Finite Group Databasebeta

← #29571817

Subgroups — #29571817

Order 15616.

1364 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840324C211yesmaximal, characteristic
2#21446136C212yesmaximal, complemented
3C2×#16839495C212yesmaximal, complemented
4C2×#16839486C211yesmaximal, characteristic, complemented
5C61×#1439C211yesmaximal, characteristic, complemented
6C23×C22⋊D61—21no—
7#16840324—24no—
8#16840324—22no—
9#16840324C414yescomplemented
10C2×#14519761—22no—
11C2×#14519751—21no—
12C2×#14519751C2212yes—
13C2×#14519747—22no—
14C2×#14519697—22no—
15C2×C22⋊#92418C2212yescomplemented
16#16839524—22no—
17#16839495—24no—
18#16839486—22no—
19C22×C4⋊C4×C61C2211yescharacteristic, complemented
20C22×C22⋊C4×C61—21no—
21C22×C22⋊C4×C61C2211yescharacteristic
22C2×#95×C61—21no—
23C2×#95×C61C2211yescharacteristic, complemented
24C24×D61—21no—
25C22×D244—21no—
26C22×D244D412yes—
27C22×C22⋊D61—22no—
28C22×C22⋊D61—24no—
29C22×C22⋊D61—24no—
30C22×C22⋊D61—22no—
31C22×C22⋊D61—42no—
32C22×C22⋊D61—42no—
33C22×C22⋊D61—42no—
34C22×C22⋊D61—42no—
35C23×Dic61—21no—
36C22×#92418—22no—
37#14519761—28no—
38#14519761—28no—
39#14519751—24no—
40#14519751—24no—
41#14519751—22no—
42#14519751—22no—
43#14519751C2×C414yes—
44#14519751D414yes—
45#14519747—28no—
46#14519747—28no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61D412yes—
49C2×#11758198—22no—
50C2×#11758198—22no—
51#14519697—44no—
52C22⋊#92418—22no—
53C22⋊#92418—22no—
54C25×C61D411yescharacteristic
55C23×C4×C61—21no—
56C23×C4×C61—21no—
57C23×C4×C61D411yescharacteristic
58C23×C4×C61C2311yescharacteristic
59C2×C4⋊C4×C61—22no—
60C2×C4⋊C4×C61—42no—
61C2×C22⋊C4×C61—22no—
62C2×C22⋊C4×C61—22no—
63C2×C22⋊C4×C61—42no—
64C2×C22⋊C4×C61—42no—
65C2×C22⋊C4×C61—42no—
66C2×C22⋊C4×C61C2×C412yes—
67C2×C22⋊C4×C61D412yes—
68#95×C61—22no—
69#95×C61—42no—
70C24×C61—21no—
71C24×C61—21no—
72C24×C61—21no—
73C24×C61—21no—
74C24×C61—41no—
75C24×C61—41no—
76C24×C61—41no—
77C24×C61—41no—
78C24×C61C22⋊C412yes—
79C24×C61D812yes—
80C24×C61SD1612yes—
81C24×C61C2×D411yescharacteristic
82C23×D61—22no—
83C23×D61—24no—
84C23×D61—42no—
85C23×D61—42no—
86C23×D61—81no—
87C23×D61—81no—
88C23×D61—81no—
89C23×D61—81no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61C22⋊C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C2×C22⋊D61—28no—
116C2×C22⋊D61—24no—
117C2×C22⋊D61—24no—
118C2×C22⋊D61—48no—
119C2×C22⋊D61—48no—
120C2×C22⋊D61—48no—
121C2×C22⋊D61—44no—
122C2×C22⋊D61—44no—
123C2×C22⋊D61—48no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—24no—
129C22×Dic61—22no—
130C22×Dic61—42no—
131C22×Dic61—42no—
132C2×D244—24no—
133C2×D244—22no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—81no—
137C2×D244—81no—
138C2×D244C22⋊C414yescomplemented
139C4⋊C4×C61—24no—
140C4⋊C4×C61—42no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—48no—
144#11758198—28no—
145#11758198—24no—
146#11758198—24no—
147C4⋊Dic61—24no—
148C4⋊Dic61—22no—
149C4⋊Dic61C22⋊C414yescomplemented
150C2×#92418—42no—
151C2×#92418—42no—
152C2×#92418—42no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—22no—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—81no—
174C23×C61—81no—
175C23×C61—81no—
176C23×C61—81no—
177C23×C61—81no—
178C23×C61—81no—
179C23×C61#10214yescomplemented
180C23×C61C2×C22⋊C411yescharacteristic
181C23×C61#12011yescharacteristic
182C23×C61#12011yescharacteristic
183C23×C61C2×D811yescharacteristic
184C23×C61C2×SD1611yescharacteristic
185C23×C61AGL(1,ℤ/8)11yescharacteristic
186C23×C61AGL(1,ℤ/8)11yescharacteristic
187C22×D61—24no—
188C22×D61—48no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—42no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—84no—
196C22×D61—84no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61—84no—
200C22×D61—84no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—81no—
217C2×C4×C61—81no—
218C2×C4×C61—81no—
219C2×C4×C61—81no—
220C2×C4×C61—81no—
221C2×C4×C61—81no—
222C2×C4×C61—81no—
223C2×C4×C61—81no—
224C2×C4×C61—81no—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61#12012yes—
227C22⋊D61—416no—
228C22⋊D61—44no—
229C22⋊D61—44no—
230C22⋊D61—44no—
231C22⋊D61—44no—
232C2×Dic61—24no—
233C2×Dic61—48no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236D244—44no—
237D244—82no—
238D244—84no—
239#92418—82no—
240#8403—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—24no—
254C22×C61—44no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61#37911yescharacteristic
265C22×C61C2×#10211yescharacteristic
266C22×C61#41811yescharacteristic
267C22×C61#44711yescharacteristic
268C22×C61#44711yescharacteristic
269C22×C61#45011yescharacteristic
270C22×C61#45011yescharacteristic
271C22×C61#45311yescharacteristic
272C22×C61#45511yescharacteristic
273C22×C61B(4,𝔽2)11yescharacteristic
274C22×C61#45811yescharacteristic
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—816no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61—82no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—82no—
287C4×C61—82no—
288Dic61—44no—
289C2×#534—611no—
290#1874—612no—
291C2×#327—612no—
292C2×#323—611no—
293#1439D6111yescharacteristic, complemented
294C2×C61—21no—
295C2×C61—21no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—44no—
299C2×C61#159511yescharacteristic
300C2×C61#160111yescharacteristic
301C2×C61#160711yescharacteristic
302C2×C61#161311yescharacteristic
303C2×C61#178611yescharacteristic
304C2×C61#179011yescharacteristic
305C2×C61#179211yescharacteristic
306D61—84no—
307C23×D4—1221no—
308#534—614no—
309#534—1224no—
310#534—1222no—
311C2×#123—1222no—
312C2×#121—612no—
313C2×#121—1221no—
314C2×#120—1222no—
315C22×C4⋊C4C2×D6111yescharacteristic, complemented
316C22×C22⋊C4—21no—
317C22×C22⋊C4C2×D6111yescharacteristic
318C2×#102—1222no—
319C2×C22⋊C8—612no—
320C2×#95—21no—
321C2×#95C2×D6111yescharacteristic, complemented
322#340—1222no—
323#327—1224no—
324#323—1222no—
325C61#840311yescharacteristic, complemented, Sylow
326C25—1221no—
327C25C22⋊D6111yescharacteristic
328C22×D4—612no—
329C22×D4—1221no—
330C22×D4—1222no—
331C22×D4—1224no—
332C22×D4—1224no—
333C22×D4—1222no—
334C22×D4—2442no—
335C22×D4—2442no—
336C22×D4—2442no—
337C22×D4—2442no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4—1221no—
341C23×C4C22⋊D6111yescharacteristic
342C23×C4C22×D6111yescharacteristic
343C22×C8—1222no—
344#123—1228no—
345#123—1228no—
346#121—614no—
347#121—614no—
348#121—1224no—
349#121—1224no—
350#121—1222no—
351#121—1222no—
352#120—1228no—
353#120—1228no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—1221no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4C4×D6112yes—
366C2×C22⋊C4D24412yes—
367#102—2444no—
368C22⋊C8—1222no—
369C22⋊C8—1222no—
370#95—22no—
371#95—42no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—1224no—
381C24—1222no—
382C24—4881no—
383C24—4881no—
384C24—4881no—
385C24—4881no—
386C24—2442no—
387C24—2442no—
388C24#1175819812yes—
389C24#1175819912yes—
390C24#1175820112yes—
391C24C2×C22⋊D6111yescharacteristic
392C2×D4—614no—
393C2×D4—1224no—
394C2×D4—1228no—
395C2×D4—1224no—
396C2×D4—1224no—
397C2×D4—1222no—
398C2×D4—4881no—
399C2×D4—4881no—
400C2×D4—2442no—
401C2×D4—2442no—
402C2×D4—2448no—
403C2×D4—2448no—
404C2×D4—2448no—
405C2×D4—2444no—
406C2×D4—2444no—
407C2×D4—2448no—
408C2×D4—2442no—
409C2×D4—2442no—
410C2×D4—2442no—
411C2×D4—2442no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—1224no—
433C22×C4—1222no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4#1175819812yes—
437C22×C4C2×C4×D6111yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4D4×D6112yes—
440C22×C4D4×D6112yes—
441C2×C8—2442no—
442C2×C8—2442no—
443C2×C8—2442no—
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—614no—
447C4⋊C4—1224no—
448C4⋊C4—1222no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—48no—
452C22⋊C4—1228no—
453C22⋊C4—1224no—
454C22⋊C4—1224no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—1224no—
482C23—4884no—
483C23—4884no—
484C23—4884no—
485C23—4884no—
486C23—4884no—
487C23—4884no—
488C23—2448no—
489C23—2442no—
490C23—2442no—
491C23—2442no—
492C23—2442no—
493C23—2442no—
494C23—2442no—
495C23#1451969714yescomplemented
496C23C2×#1175819811yescharacteristic
497C23#1451974511yescharacteristic
498C23#1451974511yescharacteristic
499C23C2×#1175819911yescharacteristic
500C23C2×#1175820111yescharacteristic
501C23#1451980211yescharacteristic
502C23#1451980211yescharacteristic
503D4—4882no—
504D4—4884no—
505D4—2444no—
506D4—24416no—
507D4—2444no—
508D4—2444no—
509D4—2444no—
510D4—2444no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—1224no—
536C2×C4—2448no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4C22⋊C4×D6112yes—
540C2×C4#1451974712yes—
541C8—4882no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—24no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—48816no—
566C22—4882no—
567C22—4882no—
568C22—4882no—
569C22—4882no—
570C22—2444no—
571C22—2444no—
572C22—2444no—
573C22#1683961711yescharacteristic
574C22C2×#1451969711yescharacteristic
575C22#1683976611yescharacteristic
576C22#1683989511yescharacteristic
577C22#1683989511yescharacteristic
578C22#1683991211yescharacteristic
579C22#1683991211yescharacteristic
580C22#1683992911yescharacteristic
581C22#1683993911yescharacteristic
582C22#1683994811yescharacteristic
583C22#1683995111yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—2444no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2—4884no—
596C2#2144473611yescharacteristic
597C2#2144477011yescharacteristic
598C2#2144480411yescharacteristic
599C2#2144483811yescharacteristic
600C2#2144574911yescharacteristic
601C2#2144576611yescharacteristic
602C2#2144577211yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1439C2
socleC23×C61#120
prime radicalC2×#16840324C2
cosocleC23×C4×C61C23
hypercenter#1439D61
hypocenterC61#8403
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. #21446136C2semidirect
3. C2×#16839495C2semidirect
4. C2×#16839486C2semidirect
5. C61×#1439C2semidirect
9. #16840324C4semidirect
15. C2×C22⋊#92418C22semidirect
19. C22×C4⋊C4×C61C22semidirect
23. C2×#95×C61C22semidirect
138. C2×D244C22⋊C4semidirect
149. C4⋊Dic61C22⋊C4semidirect
179. C23×C61#102semidirect
293. #1439D61semidirect
315. C22×C4⋊C4C2×D61semidirect
321. C2×#95C2×D61semidirect
325. C61#8403semidirect
495. C23#14519697semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214461362
C2×#168394952
C2×#168394862
C61×#14392
#840361