yescomplemented
30#16839486C2214yescomplemented
31C22×C22⋊C4×C61C2211yescharacteristic
32C61×#407C2212yes—
33C61×#323C2214yescomplemented
34C24×D61C2×C411yescharacteristic
35C22×C22⋊D61—22no—
36C22×C22⋊D61—22no—
37C22×C22⋊D61—28no—
38C22×C22⋊D61C2×C412yes—
39C22×C22⋊D61C2×C412yes—
40C23×Dic61—21no—
41C23×Dic61—21no—
42C23×Dic61—21no—
43C23×Dic61—21no—
44C23×Dic61—44no—
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C412yes—
49C23×Dic61C2×C412yes—
50C22×C4×D61—44no—
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#14519737—24no—
64#14519737—24no—
65#14519737—24no—
66#14519737—24no—
67#14519737—24no—
68#14519737—24no—
69#14519737—24no—
70#14519737—24no—
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196C2×C412yes—
74C2×#11758196C2312yes—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198C2×C412yes—
78C2×#11758198C2×C412yes—
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—
89C25×C61C2×C411yescharacteristic
90C23×C4×C61C2×C411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61C2×C412yes—
96C2×C22⋊C4×C61C2312yes—
97C22⋊C8×C61—22no—
98C22⋊C8×C61C2314yes—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61—24no—
102C24×C61—42no—
103C24×C61—42no—
104C24×C61C22×C411yescharacteristic
105C24×C61C22×C412yes—
106C23×D61—24no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61C22⋊C412yes—
110C23×D61C22⋊C412yes—
111C23×D61C22×C411yescharacteristic
112C23×D61C22×C411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—42no—
116C22×C4×C61—42no—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—216no—
123C2×C22⋊D61—216no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61M1618yescomplemented
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—24no—
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—44no—
141C22×Dic61—44no—
142C22×Dic61—44no—
143C22×Dic61—44no—
144C22×Dic61—44no—
145C22×Dic61—44no—
146C22×Dic61—42no—
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C412yes—
154C22×Dic61C22×C412yes—
155C2×C4×D61—28no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
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—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175#11758198—24no—
176#11758198—24no—
177#11758196—24no—
178#11758196—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×C61—24no—
191C23×C61—24no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—416no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C2×C22⋊C411yescharacteristic
202C23×C61C2×C22⋊C411yescharacteristic
203C23×C61C2×M1614yescomplemented
204C23×C61C23×C411yescharacteristic
205C22×D61—22no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—42no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—416no—
217C22×D61C2×C22⋊C412yes—
218C22×D61C2×C22⋊C412yes—
219C22×D61C2×M1614yes—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C22×D412yes—
230C22⋊D61—216no—
231C22⋊D61—216no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—44no—
243C2×Dic61—44no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—42no—
249C2×Dic61#9914yes—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×M1614yescomplemented
255C4×D61—24no—
256C4×D61—24no—
257C4×D61—24no—
258C4×D61—24no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61—42no—
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×C61—24no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—48no—
277C22×C61—48no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61C2×#9912yes—
281C22×C61#41312yes—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C22×M1612yes—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—44no—
290C2×D61—48no—
291C2×D61—48no—
292C2×D61—44no—
293C2×D61#40714yes—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—42no—
297Dic61—24no—
298Dic61—21no—
299Dic61—21no—
300Dic61—42no—
301Dic61C2×#9912yes—
302Dic61C2×#9912yes—
303C22×C4×D4—611no—
304C2×#407—612no—
305#1465—614no—
306#1463—612no—
307#1462—611no—
308#1462D6111yescharacteristic, complemented
309#1459—614no—
310C2×C61—22no—
311C2×C61—22no—
312C2×C61—22no—
313C2×C61—44no—
314C2×C61C2×#40711yescharacteristic
315C2×C61C22×#9911yescharacteristic
316C2×C61C2×#41311yescharacteristic
317D61—24no—
318D61—42no—
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⋊C4—611no—
332C22×C22⋊C4—611no—
333C22×C22⋊C4C2×D6111yescharacteristic
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—618no—
343#323—614no—
344#323C2×D6114yescomplemented
345C61#1736211yescharacteristic, complemented, Sylow
346C25—611no—
347C25C4×D6111yescharacteristic
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—611no—
364C23×C4—612no—
365C23×C4—612no—
366C23×C4—1221no—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4—1221no—
370C23×C4—2444no—
371C23×C4—2444no—
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—612no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—612no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4C4×D6112yes—
398C2×C22⋊C4C22×D6112yes—
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—21no—
412C24—21no—
413C24—24no—
414C24—42no—
415C24—42no—
416C24—611no—
417C24—612no—
418C24—612no—
419C24—611no—
420C24—1224no—
421C24—2442no—
422C24—2442no—
423C24C2×C4×D6111yescharacteristic
424C24C2×C4×D6112yes—
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—42no—
436C22×C4—42no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—1228no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
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—2444no—
477C22×C4—2442no—
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—1224no—
490C4⋊C4—1224no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—1224no—
494C22⋊C4—1224no—
495C42—1222no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—416no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—612no—
514C23—614no—
515C23—612no—
516C23—1224no—
517C23—1224no—
518C23—1224no—
519C23—1224no—
520C23—1222no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2441no—
525C23—2441no—
526C23—2442no—
527C23—24416no—
528C23C22⋊C4×D6111yescharacteristic
529C23C22⋊C4×D6111yescharacteristic
530C23M16×D6114yescomplemented
531C23C22×C4×D6111yescharacteristic
532D4—12216no—
533D4—12216no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—614no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—614no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1224no—
557C2×C4—1224no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2444no—
567C2×C4—2444no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C2×D4×D6112yes—
575C8—44no—
576C8—2444no—
577C22—28no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—24no—
585C22—44no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—614no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22—2444no—
598C22—2448no—
599C22—2448no—
600C22—2444no—
601C22#99×D6112yes—
602C22#1683974312yes—
603C22C2×C22⋊C4×D6111yescharacteristic
604C22C2×M16×D6112yes—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—612no—
609C4—612no—
610C4—1224no—
611C4—1221no—
612C4—1221no—
613C4—2442no—
614C2—22no—
615C2—22no—
616C2—22no—
617C2—44no—
618C2—1224no—
619C2—2442no—
620C2#407×D6111yescharacteristic
621C2C2×#99×D6111yescharacteristic
622C2C2×#1683974311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1462C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenter#1462D61
hypocenterC61#17362
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. #21444154C2semidirect
4. #21444148C2semidirect
5. #21444146C2semidirect
6. #21444134C2semidirect
7. C61×#1462C2semidirect
26. #16839724C22semidirect
28. C22⋊C8×D61C22semidirect
29. #16839487C22semidirect
30. #16839486C22semidirect
33. C61×#323C22semidirect
129. C2×C22⋊D61M16semidirect
203. C23×C61C2×M16semidirect
254. C2×Dic61C2×M16semidirect
308. #1462D61semidirect
344. #323C2×D61semidirect
345. C61#17362semidirect
530. C23M16×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197372
#407×D612
#214441542
#214441482
#214441462
#214441342
C61×#14622
#1736261