Finite Group Databasebeta

← #29685953

Subgroups — #29685953

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839935C211yesmaximal, characteristic
2C4×#14519802C212yesmaximal, complemented
3#21448526C211yesmaximal, characteristic, complemented
4#21446222C211yesmaximal, characteristic, complemented
5#21446195C212yesmaximal, complemented
6#21446045C212yesmaximal, complemented
7#21446034C211yesmaximal, characteristic, complemented
8#21445778C211yesmaximal, characteristic, complemented
9#21445629C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C4×2−1+4×C61C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C2×#14519766C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16839935C418yescomplemented
16#16839872—22no—
17#16839860—22no—
18C4×#11758201—22no—
19C4×#11758199—22no—
20#16839805C2212yes—
21C2×#14519704C2212yes—
22C2×#14519704C2212yes—
23#16839766C2212yes—
24#16839763C2212yes—
25C2×#14519696C2211yescharacteristic
26C2×#14519696C2211yescharacteristic
27C4×#11758194C2212yes—
28#16839669C2211yescharacteristic
29#16839663C2211yescharacteristic
30#16839653—22no—
31#16839633—22no—
32#16839631C2211yescharacteristic
33#16839608C2211yescharacteristic
34#16839531C2212yes—
35#16839521C2212yes—
36#16839521C2212yes—
37C2×2−1+4×C61C2211yescharacteristic
38C61×#520—22no—
39C61×#520—22no—
40C61×#520C2212yes—
41C61×#519—22no—
42C61×#519C2211yescharacteristic
43C4×#41×C61—22no—
44C4×#41×C61—22no—
45C4×#41×C61C2212yes—
46C2×C4×Q8×C61—22no—
47C2×C4×Q8×C61C2211yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50#14519802—24no—
51#14519802C2×C418yescomplemented
52C2×#11758201—22no—
53C2×#11758199—22no—
54C2×#11758194C2312yes—
55C2×#11758194C2312yes—
56#14519766—22no—
57#14519766C2×C414yescomplemented
58C4×D244—22no—
59C4×D244—22no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—22no—
62C2×#11758198—22no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×C4×Dic61C2311yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67#14519710—22no—
68#14519708—22no—
69#14519704—24no—
70#14519704C2×C418yescomplemented
71#14519701—22no—
72#14519698—22no—
73#14519697—22no—
74#14519697—22no—
75#14519696—22no—
76#14519696C2×C414yescomplemented
77#14519688—22no—
78C4×#92418—22no—
79#14519684—21no—
80#14519684—21no—
81#14519684—22no—
822−1+4×C61—24no—
832−1+4×C61—22no—
842−1+4×C61C2×C414yescomplemented
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61C2312yes—
88C22×Q8×C61—22no—
89C22×Q8×C61C2311yescharacteristic
90C4×Q8×C61—22no—
91C4×Q8×C61—24no—
92C4×Q8×C61—22no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61D412yes—
99C4×Q8×C61D412yes—
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—24no—
106C4×D4×C61—22no—
107C4×D4×C61D412yes—
108C4×D4×C61D412yes—
109#117×C61—24no—
110#117×C61—24no—
111#117×C61—22no—
112#117×C61—22no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61—24no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61D412yes—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—24no—
120C2×C42×C61—22no—
121C2×C42×C61D412yes—
122C2×C42×C61C2311yescharacteristic
123C23×D61—21no—
124#41×C61—28no—
125#41×C61—28no—
126#41×C61—28no—
127#41×C61—28no—
128#41×C61—22no—
129#41×C61—22no—
130#41×C61C22×C414yes—
131#41×C61#4114yes—
132C2×Q8×C61—22no—
133C2×Q8×C61—28no—
134C2×Q8×C61—28no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61C22×C412yes—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61#4112yes—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61C2×D412yes—
143C22×C4×C61—24no—
144C22×C4×C61—22no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×C4×D61—42no—
161C2×C4×D61—82no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61—24no—
176C42×C61—24no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61C2×D412yes—
183C42×C61C2×D412yes—
184#11758201—28no—
185#11758199—28no—
186#11758198—28no—
187#11758198—41no—
188#11758198—41no—
189#11758198—82no—
190C4⋊Dic61—22no—
191#11758196—82no—
192C4×Dic61—21no—
193C4×Dic61—22no—
194C4×Dic61C22×C412yes—
195#11758194—22no—
196#11758194—22no—
197#11758194—22no—
198#11758194C22×C414yes—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—22no—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—41no—
208Q8×C61—28no—
209Q8×C61—24no—
210Q8×C61—22no—
211Q8×C61—24no—
212Q8×C61C4×D414yes—
213D4×C61—28no—
214D4×C61—24no—
215D4×C61—24no—
216D4×C61—22no—
217D4×C61C4×D414yes—
218C2×C4×C61—216no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—44no—
230C2×C4×C61—42no—
231C2×C4×C61—82no—
232C2×C4×C61—84no—
233C2×C4×C61C4×D414yes—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12012yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D412yes—
239C2×C4×C61C2×#4112yes—
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249D244—24no—
250D244—42no—
251C4×D61—84no—
252#92418—24no—
253#92418—42no—
254#21475—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—24no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61C2×C4×D411yescharacteristic
261C22×C61C2×#12011yescharacteristic
262C22×C61#52511yescharacteristic
263C2×D61—44no—
264C2×D61—42no—
265C2×D61—82no—
266C2×D61—81no—
267C2×D61—81no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—24no—
272C4×C61—24no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61C2×C4×D412yes—
277C4×C61#52512yes—
278Dic61—42no—
279Dic61—81no—
280Dic61—81no—
281C4×2−1+4D6111yescharacteristic, complemented
282C2×#454—611no—
283C4×AGL(1,ℤ/8)—612no—
284#2326—611no—
285#1888—611no—
286#1883—612no—
287#1859—612no—
288#1857—611no—
289#1793—611no—
290#1759—612no—
291C4×#101—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—44no—
295C2×C61C4×#12011yescharacteristic
296C2×C61#302011yescharacteristic
297C2×C61#302111yescharacteristic
298D61—82no—
299C2×2−1+4C2×D6111yescharacteristic
300C2×AGL(1,ℤ/8)—612no—
301C2×#124—611no—
302#520—22no—
303#520—22no—
304#520C2×D6112yes—
305#519—22no—
306#519C2×D6111yescharacteristic
307C4×#41—22no—
308C4×#41—22no—
309C4×#41C2×D6112yes—
310C2×C4×Q8—22no—
311C2×C4×Q8C2×D6111yescharacteristic
312C2×C4×D4—611no—
313#454—618no—
314#442—1222no—
315#440—1222no—
316C4×SD16—1222no—
317C4×D8—1222no—
318#428—612no—
319C2×#104—612no—
320C2×#104—612no—
321#418—612no—
322#417—612no—
323C2×#101—611no—
324C2×#101—611no—
325C4×M16—612no—
326#391—611no—
327#390—611no—
328#388—1222no—
329#385—1222no—
330#384—611no—
331#376—611no—
332#343—612no—
333#339—612no—
334#339—612no—
335C61#2147511yescharacteristic, complemented, Sylow
3362−1+4—24no—
3372−1+4—22no—
3382−1+4C4×D6114yescomplemented
339C2×#41—22no—
340C2×#41—22no—
341C2×#41C22×D6112yes—
342C22×Q8—22no—
343C22×Q8C22×D6111yescharacteristic
344C22×D4—611no—
345C23×C4—1221no—
346AGL(1,ℤ/8)—618no—
347AGL(1,ℤ/8)—1224no—
348C2×SD16—1222no—
349C2×D8—1222no—
350C2×M16—612no—
351C2×M16—612no—
352#124—614no—
353#124—1222no—
354C4×Q8—22no—
355C4×Q8—24no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8C22⋊D6112yes—
363C4×Q8C22⋊D6112yes—
364C4×D4—22no—
365C4×D4—24no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—24no—
370C4×D4—22no—
371C4×D4—1222no—
372C4×D4—1222no—
373C4×D4C22⋊D6112yes—
374C4×D4C22⋊D6112yes—
375#117—24no—
376#117—24no—
377#117—22no—
378#117—22no—
379#117—22no—
380#117C22×D6112yes—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4C22⋊D6112yes—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1221no—
390C2×C42—24no—
391C2×C42—22no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42C22⋊D6112yes—
395C2×C42C22×D6111yescharacteristic
396#107—1222no—
397#106—1222no—
398#104—618no—
399#104—1224no—
400#103—1222no—
401#102—1222no—
402#102—1222no—
403#102—1222no—
404#101—614no—
405#101—1222no—
406#97—1222no—
407C4×C8—1222no—
408#95—1221no—
409#95—1221no—
410#95—1222no—
411C24—1221no—
412#41—28no—
413#41—28no—
414#41—28no—
415#41—22no—
416#41—22no—
417#41—28no—
418#41C2×C4×D6114yes—
419#41#1175822214yes—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8C2×C4×D6112yes—
426C2×Q8#1175822212yes—
427C2×Q8C2×C22⋊D6112yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—612no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—1221no—
434C2×D4C2×C22⋊D6112yes—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—4882no—
444C22×C4—2442no—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6112yes—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450SD16—1228no—
451D8—1228no—
452M16—614no—
453M16—1222no—
454M16—1222no—
455M16—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—1222no—
466C4⋊C4—4882no—
467C4⋊C4C2×C22⋊D6112yes—
468C4⋊C4C2×C22⋊D6112yes—
469C22⋊C4—24no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—1228no—
474C22⋊C4—4882no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C42—24no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—612no—
485C42—1221no—
486C42—1222no—
487C42C2×C22⋊D6112yes—
488C42C2×C22⋊D6112yes—
489C23—22no—
490C23—1222no—
491C23—2442no—
492C23—2442no—
493C23—2441no—
494C23—2441no—
495C23C22×C22⋊D6111yescharacteristic
496Q8—28no—
497Q8—24no—
498Q8—24no—
499Q8—22no—
500Q8C4×C22⋊D6114yes—
501D4—28no—
502D4—24no—
503D4—24no—
504D4—22no—
505D4—1224no—
506D4—2442no—
507D4C4×C22⋊D6114yes—
508C2×C4—216no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—1222no—
524C2×C4—4884no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4C4×C22⋊D6114yes—
533C2×C4#1451974611yescharacteristic
534C2×C4#1451974611yescharacteristic
535C2×C4#1451974612yes—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C22×C22⋊D6112yes—
538C2×C4C2×#1175822212yes—
539C2×C4C2×#1175822211yescharacteristic
540C8—1224no—
541C8—2442no—
542C22—21no—
543C22—21no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—4882no—
548C22—4881no—
549C22—4881no—
550C22—2444no—
551C22—2442no—
552C22C2×C4×C22⋊D6111yescharacteristic
553C22C2×#1451974611yescharacteristic
554C22#1684027311yescharacteristic
555C4—21no—
556C4—21no—
557C4—22no—
558C4—24no—
559C4—24no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—4881no—
564C4—4881no—
565C4—2442no—
566C4C2×C4×C22⋊D6112yes—
567C4#1684027312yes—
568C2—21no—
569C2—21no—
570C2—44no—
571C2—4882no—
572C2C4×#1451974611yescharacteristic
573C2#2145518311yescharacteristic
574C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×2−1+4×C61C2
socleC22×C61C2×#120
prime radicalC2×#16839935C2
cosocleC22×C4×C61C24
hypercenterC4×2−1+4D61
hypocenterC61#21475
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
2. C4×#14519802C2semidirect
3. #21448526C2semidirect
4. #21446222C2semidirect
5. #21446195C2semidirect
6. #21446045C2semidirect
7. #21446034C2semidirect
8. #21445778C2semidirect
9. #21445629C2semidirect
10. C4×#14519696C2semidirect
11. C4×2−1+4×C61C2semidirect
15. #16839935C4semidirect
51. #14519802C2×C4semidirect
57. #14519766C2×C4semidirect
70. #14519704C2×C4semidirect
76. #14519696C2×C4semidirect
84. 2−1+4×C61C2×C4semidirect
281. C4×2−1+4D61semidirect
335. C61#21475semidirect
338. 2−1+4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399352
C4×#145198022
#214485262
#214462222
#214461952
#214460452
#214460342
#214457782
#214456292
C4×#145196962
C4×2−1+4×C612
#2147561