Finite Group Databasebeta

← #29732199

Subgroups — #29732199

Order 15616.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16839991C212yesmaximal, complemented
3#21454158C211yesmaximal, characteristic, complemented
4#21447153C211yesmaximal, characteristic, complemented
5#21447054C211yesmaximal, characteristic, complemented
6#21446961C211yesmaximal, characteristic, complemented
7#21446909C211yesmaximal, characteristic, complemented
8#21446247C212yesmaximal, complemented
9#21446152C212yesmaximal, complemented
10#21446083C212yesmaximal, complemented
11C2×C61×#336C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758190C2212yes—
14#16840312—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17C2×#14519765—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16839991—28no—
24C2×#14519709C2212yes—
25C2×#14519702C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29#16839668C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839646—21no—
34#16839646—21no—
35#16839635—21no—
36#16839625—21no—
37#16839622C2211yescharacteristic
38#16839525C2212yes—
39#16839525C2212yes—
40C23×C8×C61C2212yes—
41C22×C42×C61C2211yescharacteristic
42C61×#336C2212yes—
43C61×#336C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122—21no—
49C22×Dic122C2311yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244C2312yes—
53C22×C22⋊D61—22no—
54C22×C4×D61—22no—
55C2×#11758190—24no—
56C2×#11758190—24no—
57C2×#11758190—42no—
58#14519777—24no—
59#14519774—24no—
60#14519765—28no—
61#14519756—24no—
62#14519751—28no—
63#14519751—24no—
64#14519751D418yes—
65C4×D244—28no—
66C4×D244—24no—
67C4×D244—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80#14519709—24no—
81#14519702—24no—
82#14519699—24no—
83#14519699—24no—
84#14519699—42no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61—24no—
96C22×C8×C61—24no—
97C22×C8×C61—42no—
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C2×C42×C61—24no—
101C2×C42×C61D411yescharacteristic
102C2×C42×C61D411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C23×D61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—41no—
115C22×C4×C61SD1614yescomplemented
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D414yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—28no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131#11758222—432no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—24no—
137C2×D244—24no—
138C2×D244—22no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—42no—
142C2×D244—42no—
143C2×D244C2×D414yes—
144C2×C4×D61—28no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×Dic122—24no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—48no—
160C2×C8×C61#4114yescomplemented
161C42×C61—22no—
162C42×C61—22no—
163C42×C61—22no—
164C42×C61—22no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167#11758198—28no—
168#11758198—28no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61C2×D414yes—
176#11758196—42no—
177#11758196—42no—
178#11758196—42no—
179#11758190—48no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61#12012yes—
185C23×C61C2×SD1612yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61C22×D411yescharacteristic
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—28no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—44no—
210C2×C4×C61#12012yes—
211C2×C4×C61C2×SD1612yes—
212C2×C4×C61C2×SD1612yes—
213C2×C4×C61#13512yes—
214C2×C4×C61C22×D412yes—
215C2×C4×C61C2×#4112yes—
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C22⋊D61—416no—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234D244—24no—
235D244—24no—
236D244—48no—
237D244—48no—
238D244—42no—
239D244—42no—
240C4×D61—416no—
241Dic122—48no—
242Dic122—42no—
243Dic122—42no—
244C8×C61—24no—
245C8×C61—42no—
246#25495—611nomaximal, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61#45014yes—
254C22×C61C2×#12011yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61#53711yescharacteristic
257C22×C61C22×SD1611yescharacteristic
258C22×C61C2×#13511yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57511yescharacteristic
261C2×D61—44no—
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—82no—
265C2×D61—82no—
266C2×D61—82no—
267C2×D61—82no—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61#54612yes—
274C4×C61#54712yes—
275Dic61—44no—
276Dic61—81no—
277Dic61—81no—
278Dic61—81no—
279Dic61—81no—
280C2×#532—611no—
281C2×#465—612no—
282#2855—611no—
283#2058—611no—
284#2041—611no—
285#2024—611no—
286#2014—611no—
287#1893—612no—
288#1877—612no—
289#1864—612no—
290C2×#336D6111yescharacteristic, complemented
291C2×C61—24no—
292C2×C61#243011yescharacteristic
293C2×C61C2×#45011yescharacteristic
294C2×C61#300211yescharacteristic
295C2×C61#316711yescharacteristic
296C2×C61#318011yescharacteristic
297C2×C61#324611yescharacteristic
298C2×C61#324711yescharacteristic
299D61—84no—
300C22×#41—1221no—
301C22×SD16—612no—
302C23×C8C2×D6112yes—
303#532—1228no—
304C2×C4⋊Q8—611no—
305C2×#127—611no—
306C2×#124—1221no—
307C2×C22⋊Q8—611no—
308C2×#121—612no—
309C2×#121—1221no—
310C2×C4×D4—612no—
311C2×C4×D4—1221no—
312C22×C42C2×D6111yescharacteristic
313#465—1228no—
314C2×#106—612no—
315C2×#103—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102—612no—
319#391—611no—
320#391—611no—
321#389—611no—
322#389—611no—
323#387—1221no—
324#387—1221no—
325#385—1221no—
326#381—1221no—
327#380—611no—
328#340—612no—
329#340—612no—
330#336C2×D6111yescharacteristic
331#336C2×D6112yes—
332#336C2×D6111yescharacteristic
333C61#2549511yescharacteristic, complemented, Sylow
334C2×#41—1228no—
335C2×#41—2444no—
336C2×#41—2444no—
337C22×Q8—611no—
338C22×Q8—1221no—
339C22×D4—612no—
340C22×D4—1222no—
341C22×D4—1221no—
342C22×D4—1221no—
343C23×C4—1222no—
344C23×C4D24411yescharacteristic
345C23×C4D24411yescharacteristic
346C23×C4C22×D6111yescharacteristic
347C2×SD16—1224no—
348C2×SD16—1224no—
349C2×SD16—2442no—
350C22×C8—24no—
351C22×C8—24no—
352C22×C8—42no—
353C22×C8C22×D6112yes—
354C22×C8C22×D6112yes—
355C4⋊Q8—1224no—
356#127—1224no—
357#124—1228no—
358C22⋊Q8—1224no—
359#121—618no—
360#121—1228no—
361#121—1224no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1228no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C42—24no—
378C2×C42D24411yescharacteristic
379C2×C42D24411yescharacteristic
380C2×C42C22×D6111yescharacteristic
381C2×C42C22×D6111yescharacteristic
382#106—1224no—
383#103—1224no—
384#102—1224no—
385#102—1224no—
386#102—2442no—
387#95—1221no—
388#95—1221no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392#95—1221no—
393#95—1221no—
394C24—1222no—
395C24C2×D24411yescharacteristic
396#41—24432no—
397C2×Q8—1224no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×D4—614no—
405C2×D4—1224no—
406C2×D4—1224no—
407C2×D4—1228no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4—2444no—
416C2×D4—2444no—
417C2×D4—2442no—
418C2×D4—2442no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—1228no—
428C22×C4—1222no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4#1175819014yescomplemented
436C22×C4C2×D24411yescharacteristic
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4C2×D24411yescharacteristic
440C22×C4C2×D24411yescharacteristic
441C22×C4D4×D6114yes—
442C22×C4C23×D6111yescharacteristic
443SD16—2448no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—48no—
450C2×C8#1175822214yescomplemented
451C4⋊C4—614no—
452C4⋊C4—1224no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2442no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C22⋊C4—1228no—
462C22⋊C4—1228no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—1224no—
474C23—2442no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23#1451974712yes—
481C23C2×#1175819012yes—
482C23#1451980512yes—
483C23C22×D24411yescharacteristic
484Q8—2448no—
485Q8—2442no—
486Q8—2442no—
487D4—1224no—
488D4—1224no—
489D4—2448no—
490D4—2448no—
491D4—24416no—
492D4—2442no—
493D4—2442no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—1224no—
510C2×C4—24416no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4#1451974712yes—
526C2×C4C2×#1175819012yes—
527C2×C4C2×#1175819012yes—
528C2×C4#1451979912yes—
529C2×C4C2×D4×D6112yes—
530C2×C4C2×#1175822212yes—
531C2×C4#1451982211yescharacteristic
532C2×C4#1451982511yescharacteristic
533C8—24no—
534C8—42no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—4882no—
542C22—4882no—
543C22—4882no—
544C22—4882no—
545C22—2444no—
546C22—2444no—
547C22—2444no—
548C22#1683991514yes—
549C22C2×#1451974711yescharacteristic
550C22#1684031211yescharacteristic
551C22#1684034311yescharacteristic
552C22C22×#1175819011yescharacteristic
553C22C2×#1451979911yescharacteristic
554C22C2×#1451980511yescharacteristic
555C22#1684056211yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—4881no—
562C4—4881no—
563C4—4881no—
564C4—4881no—
565C4—2444no—
566C4#1684040212yes—
567C4#1684041312yes—
568C2—24no—
569C2—4884no—
570C2#2144942911yescharacteristic
571C2C2×#1683991511yescharacteristic
572C2#2145504111yescharacteristic
573C2#2145637811yescharacteristic
574C2#2145651711yescharacteristic
575C2#2145713611yescharacteristic
576C2#2145714611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#336C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenterC2×#336D61
hypocenterC61#25495
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
2. C2×#16839991C2semidirect
3. #21454158C2semidirect
4. #21447153C2semidirect
5. #21447054C2semidirect
6. #21446961C2semidirect
7. #21446909C2semidirect
8. #21446247C2semidirect
9. #21446152C2semidirect
10. #21446083C2semidirect
11. C2×C61×#336C2semidirect
115. C22×C4×C61SD16semidirect
160. C2×C8×C61#41semidirect
290. C2×#336D61semidirect
333. C61#25495semidirect
435. C22×C4#11758190semidirect
450. C2×C8#11758222semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168399912
#214541582
#214471532
#214470542
#214469612
#214469092
#214462472
#214461522
#214460832
C2×C61×#3362
#2549561