Finite Group Databasebeta

← #29652282

Subgroups — #29652282

Order 15616.

1292 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C212yesmaximal, complemented
2#21449055C211yesmaximal, characteristic, complemented
3#21444822C212yesmaximal, complemented
4#21444805C212yesmaximal, complemented
5#21444754C212yesmaximal, complemented
6#21444737C212yesmaximal, complemented
7#21444314C212yesmaximal, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#3449C211yesmaximal, characteristic, complemented
10C2×C4×D244C2211yescharacteristic, complemented
11C2×C4×C22⋊D61—21no—
12#16840073C2214yescomplemented
13#16840039—22no—
14#16839954—22no—
15#16839895C2214yescomplemented
16C4×#11758199—24no—
17#16839815C2212yescomplemented
18#16839692C2211yescharacteristic, complemented
19#16839687C2211yescharacteristic
20#16839684—21no—
21#16839679C2211yescharacteristic, complemented
22#16839678—21no—
23#16839671C2211yescharacteristic
24C4×C4⋊Dic61C2211yescharacteristic, complemented
25C4×#11758198—21no—
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×D244C2311yescharacteristic
42C22×C22⋊D61—21no—
43C22×C4×D61—21no—
44C2×#11758199—24no—
45C4×D244—22no—
46C4×D244—22no—
47C4×C22⋊D61—44no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2311yescharacteristic
50C2×#11758196—21no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×C4×Dic61—21no—
57#14519710—24no—
58C4⋊#92418—22no—
59#14519698—24no—
60#14519697—24no—
61C22⋊#92418C2314yes—
62C4×#92418—22no—
63#14519684—21no—
64#14519684—21no—
65#14519684C2311yescharacteristic
66#14519684C2311yescharacteristic
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C2×#41×C61—44no—
72C2×#41×C61—44no—
73C22×Q8×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61—22no—
76C23×C4×C61—22no—
77C23×C4×C61D411yescharacteristic
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61D412yescomplemented
80#127×C61—21no—
81#127×C61D412yescomplemented
82#124×C61—22no—
83#124×C61—22no—
84#123×C61—24no—
85#123×C61—24no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88#121×C61—24no—
89#121×C61—24no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61C2314yes—
94#120×C61—24no—
95#120×C61—24no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61D414yescomplemented
100#117×C61—22no—
101#117×C61—22no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61C2311yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C24×C61—22no—
109C24×C61C2×D411yescharacteristic
110C23×D61—21no—
111#41×C61—44no—
112#41×C61—48no—
113#41×C61—48no—
114#41×C61—44no—
115#41×C61—44no—
116#41×C61—44no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—44no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61C2×D414yes—
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×C61D814yescomplemented
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—22no—
152C2×D244—42no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C2×D414yes—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—24no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61C2×D412yes—
172C42×C61C2×D412yes—
173C22⋊Dic61—42no—
174#11758199—48no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179C4⋊Dic61—22no—
180C4⋊Dic61—42no—
181#11758196—42no—
182C4×Dic61—42no—
183C2×#92418—24no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—24no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—44no—
190C23×C61—82no—
191C23×C61—82no—
192C23×C61C2×D812yes—
193C23×C61C22×D411yescharacteristic
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201Q8×C61—44no—
202Q8×C61—44no—
203D4×C61—44no—
204D4×C61—44no—
205D4×C61—48no—
206D4×C61—44no—
207D4×C61—44no—
208D4×C61—44no—
209D4×C61—82no—
210D4×C61—82no—
211D4×C61—82no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
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—44no—
223C2×C4×C61—48no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61#12012yes—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61C2×D812yes—
230C2×C4×C61C2×D812yes—
231C2×C4×C61AGL(1,ℤ/8)12yes—
232C2×C4×C61C22×D412yes—
233C22⋊D61—48no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243D244—24no—
244D244—42no—
245C4×D61—82no—
246C4×D61—82no—
247#92418—44no—
248#18122—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—22no—
252C22×C61—42no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—81no—
259C22×C61—88no—
260C22×C61C2×#12011yescharacteristic
261C22×C61C22×D811yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—44no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61#44712yes—
277C4×C61#44712yes—
278Dic61—42no—
279Dic61—81no—
280Dic61—81no—
281#3449D6111yescharacteristic, complemented
282#3100—612no—
283#2392—611no—
284#1610—612no—
285#1607—612no—
286#1598—612no—
287#1595—612no—
288#1502—612no—
289#1458—611no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61—42no—
293C2×C61—84no—
294C2×C61C2×#44711yescharacteristic
295C2×C61#301811yescharacteristic
296C2×C61#302511yescharacteristic
297D61—82no—
298C22×#41—21no—
299#537—21no—
300#537C2×D6112yes—
301#536—22no—
302#535—22no—
303#534—21no—
304#534C2×D6112yes—
305#532—21no—
306#532C2×D6112yes—
307#525—24no—
308C2×C4×D4—611no—
309C2×C4×D4—1221no—
310C2×#117C2×D6111yescharacteristic
311#480—614no—
312#474—1222no—
313#459—1222no—
314#447—614no—
315C4×D8—1224no—
316#431—612no—
317#398—611no—
318#397—611no—
319#396—1221no—
320#394—611no—
321#394—1221no—
322#392—611no—
323C4×C4⋊C4—611no—
324C4×C22⋊C4—1221no—
325#327—614no—
326#327—614no—
327#324—612no—
328#323—612no—
329C61#1812211yescharacteristic, complemented, Sylow
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—44no—
335C2×#41—44no—
336C22×Q8—21no—
337C22×D4—21no—
338C22×D4—22no—
339C22×D4—611no—
340C22×D4—1221no—
341C23×C4—22no—
342C23×C4—1221no—
343C23×C4C22⋊D6111yescharacteristic
344C2×D8—1224no—
345C4⋊Q8—21no—
346C4⋊Q8C22⋊D6112yescomplemented
347#127—21no—
348#127C22⋊D6112yescomplemented
349#124—22no—
350#124—22no—
351#123—24no—
352#123—24no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355#121—24no—
356#121—24no—
357#121—22no—
358#121—22no—
359#121—22no—
360#121C22×D6114yes—
361#120—24no—
362#120—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—1222no—
367C4×D4—1222no—
368C4×D4—2444no—
369C4×D4C22⋊D6114yescomplemented
370#117—22no—
371#117—22no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4C22⋊D6111yescharacteristic
383C2×C22⋊C4C22×D6111yescharacteristic
384C2×C42—1221no—
385C2×C42C22⋊D6111yescharacteristic
386C2×C42C22×D6111yescharacteristic
387#107—1224no—
388C4⋊C8—1222no—
389#102—1224no—
390#102—1224no—
391C22⋊C8—614no—
392C4×C8—1222no—
393#95—611no—
394#95—611no—
395#95—1221no—
396#95—1221no—
397C24—22no—
398C24—1221no—
399C24C2×C22⋊D6111yescharacteristic
400#41—44no—
401#41—48no—
402#41—48no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—1222no—
420C2×D4—2442no—
421C2×D4—2442no—
422C2×D4—2442no—
423C2×D4—2442no—
424C2×D4C2×C22⋊D6114yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4#1175819914yescomplemented
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446D8—2448no—
447C2×C8—1224no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C4⋊C4C2×C22⋊D6114yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—24no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C22⋊C4—2442no—
466C22⋊C4—2442no—
467C22⋊C4—2442no—
468C42—21no—
469C42—21no—
470C42—2442no—
471C42C2×C22⋊D6112yes—
472C42C2×C22⋊D6112yes—
473C23—21no—
474C23—21no—
475C23—24no—
476C23—41no—
477C23—41no—
478C23—44no—
479C23—82no—
480C23—82no—
481C23—1222no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23C2×#1175819912yes—
489C23C22×C22⋊D6111yescharacteristic
490Q8—44no—
491Q8—44no—
492D4—44no—
493D4—44no—
494D4—48no—
495D4—44no—
496D4—44no—
497D4—44no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—1224no—
502D4—2442no—
503D4—2448no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—1222no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4#1451974612yes—
530C2×C4#1451974611yescharacteristic
531C2×C4#1451974611yescharacteristic
532C2×C4C2×#1175819912yes—
533C2×C4C2×#1175819912yes—
534C2×C4#1451980312yes—
535C2×C4C22×C22⋊D6112yes—
536C8—2444no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—81no—
547C22—88no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×#1451974611yescharacteristic
556C22C22×#1175819911yescharacteristic
557C22C2×#1451980311yescharacteristic
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—44no—
564C4—4881no—
565C4—4881no—
566C4—2442no—
567C4#1683989612yes—
568C4#1683989612yes—
569C2—21no—
570C2—21no—
571C2—42no—
572C2—84no—
573C2—4882no—
574C2C2×#1683989611yescharacteristic
575C2#2145516811yescharacteristic
576C2#2145521111yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#18122
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21455815C2semidirect
2. #21449055C2semidirect
3. #21444822C2semidirect
4. #21444805C2semidirect
5. #21444754C2semidirect
6. #21444737C2semidirect
7. #21444314C2semidirect
8. #21444133C2semidirect
9. C61×#3449C2semidirect
10. C2×C4×D244C22semidirect
12. #16840073C22semidirect
15. #16839895C22semidirect
17. #16839815C22semidirect
18. #16839692C22semidirect
21. #16839679C22semidirect
24. C4×C4⋊Dic61C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
28. #16839488C22semidirect
79. C4⋊Q8×C61D4semidirect
81. #127×C61D4semidirect
99. C4×D4×C61D4semidirect
138. C22×C4×C61D8semidirect
281. #3449D61semidirect
329. C61#18122semidirect
346. C4⋊Q8C22⋊D61semidirect
348. #127C22⋊D61semidirect
369. C4×D4C22⋊D61semidirect
439. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214558152
#214490552
#214448222
#214448052
#214447542
#214447372
#214443142
#214441332
C61×#34492
#1812261