Finite Group Databasebeta

← #29657885

Subgroups — #29657885

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2#21456843C211yesmaximal, characteristic, complemented
3C2×#16840112C211yesmaximal, characteristic, complemented
4#21445047C212yesmaximal, complemented
5#21445040C212yesmaximal, complemented
6#21445033C212yesmaximal, complemented
7#21445026C212yesmaximal, complemented
8#21444545C211yesmaximal, characteristic, complemented
9#21444279C211yesmaximal, characteristic, complemented
10#21444279C211yesmaximal, characteristic, complemented
11C61×#3544C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—21no—
14#16840157—21no—
15#16840132—21no—
16#16840121—21no—
17#16840112—22no—
18#16840112C2212yes—
19#16840112C2212yes—
20#16840095—21no—
21#16840078—21no—
22#16840047C2212yes—
23#16840042C2212yes—
24#16839972C2212yes—
25#16839967C2212yes—
26#16839917—21no—
27#16839900—21no—
28#16839819C2211yescharacteristic
29#16839819C2211yescharacteristic
30C2×#14519701—22no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yes—
33#16839511C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#563—22no—
39Q82×C61C2211yescharacteristic
40C61×#555C2211yescharacteristic
41C61×#554C2211yescharacteristic
42C61×#554C2211yescharacteristic
43C61×#552—21no—
44C61×#552—21no—
45C61×#547—21no—
46C61×#547—21no—
47C61×#544—21no—
48C61×#544—21no—
49C61×#542—21no—
50C61×#542—21no—
51C61×#536—22no—
52C2×C4⋊Q8×C61C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C61×#529C2211yescharacteristic
55C22×Dic122—21no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C22×#92418—21no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2312yes—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65#14519710—22no—
66#14519708—22no—
67C4⋊#92418C2312yes—
68C4⋊#92418C2312yes—
69#14519701—24no—
70#14519701—24no—
71#14519701—22no—
72#14519700—22no—
73#14519700—22no—
74#14519698—22no—
75C22⋊#92418—21no—
76C22⋊#92418—21no—
77#14519688—21no—
78#14519688—21no—
79C4×#92418—21no—
80C4×#92418C2312yes—
81C2×#41×C61—22no—
82C22×Q8×C61D412yes—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61D412yes—
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#126×C61—22no—
93#126×C61—22no—
94#126×C61—21no—
95#126×C61—21no—
96#125×C61—22no—
97#125×C61—22no—
98#125×C61—21no—
99#125×C61—21no—
100#125×C61D411yescharacteristic
101#125×C61D411yescharacteristic
102#124×C61—22no—
103#124×C61—22no—
104#124×C61D411yescharacteristic
105#124×C61D411yescharacteristic
106#123×C61—22no—
107#123×C61—22no—
108#123×C61—22no—
109#123×C61—22no—
110#123×C61—21no—
111#123×C61—21no—
112C22⋊Q8×C61—22no—
113C22⋊Q8×C61—22no—
114C22⋊Q8×C61—22no—
115C22⋊Q8×C61—22no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C22⋊Q8×C61D411yescharacteristic
121C22⋊Q8×C61D411yescharacteristic
122#121×C61—21no—
123#121×C61—21no—
124#121×C61D411yescharacteristic
125#121×C61D411yescharacteristic
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#117×C61—22no—
137#117×C61—21no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61—21no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61—21no—
144C2×C4⋊C4×C61—21no—
145C2×C42×C61C2311yescharacteristic
146#41×C61—84no—
147C2×Q8×C61—22no—
148C2×Q8×C61—22no—
149C2×Q8×C61—44no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D412yes—
153C2×Q8×C61C2×D412yes—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—22no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×Dic61—41no—
168C2×Dic122—22no—
169C2×Dic122—41no—
170C2×Dic122—82no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—21no—
209C42×C61—22no—
210C42×C61—22no—
211C42×C61—21no—
212C42×C61—21no—
213C42×C61C2×D411yescharacteristic
214C42×C61C2×D411yescharacteristic
215C42×C61C2×D411yescharacteristic
216C42×C61C2411yescharacteristic
217#11758202—82no—
218#11758200—82no—
219C4⋊Dic61—22no—
220C4⋊Dic61—22no—
221C4⋊Dic61—41no—
222C4⋊Dic61—41no—
223C2×#92418—22no—
224C2×#92418—22no—
225C2×#92418—22no—
226C2×#92418—42no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61#12011yescharacteristic
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—42no—
233Q8×C61—42no—
234Q8×C61—82no—
235Q8×C61—82no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—42no—
239D4×C61—42no—
240D4×C61—82no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—44no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—82no—
256C2×C4×C61—82no—
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×Dic61—42no—
264C2×Dic61—82no—
265Dic122—42no—
266Dic122—42no—
267Dic122—82no—
268#92418—42no—
269#92418—42no—
270#92418—42no—
271#18690—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61#45512yes—
281C22×C61C2×#12011yescharacteristic
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61#57712yes—
295C4×C61#57812yes—
296Dic61—82no—
297#3544D6111yescharacteristic
298#3239—611no—
299#3218—611no—
300C2×#487—611no—
301#1646—612no—
302#1645—612no—
303#1644—612no—
304#1643—612no—
305#1558—611no—
306#1494—611no—
307#1494—611no—
308C2×C61—22no—
309C2×C61—81no—
310C2×C61—81no—
311C2×C61#301011yescharacteristic
312C2×C61#301111yescharacteristic
313C2×C61C2×#45511yescharacteristic
314#563—22no—
315Q82C2×D6111yescharacteristic
316#555C2×D6111yescharacteristic
317#554C2×D6111yescharacteristic
318#554C2×D6111yescharacteristic
319#552—21no—
320#552—21no—
321#547—21no—
322#547—21no—
323#544—21no—
324#544—21no—
325#542—21no—
326#542—21no—
327#536—22no—
328C2×C4⋊Q8—611no—
329C2×C4⋊Q8C2×D6111yescharacteristic
330#529C2×D6111yescharacteristic
331#529C2×D6111yescharacteristic
332#501—1221no—
333#497—1221no—
334#491—1221no—
335#489—1221no—
336#487—612no—
337#487—612no—
338#487—1222no—
339#484—1221no—
340#481—1221no—
341#475—612no—
342#474—612no—
343#462—612no—
344#461—612no—
345#451—1221no—
346#448—1221no—
347#432—611no—
348#432—611no—
349C2×#103—1222no—
350C2×C4×C8—611no—
351#333—612no—
352#333—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355D4⋊C8—612no—
356D4⋊C8—612no—
357C61#1869011yescharacteristic, complemented, Sylow
358C2×#41—22no—
359C22×Q8—1221no—
360C22×Q8C22⋊D6112yes—
361C2×Q16—1222no—
362C2×SD16—1222no—
363C22×C8—1221no—
364C4⋊Q8—22no—
365C4⋊Q8—22no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—21no—
369C4⋊Q8—21no—
370C4⋊Q8—612no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8C22⋊D6112yes—
375C4⋊Q8C22×D6111yescharacteristic
376C4⋊Q8C22×D6111yescharacteristic
377#126—22no—
378#126—22no—
379#126—21no—
380#126—21no—
381#125—22no—
382#125—22no—
383#125—21no—
384#125—21no—
385#125C22⋊D6111yescharacteristic
386#125C22⋊D6111yescharacteristic
387#124—22no—
388#124—22no—
389#124C22⋊D6111yescharacteristic
390#124C22⋊D6111yescharacteristic
391#123—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—21no—
396#123—21no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8C22⋊D6111yescharacteristic
406C22⋊Q8C22⋊D6111yescharacteristic
407#121—21no—
408#121—21no—
409#121C22⋊D6111yescharacteristic
410#121C22⋊D6111yescharacteristic
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C22×D6111yescharacteristic
414C4×Q8C22×D6111yescharacteristic
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4C22×D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421#117—22no—
422#117—21no—
423#117—21no—
424#117—21no—
425#117—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C42C22×D6111yescharacteristic
433#107—1222no—
434#106—1222no—
435C4⋊C8—612no—
436C4⋊C8—612no—
437#103—1224no—
438#103—1224no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#102—1222no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445#97—1221no—
446#97—1221no—
447C4×C8—612no—
448C4×C8—1221no—
449#41—84no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—44no—
453C2×Q8—1222no—
454C2×Q8—4882no—
455C2×Q8—2441no—
456C2×Q8C2×C22⋊D6111yescharacteristic
457C2×Q8C2×C22⋊D6111yescharacteristic
458C2×Q8C2×C22⋊D6112yes—
459C2×Q8C2×C22⋊D6112yes—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4C2×C22⋊D6111yescharacteristic
464C2×D4C2×C22⋊D6111yescharacteristic
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—2441no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474Q16—4882no—
475SD16—4882no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—2442no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42C2×C22⋊D6111yescharacteristic
527C42C2×C22⋊D6111yescharacteristic
528C42C2×C22⋊D6111yescharacteristic
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23#1451974611yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—42no—
537Q8—82no—
538Q8—82no—
539Q8—4882no—
540Q8—2442no—
541Q8—2442no—
542D4—22no—
543D4—22no—
544D4—42no—
545D4—42no—
546D4—82no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—4882no—
564C2×C4—2442no—
565C2×C4#1451974611yescharacteristic
566C2×C4#1451974611yescharacteristic
567C2×C4#1451974611yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4C22×C22⋊D6111yescharacteristic
570C2×C4C22×C22⋊D6111yescharacteristic
571C8—2442no—
572C8—2442no—
573C8—2442no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22#1683994012yes—
583C22C2×#1451974611yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—4882no—
597C4#1684057012yes—
598C4#1684057712yes—
599C2—22no—
600C2—81no—
601C2—81no—
602C2#2145510511yescharacteristic
603C2#2145511411yescharacteristic
604C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3544C2
socleC22×C61C2×#120
hypercenter#3544D61
hypocenterC61#18690
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21456843C2semidirect
3. C2×#16840112C2semidirect
4. #21445047C2semidirect
5. #21445040C2semidirect
6. #21445033C2semidirect
7. #21445026C2semidirect
8. #21444545C2semidirect
9. #21444279C2semidirect
10. #21444279C2semidirect
357. C61#18690semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214568432
C2×#168401122
#214450472
#214450402
#214450332
#214450262
#214445452
#214442792
C61×#35442
#1869061