Finite Group Databasebeta

← #29715688

Subgroups — #29715688

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×#11758201C211yesmaximal, characteristic, complemented
2C2×#16839993C212yesmaximal, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21449175C211yesmaximal, characteristic
5#21447134C212yesmaximal, complemented
6#21446820C212yesmaximal, complemented
7#21446676C212yesmaximal, complemented
8#21445842C211yesmaximal, characteristic, complemented
9#21445754C211yesmaximal, characteristic, complemented
10#21445749C211yesmaximal, characteristic, complemented
11C61×#2409C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201—28no—
14C22×#11758201—24no—
15C22×#11758201C2212yes—
16C22×#11758201C2212yes—
17C23×#92418C2211yescharacteristic
18C23×#92418C2211yescharacteristic, complemented
19C2×#14519751C2212yes—
20C2×#14519745—21no—
21#16839993—28no—
22D4×#92418—24no—
23C2×#14519708C2212yescomplemented
24C2×C4⋊#92418C2211yescharacteristic, complemented
25C2×#14519701C2212yescomplemented
26C2×#14519701C2212yescomplemented
27C2×#14519697C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic, complemented
31C2×C4×#92418C2211yescharacteristic, complemented
32#16839679—21no—
33#16839671—21no—
34#16839664C2212yes—
35#16839622C2211yescharacteristic
36#16839617—21no—
37#16839524C2212yescomplemented
38C23×Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yescomplemented
40C2×C4×D4×C61C2211yescharacteristic
41C61×#395—21no—
42C61×#393—21no—
43C61×#391C2212yescomplemented
44C61×#386—21no—
45C61×#385—21no—
46C61×#380C2211yescharacteristic
47C24×D61—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—22no—
51C22×D244—21no—
52C22×D244—42no—
53C22×D244C2312yes—
54C22×C22⋊D61—22no—
55C2×#11758201—232no—
56C2×#11758201—24no—
57C2×#11758201—24no—
58C2×#11758201—24no—
59C2×#11758201—24no—
60C2×#11758201—24no—
61C2×#11758201D418yescomplemented
62C22×#92418—24no—
63C22×#92418—24no—
64C22×#92418—22no—
65C22×#92418D414yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519751—24no—
71#14519751—42no—
72#14519745—44no—
73C2×C4⋊Dic61C2312yes—
74C2×C22⋊Dic61—21no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78#14519708—24no—
79C4⋊#92418—22no—
80#14519701—24no—
81#14519701—42no—
82#14519697—24no—
83#14519697—42no—
84C22⋊#92418—22no—
85C22⋊#92418—22no—
86C4×#92418—22no—
87#14519684—21no—
88C22×Q8×C61—21no—
89C22×Q8×C61—24no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61—42no—
93C22×Q8×C61C2312yes—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—42no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61C2312yes—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61D411yescharacteristic
109C2×C42×C61C2311yescharacteristic
110#95×C61—21no—
111#95×C61—22no—
112#95×C61—22no—
113C24×C61C2×D411yescharacteristic
114C24×C61C2×D411yescharacteristic
115C23×D61—22no—
116C23×D61—44no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—82no—
121C2×Q8×C61—22no—
122C2×Q8×C61—28no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C2×Q8×C61—48no—
129C2×Q8×C61—44no—
130C2×Q8×C61—42no—
131C2×Q8×C61C2×D414yes—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61SD1614yescomplemented
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—44no—
144C22×C4×C61—42no—
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
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—82no—
155C22×Dic61—22no—
156C2×D244—28no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—22no—
160C2×D244—22no—
161C2×D244—22no—
162C2×D244—48no—
163C2×D244—44no—
164C2×D244—44no—
165C2×D244—42no—
166C2×D244C2×D414yes—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—44no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—44no—
174C22⋊C4×C61—42no—
175C42×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—42no—
178#11758201—216no—
179#11758201—216no—
180#11758198—44no—
181#11758198—44no—
182#11758198—82no—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
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#12112yes—
202C23×C61C2×SD1612yes—
203C23×C61C2×SD1612yescomplemented
204C23×C61#13512yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—24no—
207C22×D61—48no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—88no—
215C22×D61—88no—
216C22×D61—84no—
217C22×D61—84no—
218Q8×C61—24no—
219Q8×C61—24no—
220Q8×C61—48no—
221Q8×C61—42no—
222Q8×C61—42no—
223D4×C61—28no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—22no—
235C2×C4×C61—48no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61#12112yes—
245C2×C4×C61C2×SD1612yescomplemented
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D412yes—
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)12yes—
250C22⋊D61—88no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254D244—24no—
255D244—24no—
256D244—48no—
257D244—42no—
258D244—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#46514yescomplemented
274C22×C61C2×#12111yescharacteristic
275C22×C61D4212yes—
276C22×C61C22×SD1611yescharacteristic
277C22×C61C2×#13511yescharacteristic
278C22×C61#57712yes—
279C2×D61—44no—
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—816no—
283C2×D61—82no—
284C2×D61—82no—
285C2×D61—82no—
286C2×D61—82no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—44no—
291C4×C61—41no—
292C4×C61—82no—
293C4×C61C2×#12011yescharacteristic
294C4×C61#53511yescharacteristic
295C4×C61#54612yes—
296Dic61—82no—
297C23×SD16—611no—
298C2×#465—612no—
299C2×C8×D4—611no—
300#2409D6111yescharacteristic, complemented
301#2408—611no—
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#240811yescharacteristic
311C2×C61C2×#46511yescharacteristic
312C2×C61#305711yescharacteristic
313C2×C61D4×SD1612yes—
314C2×C61#328212yes—
315D61—84no—
316C23×Q8C2×D6111yescharacteristic
317C23×D4—611no—
318C22×SD16—612no—
319C22×SD16—612no—
320C22×SD16—1224no—
321C22×SD16—1228no—
322C23×C8—611no—
323C23×C8—611no—
324C2×C22⋊Q8C2×D6112yescomplemented
325C2×#121—612no—
326C2×#120—1221no—
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—21no—
340#394—1221no—
341#393—21no—
342#392—1221no—
343#391C2×D6112yescomplemented
344#390—612no—
345#386—21no—
346#385—21no—
347#380—611no—
348#380C2×D6111yescharacteristic
349#379—1221no—
350#340—612no—
351C61#2426411yescharacteristic, complemented, Sylow
352C25—1221no—
353C22×Q8—21no—
354C22×Q8—24no—
355C22×Q8—22no—
356C22×Q8—22no—
357C22×Q8—42no—
358C22×Q8C22×D6112yes—
359C22×D4—612no—
360C22×D4—1224no—
361C22×D4—1222no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1221no—
365C22×D4—2442no—
366C22×D4C22⋊D6111yescharacteristic
367C23×C4—21no—
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—24no—
386C22⋊Q8—42no—
387#121—1224no—
388#121—2442no—
389#120—2444no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4C22⋊D6111yescharacteristic
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1222no—
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—22no—
416#95—22no—
417#95—1221no—
418C24—1222no—
419C24—4882no—
420C24—2444no—
421C24—2442no—
422C24—2442no—
423C24—2442no—
424C24C2×C22⋊D6111yescharacteristic
425C24C2×C22⋊D6111yescharacteristic
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8D4×D6114yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—614no—
440C2×D4—1228no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—4882no—
447C2×D4—2448no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4—2444no—
452C2×D4—2442no—
453C2×D4#1175820114yescomplemented
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—1222no—
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—44no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—4882no—
495C22⋊C4—2444no—
496C22⋊C4—2442no—
497C22⋊C4—2444no—
498C42—21no—
499C42—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—1224no—
507C23—4884no—
508C23—4888no—
509C23—4888no—
510C23—4884no—
511C23—2448no—
512C23—2442no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23#1451974912yes—
519C23C2×#1175820112yes—
520C23C2×#1175820112yescomplemented
521C23#1451979712yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—42no—
527Q8—42no—
528D4—28no—
529D4—1224no—
530D4—1224no—
531D4—4888no—
532D4—2448no—
533D4—2442no—
534D4—2442no—
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—24no—
545C2×C4—22no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4#1451974912yes—
559C2×C4C2×#1175820112yescomplemented
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—48816no—
578C22—4882no—
579C22—4882no—
580C22—4882no—
581C22—4882no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22#1683999314yescomplemented
586C22C2×#1451974911yescharacteristic
587C22D4×C22⋊D6112yes—
588C22C22×#1175820111yescharacteristic
589C22C2×#1451979711yescharacteristic
590C22#1684057012yes—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—44no—
596C4—82no—
597C4—4882no—
598C4#120×D6111yescharacteristic
599C4#1684032711yescharacteristic
600C4#1684039912yes—
601C2—24no—
602C2—42no—
603C2—4884no—
604C2#2144917211yescharacteristic
605C2C2×#1683999311yescharacteristic
606C2#2145546011yescharacteristic
607C2D4×#1175820112yes—
608C2#2145752712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC23×C61C22×D4
prime radical#21449175C2
hypercenter#2409D61
hypocenterC61#24264
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#11758201C2semidirect
2. C2×#16839993C2semidirect
3. C2×D4×#92418C2semidirect
5. #21447134C2semidirect
6. #21446820C2semidirect
7. #21446676C2semidirect
8. #21445842C2semidirect
9. #21445754C2semidirect
10. #21445749C2semidirect
11. C61×#2409C2semidirect
18. C23×#92418C22semidirect
23. C2×#14519708C22semidirect
24. C2×C4⋊#92418C22semidirect
25. C2×#14519701C22semidirect
26. C2×#14519701C22semidirect
30. C2×C22⋊#92418C22semidirect
31. C2×C4×#92418C22semidirect
37. #16839524C22semidirect
39. C2×C22⋊Q8×C61C22semidirect
43. C61×#391C22semidirect
61. C2×#11758201D4semidirect
65. C22×#92418D4semidirect
134. C2×D4×C61SD16semidirect
203. C23×C61C2×SD16semidirect
245. C2×C4×C61C2×SD16semidirect
262. #92418#120semidirect
273. C22×C61#465semidirect
300. #2409D61semidirect
324. C2×C22⋊Q8C2×D61semidirect
343. #391C2×D61semidirect
351. C61#24264semidirect
453. C2×D4#11758201semidirect
520. C23C2×#11758201semidirect
559. C2×C4C2×#11758201semidirect
585. C22#16839993semidirect

Maximal subgroups

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

subgroupindex
C23×#117582012
C2×#168399932
C2×D4×#924182
#214491752
#214471342
#214468202
#214466762
#214458422
#214457542
#214457492
C61×#24092
#2426461