Finite Group Databasebeta

← #29734330

Subgroups — #29734330

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16840171C211yesmaximal, characteristic, complemented
3C2×#16840127C211yesmaximal, characteristic, complemented
4#21454129C211yesmaximal, characteristic, complemented
5#21447009C212yesmaximal, complemented
6#21447002C212yesmaximal, complemented
7#21446792C211yesmaximal, characteristic, complemented
8#21446643C211yesmaximal, characteristic, complemented
9#21446083C212yesmaximal, complemented
10#21446069C212yesmaximal, complemented
11C61×#1749C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840312—24no—
14#16840312—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519765—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2212yes—
23#16840171—24no—
24#16840127—24no—
25C2×#14519711C2212yes—
26C2×#14519709C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839646—21no—
34#16839646—21no—
35#16839635—21no—
36#16839635—21no—
37#16839628C2211yescharacteristic
38#16839622C2211yescharacteristic
39#16839622C2211yescharacteristic
40#16839525C2212yes—
41#16839525C2212yes—
42C22×C42×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C2×#97×C61C2211yescharacteristic
46C61×#336C2212yes—
47C2×#11758222—28no—
48C2×#11758222—44no—
49C2×#11758222—44no—
50C22×Dic122—21no—
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C22×C4×D61—22no—
56#14519777—22no—
57#14519777D414yes—
58#14519774—22no—
59#14519774D414yes—
60#14519769—22no—
61#14519769—22no—
62#14519765—28no—
63#14519751—28no—
64#14519751—24no—
65#14519751—24no—
66C4×D244—28no—
67C4×D244—24no—
68C4×D244—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83#14519711—24no—
84#14519709—24no—
85#14519699—24no—
86#14519699—24no—
87#14519699—24no—
88#14519699—24no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C23×C4×C61D411yescharacteristic
96C23×C4×C61D411yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C2×C42×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61D411yescharacteristic
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C22⋊C8×C61—24no—
107#97×C61—22no—
108#97×C61—22no—
109C24×C61C2×D411yescharacteristic
110C23×D61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—28no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C22×Dic61—22no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135#11758222—432no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—24no—
141C2×D244—24no—
142C2×D244—22no—
143C2×D244—42no—
144C2×D244—42no—
145C2×D244—42no—
146C2×D244—42no—
147C2×D244C2×D414yes—
148C2×C4×D61—28no—
149C2×C4×D61—44no—
150C2×C4×D61—44no—
151C2×Dic122—24no—
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#4114yescomplemented
160C42×C61—22no—
161C42×C61—22no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168#11758198—28no—
169#11758198—28no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61C2×D414yes—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182#11758196—42no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61AGL(1,ℤ/8)12yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—24no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C2×C4×C61—28no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—44no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4112yes—
220C2×C4×C612−1+412yes—
221C22⋊D61—416no—
222C2×Dic61—24no—
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—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237D244—24no—
238D244—24no—
239D244—48no—
240D244—48no—
241D244—42no—
242D244—42no—
243C4×D61—416no—
244Dic122—48no—
245C8×C61—24no—
246C8×C61—42no—
247#25676—611nomaximal, Sylow
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61C2×#12011yescharacteristic
256C22×C61#53211yescharacteristic
257C22×C61#53611yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57511yescharacteristic
261C22×C61#57511yescharacteristic
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—44no—
265C2×D61—82no—
266C2×D61—82no—
267C2×D61—82no—
268C2×D61—82no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61#54712yes—
275C4×C61#54812yes—
276Dic61—44no—
277Dic61—81no—
278Dic61—81no—
279Dic61—81no—
280Dic61—81no—
281C2×#532—611no—
282C2×#501—611no—
283C2×#490—611no—
284#2851—611no—
285#2034—612no—
286#2033—612no—
287#1993—611no—
288#1961—611no—
289#1864—612no—
290#1862—612no—
291#1749D6111yescharacteristic, complemented
292C2×C61—42no—
293C2×C61#243211yescharacteristic
294C2×C61#300311yescharacteristic
295C2×C61#300511yescharacteristic
296C2×C61#317111yescharacteristic
297C2×C61#318011yescharacteristic
298C2×C61#323411yescharacteristic
299C2×C61#323511yescharacteristic
300D61—84no—
301C22×#41—1221no—
302#532—1224no—
303#532—1224no—
304C2×C4⋊Q8—611no—
305C2×#127—611no—
306C2×#125—611no—
307C2×#124—1221no—
308C2×#121—612no—
309C2×#121—1221no—
310C2×C4×D4—612no—
311C2×C4×D4—1221no—
312C22×C42C2×D6111yescharacteristic
313#501—1224no—
314#490—1224no—
315C2×#107—612no—
316C2×#106—612no—
317C2×#102—612no—
318C2×#102—612no—
319C2×#102—612no—
320C2×#102—612no—
321C2×C22⋊C8C2×D6112yes—
322C2×#97C2×D6111yescharacteristic
323C2×#97C2×D6111yescharacteristic
324#389—611no—
325#389—611no—
326#387—1221no—
327#387—1221no—
328#385—1221no—
329#385—1221no—
330#382—611no—
331#380—611no—
332#380—611no—
333#340—612no—
334#340—612no—
335#336C2×D6112yes—
336C61#2567611yescharacteristic, complemented, Sylow
337C2×#41—1228no—
338C2×#41—2444no—
339C2×#41—2444no—
340C22×Q8—1221no—
341C22×D4—612no—
342C22×D4—1222no—
343C22×D4—1221no—
344C22×D4—1221no—
345C23×C4—1222no—
346C23×C4D24411yescharacteristic
347C23×C4D24411yescharacteristic
348C23×C4C22×D6111yescharacteristic
349C22×C8C22×D6112yes—
350C22×C8C22×D6112yes—
351C4⋊Q8—614no—
352C4⋊Q8—1222no—
353#127—614no—
354#127—1222no—
355#125—1222no—
356#125—1222no—
357#124—1228no—
358#121—1224no—
359#121—1224no—
360#121—1228no—
361C4×D4—1224no—
362C4×D4—1224no—
363C4×D4—1228no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
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×C4⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42D24411yescharacteristic
381C2×C42D24411yescharacteristic
382C2×C42C22×D6111yescharacteristic
383C2×C42C22×D6111yescharacteristic
384#107—1224no—
385#106—1224no—
386#102—1224no—
387#102—1224no—
388#102—1224no—
389#102—1224no—
390C22⋊C8—24no—
391#97—22no—
392#97—22no—
393#95—1221no—
394#95—1221no—
395#95—1221no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24—1222no—
400C24C2×D24411yescharacteristic
401#41—24432no—
402C2×Q8—1224no—
403C2×Q8—2442no—
404C2×Q8—2442no—
405C2×D4—614no—
406C2×D4—1228no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—2442no—
415C2×D4—2442no—
416C2×D4—2442no—
417C2×D4—2442no—
418C2×D4—2444no—
419C2×D4—2444no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—1228no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—2444no—
437C22×C4—2444no—
438C22×C4C2×D24411yescharacteristic
439C22×C4C2×D24411yescharacteristic
440C22×C4C2×D24411yescharacteristic
441C22×C4C2×D24411yescharacteristic
442C22×C4C2×D24411yescharacteristic
443C22×C4C23×D6111yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8#1175822214yescomplemented
450C4⋊C4—614no—
451C4⋊C4—1222no—
452C4⋊C4—1224no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C4⋊C4—2442no—
462C4⋊C4—2442no—
463C22⋊C4—1228no—
464C22⋊C4—1228no—
465C22⋊C4—2442no—
466C22⋊C4—2442no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—22no—
471C42D4×D6112yes—
472C42D4×D6112yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—1224no—
480C23—2442no—
481C23—2442no—
482C23—2442no—
483C23—2442no—
484C23—2442no—
485C23—2442no—
486C23#1451980512yes—
487C23C22×D24411yescharacteristic
488Q8—2448no—
489D4—1224no—
490D4—1224no—
491D4—2448no—
492D4—2448no—
493D4—24416no—
494D4—2442no—
495D4—2442no—
496C2×C4—28no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—1224no—
513C2×C4—24416no—
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—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4#1451974712yes—
529C2×C4#1451974712yes—
530C2×C4#1451980512yes—
531C2×C4C2×D4×D6111yescharacteristic
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4C2×#1175822212yes—
534C2×C4#1451982412yes—
535C8—24no—
536C8—42no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—4882no—
545C22—4882no—
546C22—4882no—
547C22—4882no—
548C22—2444no—
549C22—2444no—
550C22—2444no—
551C22C2×#1451974711yescharacteristic
552C22#1684031211yescharacteristic
553C22#1684033811yescharacteristic
554C22C2×#1451980511yescharacteristic
555C22C2×#1451980511yescharacteristic
556C22#1684056211yescharacteristic
557C22#1684056211yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—4881no—
564C4—4881no—
565C4—4881no—
566C4—4881no—
567C4—2444no—
568C4#1684041712yes—
569C4#1684042112yes—
570C2—42no—
571C2—4884no—
572C2#2144944711yescharacteristic
573C2#2145504911yescharacteristic
574C2#2145506411yescharacteristic
575C2#2145641411yescharacteristic
576C2#2145651711yescharacteristic
577C2#2145701111yescharacteristic
578C2#2145701911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenter#1749D61
hypocenterC61#25676
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#16840171C2semidirect
3. C2×#16840127C2semidirect
4. #21454129C2semidirect
5. #21447009C2semidirect
6. #21447002C2semidirect
7. #21446792C2semidirect
8. #21446643C2semidirect
9. #21446083C2semidirect
10. #21446069C2semidirect
11. C61×#1749C2semidirect
159. C2×C8×C61#41semidirect
291. #1749D61semidirect
336. C61#25676semidirect
449. 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×#168401712
C2×#168401272
#214541292
#214470092
#214470022
#214467922
#214466432
#214460832
#214460692
C61×#17492
#2567661