Finite Group Databasebeta

← #29713283

Subgroups — #29713283

Order 15616.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2C2×#16840079C212yesmaximal, complemented
3#389×D61C211yesmaximal, characteristic, complemented
4#21447153C211yesmaximal, characteristic, complemented
5#21447152C211yesmaximal, characteristic, complemented
6#21447050C211yesmaximal, characteristic, complemented
7#21446133C212yesmaximal, complemented
8#21446082C212yesmaximal, complemented
9#21446078C212yesmaximal, complemented
10#21445886C211yesmaximal, characteristic, complemented
11C61×#2041C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13#16840310—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519764—21no—
16C2×#14519748—21no—
17C2×#14519748C2212yes—
18C2×D4×Dic61—21no—
19C2×D4×Dic61C2212yes—
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16840079—28no—
25C2×#14519709C2212yes—
26C2×#14519708C2212yes—
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519689C2212yes—
30#16839661C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839659—21no—
33#16839658C2211yescharacteristic
34#16839657—21no—
35#95×D61—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C2×#102×C61C2212yes—
43C61×#389C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C2×#11758224—28no—
46C2×#11758224—44no—
47C2×#11758224—44no—
48C22×Dic122—21no—
49C22×C22⋊D61—22no—
50C23×Dic61—22no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×#92418C2312yes—
57#14519777—24no—
58#14519764—28no—
59#14519748—28no—
60#14519748—24no—
61#14519748D418yes—
62D4×Dic61—28no—
63D4×Dic61—24no—
64D4×Dic61—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—22no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—44no—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—22no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—24no—
84#14519709—24no—
85#14519708—24no—
86#14519698—24no—
87#14519698—24no—
88#14519689—24no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61C2312yes—
95C22×C8×C61C2312yes—
96#127×C61—24no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C42×C61C2311yescharacteristic
101#102×C61—24no—
102#102×C61—24no—
103#95×C61—21no—
104C24×C61—22no—
105C23×D61C2×D411yescharacteristic
106C2×D4×C61—24no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61C2×D414yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
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—28no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—44no—
131C22×Dic61—44no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135#11758224—432no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—24no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—41no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61SD1614yes—
151C2×C4×D61C2×D414yes—
152C2×Dic122—24no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61#4112yescomplemented
162C42×C61—22no—
163C22⋊Dic61—28no—
164C22⋊Dic61—28no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61C2×D414yes—
171C4⋊Dic61#4112yescomplemented
172#11758196—24no—
173#11758196—42no—
174#11758196—42no—
175#11758196—42no—
176C4×Dic61—22no—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—24no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61C22×D411yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61#12012yes—
195C22×D61C2×SD1612yes—
196C22×D61AGL(1,ℤ/8)12yes—
197D4×C61—24no—
198D4×C61—24no—
199D4×C61—48no—
200D4×C61—48no—
201D4×C61—42no—
202D4×C61—42no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61C2×SD1612yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D412yes—
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C612−1+412yes—
219C22⋊D61—416no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—24no—
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—416no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#12012yes—
238C2×Dic61C2×SD1612yes—
239C4×D61—28no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—44no—
243C4×D61—48no—
244C4×D61—84no—
245Dic122—48no—
246C8×C61—42no—
247#92418—42no—
248#24061—611nomaximal, Sylow
249C22×C61—44no—
250C22×C61—44no—
251C22×C61—44no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#52511yescharacteristic
258C22×C61#53611yescharacteristic
259C22×C61C22×SD1611yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61#57511yescharacteristic
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61#45014yes—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61#54712yes—
275C4×C61#54712yes—
276Dic61—22no—
277Dic61—22no—
278Dic61—42no—
279Dic61—44no—
280Dic61—41no—
281Dic61—81no—
282C2×#532—611no—
283C2×#481—612no—
284C2×#389—611no—
285#2058—611no—
286#2058—611no—
287#2041—611no—
288#2041D6111yescharacteristic, complemented
289#1874—612no—
290#1864—612no—
291#1864—612no—
292#1817—611no—
293C2×C61—84no—
294C2×C61#240611yescharacteristic
295C2×C61C2×#45011yescharacteristic
296C2×C61#300611yescharacteristic
297C2×C61#310311yescharacteristic
298C2×C61#310911yescharacteristic
299C2×C61#322711yescharacteristic
300C2×C61#323411yescharacteristic
301D61—24no—
302C22×#41—1221no—
303#532—1228no—
304C2×C4⋊Q8—611no—
305C2×#127C2×D6111yescharacteristic
306C2×#124—1221no—
307C2×#121—612no—
308C2×#121—1221no—
309C2×C4×D4—612no—
310C2×C4×D4—1221no—
311C22×C4⋊C4—611no—
312C22×C4⋊C4—611no—
313C22×C4⋊C4—1221no—
314C22×C42—611no—
315#481—1228no—
316C2×#106—612no—
317C2×#106—612no—
318C2×#102—612no—
319C2×#102—612no—
320C2×#102C2×D6112yes—
321C2×#102C2×D6112yes—
322C2×C22⋊C8—612no—
323#389—611no—
324#389—611no—
325#389—611no—
326#389—1221no—
327#389—1221no—
328#389C2×D6111yescharacteristic
329C2×#95—1221no—
330#340—612no—
331#340—612no—
332#336—612no—
333#336—611no—
334#336C2×D6111yescharacteristic
335C61#2406111yescharacteristic, complemented, Sylow
336C2×#41—1228no—
337C2×#41—2444no—
338C2×#41—2444no—
339C22×Q8—1221no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4—1222no—
343C22×D4C22×D6112yes—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—1221no—
348C23×C4—1221no—
349C23×C4—1222no—
350C22×C8—612no—
351C22×C8C22×D6112yes—
352C4⋊Q8—1224no—
353#127—24no—
354#124—1228no—
355#121—618no—
356#121—1224no—
357#121—1228no—
358C4×D4—1224no—
359C4×D4—1228no—
360C4×D4—1224no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—2444no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—1224no—
383C2×C42C22×D6111yescharacteristic
384#106—1224no—
385#106—1224no—
386#102—24no—
387#102—24no—
388#102—1224no—
389#102—1224no—
390C22⋊C8—1224no—
391#95—21no—
392#95—1221no—
393#95—1221no—
394#95—1221no—
395C24—22no—
396C24—611no—
397#41—24432no—
398C2×Q8—1224no—
399C2×Q8—2442no—
400C2×Q8—2442no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—1228no—
413C2×D4—2444no—
414C2×D4—2444no—
415C2×D4D4×D6114yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—611no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1228no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—2441no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4—2444no—
442C22×C4—2444no—
443C22×C4D4×D6111yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—1222no—
450C2×C8—1222no—
451C2×C8—1222no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—614no—
456C4⋊C4—612no—
457C4⋊C4—1224no—
458C4⋊C4—1224no—
459C4⋊C4—1222no—
460C4⋊C4—1221no—
461C4⋊C4—1222no—
462C4⋊C4—2442no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C4⋊C4#1175822412yescomplemented
467C22⋊C4—1228no—
468C22⋊C4—1228no—
469C42—22no—
470C42—1222no—
471C42—1222no—
472C42—1222no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—612no—
481C23—612no—
482C23—612no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23—1221no—
487C23C2×D4×D6111yescharacteristic
488Q8—2448no—
489D4—24no—
490D4—24no—
491D4—48no—
492D4—48no—
493D4—42no—
494D4—42no—
495D4—24416no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—1228no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1224no—
514C2×C4—1224no—
515C2×C4—1221no—
516C2×C4—1224no—
517C2×C4—1222no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—4884no—
521C2×C4—2444no—
522C2×C4—2448no—
523C2×C4—24416no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4SD16×D6112yes—
532C2×C4#1451980112yes—
533C2×C4C2×D4×D6112yes—
534C2×C4#41×D6111yescharacteristic
535C2×C4C2×#1175822411yescharacteristic
536C2×C4#1451982412yes—
537C8—42no—
538C8—2442no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—614no—
547C22—1222no—
548C22—1222no—
549C22—1222no—
550C22—1222no—
551C22—1222no—
552C22—1222no—
553C22#120×D6111yescharacteristic
554C22#1684027211yescharacteristic
555C22#1684033211yescharacteristic
556C22C2×SD16×D6111yescharacteristic
557C22AGL(1,ℤ/8)×D6111yescharacteristic
558C22C2×#1451980111yescharacteristic
559C22#1684055811yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—1222no—
565C4—1222no—
566C4—4881no—
567C4—2442no—
568C4—2444no—
569C4—2441no—
570C4#1684041012yes—
571C4#1684041212yes—
572C2—84no—
573C2—1224no—
574C2#2144914911yescharacteristic
575C2#450×D6111yescharacteristic
576C2#2145506611yescharacteristic
577C2#2145584911yescharacteristic
578C2#2145590911yescharacteristic
579C2#2145694311yescharacteristic
580C2#2145700411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#16840310C2
hypercenter#2041D61
hypocenterC61#24061
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#16840079C2semidirect
3. #389×D61C2semidirect
4. #21447153C2semidirect
5. #21447152C2semidirect
6. #21447050C2semidirect
7. #21446133C2semidirect
8. #21446082C2semidirect
9. #21446078C2semidirect
10. #21445886C2semidirect
11. C61×#2041C2semidirect
161. C4⋊C4×C61#41semidirect
171. C4⋊Dic61#41semidirect
288. #2041D61semidirect
335. C61#24061semidirect
466. C4⋊C4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#168400792
#389×D612
#214471532
#214471522
#214470502
#214461332
#214460822
#214460782
#214458862
C61×#20412
#2406161