Finite Group Databasebeta

← #29610702

Subgroups — #29610702

Order 15616.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839675C211yesmaximal, characteristic
2C2×#16839667C211yesmaximal, characteristic, complemented
3C2×#16839651C211yesmaximal, characteristic
4#21444089C211yesmaximal, characteristic, complemented
5#21444078C212yesmaximal, complemented
6#21444077C212yesmaximal, complemented
7#21444067C212yesmaximal
8#21444061C211yesmaximal, characteristic
9#21444061C212yesmaximal
10#21444044C211yesmaximal, characteristic, complemented
11C61×#1444C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×#11758196C2211yescharacteristic
14C22×#11758198C2211yescharacteristic
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×#11758198C2212yes—
18C22×#11758198C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20#16839675C418yescomplemented
21#16839667—24no—
22#16839651—24no—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2212yes—
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C24×D61C2311yescharacteristic
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—42no—
42C22×Dic122—42no—
43C22×Dic122C2×C414yes—
44C23×Dic61C2312yes—
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2312yes—
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2311yescharacteristic
49C2×#11758196—22no—
50C2×#11758196—42no—
51C2×#11758198—22no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×#11758198—42no—
56C2×#11758198—42no—
57C2×#11758198—42no—
58C2×#11758198—42no—
59C2×#11758198—42no—
60C2×#11758198—42no—
61C2×#11758198—42no—
62C2×#11758198C2×C414yes—
63C2×#11758198C2×C418yes—
64C2×C4×Dic61—42no—
65C2×C4×Dic61D414yes—
66#14519684—22no—
67#14519684—22no—
68#14519684—24no—
69#14519684—24no—
70#14519684—42no—
71#14519684—42no—
72#14519684—42no—
73#14519684—42no—
74#14519684—42no—
75#14519684C2×C418yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2312yes—
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—42no—
83C2×C4⋊C4×C61—42no—
84C2×C4⋊C4×C61—42no—
85C2×C4⋊C4×C61—42no—
86#95×C61—22no—
87#95×C61—42no—
88#95×C61C2×C414yes—
89C24×C61C2411yescharacteristic
90C23×D61—21no—
91C23×D61—21no—
92C23×D61—22no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—21no—
99C23×D61—22no—
100C23×D61C22×C412yes—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22×C414yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D412yes—
159C2×Dic122—28no—
160C2×Dic122—48no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—48no—
164C4⋊C4×C61—44no—
165C4⋊C4×C61—48no—
166#11758198—28no—
167#11758198—48no—
168#11758198—44no—
169#11758198—44no—
170#11758198—48no—
171#11758196—44no—
172C4×Dic61—44no—
173C23×C61C23×C411yescharacteristic
174C23×C61C22×D411yescharacteristic
175C23×C61C2×#4111yescharacteristic
176C23×C61H(5,𝔽2)12yes—
177C23×C61H(5,𝔽2)12yes—
178C23×C61H(5,𝔽2)12yes—
179C23×C61H(5,𝔽2)11yescharacteristic
180C23×C612−1+411yescharacteristic
181C23×C612−1+411yescharacteristic
182C23×C612−1+411yescharacteristic
183C23×C612−1+412yes—
184C22×D61—22no—
185C22×D61—24no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—42no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61C4×D414yes—
236C2×Dic61—22no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—42no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265Dic122—416no—
266#14625—611nomaximal, Sylow
267C22×C61C2×C4×D411yescharacteristic
268C22×C61#51812yes—
269C22×C61#51911yescharacteristic
270C22×C61#52011yescharacteristic
271C22×C61#52012yes—
272C22×C61#53411yescharacteristic
273C22×C61#53511yescharacteristic
274C22×C61#53511yescharacteristic
275C22×C61#53611yescharacteristic
276C22×C61#53611yescharacteristic
277C22×C61#53711yescharacteristic
278C22×C61#53812yes—
279C22×C61#54111yescharacteristic
280C22×C61#54212yes—
281C22×C61#54211yescharacteristic
282C22×C61H(3,𝔽4)12yes—
283C22×C61#56212yes—
284C22×C61#56212yes—
285C22×C61#56212yes—
286C22×C61#56212yes—
287C22×C61#56312yes—
288C22×C61#56312yes—
289C22×C61#56312yes—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—82no—
299C2×D61—82no—
300C2×D61—81no—
301C4×C61—24no—
302C4×C61—44no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61—82no—
306Dic61—42no—
307Dic61—44no—
308Dic61—42no—
309Dic61—42no—
310Dic61—82no—
311C2×#393—611no—
312C2×#391—611no—
313C2×#388—611no—
314#1446—611no—
315#1444D6111yescharacteristic, complemented
316#1442—612no—
317#1442—612no—
318#1440—612no—
319#1439—611no—
320#1439—612no—
321#1434—611no—
322C2×C61#232111yescharacteristic
323C2×C61#232311yescharacteristic
324C2×C61#232611yescharacteristic
325C2×C61#237812yes—
326C2×C61#237912yes—
327C2×C61#281411yescharacteristic
328C2×C61#281611yescharacteristic
329C2×C61#281911yescharacteristic
330C2×C61#282111yescharacteristic
331C2×C61#283112yes—
332C2×C61#283212yes—
333D61—82no—
334C23×Q8—611no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4C2×D6111yescharacteristic
337C22×C4⋊C4C2×D6112yes—
338C22×C22⋊C4—611no—
339C22×C22⋊C4—612no—
340C22×C22⋊C4—611no—
341C22×C22⋊C4—611no—
342C22×C22⋊C4—612no—
343C22×C42—611no—
344#393—618no—
345#391—1224no—
346#388—1224no—
347C2×#95—612no—
348C2×#95—611no—
349C2×#95—612no—
350C2×#95—612no—
351C2×#95—611no—
352C2×#95—612no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95C2×D6111yescharacteristic
358C2×#95C2×D6112yes—
359C2×#95C2×D6111yescharacteristic
360C61#1462511yescharacteristic, complemented, Sylow
361C25—611no—
362C22×Q8—614no—
363C22×Q8—1222no—
364C22×Q8—1222no—
365C22×Q8—2442no—
366C22×Q8—2442no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—612no—
370C23×C4—611no—
371C23×C4—612no—
372C23×C4C22×D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6112yes—
375C23×C4C22×D6112yes—
376C23×C4C22×D6111yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—42no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—2442no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—618no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—2442no—
391C2×C22⋊C4—2442no—
392C2×C22⋊C4—2442no—
393C2×C22⋊C4—2442no—
394C2×C22⋊C4—2442no—
395C2×C22⋊C4—2442no—
396C2×C22⋊C4—2442no—
397C2×C42—614no—
398C2×C42—2442no—
399#95—22no—
400#95—42no—
401#95—618no—
402#95—1222no—
403#95—1224no—
404#95—1224no—
405#95—1222no—
406#95—2442no—
407#95—2442no—
408#95—2442no—
409#95—2442no—
410#95—2442no—
411#95C4×D6114yes—
412C24—612no—
413C24—1221no—
414C24—1221no—
415C24—1222no—
416C24—1221no—
417C24—1222no—
418C24—1222no—
419C24—1221no—
420C24—1221no—
421C24—1221no—
422C24—1222no—
423C24C23×D6111yescharacteristic
424C2×Q8—1228no—
425C2×Q8—2448no—
426C2×Q8—2444no—
427C2×Q8—2444no—
428C2×Q8—2448no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—612no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4C2×C4×D6114yes—
484C22×C4C2×C4×D6112yes—
485C22×C4D4×D6112yes—
486C22×C4#1175822612yes—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—2444no—
490C22⋊C4—1228no—
491C22⋊C4—2448no—
492C22⋊C4—2444no—
493C22⋊C4—2444no—
494C22⋊C4—2448no—
495C42—2444no—
496C23—1222no—
497C23—1224no—
498C23—1222no—
499C23—1222no—
500C23—1224no—
501C23—2442no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2442no—
516C23—2441no—
517C23—2442no—
518C23—2441no—
519C23C22×C4×D6111yescharacteristic
520C23C2×D4×D6111yescharacteristic
521C23C2×#1175822611yescharacteristic
522C23#1451982212yes—
523C23#1451982211yescharacteristic
524C23#1451982212yes—
525C23#1451982312yes—
526C23#1451982411yescharacteristic
527C23#1451982411yescharacteristic
528C23#1451982511yescharacteristic
529C23#1451982512yes—
530Q8—24416no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—1222no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2441no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4#1451973914yes—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—4881no—
593C22—4882no—
594C22—4882no—
595C22—2444no—
596C22—2442no—
597C22—2444no—
598C22—2442no—
599C22—2442no—
600C22C2×#1451973911yescharacteristic
601C22#1684023412yes—
602C22#1684023811yescharacteristic
603C22#1684024111yescharacteristic
604C22#1684024312yes—
605C22#1684032311yescharacteristic
606C22#1684032511yescharacteristic
607C22#1684032711yescharacteristic
608C22#1684033511yescharacteristic
609C22#1684033711yescharacteristic
610C22#1684034111yescharacteristic
611C22#1684035212yes—
612C22#1684037511yescharacteristic
613C22#1684038611yescharacteristic
614C22#1684038612yes—
615C22#1684050512yes—
616C22#1684050712yes—
617C22#1684050812yes—
618C22#1684050812yes—
619C22#1684050912yes—
620C22#1684051112yes—
621C22#1684051312yes—
622C22#1684051612yes—
623C4—24no—
624C4—44no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—4882no—
629C4—2442no—
630C4—2442no—
631C4—2444no—
632C4—2442no—
633C2—4882no—
634C2#2144847611yescharacteristic
635C2#2144849911yescharacteristic
636C2#2144852711yescharacteristic
637C2#2144898112yes—
638C2#2144899212yes—
639C2#2145383511yescharacteristic
640C2#2145384811yescharacteristic
641C2#2145386611yescharacteristic
642C2#2145388311yescharacteristic
643C2#2145398112yes—
644C2#2145399112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1444D61
hypocenterC61#14625
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
2. C2×#16839667C2semidirect
4. #21444089C2semidirect
5. #21444078C2semidirect
6. #21444077C2semidirect
10. #21444044C2semidirect
11. C61×#1444C2semidirect
20. #16839675C4semidirect
315. #1444D61semidirect
360. C61#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#168396752
C2×#168396672
C2×#168396512
#214440892
#214440782
#214440772
#214440672
#214440612
#214440442
C61×#14442
#1462561