Finite Group Databasebeta

← #29654256

Subgroups — #29654256

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456802C212yesmaximal, complemented
2#21451506C211yesmaximal, characteristic, complemented
3#21444788C212yesmaximal, complemented
4#21444771C212yesmaximal, complemented
5#21444754C212yesmaximal, complemented
6#21444737C212yesmaximal, complemented
7#21444327C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#3449C211yesmaximal, characteristic, complemented
10C2×#14519765C2211yescharacteristic, complemented
11#16840149—22no—
12#16840124—22no—
13#16839977—24no—
14#16839900C2214yescomplemented
15#16839895C2214yescomplemented
16#16839819C2212yescomplemented
17#16839682C2211yescharacteristic, complemented
18#16839679C2211yescharacteristic, complemented
19#16839674C2211yescharacteristic
20#16839671C2211yescharacteristic
21#16839662—21no—
22#16839653—21no—
23#16839652—21no—
24#16839641—21no—
25#16839608C2211yescharacteristic, complemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C22×#41×C61—21no—
31C61×#537—21no—
32C61×#537C2212yes—
33C61×#536—22no—
34C61×#535—22no—
35C61×#534—21no—
36C61×#534C2212yes—
37C61×#532—21no—
38C61×#532C2212yes—
39C61×#525—24no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122C2311yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C22⋊D61—21no—
44C2×#11758200—24no—
45C2×#11758199—24no—
46#14519765—22no—
47#14519765—22no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×C4×Dic61—21no—
56C4⋊#92418—22no—
57#14519700—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—44no—
71C2×#41×C61—44no—
72C22×Q8×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—22no—
75C23×C4×C61—22no—
76C23×C4×C61D411yescharacteristic
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yescomplemented
79#127×C61—21no—
80#127×C61D412yescomplemented
81#124×C61—22no—
82#124×C61—22no—
83#123×C61—24no—
84#123×C61—24no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87#121×C61—24no—
88#121×C61—24no—
89#121×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61C2314yes—
93#120×C61—24no—
94#120×C61—24no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61D414yescomplemented
99#117×C61—22no—
100#117×C61—22no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61C2311yescharacteristic
105C2×C42×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C24×C61—22no—
108C24×C61C2×D411yescharacteristic
109C23×D61—21no—
110#41×C61—48no—
111#41×C61—48no—
112#41×C61—44no—
113#41×C61—44no—
114#41×C61—44no—
115#41×C61—44no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—44no—
119C2×D4×C61—24no—
120C2×D4×C61—22no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—44no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61C2×D414yes—
130C22×C4×C61—22no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—48no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—81no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—22no—
151C2×D244—81no—
152C2×Dic122—22no—
153C2×Dic122—81no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D414yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—24no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170C22⋊Dic61—81no—
171#11758200—84no—
172#11758199—84no—
173#11758198—42no—
174#11758198—41no—
175#11758198—41no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179#11758196—81no—
180C4×Dic61—42no—
181C2×#92418—24no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—24no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—44no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61AGL(1,ℤ/8)12yes—
191C23×C61C22×D411yescharacteristic
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199Q8×C61—44no—
200Q8×C61—44no—
201D4×C61—44no—
202D4×C61—44no—
203D4×C61—44no—
204D4×C61—48no—
205D4×C61—44no—
206D4×C61—44no—
207D4×C61—82no—
208D4×C61—82no—
209D4×C61—82no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—24no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—48no—
222C2×C4×C61—48no—
223C2×C4×C61—44no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12012yes—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61C22×D412yes—
231C22⋊D61—84no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241D244—44no—
242Dic122—44no—
243#92418—44no—
244#18332—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—42no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—44no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—81no—
255C22×C61—88no—
256C22×C61C2×#12011yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—21no—
267C4×C61—22no—
268C4×C61—21no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—44no—
272C4×C61—41no—
273C4×C61—41no—
274Dic61—42no—
275Dic61—81no—
276Dic61—81no—
277#3449D6111yescharacteristic, complemented
278#3213—612no—
279#2610—611no—
280#1604—612no—
281#1601—612no—
282#1598—612no—
283#1595—612no—
284#1505—612no—
285#1464—611no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—42no—
289C2×C61—84no—
290C2×C61#300311yescharacteristic
291C2×C61#301811yescharacteristic
292C2×C61#302711yescharacteristic
293D61—161no—
294C22×#41—21no—
295#537—21no—
296#537C2×D6112yes—
297#536—22no—
298#535—22no—
299#534—21no—
300#534C2×D6112yes—
301#532—21no—
302#532C2×D6112yes—
303C2×#124—611no—
304#525—24no—
305C2×#117C2×D6111yescharacteristic
306#496—1222no—
307#489—1222no—
308#463—1224no—
309#448—614no—
310#447—614no—
311#432—612no—
312#395—611no—
313#394—611no—
314#393—611no—
315#392—611no—
316#390—1221no—
317#388—1221no—
318#388—1221no—
319#386—1221no—
320#376—611no—
321#327—614no—
322#327—614no—
323#324—612no—
324#323—612no—
325C61#1833211yescharacteristic, complemented, Sylow
326C2×#41—22no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×Q8—611no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—611no—
337C22×D4—1221no—
338C23×C4—22no—
339C23×C4C22⋊D6111yescharacteristic
340C2×SD16—1224no—
341C2×D8—1224no—
342C4⋊Q8—21no—
343C4⋊Q8C22⋊D6112yescomplemented
344#127—21no—
345#127C22⋊D6112yescomplemented
346#124—22no—
347#124—22no—
348#124—1222no—
349#124—1222no—
350#123—24no—
351#123—24no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354#121—24no—
355#121—24no—
356#121—22no—
357#121—22no—
358#121—22no—
359#121C22×D6114yes—
360#120—24no—
361#120—24no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4C22⋊D6114yescomplemented
366#117—22no—
367#117—22no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22⋊D6111yescharacteristic
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4C22⋊D6111yescharacteristic
378C2×C22⋊C4C22×D6111yescharacteristic
379C2×C42—1221no—
380C2×C42C22⋊D6111yescharacteristic
381C2×C42C22×D6111yescharacteristic
382C4⋊C8—1222no—
383#103—1224no—
384#102—1224no—
385C22⋊C8—614no—
386#97—1222no—
387#95—611no—
388#95—611no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392C24—22no—
393C24—1221no—
394C24C2×C22⋊D6111yescharacteristic
395#41—48no—
396#41—48no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—1222no—
405C2×Q8—4881no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—1222no—
417C2×D4—4881no—
418C2×D4—4881no—
419C2×D4—2442no—
420C2×D4—2442no—
421C2×D4C2×C22⋊D6114yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439SD16—4884no—
440D8—4884no—
441C2×C8—1224no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—4881no—
448C4⋊C4C2×C22⋊D6114yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—4881no—
456C22⋊C4—4881no—
457C22⋊C4—2442no—
458C22⋊C4—2441no—
459C22⋊C4—2441no—
460C22⋊C4—2442no—
461C22⋊C4—2442no—
462C42—21no—
463C42—21no—
464C42—2442no—
465C42C2×C22⋊D6112yes—
466C42C2×C22⋊D6112yes—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—82no—
474C23—82no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23#1451980312yes—
483C23C22×C22⋊D6111yescharacteristic
484Q8—44no—
485Q8—44no—
486Q8—2444no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—4884no—
497D4—2444no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—1222no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4#1451974611yescharacteristic
524C2×C4#1451974611yescharacteristic
525C2×C4#1451974612yes—
526C2×C4#1451980312yes—
527C2×C4C22×C22⋊D6112yes—
528C8—2444no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—81no—
539C22—88no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22—4881no—
547C22C2×#1451974611yescharacteristic
548C22C2×#1451980311yescharacteristic
549C22C2×#1451980311yescharacteristic
550C4—22no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—44no—
558C4—4881no—
559C4—4881no—
560C4—2442no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—84no—
565C2—9761no—
566C2#2145504611yescharacteristic
567C2#2145516811yescharacteristic
568C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#18332
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21456802C2semidirect
2. #21451506C2semidirect
3. #21444788C2semidirect
4. #21444771C2semidirect
5. #21444754C2semidirect
6. #21444737C2semidirect
7. #21444327C2semidirect
8. #21444153C2semidirect
9. C61×#3449C2semidirect
10. C2×#14519765C22semidirect
14. #16839900C22semidirect
15. #16839895C22semidirect
16. #16839819C22semidirect
17. #16839682C22semidirect
18. #16839679C22semidirect
25. #16839608C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
78. C4⋊Q8×C61D4semidirect
80. #127×C61D4semidirect
98. C4×D4×C61D4semidirect
277. #3449D61semidirect
325. C61#18332semidirect
343. C4⋊Q8C22⋊D61semidirect
345. #127C22⋊D61semidirect
365. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214568022
#214515062
#214447882
#214447712
#214447542
#214447372
#214443272
#214441532
C61×#34492
#1833261