Finite Group Databasebeta

← #29714772

Subgroups — #29714772

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3#21450741C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21446893C212yesmaximal, complemented
6#21446755C211yesmaximal, characteristic, complemented
7#21446625C211yesmaximal, characteristic, complemented
8#21446164C212yesmaximal, complemented
9#21446077C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C4×SD16×C61C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13#16840497—24no—
14#16840497—24no—
15C2×#14519774C2211yescharacteristic
16C2×#14519766C2212yes—
17C2×#14519765—21no—
18C2×#14519762—22no—
19C2×#14519756—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244—21no—
22C2×#14519739C2212yes—
23#16839957—24no—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29#16839685—22no—
30#16839665C2212yes—
31#16839658C2212yes—
32#16839632C2211yescharacteristic
33#16839622—21no—
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×SD16×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C4×SD16×C61—24no—
41C2×#106×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50C2×#11758199—24no—
51C2×#11758199—24no—
52C2×#11758199—42no—
53C22×#92418C2312yes—
54#14519774—22no—
55#14519774D414yes—
56#14519766—24no—
57#14519766—24no—
58#14519765—28no—
59#14519762—216no—
60#14519756—44no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×D244—28no—
64#14519739—24no—
65#14519739—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196D412yes—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×C4×Dic61C2312yes—
78C4⋊#92418—22no—
79#14519701—42no—
80#14519699—24no—
81#14519699—42no—
82#14519697—24no—
83#14519684—22no—
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C2×SD16×C61—22no—
88C2×SD16×C61—22no—
89C2×SD16×C61—22no—
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C4×Q8×C61—22no—
93C4×Q8×C61D414yes—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—22no—
103#103×C61—22no—
104#102×C61—22no—
105C4×C8×C61—22no—
106C24×C61—21no—
107C23×D61—22no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61C2×D412yes—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61#4112yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61C2×D412yes—
131C22×Dic61C2×D412yes—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—24no—
137C2×D244—22no—
138C2×D244—24no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—81no—
142C2×D244—81no—
143C2×D244C2×D414yes—
144C2×C4×D61—28no—
145C2×C4×D61—82no—
146C2×C4×D61—82no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150SD16×C61—28no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—42no—
163C42×C61—24no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—44no—
167C22⋊Dic61—82no—
168#11758199—48no—
169#11758198—28no—
170#11758198—28no—
171#11758198—28no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—42no—
176C4⋊Dic61—81no—
177#11758196—24no—
178#11758196—44no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61—22no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199Q8×C61—22no—
200Q8×C61—22no—
201Q8×C61—41no—
202Q8×C61—41no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—41no—
206D4×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—84no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61#13512yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238D244—24no—
239D244—24no—
240D244—42no—
241D244—42no—
242D244—48no—
243D244—84no—
244C4×D61—88no—
245Dic122—44no—
246C8×C61—22no—
247C8×C61—41no—
248#92418—42no—
249#24186—611nomaximal, Sylow
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#54811yescharacteristic
259C22×C61#55011yescharacteristic
260C22×C61C2×#13511yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61#57611yescharacteristic
263C22×C61#57711yescharacteristic
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—81no—
275C4×C61#44914yes—
276C4×C61#52511yescharacteristic
277C4×C61#53711yescharacteristic
278C4×C61#54611yescharacteristic
279C4×C61#54711yescharacteristic
280Dic61—42no—
281Dic61—42no—
282Dic61—82no—
283C2×#559—611no—
284C2×#459—611no—
285C2×C4×SD16D6111yescharacteristic, complemented
286#2538—611no—
287#2042—612no—
288#2012—612no—
289#1984—611no—
290#1957—611no—
291#1879—612no—
292#1864—612no—
293#1803—611no—
294C2×C61—82no—
295C2×C61#240711yescharacteristic
296C2×C61C2×#44911yescharacteristic
297C2×C61#301211yescharacteristic
298C2×C61#336111yescharacteristic
299C2×C61#337311yescharacteristic
300C2×C61#339411yescharacteristic
301C2×C61#339911yescharacteristic
302D61—84no—
303C22×SD16C2×D6111yescharacteristic
304C22×D8—612no—
305#559—1224no—
306#559—1224no—
307C2×#127—611no—
308C2×#124—612no—
309C2×#124—1221no—
310C2×#123—1222no—
311C2×C22⋊Q8—1221no—
312C2×C4×Q8—611no—
313C2×C4×Q8C2×D6111yescharacteristic
314C2×C4×D4—612no—
315C2×C4×D4—1221no—
316C2×C4×D4C2×D6111yescharacteristic
317#459—1224no—
318C4×SD16—24no—
319C2×#106C2×D6111yescharacteristic
320C2×C4⋊C8—611no—
321C2×#103—612no—
322C2×#103C2×D6111yescharacteristic
323C2×#102—612no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102C2×D6111yescharacteristic
327C2×C4×C8C2×D6111yescharacteristic
328#396—1222no—
329#390—612no—
330#389—612no—
331#384—611no—
332#380—1221no—
333#340—612no—
334#336—612no—
335#336—612no—
336C61#2418611yescharacteristic, complemented, Sylow
337C22×Q8—611no—
338C22×Q8C22×D6111yescharacteristic
339C22×D4—612no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—21no—
344C23×C4—1222no—
345C2×SD16—22no—
346C2×SD16—22no—
347C2×SD16—22no—
348C2×D8—1224no—
349C2×D8—1224no—
350C2×D8—2442no—
351C22×C8—612no—
352C22×C8C22×D6111yescharacteristic
353C22×C8C22×D6111yescharacteristic
354#127—614no—
355#127—1222no—
356#124—1228no—
357#124—1224no—
358#124—1224no—
359#123—12216no—
360C22⋊Q8—2444no—
361C4×Q8—22no—
362C4×Q8—1222no—
363C4×Q8—1222no—
364C4×Q8C22⋊D6114yes—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—1228no—
368C4×D4—1224no—
369C4×D4—1224no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C42—21no—
386C2×C42—612no—
387C2×C42C22×D6111yescharacteristic
388#106—22no—
389C4⋊C8—1222no—
390#103—22no—
391#103—2442no—
392#102—22no—
393#102—1224no—
394#102—1224no—
395#102—2442no—
396C4×C8—22no—
397#95—1222no—
398C24—21no—
399C24—1222no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8C2×C22⋊D6112yes—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—614no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—4881no—
423C2×D4—4881no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4#1175822212yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—1228no—
436C22×C4—1222no—
437C22×C4—4882no—
438C22×C4—4882no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442SD16—28no—
443D8—2448no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8#1175822212yes—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—4881no—
462C4⋊C4—2444no—
463C4⋊C4—2442no—
464C4⋊C4C2×C22⋊D6112yes—
465C22⋊C4—42no—
466C22⋊C4—1228no—
467C22⋊C4—1228no—
468C22⋊C4—1228no—
469C22⋊C4—4882no—
470C22⋊C4—2444no—
471C42—24no—
472C42—1222no—
473C42—1222no—
474C42D4×D6112yes—
475C42C2×C22⋊D6112yes—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—1224no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23—2442no—
488C23—2442no—
489C23—2442no—
490C23C2×D4×D6111yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—41no—
495Q8—2444no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500D4—1224no—
501D4—1224no—
502D4—4884no—
503D4—2442no—
504D4—2442no—
505D4—2448no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—84no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—4888no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4#1451974512yes—
529C2×C4#1451974512yes—
530C2×C4#1451979812yes—
531C2×C4#1451980412yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C2×#1175822211yescharacteristic
535C2×C4C2×#1175822211yescharacteristic
536C2×C4#1451982311yescharacteristic
537C2×C4#1451982411yescharacteristic
538C8—22no—
539C8—41no—
540C8—2442no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—4882no—
549C22—4882no—
550C22—4882no—
551C22—4882no—
552C22—2444no—
553C22—2444no—
554C22—2444no—
555C22C2×#1451974511yescharacteristic
556C22#1684042111yescharacteristic
557C22#1684043311yescharacteristic
558C22C2×#1451979811yescharacteristic
559C22C2×#1451980411yescharacteristic
560C22#1684056411yescharacteristic
561C22#1684056911yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—4882no—
567C4—2442no—
568C4—2442no—
569C4#1683990614yes—
570C4#1684027311yescharacteristic
571C4#1684034011yescharacteristic
572C4#1684040811yescharacteristic
573C4#1684041711yescharacteristic
574C2—82no—
575C2—4884no—
576C2#2144916411yescharacteristic
577C2C2×#1683990611yescharacteristic
578C2#2145511911yescharacteristic
579C2#2145847711yescharacteristic
580C2#2145860911yescharacteristic
581C2#2145881311yescharacteristic
582C2#2145885911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenterC2×C4×SD16D61
hypocenterC61#24186
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16840497C2semidirect
3. #21450741C2semidirect
4. #21447056C2semidirect
5. #21446893C2semidirect
6. #21446755C2semidirect
7. #21446625C2semidirect
8. #21446164C2semidirect
9. #21446077C2semidirect
10. #21445824C2semidirect
11. C2×C4×SD16×C61C2semidirect
285. C2×C4×SD16D61semidirect
336. C61#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168399572
#214507412
#214470562
#214468932
#214467552
#214466252
#214461642
#214460772
#214458242
C2×C4×SD16×C612
#2418661