Finite Group Databasebeta

← #29713917

Subgroups — #29713917

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic, complemented
4#21447006C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446780C211yesmaximal, characteristic, complemented
7#21446613C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446015C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×D8×C61C211yesmaximal, characteristic, complemented
12#16840477—24no—
13#16840477—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751—21no—
17C2×#14519751—21no—
18C2×#14519748C2212yes—
19C2×#14519748C2212yes—
20C2×C4×D244C2211yescharacteristic
21C2×#14519736—21no—
22C2×#14519735C2212yes—
23#16840052—24no—
24C2×#14519708C2212yes—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×#14519689C2212yes—
29C2×#14519689C2212yes—
30#16839678—22no—
31#16839632C2211yescharacteristic
32#16839622—21no—
33#16839619C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×D8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C4×D8×C61—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C22×C4×D61C2312yes—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yescomplemented
53#14519763—216no—
54#14519751—28no—
55#14519751—44no—
56#14519748—24no—
57#14519748—24no—
58#14519748—24no—
59#14519748—42no—
60C4×D244—22no—
61C4×D244—22no—
62#14519736—28no—
63#14519735—24no—
64#14519735—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2312yes—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198D412yes—
76C2×C4×Dic61—22no—
77#14519708—24no—
78C4⋊#92418—22no—
79#14519698—24no—
80#14519698—42no—
81#14519689—24no—
82#14519689—42no—
83#14519684—22no—
84C22×D4×C61C2311yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C2×D8×C61—22no—
89C2×D8×C61—22no—
90C2×D8×C61—22no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61D414yescomplemented
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#102×C61—22no—
104#102×C61—22no—
105C4×C8×C61—22no—
106C24×C61—21no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61C2×D412yes—
120C2×D4×C61#4112yes—
121C22×C4×C61—21no—
122C22×C4×C61—24no—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—28no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C2×C22⋊D61—82no—
137C2×C22⋊D61—82no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61C2×D412yes—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×Dic122—24no—
151C2×Dic122—81no—
152C2×Dic122—81no—
153D8×C61—28no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61#4112yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—42no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—28no—
169C22⋊Dic61—44no—
170#11758198—28no—
171#11758198—24no—
172#11758198—44no—
173C4⋊Dic61—24no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—42no—
179C4⋊Dic61—81no—
180C4⋊Dic61C2×D414yes—
181C4×Dic61—28no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—48no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—84no—
225C2×C4×C61#12012yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#13512yes—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C22⋊D61—88no—
236C22⋊D61—88no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246D244—44no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251Dic122—84no—
252C8×C61—22no—
253C8×C61—41no—
254#92418—42no—
255#24117—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12011yescharacteristic
271C22×C61#54611yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61#44914yes—
288C4×C61#52511yescharacteristic
289C4×C61#53711yescharacteristic
290C4×C61#54811yescharacteristic
291C4×C61#55011yescharacteristic
292Dic61—42no—
293Dic61—44no—
294Dic61—82no—
295C2×#555—611no—
296C2×#476—611no—
297C2×C4×D8D6111yescharacteristic, complemented
298#2537—611no—
299#2034—612no—
300#1999—612no—
301#1990—611no—
302#1954—611no—
303#1875—612no—
304#1854—612no—
305#1802—611no—
306C2×C61—82no—
307C2×C61—82no—
308C2×C61#240711yescharacteristic
309C2×C61C2×#44911yescharacteristic
310C2×C61#301211yescharacteristic
311C2×C61#330311yescharacteristic
312C2×C61#332511yescharacteristic
313C2×C61#334111yescharacteristic
314C2×C61#335411yescharacteristic
315D61—82no—
316C22×D8C2×D6111yescharacteristic
317#555—1224no—
318#555—1224no—
319C2×C4⋊Q8—611no—
320C2×#124—1222no—
321C2×#121—612no—
322C2×#121—612no—
323C2×#121—1221no—
324C2×#121—1221no—
325C2×C4×D4—611no—
326C2×C4×D4C2×D6111yescharacteristic
327C2×C4×D4C2×D6111yescharacteristic
328C2×#117—612no—
329C2×#117—1221no—
330#476—1224no—
331C4×D8—24no—
332C2×#107C2×D6111yescharacteristic
333C2×#106—612no—
334C2×C4⋊C8—611no—
335C2×#102—612no—
336C2×#102—612no—
337C2×#102C2×D6111yescharacteristic
338C2×#102C2×D6111yescharacteristic
339C2×C22⋊C8—612no—
340C2×C22⋊C8—612no—
341C2×C4×C8C2×D6111yescharacteristic
342#394—1222no—
343#384—611no—
344#380—612no—
345#380—1221no—
346#340—612no—
347#340—612no—
348#340—612no—
349C61#2411711yescharacteristic, complemented, Sylow
350C22×Q8—1221no—
351C22×D4—611no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4C22×D6111yescharacteristic
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—21no—
358C23×C4—612no—
359C2×D8—22no—
360C2×D8—22no—
361C2×D8—22no—
362C22×C8—612no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C4⋊Q8—614no—
366C4⋊Q8—1222no—
367#124—12216no—
368#121—1224no—
369#121—1228no—
370#121—1224no—
371#121—1224no—
372#121—2442no—
373#121—2444no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—1222no—
378C4×D4—1222no—
379C4×D4C22⋊D6114yescomplemented
380#117—1224no—
381#117—1228no—
382#117—1224no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C42—1222no—
399C2×C42C22×D6111yescharacteristic
400#107—22no—
401#106—1224no—
402C4⋊C8—1222no—
403#102—22no—
404#102—22no—
405#102—1224no—
406#102—2442no—
407C22⋊C8—1224no—
408C22⋊C8—2442no—
409C4×C8—22no—
410#95—1222no—
411C24—21no—
412C24—21no—
413C24—612no—
414C2×Q8—1224no—
415C2×Q8—4881no—
416C2×Q8—4881no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—1228no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—4882no—
432C2×D4—4882no—
433C2×D4—4882no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4#1175822212yes—
437C2×D4C2×C22⋊D6112yes—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—612no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458D8—28no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8#1175822212yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—614no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1224no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—4881no—
478C4⋊C4—2442no—
479C4⋊C4C2×C22⋊D6112yes—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—1228no—
483C22⋊C4—1228no—
484C22⋊C4—1224no—
485C22⋊C4—2444no—
486C22⋊C4—2444no—
487C42—1228no—
488C42D4×D6112yes—
489C42C2×C22⋊D6112yes—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23C2×D4×D6111yescharacteristic
512Q8—4884no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4—22no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—4888no—
522D4—4888no—
523D4—2444no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—48no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974512yes—
548C2×C4#1451974512yes—
549C2×C4#1451979612yes—
550C2×C4#1451980112yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4C2×#1175822211yescharacteristic
555C2×C4#1451982311yescharacteristic
556C2×C4#1451982411yescharacteristic
557C8—22no—
558C8—41no—
559C8—2442no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—2442no—
575C22—2442no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×#1451974511yescharacteristic
582C22#1684040811yescharacteristic
583C22#1684041711yescharacteristic
584C22C2×#1451979611yescharacteristic
585C22C2×#1451980111yescharacteristic
586C22#1684056411yescharacteristic
587C22#1684056911yescharacteristic
588C4—22no—
589C4—42no—
590C4—81no—
591C4—4882no—
592C4—2444no—
593C4—2442no—
594C4#1683990414yes—
595C4#1684027311yescharacteristic
596C4#1684034011yescharacteristic
597C4#1684042111yescharacteristic
598C4#1684043311yescharacteristic
599C2—82no—
600C2—82no—
601C2—4882no—
602C2#2144916411yescharacteristic
603C2C2×#1683990411yescharacteristic
604C2#2145511911yescharacteristic
605C2#2145774911yescharacteristic
606C2#2145806311yescharacteristic
607C2#2145824311yescharacteristic
608C2#2145841611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
hypercenterC2×C4×D8D61
hypocenterC61#24117
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16840477C2semidirect
2. C2×#16840052C2semidirect
3. #21450732C2semidirect
4. #21447006C2semidirect
5. #21446814C2semidirect
6. #21446780C2semidirect
7. #21446613C2semidirect
8. #21446138C2semidirect
9. #21446015C2semidirect
10. #21445819C2semidirect
11. C2×C4×D8×C61C2semidirect
52. #14519777D4semidirect
96. C4×D4×C61D4semidirect
297. C2×C4×D8D61semidirect
349. C61#24117semidirect
379. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168400522
#214507322
#214470062
#214468142
#214467802
#214466132
#214461382
#214460152
#214458192
C2×C4×D8×C612
#2411761