Finite Group Databasebeta

← #29715683

Subgroups — #29715683

Order 15616.

1606 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758200C211yesmaximal, characteristic, complemented
2C2×#16839989C212yesmaximal, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21449184C211yesmaximal, characteristic, complemented
5#21447136C212yesmaximal, complemented
6#21446822C212yesmaximal, complemented
7#21446672C212yesmaximal, complemented
8#21445842C211yesmaximal, characteristic, complemented
9#21445753C211yesmaximal, characteristic, complemented
10#21445750C211yesmaximal, characteristic, complemented
11C61×#2408C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200—28no—
14C22×#11758200—24no—
15C22×#11758200C2212yes—
16C22×#11758200C2212yes—
17C23×#92418C2211yescharacteristic
18C23×#92418C2211yescharacteristic, complemented
19C2×#14519756C2212yescomplemented
20#16839989—28no—
21D4×#92418—24no—
22C2×#14519708C2212yescomplemented
23C2×C4⋊#92418C2211yescharacteristic, complemented
24C2×#14519700C2212yescomplemented
25C2×#14519700C2212yescomplemented
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic, complemented
30C2×C4×#92418C2211yescharacteristic, complemented
31#16839682—21no—
32#16839674—21no—
33#16839668C2212yescomplemented
34#16839642—21no—
35#16839634—21no—
36#16839622C2211yescharacteristic
37#16839524C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#121×C61C2212yes—
40C2×#120×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C61×#394—21no—
43C61×#392—21no—
44C61×#390C2212yes—
45C61×#380C2211yescharacteristic
46C61×#379—21no—
47C22×Dic122—22no—
48C22×Dic122—24no—
49C22×Dic122—22no—
50C22×Dic122—21no—
51C22×Dic122—42no—
52C22×Dic122C2312yes—
53C23×Dic61—21no—
54C2×#11758200—232no—
55C2×#11758200—24no—
56C2×#11758200—24no—
57C2×#11758200—24no—
58C2×#11758200—24no—
59C2×#11758200—24no—
60C2×#11758200D418yescomplemented
61C22×#92418—24no—
62C22×#92418—24no—
63C22×#92418—22no—
64C22×#92418D414yescomplemented
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519756—24no—
70#14519756—42no—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—21no—
75#14519708—24no—
76C4⋊#92418—22no—
77#14519700—24no—
78#14519700—42no—
79#14519698—24no—
80#14519698—42no—
81C22⋊#92418—22no—
82C22⋊#92418—22no—
83C4×#92418—22no—
84#14519684—21no—
85#14519684—22no—
86#14519684—22no—
87C25×C61—21no—
88C22×D4×C61—24no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61—21no—
92C22×D4×C61—22no—
93C22×D4×C61—42no—
94C22×D4×C61D411yescharacteristic
95C22×D4×C61C2312yes—
96C23×C4×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98#121×C61—24no—
99#121×C61—42no—
100#120×C61—44no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—22no—
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#95×C61—21no—
113C24×C61—22no—
114C24×C61—44no—
115C24×C61—42no—
116C24×C61—42no—
117C24×C61—42no—
118C24×C61—82no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—28no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—48no—
130C2×D4×C61—44no—
131C2×D4×C61—44no—
132C2×D4×C61—44no—
133C2×D4×C61—44no—
134C2×D4×C61—42no—
135C2×D4×C61—82no—
136C2×D4×C61SD1614yescomplemented
137C2×D4×C61C2×D414yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61#4112yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—44no—
155C22×Dic61—42no—
156C2×Dic122—28no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—48no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122—42no—
166C2×Dic122C2×D414yes—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—44no—
173C22⋊C4×C61—42no—
174C22⋊C4×C61—44no—
175C22⋊C4×C61—82no—
176C42×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—44no—
179C22⋊Dic61—42no—
180#11758200—216no—
181#11758200—216no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184#11758196—44no—
185C2×#92418—22no—
186C2×#92418—28no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—44no—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×D414yes—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—48no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—88no—
210C23×C61—88no—
211C23×C61—84no—
212C23×C61—84no—
213C23×C61#12112yes—
214C23×C61C2×SD1612yes—
215C23×C61C2×SD1612yescomplemented
216C23×C61#13512yes—
217C23×C61C22×D411yescharacteristic
218D4×C61—28no—
219D4×C61—24no—
220D4×C61—24no—
221D4×C61—48no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—88no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61#12112yes—
242C2×C4×C61C2×SD1612yescomplemented
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D412yes—
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)12yes—
247C2×Dic61—24no—
248C2×Dic61—48no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254Dic122—24no—
255Dic122—24no—
256Dic122—48no—
257Dic122—42no—
258Dic122—42no—
259#92418—22no—
260#92418—41no—
261#92418—41no—
262#92418#12014yescomplemented
263#24264—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—22no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—816no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61#46514yescomplemented
282C22×C61C2×#12111yescharacteristic
283C22×C61D4212yes—
284C22×C61C22×SD1611yescharacteristic
285C22×C61C2×#13511yescharacteristic
286C22×C61#57712yes—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—82no—
292C4×C61C2×#12011yescharacteristic
293C4×C61#53511yescharacteristic
294C4×C61#54612yes—
295Dic61—44no—
296Dic61—82no—
297C23×SD16—611no—
298C2×#465—612no—
299C2×C8×D4—611no—
300#2409—611no—
301#2408D6111yescharacteristic
302#2055—612no—
303#2000—612no—
304#1968—612no—
305#1807—611no—
306#1787—611no—
307#1786—611no—
308C2×C61—24no—
309C2×C61—42no—
310C2×C61—84no—
311C2×C61#240811yescharacteristic
312C2×C61C2×#46511yescharacteristic
313C2×C61#305711yescharacteristic
314C2×C61D4×SD1612yes—
315C2×C61#328212yes—
316C23×Q8—611no—
317C23×D4C2×D6111yescharacteristic
318C22×SD16—612no—
319C22×SD16—612no—
320C22×SD16—1224no—
321C22×SD16—1228no—
322C23×C8—611no—
323C23×C8—611no—
324C2×C22⋊Q8—612no—
325C2×#121C2×D6112yes—
326C2×#120—21no—
327C2×C4×D4C2×D6111yescharacteristic
328#465—1228no—
329C8×D4—1224no—
330C2×#106—612no—
331C2×C4⋊C8—611no—
332C2×#103—612no—
333C2×#103—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×C22⋊C8—611no—
337C2×C22⋊C8—611no—
338C2×C4×C8—611no—
339#395—1221no—
340#394—21no—
341#393—1221no—
342#392—21no—
343#391—612no—
344#390C2×D6112yes—
345#386—1221no—
346#385—1221no—
347#380—611no—
348#380C2×D6111yescharacteristic
349#379—21no—
350#340—612no—
351C61#2426411yescharacteristic, complemented, Sylow
352C25—21no—
353C22×Q8—612no—
354C22×Q8—1224no—
355C22×Q8—1221no—
356C22×Q8—1222no—
357C22×Q8—1222no—
358C22×Q8—2442no—
359C22×D4—24no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—21no—
364C22×D4—42no—
365C22×D4C22⋊D6111yescharacteristic
366C22×D4C22×D6112yes—
367C23×C4—1221no—
368C23×C4C22×D6111yescharacteristic
369C23×C4C22×D6111yescharacteristic
370C2×SD16—618no—
371C2×SD16—1224no—
372C2×SD16—1224no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C2×SD16—12232no—
376C2×SD16—1224no—
377C22×C8—611no—
378C22×C8—614no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8—1224no—
383C22×C8—1222no—
384C22×C8—1224no—
385C22⋊Q8—1224no—
386C22⋊Q8—2442no—
387#121—24no—
388#121—42no—
389#120—44no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4C22⋊D6111yescharacteristic
403C2×C22⋊C4C22⋊D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#106—1224no—
406C4⋊C8—1222no—
407#103—1224no—
408#103—2442no—
409#102—1224no—
410#102—2442no—
411C22⋊C8—1222no—
412C22⋊C8—1222no—
413C4×C8—1222no—
414#95—21no—
415#95—1221no—
416#95—1222no—
417#95—1222no—
418C24—22no—
419C24—44no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—82no—
424C24C2×C22⋊D6111yescharacteristic
425C24C2×C22⋊D6111yescharacteristic
426C2×Q8—614no—
427C2×Q8—1228no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C2×Q8—2448no—
435C2×Q8—2444no—
436C2×Q8—2442no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—48no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—82no—
452C2×D4#1175820014yescomplemented
453C2×D4D4×D6114yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—2444no—
464C22×C4—2442no—
465C22×C4D4×D6112yes—
466C22×C4#1175822412yes—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—12216no—
473SD16—12216no—
474C2×C8—612no—
475C2×C8—614no—
476C2×C8—612no—
477C2×C8—1228no—
478C2×C8—1222no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—2444no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—2444no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C22⋊C4—2444no—
497C22⋊C4—2442no—
498C42—21no—
499C42—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—88no—
515C23—88no—
516C23—84no—
517C23—84no—
518C23#1451974912yes—
519C23C2×#1175820012yes—
520C23C2×#1175820012yescomplemented
521C23#1451979712yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—1224no—
524Q8—1224no—
525Q8—2448no—
526Q8—2442no—
527Q8—2442no—
528D4—28no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—42no—
533D4—42no—
534D4—88no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—1224no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2448no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4#1451974912yes—
559C2×C4C2×#1175820012yescomplemented
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6112yes—
562C2×C4C2×#1175822411yescharacteristic
563C2×C4#1451982212yes—
564C8—614no—
565C8—1222no—
566C8—2441no—
567C8—2441no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—41no—
575C22—42no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—816no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#1683998914yescomplemented
586C22C2×#1451974911yescharacteristic
587C22D4×C22⋊D6112yes—
588C22C22×#1175820011yescharacteristic
589C22C2×#1451979711yescharacteristic
590C22#1684057312yes—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—82no—
596C4—4882no—
597C4—2444no—
598C4#120×D6111yescharacteristic
599C4#1684032711yescharacteristic
600C4#1684039912yes—
601C2—24no—
602C2—42no—
603C2—84no—
604C2#2144917211yescharacteristic
605C2C2×#1683998911yescharacteristic
606C2#2145545511yescharacteristic
607C2D4×#1175820012yes—
608C2#2145752212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2408C2
socleC23×C61C22×D4
hypercenter#2408D61
hypocenterC61#24264
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C23×#11758200C2semidirect
2. C2×#16839989C2semidirect
3. C2×D4×#92418C2semidirect
4. #21449184C2semidirect
5. #21447136C2semidirect
6. #21446822C2semidirect
7. #21446672C2semidirect
8. #21445842C2semidirect
9. #21445753C2semidirect
10. #21445750C2semidirect
18. C23×#92418C22semidirect
19. C2×#14519756C22semidirect
22. C2×#14519708C22semidirect
23. C2×C4⋊#92418C22semidirect
24. C2×#14519700C22semidirect
25. C2×#14519700C22semidirect
29. C2×C22⋊#92418C22semidirect
30. C2×C4×#92418C22semidirect
33. #16839668C22semidirect
37. #16839524C22semidirect
60. C2×#11758200D4semidirect
64. C22×#92418D4semidirect
136. C2×D4×C61SD16semidirect
215. C23×C61C2×SD16semidirect
242. C2×C4×C61C2×SD16semidirect
262. #92418#120semidirect
281. C22×C61#465semidirect
351. C61#24264semidirect
452. C2×D4#11758200semidirect
520. C23C2×#11758200semidirect
559. C2×C4C2×#11758200semidirect
585. C22#16839989semidirect

Maximal subgroups

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

subgroupindex
C23×#117582002
C2×#168399892
C2×D4×#924182
#214491842
#214471362
#214468222
#214466722
#214458422
#214457532
#214457502
C61×#24082
#2426461