Finite Group Databasebeta

← #29721545

Subgroups — #29721545

Order 15616.

1222 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×#16839955C212yesmaximal, complemented
2C2×C4×#11758199C211yesmaximal, characteristic, complemented
3#21450766C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447203C212yesmaximal, complemented
6#21447052C212yesmaximal, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446535C211yesmaximal, characteristic, complemented
9#21446169C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic, complemented
12C22×#11758199C2211yescharacteristic
13C2×#14519773C2212yes—
14C2×#14519748—21no—
15C2×#14519748—21no—
16C2×C4×D244C2211yescharacteristic
17C2×D4×Dic61C2212yes—
18C22×C4⋊Dic61—21no—
19#16839955—28no—
20C4×#11758199—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2212yes—
23C2×#14519699C2212yes—
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839670—22no—
30#16839665C2212yes—
31#16839661C2212yes—
32#16839637—22no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839516C2212yes—
38C22×D8×C61C2212yes—
39C2×#127×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#459—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C23×Dic61—22no—
48C22×C4×D61—21no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51C2×#11758199D414yescomplemented
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519773—24no—
55#14519773—42no—
56#14519748—44no—
57#14519748—44no—
58C4×D244—22no—
59C4×D244—22no—
60D4×Dic61—24no—
61D4×Dic61—42no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—44no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—21no—
74C2×C4×Dic61C2312yes—
75#14519710—22no—
76#14519706—24no—
77#14519699—24no—
78#14519698—22no—
79#14519698—24no—
80#14519698—42no—
81#14519697—22no—
82C4×#92418—22no—
83#14519684—22no—
84#14519684D412yes—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61C2312yes—
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C2×D8×C61—24no—
91C2×D8×C61—24no—
92C2×D8×C61—24no—
93C22×C8×C61C2312yes—
94#127×C61—22no—
95#127×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61D414yescomplemented
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102#102×C61—24no—
103C24×C61—21no—
104C24×C61—22no—
105C23×D61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—81no—
120C2×D4×C61—81no—
121C2×D4×C61C2×D412yes—
122C2×D4×C61C2×D412yes—
123C2×D4×C61#4112yes—
124C22×C4×C61—21no—
125C22×C4×C61—24no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C2×C22⋊D61—82no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141C22×Dic61—82no—
142C22×Dic61C2×D412yes—
143C22×Dic61C2×D412yes—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C2×D412yes—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153D8×C61—48no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D412yes—
159C22⋊C4×C61—24no—
160C42×C61C2×D412yes—
161C42×C61#4112yes—
162C22⋊Dic61—44no—
163C22⋊Dic61—44no—
164C22⋊Dic61—44no—
165#11758199—28no—
166#11758198—42no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61—44no—
174C4⋊Dic61—81no—
175C4⋊Dic61D814yescomplemented
176C4×Dic61—22no—
177C4×Dic61—22no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418C2×D412yes—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206D4×C61—22no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—84no—
214D4×C61—84no—
215D4×C61#12114yescomplemented
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—48no—
222C2×C4×C61#12112yes—
223C2×C4×C61#12112yes—
224C2×C4×C61C2×D812yes—
225C2×C4×C61AGL(1,ℤ/8)12yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C22⋊D61—88no—
233C22⋊D61—88no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—88no—
246C2×Dic61C2×D814yes—
247D244—22no—
248D244—22no—
249D244—41no—
250D244—41no—
251C4×D61—84no—
252C8×C61—42no—
253#92418—22no—
254#92418—41no—
255#24726—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61C2×#12111yescharacteristic
271C22×C61D4211yescharacteristic
272C22×C61#54811yescharacteristic
273C22×C61C22×D811yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61#45914yescomplemented
287C4×C61C2×#12111yescharacteristic
288C4×C61#53711yescharacteristic
289C4×C61#54711yescharacteristic
290C4×C61#55311yescharacteristic
291Dic61—24no—
292Dic61—42no—
293Dic61—82no—
294Dic61—81no—
295C2×#459—612no—
296C2×#459D6111yescharacteristic, complemented
297C2×C4×D8—611no—
298#2541—611no—
299D4×C4⋊C4—611no—
300#2067—612no—
301#2041—612no—
302#1957—611no—
303#1939—611no—
304#1879—612no—
305#1802—611no—
306C2×C61—44no—
307C2×C61—162no—
308C2×C61#241611yescharacteristic
309C2×C61C2×#45911yescharacteristic
310C2×C61#304511yescharacteristic
311C2×C61#327211yescharacteristic
312C2×C61D4×D811yescharacteristic
313C2×C61#335811yescharacteristic
314C2×C61#336611yescharacteristic
315D61—82no—
316C22×D8—611no—
317C22×D8C2×D6112yes—
318C2×#127—612no—
319C2×#127C2×D6111yescharacteristic
320C2×#121—1221no—
321C2×#121—1221no—
322C2×C4×D4—612no—
323C2×C4×D4—611no—
324C2×C4×D4C2×D6111yescharacteristic
325C22×C4⋊C4—1221no—
326#459—24no—
327#459—1228no—
328C4×D8—1224no—
329C2×#107—611no—
330C2×C4⋊C8—612no—
331C2×C4⋊C8C2×D6111yescharacteristic
332C2×#102—612no—
333C2×#102—611no—
334C2×#102—612no—
335C2×#102—611no—
336C2×#102—612no—
337C2×#102C2×D6112yes—
338C2×C4×C8—611no—
339#392—1222no—
340#390—612no—
341#389—612no—
342#385—1222no—
343#384—611no—
344#380—1221no—
345C4×C4⋊C4—611no—
346#340—612no—
347#336—612no—
348C61#2472611yescharacteristic, complemented, Sylow
349C22×D4—21no—
350C22×D4—21no—
351C22×D4—611no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4C22×D6112yes—
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C23×C4—1222no—
358C23×C4—1221no—
359C2×D8—24no—
360C2×D8—24no—
361C2×D8—24no—
362C2×D8—614no—
363C2×D8—1222no—
364C2×D8—1222no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8C22×D6112yes—
368#127—22no—
369#127—22no—
370#127—1224no—
371#127—2442no—
372#121—2444no—
373#121—2444no—
374C4×D4—22no—
375C4×D4—1224no—
376C4×D4—1222no—
377C4×D4—1222no—
378C4×D4—2442no—
379C4×D4C22⋊D6114yescomplemented
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—2444no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1221no—
394C2×C42—612no—
395C2×C42C22×D6111yescharacteristic
396#107—1222no—
397C4⋊C8—22no—
398C4⋊C8—1224no—
399#102—24no—
400#102—1224no—
401#102—1224no—
402#102—1222no—
403#102—1222no—
404#102—2442no—
405C4×C8—1222no—
406#95—612no—
407#95—1222no—
408C24—21no—
409C24—22no—
410C24—1221no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4—612no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—4882no—
433C2×D4—4882no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4—2444no—
437C2×D4—2444no—
438C2×D4D4×D6112yes—
439C2×D4#1175822412yes—
440C2×D4C2×C22⋊D6112yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—4882no—
450C22×C4—2444no—
451C22×C4—2444no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458D8—48no—
459D8—1228no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—612no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C4⋊C4—21no—
470C4⋊C4—614no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—4881no—
477C4⋊C4—2444no—
478C4⋊C4—2442no—
479C4⋊C4C2×C22⋊D6112yes—
480C22⋊C4—24no—
481C22⋊C4—2444no—
482C22⋊C4—2444no—
483C22⋊C4—2444no—
484C22⋊C4—2442no—
485C42—1222no—
486C42—1222no—
487C42#1175822412yes—
488C42C2×C22⋊D6112yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—1222no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23C2×D4×D6111yescharacteristic
511D4—22no—
512D4—22no—
513D4—22no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—84no—
519D4—84no—
520D4—1222no—
521D4—1222no—
522D4—4888no—
523D4—4888no—
524D4—2441no—
525D4—2441no—
526D4#1451974814yescomplemented
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—48no—
533C2×C4—614no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—4888no—
540C2×C4—4884no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974912yes—
548C2×C4#1451974912yes—
549C2×C4D8×D6112yes—
550C2×C4#1451980112yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4#1451982311yescharacteristic
556C2×C4#1451982411yescharacteristic
557C8—42no—
558C8—1222no—
559C8—2441no—
560C22—24no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2442no—
579C22—2442no—
580C22—2442no—
581C22C2×#1451974911yescharacteristic
582C22#1684039611yescharacteristic
583C22#1684041811yescharacteristic
584C22C2×D8×D6111yescharacteristic
585C22C2×#1451980111yescharacteristic
586C22#1684056611yescharacteristic
587C22#1684057211yescharacteristic
588C4—22no—
589C4—42no—
590C4—1224no—
591C4—4882no—
592C4—4881no—
593C4—2442no—
594C4#1683995514yescomplemented
595C4C2×#1451974811yescharacteristic
596C4#1684034011yescharacteristic
597C4#1684041211yescharacteristic
598C4#1684046411yescharacteristic
599C2—44no—
600C2—162no—
601C2—4882no—
602C2#2144926011yescharacteristic
603C2C2×#1683995511yescharacteristic
604C2#2145536211yescharacteristic
605C2#2145741411yescharacteristic
606C2#2145749111yescharacteristic
607C2#2145844911yescharacteristic
608C2#2145853011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#459C2
socleC23×C61C22×D4
hypercenterC2×#459D61
hypocenterC61#24726
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16839955C2semidirect
2. C2×C4×#11758199C2semidirect
3. #21450766C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447203C2semidirect
6. #21447052C2semidirect
7. #21446626C2semidirect
8. #21446535C2semidirect
9. #21446169C2semidirect
10. #21445820C2semidirect
11. C2×C61×#459C2semidirect
51. C2×#11758199D4semidirect
97. C4×D4×C61D4semidirect
175. C4⋊Dic61D8semidirect
215. D4×C61#121semidirect
286. C4×C61#459semidirect
296. C2×#459D61semidirect
348. C61#24726semidirect
379. C4×D4C22⋊D61semidirect
526. D4#14519748semidirect
594. 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×C4×#117581992
#214507662
D4×C4⋊Dic612
#214472032
#214470522
#214466262
#214465352
#214461692
#214458202
C2×C61×#4592
#2472661