Finite Group Databasebeta

← #29707624

Subgroups — #29707624

Order 15616.

1586 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×#16840329C211yesmaximal, characteristic
2#21454409C211yesmaximal, characteristic, complemented
3C2×#16839778C211yesmaximal, characteristic, complemented
4#21454339C211yesmaximal, characteristic, complemented
5#21448401C211yesmaximal, characteristic, complemented
6#21447030C218yesmaximal, complemented
7#21445530C211yesmaximal, characteristic, complemented
8C22×#100×C61C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10#16840329—22no—
11#16840329—22no—
12#16840329—22no—
13#16840329C2214yes—
14C2×#14519773—21no—
15C2×#14519764—21no—
16C2×#14519748—22no—
17C2×#14519746—22no—
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839804—22no—
22#16839804C2214yescomplemented
23#16839800—22no—
24#16839778—24no—
25C2×#14519703C2214yescomplemented
26#16839772—21no—
27#16839768—21no—
28#16839764C2214yescomplemented
29#16839759—22no—
30#16839665—22no—
31#16839665C2214yescomplemented
32#16839618—22no—
33#16839523C2214yescomplemented
34C23×D4×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#100×C61—24no—
38C2×#100×C61C2214yescomplemented
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222D412yes—
46C2×#11758222C2312yes—
47C22×Dic122D411yescharacteristic
48C22×D244D411yescharacteristic
49C22×C22⋊D61—21no—
50C22×C22⋊D61—24no—
51C22×C4×D61—21no—
52#14519773—24no—
53#14519773—24no—
54#14519764—24no—
55#14519764—24no—
56#14519748—28no—
57#14519748—28no—
58#14519746—28no—
59#14519746—28no—
60#14519736—21no—
61#14519736—21no—
62#14519736—22no—
63#14519736—22no—
64#14519736—21no—
65#14519736—21no—
66#14519736—21no—
67#14519736—21no—
68#14519736D412yes—
69#14519736C2312yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D412yes—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—24no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—28no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2312yes—
81#14519711—24no—
82#14519709—24no—
83#14519703—28no—
84#14519702—24no—
85#14519699—24no—
86C25×C61D412yes—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—22no—
90C22×D4×C61—21no—
91C22×D4×C61—21no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—22no—
95C22×D4×C61D412yes—
96C22×D4×C61C2312yes—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—21no—
99C2×M16×C61—22no—
100C2×M16×C61—21no—
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61D414yescomplemented
104C2×M16×C61C2312yes—
105C2×M16×C61C2312yes—
106C22×C8×C61—21no—
107C22×C8×C61—21no—
108#100×C61—28no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—24no—
112C24×C61—24no—
113C24×C61—84no—
114C24×C61C2×D414yes—
115C24×C61C2×D411yescharacteristic
116C23×D61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—42no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—42no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—216no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—88no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758222—24no—
156#11758222—24no—
157#11758222—22no—
158#11758222—22no—
159#11758222—22no—
160#11758222—22no—
161#11758222—42no—
162#11758222—42no—
163#11758222—42no—
164#11758222—42no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—21no—
168C2×D244C2×D412yes—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—21no—
176C2×Dic122C2×D412yes—
177M16×C61—24no—
178M16×C61—22no—
179M16×C61—22no—
180M16×C61—22no—
181M16×C61—42no—
182M16×C61C2×D414yescomplemented
183C2×C8×C61—24no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61—42no—
187C22⋊Dic61—216no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—41no—
191C22⋊Dic61—41no—
192C22⋊Dic61—42no—
193C22⋊Dic61—88no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—42no—
198C4⋊Dic61C2×D414yes—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204C4×Dic61—41no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—42no—
209C23×C61—48no—
210C23×C61—48no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61—84no—
214C23×C61—88no—
215C23×C61—88no—
216C23×C61#12011yescharacteristic
217C23×C61#12011yescharacteristic
218C23×C61#12112yes—
219C23×C61#12312yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—44no—
223C22×D61—42no—
224D4×C61—42no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—44no—
229D4×C61—44no—
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61#12112yes—
235C2×C4×C61#12112yes—
236C2×C4×C61#12112yes—
237C2×C4×C61#12312yes—
238C2×C4×C61#12312yes—
239C2×C4×C61#12312yes—
240C2×C4×C61#12412yes—
241C2×C4×C61#12412yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—42no—
249C22⋊D61—42no—
250C22⋊D61—44no—
251C22⋊D61—816no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—44no—
255C2×Dic61—42no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260D244—22no—
261D244—22no—
262D244—42no—
263C4×D61—42no—
264C4×D61—42no—
265C4×D61—44no—
266Dic122—22no—
267Dic122—22no—
268Dic122—42no—
269C8×C61—24no—
270C8×C61—42no—
271#23493—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—48no—
278C22×C61—44no—
279C22×C61—48no—
280C22×C61—84no—
281C22×C61—88no—
282C22×C61—88no—
283C22×C61#39412yes—
284C22×C61#39412yes—
285C22×C61C2×#12011yescharacteristic
286C22×C61C2×#12111yescharacteristic
287C22×C61C2×#12311yescharacteristic
288C2×D61—42no—
289C2×D61—44no—
290C2×D61—84no—
291C4×C61#39412yes—
292C4×C61#39412yes—
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12111yescharacteristic
295C4×C61C2×#12311yescharacteristic
296C4×C61C2×#12411yescharacteristic
297Dic61—42no—
298Dic61—44no—
299Dic61—82no—
300Dic61—82no—
301C2×#535—611no—
302#2911—611no—
303C2×#421—611no—
304#2896—611no—
305C22×#100D6111yescharacteristic, complemented
306#2309—611no—
307#2037—618no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61—84no—
314C2×C61C2×#39411yescharacteristic
315C2×C61#301912yes—
316D61—82no—
317C22×#41—611no—
318C23×D4C2×D6111yescharacteristic
319C22×M16C2×D6111yescharacteristic
320C22×M16C2×D6111yescharacteristic
321#535—614no—
322#535—1222no—
323#535—1222no—
324#535—1222no—
325C2×#127—1221no—
326C2×#124—1221no—
327C2×#121—1222no—
328C2×#120—1222no—
329C2×#117—611no—
330C2×#117—611no—
331C2×#117—611no—
332#428—614no—
333#428—1222no—
334#427—1222no—
335#421—1224no—
336C2×#104—614no—
337#419—1221no—
338#418—1221no—
339#417—614no—
340#416—1222no—
341C2×#100—24no—
342C2×#100C2×D6114yescomplemented
343#390—614no—
344#390—1222no—
345#379—1222no—
346#339—614no—
347C61#2349311yescharacteristic, complemented, Sylow
348C25D24412yes—
349C2×#41—612no—
350C2×#41—612no—
351C2×#41—1221no—
352C2×#41—1221no—
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1222no—
357C22×Q8—611no—
358C22×D4—22no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—21no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—611no—
367C22×D4—1224no—
368C22×D4—1221no—
369C22×D4D24412yes—
370C22×D4C22×D6112yes—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—21no—
374C2×M16—22no—
375C2×M16—21no—
376C2×M16—22no—
377C2×M16—22no—
378C2×M16C22⋊D6114yescomplemented
379C2×M16C22×D6112yes—
380C2×M16C22×D6112yes—
381C22×C8—21no—
382C22×C8—21no—
383#127—1224no—
384#127—1224no—
385#124—1224no—
386#124—1224no—
387#121—1228no—
388#121—1228no—
389#120—1228no—
390#120—1228no—
391#117—612no—
392#117—612no—
393#117—1221no—
394#117—1221no—
395#117—1222no—
396#117—1222no—
397#117—1221no—
398#117—1221no—
399#117—1221no—
400#117—1221no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1221no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1228no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—612no—
410C2×C42—1221no—
411C2×C42—1221no—
412#107—1224no—
413#106—1224no—
414#104—1228no—
415#103—1224no—
416#102—1224no—
417#100—28no—
418C24—22no—
419C24—22no—
420C24—24no—
421C24—24no—
422C24—84no—
423C24—1221no—
424C24C2×D24414yes—
425C24C2×D24411yescharacteristic
426#41—1222no—
427#41—1222no—
428#41—1222no—
429#41—1222no—
430#41—1224no—
431#41—1224no—
432#41—2442no—
433#41—2442no—
434#41—2442no—
435#41—2442no—
436C2×Q8—612no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—612no—
459C2×D4—12216no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—4888no—
466C2×D4—2442no—
467C2×D4—2442no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4C2×D24411yescharacteristic
477C22×C4#1175822212yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4#1175822412yes—
483C22×C4#1175822412yes—
484C22×C4C2×C22⋊D6112yes—
485C22×C4C23×D6111yescharacteristic
486M16—22no—
487M16—24no—
488M16—22no—
489M16—22no—
490M16—42no—
491M16C2×C22⋊D6114yescomplemented
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—42no—
496C4⋊C4—614no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—2442no—
501C22⋊C4—12216no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—4888no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2442no—
508C42—1222no—
509C42—1222no—
510C42—1222no—
511C42—1222no—
512C42—2441no—
513C42—2441no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—42no—
518C23—48no—
519C23—48no—
520C23—44no—
521C23—44no—
522C23—84no—
523C23—88no—
524C23—88no—
525C23—1222no—
526C23—2444no—
527C23—2442no—
528C23#1451974711yescharacteristic
529C23#1451974711yescharacteristic
530C23#1451975112yes—
531C23#1451976112yes—
532C23C22×D24411yescharacteristic
533Q8—1222no—
534Q8—1222no—
535Q8—2442no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—44no—
541D4—44no—
542D4—1222no—
543D4—1222no—
544D4—48816no—
545D4—2442no—
546D4—2442no—
547D4—2444no—
548D4—2442no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4#1451974711yescharacteristic
561C2×C4#1451974711yescharacteristic
562C2×C4#1451974812yes—
563C2×C4#1451974812yes—
564C2×C4#1451974912yes—
565C2×C4#1451974912yes—
566C2×C4#1451975112yes—
567C2×C4#1451975912yes—
568C2×C4#1451975912yes—
569C2×C4#1451976112yes—
570C2×C4#1451976312yes—
571C2×C4#1451976312yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822411yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—24no—
579C8—42no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—42no—
585C22—48no—
586C22—44no—
587C22—48no—
588C22—84no—
589C22—88no—
590C22—88no—
591C22—4884no—
592C22—2442no—
593C22—2444no—
594C22#1683967812yes—
595C22#1683967812yes—
596C22C2×#1451974711yescharacteristic
597C22C2×#1451975111yescharacteristic
598C22C2×#1451976111yescharacteristic
599C4—4882no—
600C4—4882no—
601C4—2442no—
602C4—2444no—
603C4#1683967812yes—
604C4#1683967812yes—
605C4C2×#1451974811yescharacteristic
606C4C2×#1451974911yescharacteristic
607C4C2×#1451975911yescharacteristic
608C4C2×#1451976311yescharacteristic
609C2—21no—
610C2—21no—
611C2—22no—
612C2—22no—
613C2—84no—
614C2—4882no—
615C2C2×#1683967811yescharacteristic
616C2#2145517912yes—
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#100×C61C2
socleC22×C61C2×#120
prime radicalC2×#16840329C2
hypercenterC22×#100D61
hypocenterC61#23493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
2. #21454409C2semidirect
3. C2×#16839778C2semidirect
4. #21454339C2semidirect
5. #21448401C2semidirect
6. #21447030C2semidirect
7. #21445530C2semidirect
8. C22×#100×C61C2semidirect
22. #16839804C22semidirect
25. C2×#14519703C22semidirect
28. #16839764C22semidirect
31. #16839665C22semidirect
33. #16839523C22semidirect
38. C2×#100×C61C22semidirect
103. C2×M16×C61D4semidirect
182. M16×C61C2×D4semidirect
305. C22×#100D61semidirect
342. C2×#100C2×D61semidirect
347. C61#23493semidirect
378. C2×M16C22⋊D61semidirect
491. M16C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403292
#214544092
C2×#168397782
#214543392
#214484012
#214470302
#214455302
C22×#100×C612
#2349361