Finite Group Databasebeta

← #29708176

Subgroups — #29708176

Order 15616.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic, complemented
2C2×C4×#11758190C212yesmaximal, complemented
3C4×#14519751C211yesmaximal, characteristic
4#21447153C211yesmaximal, characteristic, complemented
5#21447066C211yesmaximal, characteristic, complemented
6#21447016C212yesmaximal, complemented
7#21446956C211yesmaximal, characteristic, complemented
8#21446904C211yesmaximal, characteristic, complemented
9#21446824C212yesmaximal
10C4×#14519702C212yesmaximal, complemented
11C4×C22⋊C8×C61C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13#16840319—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519756—21no—
18C2×#14519756C2212yes—
19C2×#14519751C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2212yes—
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519736—21no—
25C4×#11758190—28no—
26C2×#14519709C2212yes—
27C2×#14519702C2212yes—
28C2×#14519702C2212yes—
29C2×#14519702C2212yes—
30C2×#14519699C2212yes—
31#16839664C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839653—21no—
34#16839641—21no—
35#16839622C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40C22×C42×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2212yes—
42C2×C4×C8×C61C2212yes—
43C61×#336C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2312yes—
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C4×D61—21no—
50C2×#11758190—24no—
51C2×#11758190—24no—
52C2×#11758190—24no—
53#14519777—24no—
54#14519777—24no—
55#14519769—28no—
56#14519756—28no—
57#14519756—24no—
58#14519756—24no—
59#14519751—22no—
60#14519751—22no—
61C4×Dic122—28no—
62C4×Dic122—24no—
63C4×Dic122D418yescomplemented
64C4×D244—24no—
65C4×C22⋊D61—44no—
66#14519736—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—21no—
85#14519709—24no—
86#14519702—24no—
87#14519702—24no—
88#14519702—24no—
89#14519699—24no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C23×C4×C61D411yescharacteristic
94C23×C4×C61D411yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C22×C8×C61C2312yes—
97C22×C8×C61C2312yes—
98C2×C42×C61—24no—
99C2×C42×C61D411yescharacteristic
100C2×C42×C61D411yescharacteristic
101C2×C42×C61C2311yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C22⋊C8×C61—24no—
104C4×C8×C61—24no—
105C24×C61C2×D411yescharacteristic
106C23×D61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—41no—
116C22×C4×C61SD1614yescomplemented
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—21no—
134C2×D244#4112yes—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—24no—
143C2×Dic122—22no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C2×Dic122C2×D414yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61#4114yes—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61—22no—
156C42×C61C2×D414yes—
157C22⋊Dic61—28no—
158C22⋊Dic61—42no—
159#11758198—24no—
160#11758198—42no—
161#11758198—42no—
162#11758198—42no—
163C4⋊Dic61—24no—
164C4⋊Dic61—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61C2×D414yes—
171C4⋊Dic61#4112yes—
172#11758196—28no—
173#11758196—28no—
174#11758196—42no—
175C4×Dic61—28no—
176C4×Dic61—42no—
177#11758190—216no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61C2×SD1612yes—
185C23×C61C22×D411yescharacteristic
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—28no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61C2×SD1612yes—
209C2×C4×C61C2×SD1612yes—
210C2×C4×C61#13512yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D412yes—
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C61C2×#4111yescharacteristic
216C22⋊D61—48no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—22no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232D244—24no—
233D244—42no—
234C4×D61—84no—
235Dic122—24no—
236Dic122—24no—
237Dic122—48no—
238Dic122—42no—
239Dic122—42no—
240C8×C61—24no—
241C8×C61—42no—
242#23568—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61C2×#12011yescharacteristic
251C22×C61#52511yescharacteristic
252C22×C61#53211yescharacteristic
253C22×C61C22×SD1611yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61#57511yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61#44814yes—
269C4×C61#52512yes—
270C4×C61#53212yes—
271Dic61—42no—
272Dic61—44no—
273Dic61—44no—
274Dic61—81no—
275Dic61—81no—
276C2×#533—611no—
277C2×C4×SD16—612no—
278C4×#121—611no—
279#2058—611no—
280#2043—611no—
281#2035—612no—
282#2023—611no—
283#2013—611no—
284#2000—612no—
285C4×#103—612no—
286C4×C22⋊C8D6111yescharacteristic, complemented
287C2×C61—42no—
288C2×C61#238811yescharacteristic
289C2×C61C2×#44811yescharacteristic
290C2×C61#300211yescharacteristic
291C2×C61#310111yescharacteristic
292C2×C61#310711yescharacteristic
293C2×C61#316711yescharacteristic
294C2×C61#317211yescharacteristic
295D61—82no—
296C22×SD16—612no—
297#533—1228no—
298C2×C4⋊Q8—611no—
299C2×C4⋊Q8—611no—
300C2×#125—1221no—
301C2×C22⋊Q8—612no—
302C2×C22⋊Q8—1221no—
303C2×#121—611no—
304C2×C4×Q8—612no—
305C2×C4×Q8—1221no—
306C2×C4×D4—611no—
307C2×C4×D4—1221no—
308C2×#117—1221no—
309C22×C42C2×D6111yescharacteristic
310C4×SD16—1228no—
311C2×#106—612no—
312C2×#103—612no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#102—612no—
316C2×C22⋊C8C2×D6112yes—
317C2×C4×C8C2×D6112yes—
318#390—611no—
319#389—611no—
320#388—1221no—
321#386—1221no—
322#380—611no—
323C4×C4⋊C4—611no—
324C4×C22⋊C4—1221no—
325#340—612no—
326#340—612no—
327#336C2×D6111yescharacteristic
328#336C2×D6111yescharacteristic
329C61#2356811yescharacteristic, complemented, Sylow
330C22×Q8—612no—
331C22×Q8—1221no—
332C22×D4—611no—
333C22×D4—1221no—
334C23×C4—1221no—
335C23×C4D24411yescharacteristic
336C23×C4D24411yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C2×SD16—1224no—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C22×C8C22×D6112yes—
342C22×C8C22×D6112yes—
343C4⋊Q8—1224no—
344C4⋊Q8—1224no—
345#125—1228no—
346C22⋊Q8—1224no—
347C22⋊Q8—1224no—
348C22⋊Q8—1228no—
349#121—1222no—
350#121—1222no—
351C4×Q8—618no—
352C4×Q8—1228no—
353C4×Q8—1224no—
354C4×D4—1224no—
355C4×D4—2444no—
356#117—1228no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C42—24no—
374C2×C42—1222no—
375C2×C42—1221no—
376C2×C42D24411yescharacteristic
377C2×C42D24411yescharacteristic
378C2×C42C22×D6111yescharacteristic
379C2×C42C22×D6111yescharacteristic
380#106—1224no—
381#103—1224no—
382#103—1224no—
383#103—1224no—
384#102—1224no—
385C22⋊C8—24no—
386C4×C8—24no—
387#95—1221no—
388#95—1221no—
389#95—1221no—
390C24—1221no—
391C24C2×D24411yescharacteristic
392C2×Q8—614no—
393C2×Q8—1224no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×Q8—2442no—
400C2×Q8—2442no—
401C2×D4—612no—
402C2×D4—1224no—
403C2×D4—1222no—
404C2×D4—1221no—
405C2×D4—1222no—
406C2×D4—2442no—
407C2×D4—2442no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4#1175819014yescomplemented
426C22×C4C2×D24411yescharacteristic
427C22×C4C2×D24411yescharacteristic
428C22×C4C2×D24411yescharacteristic
429C22×C4C2×D24411yescharacteristic
430C22×C4C2×D24411yescharacteristic
431C22×C4C23×D6111yescharacteristic
432SD16—12216no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8#1175822214yes—
439C4⋊C4—614no—
440C4⋊C4—612no—
441C4⋊C4—1228no—
442C4⋊C4—1224no—
443C4⋊C4—1224no—
444C4⋊C4—1224no—
445C4⋊C4—1222no—
446C4⋊C4—1221no—
447C4⋊C4—1228no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—2442no—
451C22⋊C4—1224no—
452C22⋊C4—1228no—
453C22⋊C4—2442no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—1228no—
461C42—2442no—
462C42D4×D6114yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—1222no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23C2×#1175819012yes—
477C23C22×D24411yescharacteristic
478Q8—1224no—
479Q8—1224no—
480Q8—2448no—
481Q8—2442no—
482Q8—2442no—
483D4—1224no—
484D4—2448no—
485D4—2442no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1224no—
500C2×C4—1224no—
501C2×C4—1222no—
502C2×C4—4884no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4#1451974712yes—
516C2×C4#1451974712yes—
517C2×C4C2×#1175819012yes—
518C2×C4C2×#1175819012yes—
519C2×C4#1451979912yes—
520C2×C4#1451981012yes—
521C2×C4C2×D4×D6112yes—
522C2×C4#41×D6111yescharacteristic
523C2×C4#41×D6111yescharacteristic
524C2×C4C2×#1175822212yes—
525C8—24no—
526C8—42no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22C2×#1451974711yescharacteristic
542C22#1684027411yescharacteristic
543C22#1684031211yescharacteristic
544C22C22×#1175819011yescharacteristic
545C22C2×#1451979911yescharacteristic
546C22C2×#1451981011yescharacteristic
547C22#1684056211yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—4881no—
553C4—4881no—
554C4—2444no—
555C4—2444no—
556C4—2442no—
557C4#1683990314yes—
558C4#1684027112yes—
559C4#1684030912yes—
560C2—42no—
561C2—4882no—
562C2#2144903511yescharacteristic
563C2C2×#1683990311yescharacteristic
564C2#2145504111yescharacteristic
565C2#2145582811yescharacteristic
566C2#2145589111yescharacteristic
567C2#2145637811yescharacteristic
568C2#2145642511yescharacteristic
subgroupG/H
centerC22×C4C2×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×C22⋊C8×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC4×C22⋊C8D61
hypocenterC61#23568
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16840319C2semidirect
2. C2×C4×#11758190C2semidirect
4. #21447153C2semidirect
5. #21447066C2semidirect
6. #21447016C2semidirect
7. #21446956C2semidirect
8. #21446904C2semidirect
10. C4×#14519702C2semidirect
11. C4×C22⋊C8×C61C2semidirect
63. C4×Dic122D4semidirect
116. C22×C4×C61SD16semidirect
286. C4×C22⋊C8D61semidirect
329. C61#23568semidirect
425. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×C4×#117581902
C4×#145197512
#214471532
#214470662
#214470162
#214469562
#214469042
#214468242
C4×#145197022
C4×C22⋊C8×C612
#2356861