Finite Group Databasebeta

← #29711999

Subgroups — #29711999

Order 15616.

1382 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840383C211yesmaximal, characteristic, complemented
2C2×#16839913C211yesmaximal, characteristic, complemented
3C2×#16839896C211yesmaximal, characteristic
4#21446955C212yesmaximal, complemented
5#21446895C212yesmaximal, complemented
6#21446177C212yesmaximal, complemented
7#21446170C212yesmaximal, complemented
8#21446141C211yesmaximal, characteristic, complemented
9#21446131C211yesmaximal, characteristic, complemented
10#21445890C211yesmaximal, characteristic, complemented
11C61×#2309C211yesmaximal, characteristic, complemented
12C22×#11758200C2212yes—
13C22×#11758199C2212yes—
14#16840383—24no—
15#16840383—24no—
16C2×#14519777C2211yescharacteristic, complemented
17C2×#14519772—21no—
18C2×#14519765C2211yescharacteristic, complemented
19C2×#14519763—21no—
20C2×#14519759—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519755—21no—
23C2×#14519751C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic, complemented
25C2×C4×D244C2211yescharacteristic, complemented
26#16839913—24no—
27#16839896—24no—
28C2×#14519700C2212yescomplemented
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yescomplemented
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839524C2212yescomplemented
35#16839524C2212yescomplemented
36#16839517C2212yescomplemented
37#16839517C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#390—21no—
42C61×#390—21no—
43C61×#390C2212yes—
44C61×#390C2212yes—
45C61×#379—21no—
46C61×#379—21no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52C2×#11758200—24no—
53C2×#11758200—24no—
54C2×#11758200—42no—
55C2×#11758199—24no—
56C2×#11758199—24no—
57C2×#11758199—42no—
58C22×#92418C2312yes—
59C22×#92418C2312yes—
60#14519777—22no—
61#14519777—22no—
62#14519772—28no—
63#14519765—22no—
64#14519765—22no—
65#14519763—28no—
66#14519759—28no—
67#14519756—22no—
68#14519756D414yes—
69#14519755—28no—
70#14519751—22no—
71#14519751D414yes—
72C4×Dic122—22no—
73C4×Dic122—22no—
74C4×D244—22no—
75C4×D244—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519700—42no—
88#14519698—24no—
89#14519698—24no—
90#14519697—42no—
91C22⋊#92418—22no—
92C22⋊#92418—22no—
93C25×C61—21no—
94C22×D4×C61—24no—
95C22×D4×C61—21no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61—42no—
99C22×D4×C61C2312yes—
100C23×C4×C61C2311yescharacteristic
101#117×C61—22no—
102#117×C61—22no—
103#117×C61—42no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C24×C61—24no—
118C24×C61—22no—
119C24×C61—42no—
120C24×C61—42no—
121C24×C61—82no—
122C24×C61—82no—
123C24×C61C2×D411yescharacteristic
124C23×D61—21no—
125C2×D4×C61—24no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—48no—
132C2×D4×C61—48no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C2×D4×C61#4114yes—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—24no—
148C2×C22⋊D61—82no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244C2×D412yes—
156C2×C4×D61—24no—
157C2×C4×D61—82no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—21no—
161C2×Dic122—24no—
162C2×Dic122—82no—
163C2×Dic122C2×D412yes—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—42no—
171C22⋊C4×C61—82no—
172C22⋊C4×C61—82no—
173C22⋊C4×C61—82no—
174C22⋊C4×C61—82no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—42no—
180C22⋊Dic61—24no—
181#11758200—48no—
182#11758199—48no—
183#11758198—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C2×D412yes—
191C4⋊Dic61C2×D412yes—
192#11758196—24no—
193#11758196—24no—
194C4×Dic61—24no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—24no—
206C23×C61—48no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—84no—
214C23×C61—84no—
215C23×C61—88no—
216C23×C61—88no—
217C23×C61#12012yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—42no—
223C22×D61—42no—
224D4×C61—24no—
225D4×C61—24no—
226D4×C61—48no—
227D4×C61—42no—
228D4×C61—42no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61#12012yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4112yes—
248C2×C4×C61H(5,𝔽2)12yes—
249C22⋊D61—84no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257D244—24no—
258D244—42no—
259C4×D61—84no—
260Dic122—24no—
261Dic122—42no—
262Dic122—84no—
263#92418—42no—
264#92418—42no—
265#23945—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—44no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—816no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61C2×#12011yescharacteristic
280C22×C61#52511yescharacteristic
281C22×C61#53511yescharacteristic
282C22×C61#57612yes—
283C22×C61#57712yes—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—82no—
288C2×D61—82no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—82no—
295C4×C61#54612yes—
296C4×C61#54612yes—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300C2×#542—611no—
301C2×#450—611no—
302C2×#447—611no—
303#2309D6111yescharacteristic, complemented
304#2023—612no—
305#2012—612no—
306#1880—612no—
307#1879—612no—
308#1875—611no—
309#1873—611no—
310#1818—611no—
311C2×C61—42no—
312C2×C61—84no—
313C2×C61#240411yescharacteristic
314C2×C61#300811yescharacteristic
315C2×C61#301011yescharacteristic
316C2×C61#311812yes—
317C2×C61#322412yes—
318D61—82no—
319C23×D4C2×D6111yescharacteristic
320C22×SD16—612no—
321C22×D8—612no—
322#542—1224no—
323#542—1224no—
324C2×C4⋊Q8—611no—
325C2×#126—1221no—
326C2×#124—611no—
327C2×#124—1221no—
328C2×#123—1221no—
329C2×C22⋊Q8—611no—
330C2×C22⋊Q8—1221no—
331C2×#121—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×#117C2×D6111yescharacteristic
335C2×#117C2×D6111yescharacteristic
336#450—1224no—
337#447—1224no—
338C2×#103—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344#390—21no—
345#390—21no—
346#390C2×D6112yes—
347#390C2×D6112yes—
348#379—21no—
349#379—21no—
350#340—612no—
351#340—612no—
352#336—612no—
353#336—612no—
354C61#2394511yescharacteristic, complemented, Sylow
355C25—21no—
356C22×Q8—611no—
357C22×Q8—1221no—
358C22×D4—24no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—21no—
362C22×D4—42no—
363C22×D4—611no—
364C22×D4—1221no—
365C22×D4C22×D6112yes—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368C2×SD16—1224no—
369C2×SD16—1224no—
370C2×SD16—2442no—
371C2×D8—1224no—
372C2×D8—1224no—
373C2×D8—2442no—
374C22×C8—612no—
375C22×C8—612no—
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378#126—1228no—
379#124—1222no—
380#124—1222no—
381#124—1228no—
382#123—1228no—
383C22⋊Q8—614no—
384C22⋊Q8—1228no—
385C22⋊Q8—1222no—
386#121—614no—
387#121—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×D4—1222no—
391C4×D4—1222no—
392#117—22no—
393#117—22no—
394#117—42no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C22⋊C4C22⋊D6111yescharacteristic
415C2×C42—21no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419#103—2442no—
420#102—1224no—
421#102—1224no—
422#102—2442no—
423C22⋊C8—1222no—
424C22⋊C8—1222no—
425C24—24no—
426C24—22no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24—1221no—
432C24C2×C22⋊D6111yescharacteristic
433C2×Q8—612no—
434C2×Q8—1224no—
435C2×Q8—1221no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—4882no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—48no—
446C2×D4—48no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—612no—
450C2×D4—1224no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—4882no—
455C2×D4#1175822214yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—4882no—
464C22×C4D4×D6112yes—
465C22×C4D4×D6112yes—
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—2448no—
473D8—2448no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1224no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—1224no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—24no—
514C23—48no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—84no—
522C23—84no—
523C23—88no—
524C23—88no—
525C23—1222no—
526C23—2441no—
527C23—2442no—
528C23—2442no—
529C23—2441no—
530C23#1451974512yes—
531C23C22×C22⋊D6111yescharacteristic
532Q8—1224no—
533Q8—4884no—
534Q8—2442no—
535D4—24no—
536D4—24no—
537D4—48no—
538D4—42no—
539D4—42no—
540D4—1224no—
541D4—4884no—
542D4—2442no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—4884no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4#1451974512yes—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×#1175822212yes—
570C2×C4#1451982212yes—
571C8—2442no—
572C8—2442no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—816no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—4882no—
587C22—4882no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22C2×#1451974511yescharacteristic
592C22#1684027311yescharacteristic
593C22#1684032911yescharacteristic
594C22#1684056512yes—
595C22#1684057312yes—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—82no—
602C4—2442no—
603C4—2442no—
604C4—2442no—
605C4#1684040212yes—
606C4#1684040212yes—
607C2—42no—
608C2—84no—
609C2—4882no—
610C2#2144913811yescharacteristic
611C2#2145508511yescharacteristic
612C2#2145510311yescharacteristic
613C2#2145600312yes—
614C2#2145691312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC23×C61C22×D4
prime radicalC2×#16839896C2
hypercenter#2309D61
hypocenterC61#23945
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. C2×#16840383C2semidirect
2. C2×#16839913C2semidirect
4. #21446955C2semidirect
5. #21446895C2semidirect
6. #21446177C2semidirect
7. #21446170C2semidirect
8. #21446141C2semidirect
9. #21446131C2semidirect
10. #21445890C2semidirect
11. C61×#2309C2semidirect
16. C2×#14519777C22semidirect
18. C2×#14519765C22semidirect
24. C2×C4×Dic122C22semidirect
25. C2×C4×D244C22semidirect
28. C2×#14519700C22semidirect
31. C2×#14519697C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
36. #16839517C22semidirect
37. #16839517C22semidirect
303. #2309D61semidirect
354. C61#23945semidirect

Maximal subgroups

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

subgroupindex
C2×#168403832
C2×#168399132
C2×#168398962
#214469552
#214468952
#214461772
#214461702
#214461412
#214461312
#214458902
C61×#23092
#2394561