Finite Group Databasebeta

← #29610693

Subgroups — #29610693

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×#16839674C211yesmaximal, characteristic
2C2×#16839668C211yesmaximal, characteristic, complemented
3#21444089C211yesmaximal, characteristic, complemented
4#21444083C211yesmaximal, characteristic, complemented
5#21444078C212yesmaximal, complemented
6#21444074C212yesmaximal, complemented
7#21444066C212yesmaximal
8#21444064C211yesmaximal, characteristic
9#21444062C212yesmaximal
10#21444046C211yesmaximal, characteristic, complemented
11C2×C61×#388C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2212yes—
17#16839674C418yescomplemented
18#16839668—24no—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2211yescharacteristic
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C22×C4⋊C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C61×#388—24no—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C22×Dic122—22no—
39C22×Dic122—22no—
40C22×Dic122—42no—
41C22×Dic122—42no—
42C22×Dic122C2×C414yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46C23×Dic61C2312yes—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—42no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196—42no—
52C2×C22⋊Dic61—42no—
53C2×C22⋊Dic61—42no—
54C2×C22⋊Dic61—42no—
55C2×C22⋊Dic61—42no—
56C2×C22⋊Dic61—42no—
57C2×C22⋊Dic61C2×C418yes—
58#14519684—24no—
59#14519684—24no—
60#14519684—22no—
61#14519684—42no—
62#14519684—42no—
63#14519684—42no—
64#14519684—42no—
65#14519684—42no—
66#14519684—42no—
67#14519684C2×C418yes—
68#14519684C2×C414yes—
69C25×C61C2311yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—42no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61—42no—
83C2×C22⋊C4×C61—42no—
84C2×C22⋊C4×C61C2×C414yes—
85C2×C42×C61—42no—
86C2×C42×C61D414yes—
87#95×C61—22no—
88#95×C61—22no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—21no—
98C24×C61—22no—
99C24×C61C22×C412yes—
100C24×C61C2411yescharacteristic
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61#4112yes—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61C22×C414yes—
158C22×Dic61C22×C414yes—
159C2×Dic122—28no—
160C2×Dic122—48no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—48no—
164C4⋊C4×C61—44no—
165C22⋊C4×C61—28no—
166C22⋊C4×C61—44no—
167C22⋊C4×C61—44no—
168C42×C61—44no—
169C22⋊Dic61—48no—
170C22⋊Dic61—48no—
171C4⋊Dic61—44no—
172#11758196—48no—
173C23×C61—22no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C23×C411yescharacteristic
197C23×C61C22×D411yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61H(5,𝔽2)12yes—
200C23×C61H(5,𝔽2)12yes—
201C23×C61H(5,𝔽2)12yes—
202C23×C61H(5,𝔽2)11yescharacteristic
203C23×C612−1+411yescharacteristic
204C23×C612−1+411yescharacteristic
205C23×C612−1+411yescharacteristic
206C23×C612−1+412yes—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C4×D414yes—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
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×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—44no—
271C22×C61—44no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—81no—
278C22×C61C2×C4×D411yescharacteristic
279C22×C61#51812yes—
280C22×C61#51911yescharacteristic
281C22×C61#52012yes—
282C22×C61#52011yescharacteristic
283C22×C61#53411yescharacteristic
284C22×C61#53511yescharacteristic
285C22×C61#53511yescharacteristic
286C22×C61#53611yescharacteristic
287C22×C61#53611yescharacteristic
288C22×C61#53711yescharacteristic
289C22×C61#53812yes—
290C22×C61#54111yescharacteristic
291C22×C61#54212yes—
292C22×C61#54211yescharacteristic
293C22×C61H(3,𝔽4)12yes—
294C22×C61#56212yes—
295C22×C61#56212yes—
296C22×C61#56212yes—
297C22×C61#56212yes—
298C22×C61#56312yes—
299C22×C61#56312yes—
300C22×C61#56312yes—
301C4×C61—24no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307Dic61—44no—
308Dic61—44no—
309Dic61—82no—
310Dic61—82no—
311C2×#393—611no—
312C2×#391—611no—
313C2×#388D6111yescharacteristic
314#1446—611no—
315#1444—611no—
316#1442—612no—
317#1442—612no—
318#1440—612no—
319#1439—612no—
320#1439—611no—
321#1434—611no—
322C2×C61—82no—
323C2×C61#232111yescharacteristic
324C2×C61#232311yescharacteristic
325C2×C61#232611yescharacteristic
326C2×C61#237812yes—
327C2×C61#237912yes—
328C2×C61#281411yescharacteristic
329C2×C61#281611yescharacteristic
330C2×C61#281911yescharacteristic
331C2×C61#282111yescharacteristic
332C2×C61#283112yes—
333C2×C61#283212yes—
334C23×Q8—611no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—612no—
337C22×C4⋊C4C2×D6111yescharacteristic
338C22×C22⋊C4—612no—
339C22×C22⋊C4—612no—
340C22×C22⋊C4C2×D6111yescharacteristic
341C22×C22⋊C4C2×D6111yescharacteristic
342C22×C22⋊C4C2×D6111yescharacteristic
343C22×C42C2×D6111yescharacteristic
344#393—618no—
345#391—1224no—
346#388—24no—
347C2×#95—611no—
348C2×#95—612no—
349C2×#95—611no—
350C2×#95—612no—
351C2×#95—612no—
352C2×#95—612no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95C2×D6111yescharacteristic
359C2×#95C2×D6111yescharacteristic
360C61#1462511yescharacteristic, complemented, Sylow
361C25C22×D6111yescharacteristic
362C22×Q8—614no—
363C22×Q8—1222no—
364C22×Q8—1222no—
365C22×Q8—2442no—
366C22×Q8—2442no—
367C23×C4—612no—
368C23×C4—612no—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4C22×D6111yescharacteristic
372C23×C4C22×D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—42no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—2442no—
381C2×C4⋊C4—2442no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4—2442no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—42no—
389C2×C22⋊C4—42no—
390C2×C22⋊C4—618no—
391C2×C22⋊C4—2442no—
392C2×C22⋊C4—2442no—
393C2×C22⋊C4—2442no—
394C2×C22⋊C4—2442no—
395C2×C22⋊C4—2442no—
396C2×C22⋊C4C4×D6114yes—
397C2×C42—42no—
398C2×C42C22⋊D6114yes—
399#95—22no—
400#95—22no—
401#95—614no—
402#95—618no—
403#95—1222no—
404#95—1224no—
405#95—1224no—
406#95—2442no—
407#95—2442no—
408#95—2442no—
409#95—2442no—
410#95—2442no—
411#95—2442no—
412C24—21no—
413C24—21no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—22no—
421C24—21no—
422C24C2×C4×D6112yes—
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—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—614no—
454C22×C4—614no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
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—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4C2×C4×D6112yes—
482C22×C4C2×C4×D6112yes—
483C22×C4#1175822212yes—
484C22×C4C2×C22⋊D6112yes—
485C22×C4C2×C22⋊D6112yes—
486C22×C4C2×C22⋊D6112yes—
487C4⋊C4—44no—
488C4⋊C4—2444no—
489C4⋊C4—2448no—
490C22⋊C4—28no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—2448no—
494C22⋊C4—2448no—
495C42—44no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23C22×C4×D6111yescharacteristic
520C23C22×C22⋊D6111yescharacteristic
521C23C2×#1175822211yescharacteristic
522C23#1451982212yes—
523C23#1451982212yes—
524C23#1451982312yes—
525C23#1451982311yescharacteristic
526C23#1451982411yescharacteristic
527C23#1451982411yescharacteristic
528C23#1451982412yes—
529C23#1451982411yescharacteristic
530Q8—24416no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
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—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4C4×C22⋊D6114yes—
589C22—42no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22C2×C4×C22⋊D6111yescharacteristic
601C22#1684023412yes—
602C22#1684023911yescharacteristic
603C22#1684024112yes—
604C22#1684024211yescharacteristic
605C22#1684032111yescharacteristic
606C22#1684032611yescharacteristic
607C22#1684032911yescharacteristic
608C22#1684033311yescharacteristic
609C22#1684033311yescharacteristic
610C22#1684034011yescharacteristic
611C22#1684034512yes—
612C22#1684037411yescharacteristic
613C22#1684037712yes—
614C22#1684038311yescharacteristic
615C22#1684050412yes—
616C22#1684050612yes—
617C22#1684050712yes—
618C22#1684050712yes—
619C22#1684050912yes—
620C22#1684051412yes—
621C22#1684051412yes—
622C22#1684051512yes—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—4882no—
630C4—4882no—
631C4—2444no—
632C4—2444no—
633C2—82no—
634C2#2144847311yescharacteristic
635C2#2144850111yescharacteristic
636C2#2144853111yescharacteristic
637C2#2144898612yes—
638C2#2144898812yes—
639C2#2145383211yescharacteristic
640C2#2145384711yescharacteristic
641C2#2145386411yescharacteristic
642C2#2145387711yescharacteristic
643C2#2145396912yes—
644C2#2145398412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#388C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#388D61
hypocenterC61#14625
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#16839668C2semidirect
3. #21444089C2semidirect
4. #21444083C2semidirect
5. #21444078C2semidirect
6. #21444074C2semidirect
10. #21444046C2semidirect
17. #16839674C4semidirect
360. C61#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#168396742
C2×#168396682
#214440892
#214440832
#214440782
#214440742
#214440662
#214440642
#214440622
#214440462
C2×C61×#3882
#1462561