Finite Group Databasebeta

← #29655109

Subgroups — #29655109

Order 15616.

1104 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic, complemented
2#21456757C211yesmaximal, characteristic, complemented
3#21451927C211yesmaximal, characteristic, complemented
4#21444765C214yesmaximal, complemented
5#21444742C214yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444171C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic, complemented
10C2×#14519774C2211yescharacteristic
11C2×#14519773—21no—
12C2×#14519766—21no—
13#16840152—21no—
14#16840136—21no—
15#16840128—21no—
16#16840110—21no—
17#16839962—22no—
18#16839962—21no—
19#16839957—21no—
20#16839912C2214yes—
21#16839912C2212yes—
22#16839895C2212yes—
23#16839819C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839671—22no—
26#16839671C2214yes—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839499C2214yes—
30#16839496C2212yes—
31#16839486C2214yes—
32C61×#563—21no—
33C61×#562—21no—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543C2212yes—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538C2212yes—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C22×C22⋊D61—24no—
51C2×#11758201—24no—
52C2×#11758201—22no—
53C2×#11758199—22no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774—21no—
58#14519773—44no—
59#14519766—44no—
60C2×#11758198—24no—
61C2×C4×Dic61—21no—
62C4⋊#92418—21no—
63C4⋊#92418—21no—
64#14519697—22no—
65#14519697—24no—
66#14519697—22no—
67C22⋊#92418C2312yes—
68C22⋊#92418C2312yes—
69#14519688—21no—
70C4×#92418—21no—
71#14519684—21no—
72#14519684C2312yes—
73C2×#41×C61—21no—
74C22×Q8×C61—21no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61D411yescharacteristic, complemented
80C4⋊Q8×C61D411yescharacteristic, complemented
81#126×C61—22no—
82#126×C61—22no—
83#126×C61—22no—
84#126×C61—21no—
85#125×C61—21no—
86#125×C61D411yescharacteristic, complemented
87#124×C61—21no—
88#124×C61—22no—
89#124×C61—22no—
90#124×C61—21no—
91#124×C61—21no—
92#124×C61—21no—
93#124×C61—21no—
94#124×C61D411yescharacteristic, complemented
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—22no—
98#123×C61—21no—
99#123×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61C2312yes—
110C22⋊Q8×C61C2311yescharacteristic
111#121×C61—21no—
112#121×C61—22no—
113#121×C61—22no—
114#121×C61C2311yescharacteristic
115#120×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61D412yescomplemented
119C4×Q8×C61D411yescharacteristic, complemented
120C4×D4×C61—21no—
121C4×D4×C61—22no—
122C4×D4×C61—22no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic, complemented
125#117×C61—22no—
126#117×C61—21no—
127#117×C61—21no—
128C2×C22⋊C4×C61—22no—
129C2×C22⋊C4×C61D412yes—
130C2×C22⋊C4×C61C2312yes—
131C2×C42×C61C2311yescharacteristic
132C24×C61C2×D412yes—
133C23×D61—22no—
134#41×C61—82no—
135C2×Q8×C61—22no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—42no—
140C2×Q8×C61—82no—
141C2×Q8×C61C2×D411yescharacteristic
142C2×Q8×C61C2×D412yes—
143C2×D4×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—42no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—48no—
159C2×C22⋊D61—84no—
160C2×C22⋊D61—84no—
161C22×Dic61—22no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—42no—
165C2×D244—42no—
166C2×D244—41no—
167C2×D244—41no—
168C2×D244—41no—
169C2×D244—41no—
170C2×D244—82no—
171C2×D244—82no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D412yes—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—21no—
203C42×C61—22no—
204C42×C61—22no—
205C42×C61—21no—
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2×D411yescharacteristic
210#11758201—82no—
211#11758201—84no—
212#11758199—82no—
213#11758198—48no—
214C4×Dic61—24no—
215C2×#92418—22no—
216C2×#92418—22no—
217C23×C61—22no—
218C23×C61—21no—
219C23×C61—24no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61C22×D411yescharacteristic
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—44no—
226C22×D61—42no—
227C22×D61—44no—
228Q8×C61—41no—
229Q8×C61—42no—
230Q8×C61—42no—
231Q8×C61—42no—
232Q8×C61—81no—
233Q8×C61—82no—
234D4×C61—41no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—81no—
238D4×C61—82no—
239D4×C61—82no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—82no—
252C2×C4×C61—82no—
253C2×C4×C61—82no—
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—816no—
261C2×Dic61—24no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264D244—42no—
265D244—42no—
266D244—44no—
267D244—82no—
268D244—82no—
269D244—82no—
270D244—82no—
271#92418—42no—
272#92418—42no—
273#18428—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—41no—
282C22×C61—84no—
283C22×C61C2×#12011yescharacteristic
284C22×C61#57712yes—
285C2×D61—84no—
286C2×D61—84no—
287C2×D61—82no—
288C2×D61—82no—
289C2×D61—82no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61#45312yes—
302Dic61—44no—
303#3561D6111yescharacteristic, complemented
304#3217—611no—
305#3208—611no—
306#2657—611no—
307#1600—614no—
308#1596—614no—
309#1506—611no—
310#1500—611no—
311#1469—611no—
312C2×C61—22no—
313C2×C61—81no—
314C2×C61—82no—
315C2×C61#301011yescharacteristic
316C2×C61C2×#45311yescharacteristic
317C2×C61#302711yescharacteristic
318D61—162no—
319#563—21no—
320#562—21no—
321#556—21no—
322#551—22no—
323D4×Q8—21no—
324#548—21no—
325#546—21no—
326#543C2×D6112yes—
327#542—21no—
328#542—22no—
329#538C2×D6112yes—
330#536—21no—
331#533C2×D6111yescharacteristic
332C2×#127—611no—
333C2×#127—1221no—
334#529C2×D6111yescharacteristic
335C2×#124—1221no—
336C2×#124C2×D6111yescharacteristic
337#496—1221no—
338#492—1221no—
339#490—1221no—
340#486—1221no—
341#460—1222no—
342#460—1221no—
343#459—1221no—
344#450—612no—
345#450—614no—
346#447—612no—
347#432—611no—
348#431—611no—
349#392—614no—
350#392—1222no—
351#376—611no—
352#328—612no—
353#328—614no—
354#327—612no—
355#323—614no—
356C61#1842811yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C22×Q8—21no—
359C22×Q8C22⋊D6111yescharacteristic
360C22×D4—612no—
361C22×D4—1224no—
362C22×D4—1221no—
363C22×D4—1221no—
364C22×D4C22⋊D6111yescharacteristic
365C2×SD16—1224no—
366C2×SD16—1222no—
367C2×D8—1222no—
368C4⋊Q8—21no—
369C4⋊Q8—22no—
370C4⋊Q8C22⋊D6111yescharacteristic, complemented
371C4⋊Q8C22⋊D6111yescharacteristic, complemented
372#127—1221no—
373#127—1221no—
374#127—1221no—
375#127—1221no—
376#127—2444no—
377#126—22no—
378#126—22no—
379#126—22no—
380#126—21no—
381#125—21no—
382#125C22⋊D6111yescharacteristic, complemented
383#124—21no—
384#124—22no—
385#124—22no—
386#124—21no—
387#124—21no—
388#124—21no—
389#124—21no—
390#124—2444no—
391#124C22⋊D6111yescharacteristic, complemented
392#123—22no—
393#123—22no—
394#123—22no—
395#123—21no—
396#123—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8C22×D6111yescharacteristic
407C22⋊Q8C22×D6112yes—
408#121—21no—
409#121—22no—
410#121—22no—
411#121C22×D6111yescharacteristic
412#120—22no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8C22⋊D6111yescharacteristic, complemented
416C4×Q8C22⋊D6112yescomplemented
417C4×D4—21no—
418C4×D4—22no—
419C4×D4—21no—
420C4×D4—22no—
421C4×D4C22⋊D6111yescharacteristic, complemented
422#117—22no—
423#117—21no—
424#117—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1224no—
427C2×C22⋊C4C22⋊D6112yes—
428C2×C22⋊C4C22×D6112yes—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433#102—1222no—
434#102—1222no—
435#102—1224no—
436C22⋊C8—612no—
437C22⋊C8—612no—
438#97—1221no—
439C4×C8—1221no—
440#95—612no—
441#95—1221no—
442C24—1222no—
443C24C2×C22⋊D6112yes—
444#41—82no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×Q8C2×C22⋊D6112yes—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—4882no—
462C2×D4—4882no—
463C2×D4—4884no—
464C2×D4—4884no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2448no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4—2441no—
471C2×D4—2441no—
472C2×D4C2×C22⋊D6111yescharacteristic
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—1222no—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6112yes—
481C22×C4C23×D6111yescharacteristic
482SD16—4882no—
483SD16—4884no—
484D8—4882no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4C2×C22⋊D6111yescharacteristic
501C4⋊C4C2×C22⋊D6111yescharacteristic
502C4⋊C4C2×C22⋊D6112yes—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—2448no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—1224no—
523C42C2×C22⋊D6111yescharacteristic
524C42C2×C22⋊D6111yescharacteristic
525C42C2×C22⋊D6111yescharacteristic
526C42C2×C22⋊D6111yescharacteristic
527C23—22no—
528C23—21no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—2442no—
533C23—2444no—
534C23—2442no—
535C23—2442no—
536C23—2444no—
537C23C22×C22⋊D6111yescharacteristic
538Q8—41no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—81no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—42no—
547D4—81no—
548D4—82no—
549D4—82no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—48816no—
554D4—4882no—
555D4—2442no—
556D4—2442no—
557D4—2444no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—1224no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4#1451974611yescharacteristic
576C2×C4#1451974611yescharacteristic
577C2×C4#1451974611yescharacteristic
578C2×C4#1451974611yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C2×C4C22×C22⋊D6111yescharacteristic
581C8—2442no—
582C8—2442no—
583C22—21no—
584C22—21no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—84no—
592C22—4884no—
593C22—4884no—
594C22—4882no—
595C22—4882no—
596C22—4882no—
597C22C2×#1451974611yescharacteristic
598C22#1684057012yes—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—2444no—
611C4#1683993012yes—
612C2—22no—
613C2—82no—
614C2—81no—
615C2—9762no—
616C2#2145510511yescharacteristic
617C2C2×#1683993011yescharacteristic
618C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#18428
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. #21456757C2semidirect
3. #21451927C2semidirect
4. #21444765C2semidirect
5. #21444742C2semidirect
6. #21444331C2semidirect
7. #21444305C2semidirect
8. #21444171C2semidirect
9. C61×#3561C2semidirect
79. C4⋊Q8×C61D4semidirect
80. C4⋊Q8×C61D4semidirect
86. #125×C61D4semidirect
94. #124×C61D4semidirect
118. C4×Q8×C61D4semidirect
119. C4×Q8×C61D4semidirect
124. C4×D4×C61D4semidirect
303. #3561D61semidirect
356. C61#18428semidirect
370. C4⋊Q8C22⋊D61semidirect
371. C4⋊Q8C22⋊D61semidirect
382. #125C22⋊D61semidirect
391. #124C22⋊D61semidirect
415. C4×Q8C22⋊D61semidirect
416. C4×Q8C22⋊D61semidirect
421. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214567572
#214519272
#214447652
#214447422
#214443312
#214443052
#214441712
C61×#35612
#1842861