Finite Group Databasebeta

← #29730469

Subgroups — #29730469

Order 15616.

1290 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839955C212yesmaximal, complemented
2C2×#16839954C211yesmaximal, characteristic, complemented
3#21450766C211yesmaximal, characteristic, complemented
4#21448884C211yesmaximal, characteristic, complemented
5#21447127C212yesmaximal, complemented
6#21446888C212yesmaximal, complemented
7#21446747C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446169C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C61×#493C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519773C2212yes—
14C2×#14519748—21no—
15C2×#14519748—21no—
16C2×C4×D244C2211yescharacteristic
17C2×D4×Dic61—21no—
18C2×D4×Dic61C2212yes—
19#16839955—28no—
20#16839954—24no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519706C2212yes—
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28#16839670—22no—
29#16839665—21no—
30#16839665C2212yes—
31#16839665C2212yes—
32#16839642—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839516C2212yes—
38C22×D8×C61C2212yes—
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#493—24no—
43C2×C4×C8×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C23×Dic61—22no—
48C22×C4×D61—21no—
49C2×#11758199—24no—
50C2×#11758199—24no—
51C2×#11758199—42no—
52C22×#92418C2312yes—
53#14519773—24no—
54#14519773—42no—
55#14519748—44no—
56#14519748—44no—
57C4×D244—22no—
58C4×D244—22no—
59D4×Dic61—24no—
60D4×Dic61—42no—
61D4×Dic61—44no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C22⋊Dic61—22no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61C2312yes—
75C4⋊#92418—22no—
76#14519706—24no—
77#14519699—22no—
78#14519699D414yescomplemented
79#14519698—24no—
80#14519698—24no—
81#14519698—42no—
82#14519684—21no—
83#14519684D412yes—
84C22×D4×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61C2312yes—
89C22×D4×C61C2312yes—
90C2×D8×C61—24no—
91C2×D8×C61—24no—
92C2×D8×C61—24no—
93C2×D8×C61—24no—
94C2×D8×C61—42no—
95C2×D8×C61—42no—
96C22×C8×C61C2311yescharacteristic
97C22×C8×C61C2311yescharacteristic
98#127×C61—22no—
99#127×C61—22no—
100#127×C61—22no—
101#127×C61D414yescomplemented
102C2×C42×C61C2311yescharacteristic
103C4×C8×C61—22no—
104C24×C61—22no—
105C24×C61—22no—
106C23×D61—21no—
107C2×D4×C61—24no—
108C2×D4×C61—24no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—81no—
127C2×D4×C61—81no—
128C2×D4×C61—81no—
129C2×D4×C61—81no—
130C2×D4×C61C2×D414yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—44no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—82no—
139C2×C22⋊D61—82no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—44no—
144C22×Dic61—44no—
145C22×Dic61—82no—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C2×D412yes—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157D8×C61—48no—
158D8×C61—48no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C2×D412yes—
164C2×C8×C61#4112yes—
165C42×C61C2×D412yes—
166C42×C61#4112yes—
167C22⋊Dic61—44no—
168C22⋊Dic61—44no—
169C22⋊Dic61—44no—
170C22⋊Dic61—44no—
171C22⋊Dic61—82no—
172#11758199—48no—
173#11758198—42no—
174C4⋊Dic61—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61D814yescomplemented
181C4⋊Dic61C2×D412yes—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—42no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—24no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210D4×C61—24no—
211D4×C61—24no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—48no—
219D4×C61—84no—
220D4×C61—84no—
221D4×C61—84no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61C2×D812yes—
227C2×C4×C61AGL(1,ℤ/8)12yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C22⋊D61—88no—
235C22⋊D61—88no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—88no—
251C2×Dic61C2×D814yes—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—84no—
257C8×C61—22no—
258C8×C61—22no—
259#92418—42no—
260#25364—611nomaximal, Sylow
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61C2×#12111yescharacteristic
276C22×C61D4211yescharacteristic
277C22×C61#55011yescharacteristic
278C22×C61C22×D811yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57611yescharacteristic
281C22×C61#57611yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—22no—
290C4×C61#45914yescomplemented
291C4×C61#53211yescharacteristic
292C4×C61#54011yescharacteristic
293C4×C61D4211yescharacteristic
294C4×C61#54611yescharacteristic
295Dic61—24no—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C2×#493D6111yescharacteristic, complemented
301C2×#459—612no—
302C2×#459—611no—
303#2541—611no—
304#2366—611no—
305#2054—612no—
306#2011—612no—
307#1982—611no—
308#1939—611no—
309#1879—612no—
310C4×#102—611no—
311C2×C61—84no—
312C2×C61—162no—
313C2×C61#242811yescharacteristic
314C2×C61C2×#45911yescharacteristic
315C2×C61#304111yescharacteristic
316C2×C61#327411yescharacteristic
317C2×C61D4×D811yescharacteristic
318C2×C61#339111yescharacteristic
319C2×C61#339911yescharacteristic
320D61—82no—
321C22×D8—612no—
322C22×D8C2×D6112yes—
323C22×D8C2×D6112yes—
324C2×#127—612no—
325C2×#127C2×D6111yescharacteristic
326C2×#127C2×D6111yescharacteristic
327C2×#121—1221no—
328C2×#121—1221no—
329C2×C4×D4—612no—
330C2×C4×D4—611no—
331C2×C4×D4—1221no—
332#493—24no—
333#459—1228no—
334#459—1224no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—611no—
337C2×#102—611no—
338C2×#102—612no—
339C2×#102—612no—
340C2×#102—611no—
341C2×#102—612no—
342C2×C4×C8C2×D6111yescharacteristic
343#392—1222no—
344#390—612no—
345#390—612no—
346#390—1221no—
347#386—1222no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350#340—611no—
351#340—611no—
352#336—612no—
353C61#2536411yescharacteristic, complemented, Sylow
354C22×D4—21no—
355C22×D4—21no—
356C22×D4—21no—
357C22×D4—21no—
358C22×D4—611no—
359C22×D4—1222no—
360C22×D4—1222no—
361C22×D4C22×D6112yes—
362C22×D4C22×D6112yes—
363C23×C4—1222no—
364C23×C4—1221no—
365C2×D8—24no—
366C2×D8—24no—
367C2×D8—24no—
368C2×D8—24no—
369C2×D8—42no—
370C2×D8—42no—
371C2×D8—1224no—
372C2×D8—1224no—
373C2×D8—2442no—
374C22×C8—612no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377#127—22no—
378#127—22no—
379#127—22no—
380#127—1224no—
381#127—2442no—
382#127C22⋊D6114yescomplemented
383#121—2444no—
384#121—2444no—
385C4×D4—1224no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—2442no—
389C4×D4—2444no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1221no—
401C2×C42—612no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404C4⋊C8—1224no—
405C4⋊C8—1222no—
406#102—614no—
407#102—1224no—
408#102—1224no—
409#102—1222no—
410#102—2442no—
411C4×C8—22no—
412#95—612no—
413#95—1221no—
414C24—22no—
415C24—22no—
416C24—1221no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—81no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—612no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—4882no—
447C2×D4—4882no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4C2×C22⋊D6114yes—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—4882no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4C23×D6111yescharacteristic
467D8—48no—
468D8—48no—
469D8—2448no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8D4×D6112yes—
478C2×C8#1175822412yes—
479C4⋊C4—614no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1222no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C22⋊C4—4882no—
488C22⋊C4—2444no—
489C22⋊C4—2444no—
490C22⋊C4—2444no—
491C22⋊C4—2444no—
492C22⋊C4—2442no—
493C42—1222no—
494C42—1222no—
495C42—2442no—
496C42#1175822412yes—
497C42C2×C22⋊D6112yes—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23C2×D4×D6111yescharacteristic
520D4—24no—
521D4—24no—
522D4—48no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—84no—
530D4—84no—
531D4—84no—
532D4—1222no—
533D4—1222no—
534D4—4888no—
535D4—4888no—
536D4—2441no—
537D4—2441no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—614no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4888no—
548C2×C4—4884no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4#1451974912yes—
558C2×C4#1451974912yes—
559C2×C4D8×D6112yes—
560C2×C4#1451980112yes—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4#1451982211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C8—22no—
568C8—22no—
569C8—2442no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22C2×#1451974911yescharacteristic
592C22#1684039711yescharacteristic
593C22#1684043211yescharacteristic
594C22C2×D8×D6111yescharacteristic
595C22C2×#1451980111yescharacteristic
596C22#1684056311yescharacteristic
597C22#1684056311yescharacteristic
598C4—22no—
599C4—1224no—
600C4—4881no—
601C4—4881no—
602C4—2442no—
603C4—2442no—
604C4#1683995514yescomplemented
605C4#1684031011yescharacteristic
606C4#1684036611yescharacteristic
607C4D4×C22⋊D6111yescharacteristic
608C4#1684040611yescharacteristic
609C2—84no—
610C2—162no—
611C2—4882no—
612C2#2144941011yescharacteristic
613C2C2×#1683995511yescharacteristic
614C2#2145531911yescharacteristic
615C2#2145743811yescharacteristic
616C2#2145749411yescharacteristic
617C2#2145879211yescharacteristic
618C2#2145886611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#493C2
socleC23×C61C22×D4
hypercenterC2×#493D61
hypocenterC61#25364
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×#16839955C2semidirect
2. C2×#16839954C2semidirect
3. #21450766C2semidirect
4. #21448884C2semidirect
5. #21447127C2semidirect
6. #21446888C2semidirect
7. #21446747C2semidirect
8. #21446537C2semidirect
9. #21446169C2semidirect
10. C4×#14519699C2semidirect
11. C2×C61×#493C2semidirect
78. #14519699D4semidirect
101. #127×C61D4semidirect
180. C4⋊Dic61D8semidirect
290. C4×C61#459semidirect
300. C2×#493D61semidirect
353. C61#25364semidirect
382. #127C22⋊D61semidirect
604. C4#16839955semidirect

Maximal subgroups

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

subgroupindex
C2×#168399552
C2×#168399542
#214507662
#214488842
#214471272
#214468882
#214467472
#214465372
#214461692
C4×#145196992
C2×C61×#4932
#2536461