Finite Group Databasebeta

← #29661451

Subgroups — #29661451

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C211yesmaximal, characteristic, complemented
2#21456272C211yesmaximal, characteristic, complemented
3C2×#16840128C211yesmaximal, characteristic, complemented
4#21445117C214yesmaximal, complemented
5#21445112C214yesmaximal, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic, complemented
10C2×#14519802—22no—
11C2×#14519774C2211yescharacteristic
12#16840152C2214yes—
13#16840152C2212yes—
14#16840138—21no—
15#16840136—21no—
16#16840128—22no—
17#16840128C2214yes—
18#16840128C2212yes—
19#16839962—21no—
20#16839957—21no—
21#16839912—21no—
22#16839895—21no—
23#16839819C2211yescharacteristic
24C2×#14519697—21no—
25C2×#14519697—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2212yescomplemented
29#16839504C2214yescomplemented
30#16839502C2212yescomplemented
31#16839502C2214yescomplemented
32C61×#563C2212yescomplemented
33C61×#562C2212yescomplemented
34C61×#554—21no—
35C61×#553—21no—
36C61×#553—21no—
37C61×#547—22no—
38C61×#547—21no—
39C61×#544—21no—
40C61×#540—21no—
41C61×#539—22no—
42C61×#539—21no—
43C61×#537—21no—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50#14519802—48no—
51C2×#11758201—24no—
52C2×#11758201—22no—
53C2×#11758199—24no—
54C2×#11758199—22no—
55C2×#11758194—21no—
56C22×#92418—21no—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774C2312yes—
61C4⋊#92418—21no—
62#14519697—22no—
63#14519697—22no—
64#14519697—24no—
65#14519697—24no—
66C22⋊#92418—21no—
67#14519688—21no—
68#14519688C2312yes—
69#14519688C2312yes—
70#14519688C2312yes—
71C4×#92418—21no—
72C2×#41×C61—22no—
73C2×#41×C61D412yes—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61D411yescharacteristic
77#127×C61D411yescharacteristic
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—21no—
81#125×C61—22no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61—21no—
85#125×C61D412yes—
86#125×C61C2312yes—
87#125×C61C2311yescharacteristic
88#124×C61C2312yes—
89#124×C61C2311yescharacteristic
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—22no—
94#123×C61—21no—
95#123×C61—22no—
96#123×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61D411yescharacteristic
104#121×C61—22no—
105#121×C61—22no—
106#121×C61—21no—
107#121×C61—21no—
108#121×C61—22no—
109#121×C61—22no—
110#121×C61—21no—
111#121×C61—21no—
112#121×C61D411yescharacteristic
113C4×Q8×C61D411yescharacteristic
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122#117×C61—21no—
123#117×C61—21no—
124C2×C4⋊C4×C61—22no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C23×D61—41no—
132#41×C61—44no—
133#41×C61—84no—
134C2×Q8×C61—22no—
135C2×Q8×C61C2×D411yescharacteristic
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C2×D411yescharacteristic
144C2×D4×C61C2×D412yes—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—44no—
159C2×D244—44no—
160C2×D244—41no—
161C2×D244—41no—
162C2×D244—41no—
163C2×D244—82no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—21no—
196C42×C61—22no—
197C42×C61—21no—
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2411yescharacteristic
202#11758201—48no—
203#11758201—82no—
204#11758199—48no—
205#11758199—82no—
206#11758194—24no—
207C2×#92418—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—42no—
211C23×C61—22no—
212C23×C61—21no—
213C23×C61#12011yescharacteristic
214C22×D61—42no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—82no—
218Q8×C61—42no—
219Q8×C61—41no—
220Q8×C61—82no—
221D4×C61—42no—
222D4×C61—41no—
223D4×C61—44no—
224D4×C61—42no—
225D4×C61—42no—
226D4×C61—81no—
227D4×C61—82no—
228D4×C61—82no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—82no—
241C2×C4×C61—81no—
242C2×C4×C61—82no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#12011yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251D244—44no—
252D244—42no—
253D244—42no—
254D244—44no—
255D244—82no—
256D244—82no—
257D244—82no—
258#92418—24no—
259#92418—42no—
260#92418—42no—
261#19016—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—42no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—41no—
270C22×C61#45312yes—
271C22×C61C2×#12011yescharacteristic
272C2×D61—82no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61—164no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61#45412yes—
288#3559D6111yescharacteristic, complemented
289#3217—611no—
290#3151—611no—
291C2×#490—611no—
292#1657—614no—
293#1656—614no—
294#1530—611no—
295#1528—611no—
296#1524—611no—
297C2×C61—22no—
298C2×C61—82no—
299C2×C61—81no—
300C2×C61C2×#45311yescharacteristic
301C2×C61C2×#45411yescharacteristic
302C2×C61#302111yescharacteristic
303D61—162no—
304C2×AGL(1,ℤ/8)—1222no—
305#563C2×D6112yescomplemented
306#562C2×D6112yescomplemented
307#554—21no—
308#553—21no—
309#553—21no—
310#547—22no—
311#547—21no—
312#544—21no—
313#540—21no—
314#539—22no—
315#539—21no—
316#537—21no—
317#532C2×D6111yescharacteristic
318C2×#127—611no—
319#529C2×D6111yescharacteristic
320C2×#125C2×D6111yescharacteristic
321#496—614no—
322#496—612no—
323#493—1221no—
324#492—1221no—
325#490—614no—
326#490—612no—
327#490—1222no—
328#460—1221no—
329#459—1221no—
330#450—1221no—
331#447—1221no—
332#432—611no—
333C2×#102—1221no—
334C2×#102—1221no—
335C4×M16—611no—
336C2×#97—611no—
337#330—614no—
338#330—612no—
339#329—612no—
340#329—614no—
341C61#1901611yescharacteristic, complemented, Sylow
342C2×#41—22no—
343C2×#41C22⋊D6112yes—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1221no—
347AGL(1,ℤ/8)—2448no—
348C2×SD16—1222no—
349C2×SD16—1224no—
350C2×D8—1222no—
351C2×D8—1224no—
352C2×M16—1221no—
353C22×C8—1221no—
354C4⋊Q8—21no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D6111yescharacteristic
357#127—612no—
358#127—1221no—
359#127—1221no—
360#127—1221no—
361#127C22⋊D6111yescharacteristic
362#126—22no—
363#126—22no—
364#126—21no—
365#125—22no—
366#125—21no—
367#125—21no—
368#125—21no—
369#125C22⋊D6112yes—
370#125C22×D6112yes—
371#125C22×D6111yescharacteristic
372#124C22×D6111yescharacteristic
373#124C22×D6112yes—
374#123—22no—
375#123—22no—
376#123—22no—
377#123—22no—
378#123—21no—
379#123—22no—
380#123—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8C22⋊D6111yescharacteristic
388#121—22no—
389#121—22no—
390#121—21no—
391#121—21no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121—21no—
396#121C22⋊D6111yescharacteristic
397C4×Q8C22⋊D6111yescharacteristic
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C22⋊D6111yescharacteristic
406#117—21no—
407#117—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C42C22×D6111yescharacteristic
415C4⋊C8—1221no—
416#102—1224no—
417#102—1224no—
418#102—1222no—
419#102—1222no—
420C22⋊C8—1221no—
421#97—612no—
422#97—612no—
423#97—612no—
424#97—1221no—
425C4×C8—1221no—
426C24—2441no—
427#41—44no—
428#41—84no—
429C2×Q8—22no—
430C2×Q8C2×C22⋊D6111yescharacteristic
431C2×Q8C2×C22⋊D6112yes—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1222no—
441C2×D4—4882no—
442C2×D4—2444no—
443C2×D4—2444no—
444C2×D4—2441no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4C2×C22⋊D6111yescharacteristic
448C2×D4C2×C22⋊D6112yes—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459SD16—4882no—
460SD16—2448no—
461D8—4882no—
462D8—2448no—
463M16—1224no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—2442no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4C2×C22⋊D6112yes—
485C4⋊C4C2×C22⋊D6112yes—
486C4⋊C4C2×C22⋊D6111yescharacteristic
487C4⋊C4C2×C22⋊D6111yescharacteristic
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C42—22no—
501C42—21no—
502C42C2×C22⋊D6111yescharacteristic
503C42C2×C22⋊D6111yescharacteristic
504C42C2×C22⋊D6111yescharacteristic
505C42C23×D6111yescharacteristic
506C23—22no—
507C23—21no—
508C23—4882no—
509C23—4882no—
510C23—4882no—
511C23—2442no—
512C23#1451974611yescharacteristic
513Q8—41no—
514Q8—42no—
515Q8—82no—
516D4—41no—
517D4—42no—
518D4—44no—
519D4—42no—
520D4—42no—
521D4—81no—
522D4—82no—
523D4—82no—
524D4—4882no—
525D4—4882no—
526D4—4882no—
527D4—2442no—
528D4—2444no—
529D4—2442no—
530D4—2444no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#1451974611yescharacteristic
548C2×C4#1451974611yescharacteristic
549C2×C4#1451974611yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C8—1224no—
554C8—2442no—
555C8—2442no—
556C22—21no—
557C22—21no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—9764no—
565C22—4882no—
566C22—4882no—
567C22—4882no—
568C22#1683993012yes—
569C22C2×#1451974611yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4#1683993512yes—
582C2—22no—
583C2—82no—
584C2—81no—
585C2—9762no—
586C2C2×#1683993011yescharacteristic
587C2C2×#1683993511yescharacteristic
588C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#19016
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. #21456272C2semidirect
3. C2×#16840128C2semidirect
4. #21445117C2semidirect
5. #21445112C2semidirect
6. #21444430C2semidirect
7. #21444421C2semidirect
8. #21444407C2semidirect
9. C61×#3559C2semidirect
28. #16839504C22semidirect
29. #16839504C22semidirect
30. #16839502C22semidirect
31. #16839502C22semidirect
32. C61×#563C22semidirect
33. C61×#562C22semidirect
288. #3559D61semidirect
305. #563C2×D61semidirect
306. #562C2×D61semidirect
341. C61#19016semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214562722
C2×#168401282
#214451172
#214451122
#214444302
#214444212
#214444072
C61×#35592
#1901661