Finite Group Databasebeta

← #29580265

Subgroups — #29580265

Order 15616.

1278 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839913C211yesmaximal, characteristic, complemented
2#21451624C211yesmaximal, characteristic, complemented
3#21445750C211yesmaximal, characteristic, complemented
4C2×#16839498C211yesmaximal, characteristic, complemented
5C2×#16839486C211yesmaximal, characteristic, complemented
6#21443698C211yesmaximal, characteristic, complemented
7C2×C61×#392C211yesmaximal, characteristic
8C22×#11758200—21no—
9C2×#14519756C2211yescharacteristic
10#16839913—22no—
11#16839913—22no—
12C2×#14519698—21no—
13C2×#14519698—22no—
14C2×C22⋊#92418C2211yescharacteristic
15C2×C22⋊#92418C2211yescharacteristic
16#16839698—21no—
17#16839685—22no—
18#16839674—21no—
19#16839635—21no—
20#16839622C2211yescharacteristic
21#16839618—21no—
22#16839498—22no—
23#16839486—22no—
24C23×D4×C61—21no—
25C23×D4×C61C2211yescharacteristic
26C22×C22⋊C4×C61—21no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C61×#392—22no—
29C61×#392—22no—
30C2×#95×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic, complemented
32C2×#11758200—24no—
33C2×#11758200—44no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61D411yescharacteristic, complemented
41C2×C4⋊Dic61D412yescomplemented
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—22no—
48#14519698—24no—
49#14519698—44no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418D412yescomplemented
54#14519684—22no—
55#14519684—21no—
56C25×C61—21no—
57C25×C61D411yescharacteristic
58C22×D4×C61—24no—
59C22×D4×C61—21no—
60C22×D4×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—44no—
65C22×D4×C61—44no—
66C22×D4×C61—44no—
67C22×D4×C61—42no—
68C22×D4×C61—42no—
69C22×D4×C61D412yes—
70C23×C4×C61—21no—
71C23×C4×C61D411yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—44no—
81#95×C61—21no—
82#95×C61D412yes—
83C24×C61—24no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61—21no—
87C24×C61—21no—
88C24×C61—21no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—42no—
92C24×C61—42no—
93C24×C61—42no—
94C24×C61—42no—
95C24×C61—42no—
96C24×C61—41no—
97C24×C61—41no—
98C24×C61—41no—
99C24×C61—41no—
100C24×C61—41no—
101C24×C61—41no—
102C24×C61SD1612yescomplemented
103C24×C61C2×D411yescharacteristic
104C2×D4×C61—24no—
105C2×D4×C61—24no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—42no—
109C2×D4×C61—42no—
110C2×D4×C61—48no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61—44no—
115C2×D4×C61—48no—
116C2×D4×C61—48no—
117C2×D4×C61—416no—
118C2×D4×C61—416no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—41no—
129C22×C4×C61—41no—
130C22×C4×C61—42no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4111yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—42no—
143C22⋊C4×C61—48no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C22⋊Dic61—82no—
153C22⋊Dic61—82no—
154#11758200—48no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—42no—
158#11758196—41no—
159#11758196—41no—
160#11758196—82no—
161C2×#92418—22no—
162C2×#92418—42no—
163C2×#92418—42no—
164C2×#92418—41no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—44no—
172C23×C61—42no—
173C23×C61—44no—
174C23×C61—44no—
175C23×C61—44no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—44no—
181C23×C61—44no—
182C23×C61—44no—
183C23×C61—41no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—44no—
192C23×C61—41no—
193C23×C61—82no—
194C23×C61—82no—
195C23×C61—82no—
196C23×C61—82no—
197C23×C61—82no—
198C23×C61—82no—
199C23×C61—82no—
200C23×C61—81no—
201C23×C61—81no—
202C23×C61—81no—
203C23×C61—81no—
204C23×C61—82no—
205C23×C61—81no—
206C23×C61—81no—
207C23×C61#12011yescharacteristic
208C23×C61#12011yescharacteristic
209C23×C61#12111yescharacteristic
210C23×C61C2×SD1611yescharacteristic
211C23×C61#13511yescharacteristic
212C23×C61AGL(1,ℤ/8)11yescharacteristic
213C23×C61#13711yescharacteristic
214D4×C61—24no—
215D4×C61—24no—
216D4×C61—42no—
217D4×C61—44no—
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—432no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—81no—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61#12711yescharacteristic
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247Dic122—42no—
248Dic122—81no—
249#92418—42no—
250#92418—81no—
251#9922—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61—42no—
265C22×C61—44no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61—84no—
271C22×C61—82no—
272C22×C61—84no—
273C22×C61—84no—
274C22×C61—82no—
275C22×C61—84no—
276C22×C61—82no—
277C22×C61—81no—
278C22×C61—82no—
279C22×C61—81no—
280C22×C61#39211yescharacteristic
281C22×C61#44811yescharacteristic
282C22×C61#44911yescharacteristic
283C22×C61#45011yescharacteristic
284C22×C61#45211yescharacteristic
285C22×C61#45312yes—
286C22×C61B(4,𝔽2)12yes—
287C22×C61#46511yescharacteristic
288C22×C61#46811yescharacteristic
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—44no—
294C4×C61—81no—
295Dic61—82no—
296Dic61—81no—
297Dic61—81no—
298C2×#450—611no—
299#2626—611no—
300C2×#392D6111yescharacteristic
301#1786—611no—
302C2×#328—611no—
303C2×#323—611no—
304#1294—611no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—82no—
310C2×C61—84no—
311C2×C61#159612yes—
312C2×C61#160012yes—
313C2×C61#200011yescharacteristic
314C2×C61#200211yescharacteristic
315C2×C61#202111yescharacteristic
316C23×D4—21no—
317C23×D4C2×D6111yescharacteristic
318C22×SD16—1221no—
319C2×C22⋊Q8—611no—
320C22×C22⋊C4—21no—
321C22×C22⋊C4C2×D6111yescharacteristic
322#450—1222no—
323#450—1222no—
324C2×#102—1221no—
325C2×#102—1222no—
326C2×C22⋊C8—611no—
327C2×C22⋊C8—611no—
328#399—1221no—
329#396—1222no—
330#393—1221no—
331#392—22no—
332#392—22no—
333#385—1221no—
334#380—611no—
335#379—1221no—
336C2×#95C2×D6111yescharacteristic
337#328—1222no—
338#323—1222no—
339C61#992211yescharacteristic, complemented, Sylow
340C25—21no—
341C25C22⋊D6111yescharacteristic
342C22×Q8—611no—
343C22×D4—24no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4—44no—
350C22×D4—44no—
351C22×D4—44no—
352C22×D4—42no—
353C22×D4—42no—
354C22×D4C22⋊D6112yes—
355C23×C4—21no—
356C23×C4C22⋊D6111yescharacteristic
357C23×C4C22×D6111yescharacteristic
358C2×SD16—1224no—
359C2×SD16—2444no—
360C22×C8—1221no—
361C22×C8—1221no—
362C22⋊Q8—1221no—
363C22⋊Q8—1221no—
364C22⋊Q8—1221no—
365C22⋊Q8—1221no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382#102—1224no—
383#102—2444no—
384C22⋊C8—612no—
385C22⋊C8—1221no—
386C22⋊C8—1221no—
387C22⋊C8—1221no—
388#95—21no—
389#95—1222no—
390#95—1221no—
391#95C22⋊D6112yes—
392C24—24no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24#1175820012yescomplemented
412C24C2×C22⋊D6111yescharacteristic
413C2×Q8—1221no—
414C2×Q8—1221no—
415C2×Q8—2442no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—416no—
431C2×D4—41no—
432C2×D4—41no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4D4×D6111yescharacteristic
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4#1175822411yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452SD16—2448no—
453C2×C8—1222no—
454C2×C8—2442no—
455C2×C8—2441no—
456C2×C8—2442no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—4882no—
460C4⋊C4—2442no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C22⋊C4—48no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—4882no—
469C22⋊C4—4882no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C22⋊C4—2441no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—82no—
503C23—81no—
504C23—82no—
505C23—81no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23#1451974511yescharacteristic
517C23#1451974611yescharacteristic
518C23#1451974911yescharacteristic
519C23C2×#1175820011yescharacteristic
520C23#1451979711yescharacteristic
521C23#1451980311yescharacteristic
522C23#1451980711yescharacteristic
523Q8—4881no—
524Q8—2442no—
525D4—24no—
526D4—24no—
527D4—44no—
528D4—42no—
529D4—44no—
530D4—42no—
531D4—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—81no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4#1451974511yescharacteristic
555C2×C4#1451974811yescharacteristic
556C2×C4#1451974911yescharacteristic
557C2×C4#1451977311yescharacteristic
558C8—4881no—
559C8—2442no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22—82no—
580C22—84no—
581C22—81no—
582C22—84no—
583C22—82no—
584C22—84no—
585C22—82no—
586C22—81no—
587C22—82no—
588C22#1683967011yescharacteristic
589C22#1683990011yescharacteristic
590C22#1683990511yescharacteristic
591C22#1683991311yescharacteristic
592C22#1683992411yescharacteristic
593C22#1683992912yes—
594C22#1683994812yes—
595C22#1683998911yescharacteristic
596C22#1684000611yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—81no—
603C4—4881no—
604C4—4882no—
605C4—4881no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—84no—
612C2#2144474112yes—
613C2#2144476312yes—
614C2#2144682211yescharacteristic
615C2#2144683411yescharacteristic
616C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
Fitting, prime radicalC2×C61×#392C2
socleC23×C61#120
hypercenterC2×#392D61
hypocenterC61#9922
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
1. C2×#16839913C2semidirect
2. #21451624C2semidirect
3. #21445750C2semidirect
4. C2×#16839498C2semidirect
5. C2×#16839486C2semidirect
6. #21443698C2semidirect
31. C22×Dic122D4semidirect
40. C2×C4⋊Dic61D4semidirect
41. C2×C4⋊Dic61D4semidirect
53. C22⋊#92418D4semidirect
102. C24×C61SD16semidirect
339. C61#9922semidirect
411. C24#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399132
#214516242
#214457502
C2×#168394982
C2×#168394862
#214436982
C2×C61×#3922
#992261