Finite Group Databasebeta

← #29658686

Subgroups — #29658686

Order 15616.

1288 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic, complemented
4#21445012C214yesmaximal, complemented
5#21444884C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840138C2212yescomplemented
13#16840108C2212yescomplemented
14#16840075—21no—
15#16840042C2212yescomplemented
16#16839984—22no—
17#16839972C2212yescomplemented
18#16839962C2212yescomplemented
19#16839957C2212yescomplemented
20#16839954—22no—
21#16839896—22no—
22#16839896—21no—
23C4×#11758199—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839506C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#550—22no—
33C61×#550—22no—
34C61×#550—22no—
35C61×#550C2214yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—22no—
39C2×#121×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519808—44no—
45C2×#11758202—22no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758199—24no—
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#14519700—22no—
63#14519698—24no—
64#14519698—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C2×#41×C61—24no—
72C2×#41×C61—24no—
73C2×#41×C61—44no—
74C2×#41×C61—44no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—22no—
77C22×D4×C61—22no—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61D412yes—
82C23×C4×C61—22no—
83C23×C4×C61—21no—
84#127×C61—24no—
85#127×C61—24no—
86#127×C61—21no—
87#127×C61—21no—
88#127×C61D412yes—
89#127×C61C2312yes—
90#121×C61—24no—
91#121×C61—28no—
92#121×C61—28no—
93#121×C61—24no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61D412yes—
97C4×Q8×C61C2312yes—
98C4×D4×C61—24no—
99C4×D4×C61—24no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D412yes—
103C4×D4×C61C2312yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—22no—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110C2×C42×C61C2311yescharacteristic
111C24×C61—22no—
112C24×C61—21no—
113#41×C61—48no—
114#41×C61—416no—
115#41×C61—48no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61D814yescomplemented
119C2×Q8×C61C2×D412yes—
120C2×D4×C61—28no—
121C2×D4×C61—28no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—42no—
129C2×D4×C61—44no—
130C2×D4×C61—48no—
131C2×D4×C61—42no—
132C2×D4×C61—42no—
133C2×D4×C61—82no—
134C2×D4×C61—84no—
135C2×D4×C61C2×D412yes—
136C2×D4×C61C2×D414yes—
137C22×C4×C61—21no—
138C22×C4×C61—28no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—48no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C2×C22⋊D61—41no—
150#11758222—42no—
151#11758222—42no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×Dic122—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—28no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2411yescharacteristic
172#11758202—44no—
173#11758201—44no—
174#11758199—44no—
175#11758199—84no—
176#11758198—41no—
177#11758198—41no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180#11758194—42no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—42no—
187C23×C61—44no—
188C23×C61—44no—
189C23×C61—42no—
190C23×C61—44no—
191C23×C61—42no—
192C23×C61#12011yescharacteristic
193C22×D61—41no—
194Q8×C61—22no—
195Q8×C61—24no—
196Q8×C61C2×D814yes—
197D4×C61—22no—
198D4×C61—22no—
199D4×C61—44no—
200D4×C61—42no—
201D4×C61—44no—
202D4×C61—416no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—44no—
206D4×C61—44no—
207D4×C61—82no—
208D4×C61—88no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—416no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—44no—
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61C2×D812yes—
225C2×C4×C61AGL(1,ℤ/8)12yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C22⋊D61—82no—
230C2×Dic61—41no—
231D244—42no—
232D244—41no—
233D244—41no—
234D244—42no—
235C4×D61—82no—
236Dic122—41no—
237Dic122—41no—
238#92418—42no—
239#92418—42no—
240#92418—42no—
241#18766—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—44no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—44no—
248C22×C61—44no—
249C22×C61—42no—
250C22×C61—84no—
251C22×C61—88no—
252C22×C61#44712yes—
253C22×C61C2×#12011yescharacteristic
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C4×C61—24no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61#44712yes—
264C4×C61C22×D811yescharacteristic
265C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
266C4×C61#57512yes—
267Dic61—81no—
268C2×#550D6111yescharacteristic, complemented
269#3205—611no—
270#3100—611no—
271#3037—611no—
272#1641—614no—
273#1621—614no—
274#1545—611no—
275#1486—611no—
276#1479—611no—
277C2×C61—22no—
278C2×C61—82no—
279C2×C61—84no—
280C2×C61C2×#44711yescharacteristic
281C2×C61#300511yescharacteristic
282C2×C61#301911yescharacteristic
283D61—161no—
284C22×#41—21no—
285C2×#137—1221no—
286#550—22no—
287#550—22no—
288#550—22no—
289#550C2×D6114yes—
290#532—611no—
291C2×#127—21no—
292C2×#127C2×D6111yescharacteristic
293C2×#121—22no—
294C2×#121—21no—
295C2×C4×Q8C2×D6111yescharacteristic
296C2×C4×D4—21no—
297C2×C4×D4C2×D6111yescharacteristic
298#493—612no—
299#486—612no—
300#480—1221no—
301#474—612no—
302#464—1222no—
303#462—612no—
304#460—612no—
305#459—612no—
306#459—1222no—
307#447—1221no—
308#447—1222no—
309C4×D8—1222no—
310#431—611no—
311#431—611no—
312#423—611no—
313#417—1221no—
314#331—614no—
315Q8⋊C8—614no—
316D4⋊C8—614no—
317C61#1876611yescharacteristic, complemented, Sylow
318C2×#41—24no—
319C2×#41—24no—
320C2×#41—44no—
321C2×#41—44no—
322C2×#41—1221no—
323C22×Q8C22⋊D6111yescharacteristic
324C22×D4—22no—
325C22×D4—22no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4C22⋊D6111yescharacteristic
329C22×D4C22⋊D6112yes—
330C23×C4—22no—
331C23×C4—21no—
332#137—2444no—
333C2×Q16—1222no—
334C2×SD16—1222no—
335C2×D8—1224no—
336C2×D8—1222no—
337C2×M16—1221no—
338C4⋊Q8—611no—
339#127—24no—
340#127—24no—
341#127—21no—
342#127—21no—
343#127—611no—
344#127C22⋊D6112yes—
345#127C22×D6112yes—
346#124—1221no—
347#121—24no—
348#121—28no—
349#121—28no—
350#121—24no—
351#121—1221no—
352#121—1221no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8C22⋊D6112yes—
356C4×Q8C22×D6112yes—
357C4×D4—24no—
358C4×D4—24no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—1221no—
362C4×D4—1221no—
363C4×D4C22⋊D6112yes—
364C4×D4C22×D6112yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4C22⋊D6111yescharacteristic
367C2×C4⋊C4C22⋊D6111yescharacteristic
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C42—21no—
371C2×C42C22×D6111yescharacteristic
372#107—1222no—
373C4⋊C8—612no—
374C4⋊C8—612no—
375C4⋊C8—1221no—
376C4⋊C8—1221no—
377#103—1222no—
378#102—1222no—
379#102—1224no—
380#102—1222no—
381#102—1222no—
382C22⋊C8—1221no—
383C22⋊C8—1221no—
384C4×C8—612no—
385C4×C8—1221no—
386C24—22no—
387C24—21no—
388#41—48no—
389#41—416no—
390#41—48no—
391#41—2442no—
392#41—2442no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—1221no—
396C2×Q8#1175819914yescomplemented
397C2×Q8C2×C22⋊D6112yes—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—82no—
412C2×D4—84no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—2441no—
417C2×D4C2×C22⋊D6112yes—
418C2×D4C2×C22⋊D6114yes—
419C22×C4—21no—
420C22×C4—28no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—2441no—
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432Q16—2444no—
433SD16—2444no—
434D8—4884no—
435D8—2444no—
436M16—2442no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4C2×C22⋊D6112yes—
447C4⋊C4C2×C22⋊D6112yes—
448C22⋊C4—28no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—2441no—
452C22⋊C4—2441no—
453C42—22no—
454C42—22no—
455C42C2×C22⋊D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C42C23×D6111yescharacteristic
459C23—24no—
460C23—22no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—2441no—
468C23#1451974611yescharacteristic
469Q8—22no—
470Q8—24no—
471Q8—2441no—
472Q8—2441no—
473Q8C2×#1175819914yes—
474D4—22no—
475D4—22no—
476D4—44no—
477D4—42no—
478D4—44no—
479D4—416no—
480D4—42no—
481D4—42no—
482D4—44no—
483D4—44no—
484D4—82no—
485D4—88no—
486D4—4882no—
487D4—2442no—
488D4—2441no—
489D4—2441no—
490D4—2442no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—416no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—4882no—
504C2×C4—2441no—
505C2×C4#1451974611yescharacteristic
506C2×C4#1451974611yescharacteristic
507C2×C4#1451974611yescharacteristic
508C2×C4C2×#1175819912yes—
509C2×C4#1451980312yes—
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C8—2442no—
514C8—2442no—
515C8—2442no—
516C22—21no—
517C22—21no—
518C22—44no—
519C22—42no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—88no—
525C22—84no—
526C22—4881no—
527C22—4881no—
528C22—4881no—
529C22#1683989612yes—
530C22C2×#1451974611yescharacteristic
531C4—21no—
532C4—21no—
533C4—24no—
534C4—21no—
535C4—42no—
536C4—42no—
537C4—4881no—
538C4#1683989612yes—
539C4C22×#1175819911yescharacteristic
540C4C2×#1451980311yescharacteristic
541C4#1684056012yes—
542C2—22no—
543C2—84no—
544C2—82no—
545C2—9761no—
546C2C2×#1683989611yescharacteristic
547C2#2145506111yescharacteristic
548C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18766
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455819C2semidirect
3. #21455291C2semidirect
4. #21445012C2semidirect
5. #21444884C2semidirect
6. #21444487C2semidirect
7. #21444244C2semidirect
8. #21444214C2semidirect
9. C2×C61×#550C2semidirect
12. #16840138C22semidirect
13. #16840108C22semidirect
15. #16840042C22semidirect
17. #16839972C22semidirect
18. #16839962C22semidirect
19. #16839957C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
118. C2×Q8×C61D8semidirect
268. C2×#550D61semidirect
317. C61#18766semidirect
396. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214558192
#214552912
#214450122
#214448842
#214444872
#214442442
#214442142
C2×C61×#5502
#1876661