Finite Group Databasebeta

← #29707623

Subgroups — #29707623

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#21454408C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4#21454336C211yesmaximal, characteristic, complemented
5C22×#14519694C211yesmaximal, characteristic, complemented
6#21447029C218yesmaximal, complemented
7#21445529C211yesmaximal, characteristic, complemented
8C61×#2309C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840329—22no—
13#16840329—22no—
14#16840329—22no—
15#16840329C2214yes—
16C2×#14519773—21no—
17C2×#14519764—21no—
18C2×#14519748—22no—
19C2×#14519746—22no—
20C2×#14519736C2211yescharacteristic
21#16839805—22no—
22#16839802C2214yescomplemented
23#16839798—22no—
24#16839780—24no—
25C2×#14519705C2214yescomplemented
26#16839770—21no—
27#16839766—21no—
28#16839761C2214yescomplemented
29#16839756—22no—
30C2×#14519694—24no—
31C2×#14519694C2214yescomplemented
32#16839522C2214yescomplemented
33C23×D4×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#390—22no—
37C61×#390C2214yescomplemented
38C61×#379—22no—
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—
52C2×#11758194—21no—
53C2×#11758194—22no—
54C2×#11758194—21no—
55C2×#11758194—22no—
56C2×#11758194—22no—
57C2×#11758194D414yescomplemented
58C2×#11758194C2312yes—
59C2×#11758194C2312yes—
60C22×#92418—21no—
61C22×#92418—21no—
62#14519773—24no—
63#14519773—24no—
64#14519764—24no—
65#14519764—24no—
66#14519748—28no—
67#14519748—28no—
68#14519746—28no—
69#14519746—28no—
70#14519736—21no—
71#14519736—21no—
72#14519736D412yes—
73#14519736C2312yes—
74C2×C4⋊Dic61D412yes—
75C2×C22⋊Dic61—24no—
76C2×C22⋊Dic61—21no—
77C2×C4×Dic61—21no—
78#14519710—24no—
79#14519708—24no—
80#14519705—28no—
81#14519700—24no—
82#14519697—24no—
83#14519694—28no—
84C25×C61D412yes—
85C22×D4×C61—22no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61—22no—
91C22×D4×C61—22no—
92C22×D4×C61—22no—
93C22×D4×C61D412yes—
94C22×D4×C61C2312yes—
95C23×C4×C61C2311yescharacteristic
96#117×C61—22no—
97#117×C61—22no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—28no—
107C2×C42×C61—21no—
108C2×C42×C61C2312yes—
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×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—22no—
149C2×C22⋊D61—216no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—88no—
153C22×Dic61—21no—
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—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—42no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—88no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—41no—
187C42×C61—41no—
188C22⋊Dic61—216no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C2×D414yes—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—24no—
196#11758194—22no—
197#11758194—22no—
198#11758194—22no—
199#11758194—42no—
200#11758194C2×D414yescomplemented
201C2×#92418—24no—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—42no—
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—44no—
231C2×C4×C61—44no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12112yes—
237C2×C4×C61#12112yes—
238C2×C4×C61#12112yes—
239C2×C4×C61#12112yes—
240C2×C4×C61#12112yes—
241C2×C4×C61#12312yes—
242C2×C4×C61#12312yes—
243C2×C4×C61#12312yes—
244C2×C4×C61#12412yes—
245C2×C4×C61#12412yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—42no—
253C22⋊D61—42no—
254C22⋊D61—44no—
255C22⋊D61—816no—
256C2×Dic61—24no—
257C2×Dic61—22no—
258C2×Dic61—44no—
259C2×Dic61—42no—
260D244—22no—
261D244—22no—
262D244—42no—
263C4×D61—42no—
264C4×D61—42no—
265C4×D61—44no—
266Dic122—22no—
267Dic122—22no—
268Dic122—42no—
269#92418—24no—
270#92418—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—82no—
292C4×C61—82no—
293C4×C61#39412yes—
294C4×C61#39412yes—
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×#12311yescharacteristic
298C4×C61C2×#12411yescharacteristic
299Dic61—42no—
300Dic61—44no—
301C2×#535—611no—
302#2911—611no—
303C2×#421—611no—
304#2896—611no—
305C22×#100—611no—
306#2309D6111yescharacteristic, complemented
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×M16—611no—
320C22×M16—611no—
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×#117C2×D6111yescharacteristic
331C2×#117C2×D6111yescharacteristic
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—614no—
342C2×#100—1224no—
343#390—22no—
344#390C2×D6114yescomplemented
345#379—22no—
346#339—614no—
347C61#2349311yescharacteristic, complemented, Sylow
348C25C22⋊D6112yes—
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—22no—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—21no—
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—611no—
367C22×D4—1224no—
368C22×D4—1221no—
369C22×D4C22⋊D6112yes—
370C22×D4C22×D6112yes—
371C23×C4—1221no—
372C23×C4C22×D6111yescharacteristic
373C2×M16—614no—
374C2×M16—612no—
375C2×M16—612no—
376C2×M16—1222no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—1222no—
380C2×M16—1222no—
381C22×C8—1221no—
382C22×C8—1221no—
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—22no—
392#117—22no—
393#117—21no—
394#117—21no—
395#117—21no—
396#117—21no—
397#117—612no—
398#117—612no—
399#117—1221no—
400#117—1221no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4C22×D6112yes—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—28no—
407C2×C22⋊C4—1224no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—1221no—
411C2×C42C22×D6112yes—
412#107—1224no—
413#106—1224no—
414#104—1228no—
415#103—1224no—
416#102—1224no—
417#100—1228no—
418C24—22no—
419C24—22no—
420C24—24no—
421C24—24no—
422C24—84no—
423C24—1221no—
424C24C2×C22⋊D6114yes—
425C24C2×C22⋊D6111yescharacteristic
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—22no—
469C22×C4—22no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—2442no—
475C22×C4—2442no—
476C22×C4#1175822212yes—
477C22×C4#1175822212yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6112yes—
483C22×C4#1175822412yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—614no—
487M16—1222no—
488M16—1224no—
489M16—1222no—
490M16—1222no—
491M16—2442no—
492C2×C8—1222no—
493C2×C8—1224no—
494C2×C8—1222no—
495C2×C8—2442no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—614no—
500C4⋊C4—1222no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—88no—
505C22⋊C4—12216no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C42—22no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—1222no—
513C42—1222no—
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#1451974511yescharacteristic
529C23#1451974511yescharacteristic
530C23#1451974912yes—
531C23#1451976012yes—
532C23C22×C22⋊D6111yescharacteristic
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—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—1224no—
554C2×C4—1222no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4#1451974511yescharacteristic
561C2×C4#1451974511yescharacteristic
562C2×C4#1451974912yes—
563C2×C4#1451975012yes—
564C2×C4#1451975012yes—
565C2×C4#1451975012yes—
566C2×C4#1451975012yes—
567C2×C4#1451975912yes—
568C2×C4#1451975912yes—
569C2×C4#1451976012yes—
570C2×C4#1451976312yes—
571C2×C4#1451976312yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822211yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—1224no—
579C8—2442no—
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#1683967712yes—
595C22#1683967712yes—
596C22C2×#1451974511yescharacteristic
597C22C2×#1451974911yescharacteristic
598C22C2×#1451976011yescharacteristic
599C4—82no—
600C4—82no—
601C4—2442no—
602C4—2444no—
603C4#1683967712yes—
604C4#1683967712yes—
605C4C2×#1451975011yescharacteristic
606C4C2×#1451975011yescharacteristic
607C4C2×#1451975911yescharacteristic
608C4C2×#1451976311yescharacteristic
609C2—21no—
610C2—21no—
611C2—22no—
612C2—22no—
613C2—84no—
614C2—4882no—
615C2C2×#1683967711yescharacteristic
616C2#2145517512yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC22×C61C2×#120
prime radicalC2×#16840329C2
hypercenter#2309D61
hypocenterC61#23493
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
3. C2×#16839780C2semidirect
4. #21454336C2semidirect
5. C22×#14519694C2semidirect
6. #21447029C2semidirect
7. #21445529C2semidirect
8. C61×#2309C2semidirect
22. #16839802C22semidirect
25. C2×#14519705C22semidirect
28. #16839761C22semidirect
31. C2×#14519694C22semidirect
32. #16839522C22semidirect
37. C61×#390C22semidirect
57. C2×#11758194D4semidirect
200. #11758194C2×D4semidirect
306. #2309D61semidirect
344. #390C2×D61semidirect
347. C61#23493semidirect

Maximal subgroups

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

subgroupindex
C2×#168403292
#214544082
C2×#168397802
#214543362
C22×#145196942
#214470292
#214455292
C61×#23092
#2349361