Finite Group Databasebeta

← #29658740

Subgroups — #29658740

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic, complemented
4#21445047C214yesmaximal, complemented
5#21444902C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#555C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840136C2212yescomplemented
13#16840112C2212yescomplemented
14#16840075—21no—
15#16840042C2212yescomplemented
16#16839977—22no—
17#16839967C2212yescomplemented
18#16839964—22no—
19#16839957C2212yescomplemented
20#16839957C2212yescomplemented
21#16839901—22no—
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839511C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C61×#555—22no—
32C61×#555—22no—
33C61×#555—22no—
34C61×#555C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#124×C61—22no—
37C2×#121×C61—21no—
38C2×#121×C61—21no—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C2×#117×C61—21no—
42C2×#117×C61—21no—
43C2×#11758222—21no—
44#14519803—44no—
45C2×#11758200—22no—
46C2×#11758201—24no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519710—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—24no—
63#14519700—22no—
64#14519698—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61D412yes—
72C22×D4×C61—22no—
73C22×D4×C61D411yescharacteristic
74C22×D4×C61D411yescharacteristic
75C23×C4×C61—21no—
76C23×C4×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61D412yes—
80C4⋊Q8×C61C2312yes—
81#124×C61—28no—
82#124×C61—28no—
83#121×C61—24no—
84#121×C61—24no—
85#121×C61—24no—
86#121×C61—24no—
87C4×D4×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61—21no—
91C4×D4×C61D412yes—
92C4×D4×C61D412yes—
93C4×D4×C61C2312yes—
94C4×D4×C61C2312yes—
95#117×C61—24no—
96#117×C61—24no—
97#117×C61—24no—
98#117×C61—24no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—22no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—82no—
113C2×Q8×C61C2×D414yes—
114C2×D4×C61—28no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—84no—
121C2×D4×C61C2×D412yes—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—44no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×C22⋊D61—41no—
137#11758222—42no—
138#11758222—42no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×C4×D61—41no—
143C2×Dic122—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—28no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C42×C61—28no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163#11758201—84no—
164#11758200—44no—
165#11758199—44no—
166#11758199—44no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61#12011yescharacteristic
184C22×D61—41no—
185Q8×C61—44no—
186Q8×C61—82no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—88no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—41no—
199C2×C4×C61—44no—
200C2×C4×C61—42no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#13512yes—
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C22⋊D61—82no—
216C2×Dic61—41no—
217D244—42no—
218D244—41no—
219D244—41no—
220D244—42no—
221C4×D61—82no—
222Dic122—41no—
223Dic122—41no—
224#92418—42no—
225#92418—42no—
226#92418—42no—
227#18772—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—84no—
237C22×C61—84no—
238C22×C61#44912yes—
239C22×C61C2×#12011yescharacteristic
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—44no—
249C4×C61#44912yes—
250C4×C61C2×#13511yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C61#57512yes—
253Dic61—81no—
254C2×#555D6111yescharacteristic, complemented
255#3206—611no—
256#3100—611no—
257#3036—611no—
258#1646—614no—
259#1624—614no—
260#1548—611no—
261#1486—611no—
262#1478—611no—
263C2×C61—22no—
264C2×C61—82no—
265C2×C61—82no—
266C2×C61C2×#44911yescharacteristic
267C2×C61#300511yescharacteristic
268C2×C61#301911yescharacteristic
269D61—161no—
270C2×AGL(1,ℤ/8)—1221no—
271#555—22no—
272#555—22no—
273#555—22no—
274#555C2×D6114yes—
275#532—611no—
276C2×C4⋊Q8C2×D6111yescharacteristic
277C2×#124—22no—
278C2×#121—21no—
279C2×#121—21no—
280C2×C4×D4C2×D6111yescharacteristic
281C2×C4×D4C2×D6111yescharacteristic
282C2×#117—21no—
283C2×#117—21no—
284#492—612no—
285#487—612no—
286#480—1221no—
287#474—612no—
288#463—1222no—
289#461—612no—
290#461—1222no—
291#459—612no—
292#459—612no—
293#448—1222no—
294#447—1221no—
295C4×D8—1222no—
296#431—611no—
297#431—611no—
298#423—611no—
299#417—1221no—
300#333—614no—
301D4⋊C8—614no—
302D4⋊C8—614no—
303C61#1877211yescharacteristic, complemented, Sylow
304C2×#41—1221no—
305C22×Q8C22⋊D6112yes—
306C22×D4—22no—
307C22×D4C22⋊D6111yescharacteristic
308C22×D4C22⋊D6111yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311AGL(1,ℤ/8)—2444no—
312C2×SD16—1224no—
313C2×SD16—1222no—
314C2×D8—1222no—
315C2×D8—1222no—
316C2×M16—1221no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8—611no—
320C4⋊Q8C22⋊D6112yes—
321C4⋊Q8C22×D6112yes—
322#127—611no—
323#124—28no—
324#124—28no—
325#124—1221no—
326#121—24no—
327#121—24no—
328#121—24no—
329#121—24no—
330#121—1221no—
331#121—1221no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—1221no—
337C4×D4—1221no—
338C4×D4C22⋊D6112yes—
339C4×D4C22⋊D6112yes—
340C4×D4C22×D6112yes—
341C4×D4C22×D6112yes—
342#117—24no—
343#117—24no—
344#117—24no—
345#117—24no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C42—22no—
355C2×C42C22×D6111yescharacteristic
356#107—1222no—
357C4⋊C8—612no—
358C4⋊C8—612no—
359C4⋊C8—1221no—
360C4⋊C8—1221no—
361#103—1224no—
362#103—1222no—
363#102—1222no—
364#102—1222no—
365#102—1222no—
366C22⋊C8—1221no—
367C22⋊C8—1221no—
368C4×C8—612no—
369C4×C8—1221no—
370C24—21no—
371C24—21no—
372#41—2442no—
373#41—2442no—
374C2×Q8—22no—
375C2×Q8—82no—
376C2×Q8—1221no—
377C2×Q8C2×C22⋊D6114yes—
378C2×D4—28no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—42no—
384C2×D4—84no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2441no—
389C2×D4C2×C22⋊D6112yes—
390C2×D4C2×C22⋊D6112yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—2441no—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405SD16—4884no—
406SD16—2444no—
407D8—2444no—
408D8—2444no—
409M16—2442no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4C2×C22⋊D6112yes—
422C4⋊C4C2×C22⋊D6112yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—2441no—
430C22⋊C4—2441no—
431C42—28no—
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C42C2×C22⋊D6111yescharacteristic
435C42C23×D6111yescharacteristic
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—2441no—
445C23#1451974611yescharacteristic
446Q8—44no—
447Q8—82no—
448Q8—2441no—
449Q8—2441no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—42no—
455D4—42no—
456D4—88no—
457D4—4882no—
458D4—2442no—
459D4—2441no—
460D4—2441no—
461D4—2442no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—4882no—
476C2×C4—2441no—
477C2×C4#1451974611yescharacteristic
478C2×C4#1451974611yescharacteristic
479C2×C4#1451974611yescharacteristic
480C2×C4#1451979712yes—
481C2×C4#1451980212yes—
482C2×C4C22×C22⋊D6111yescharacteristic
483C2×C4C22×C22⋊D6111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C8—2442no—
486C8—2442no—
487C8—2442no—
488C22—21no—
489C22—21no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—84no—
497C22—84no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22#1683990712yes—
502C22C2×#1451974611yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—44no—
507C4—42no—
508C4—42no—
509C4—4881no—
510C4#1683990712yes—
511C4C2×#1451979711yescharacteristic
512C4C2×#1451980211yescharacteristic
513C4#1684056012yes—
514C2—22no—
515C2—82no—
516C2—82no—
517C2—9761no—
518C2C2×#1683990711yescharacteristic
519C2#2145506111yescharacteristic
520C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#555C2
socleC22×C61C2×#120
hypercenterC2×#555D61
hypocenterC61#18772
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
2. #21455819C2semidirect
3. #21455283C2semidirect
4. #21445047C2semidirect
5. #21444902C2semidirect
6. #21444500C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
9. C2×C61×#555C2semidirect
12. #16840136C22semidirect
13. #16840112C22semidirect
15. #16840042C22semidirect
17. #16839967C22semidirect
19. #16839957C22semidirect
20. #16839957C22semidirect
28. #16839511C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
254. C2×#555D61semidirect
303. C61#18772semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214558192
#214552832
#214450472
#214449022
#214445002
#214442442
#214442102
C2×C61×#5552
#1877261