Finite Group Databasebeta

← #29714099

Subgroups — #29714099

Order 15616.

1336 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
1C2×#16840417C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3C2×#16839862C211yesmaximal, characteristic, complemented
4#21450729C211yesmaximal, characteristic, complemented
5#21447000C212yesmaximal, complemented
6#21446823C212yesmaximal, complemented
7#21446791C211yesmaximal, characteristic, complemented
8#21446616C211yesmaximal, characteristic, complemented
9#21446125C212yesmaximal, complemented
10#21446008C212yesmaximal, complemented
11C61×#1804C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13C22×#11758200C2211yescharacteristic
14#16840417—24no—
15#16840417—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519757C2212yes—
19C2×#14519756—21no—
20C2×#14519748C2212yes—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24#16840058—24no—
25#16839862—24no—
26C2×#14519711C2212yes—
27C2×#14519710C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839694—21no—
35#16839680—21no—
36#16839654—22no—
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#382C2211yescharacteristic
44C61×#340C2212yes—
45C2×#11758222—28no—
46C2×#11758222—48no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×D244D411yescharacteristic
50C22×C22⋊D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2312yes—
53C2×#11758200—22no—
54C2×#11758200—22no—
55C2×#11758200—22no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519777—22no—
59#14519777D414yes—
60#14519759—216no—
61#14519757—24no—
62#14519757—42no—
63#14519756—28no—
64#14519748—24no—
65#14519748—24no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68C4×D244—22no—
69C4×D244—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758196—21no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—22no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83C2×C4×Dic61—21no—
84#14519711—24no—
85#14519710—22no—
86#14519702—24no—
87#14519700—22no—
88#14519698—22no—
89#14519689—24no—
90#14519689—42no—
91#14519688—22no—
92#14519684—22no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C22×C8×C61C2312yes—
96C4×D4×C61—22no—
97C4×D4×C61D414yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103#102×C61—42no—
104#95×C61—21no—
105#95×C61—21no—
106C24×C61—21no—
107C23×D61C2×D412yes—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—44no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—28no—
122C2×C22⋊D61—44no—
123C2×C22⋊D61—44no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61C2×D412yes—
128#11758222—432no—
129C2×D244—24no—
130C2×D244—22no—
131C2×C4×D61—28no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—24no—
145C2×Dic122—21no—
146C2×Dic122—44no—
147C2×Dic122#4112yescomplemented
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—24no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—28no—
159#11758200—28no—
160#11758198—24no—
161#11758198—44no—
162#11758198—44no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—44no—
169C4⋊Dic61C2×D414yes—
170#11758196—28no—
171#11758196—42no—
172C4×Dic61—42no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418#4112yescomplemented
179C23×C61—22no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61C22×D411yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61AGL(1,ℤ/8)14yes—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—48no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12012yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C22⋊D61—416no—
213C2×Dic61—22no—
214C2×Dic61—24no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224D244—28no—
225C4×D61—416no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230Dic122—22no—
231Dic122—22no—
232Dic122—48no—
233Dic122—41no—
234Dic122—41no—
235C8×C61—42no—
236#92418—22no—
237#92418—41no—
238#24132—611nomaximal, Sylow
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—82no—
243C22×C61—82no—
244C22×C61C2×#12011yescharacteristic
245C22×C61#54611yescharacteristic
246C22×C61#54711yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)12yes—
248C22×C61#57812yes—
249C2×D61—24no—
250C2×D61—24no—
251C2×D61—24no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—82no—
260C4×C61#52511yescharacteristic
261C4×C61#53711yescharacteristic
262C4×C61#54811yescharacteristic
263C4×C61#55011yescharacteristic
264Dic61—42no—
265Dic61—42no—
266Dic61—44no—
267Dic61—81no—
268C2×#547—611no—
269C2×#477—611no—
270C2×#440—611no—
271#2536—611no—
272#2033—612no—
273#2000—612no—
274#1993—611no—
275#1955—611no—
276#1872—612no—
277#1852—612no—
278#1804D6111yescharacteristic, complemented
279C2×C61—82no—
280C2×C61#240711yescharacteristic
281C2×C61#300311yescharacteristic
282C2×C61#301111yescharacteristic
283C2×C61#330812yes—
284C2×C61#334312yes—
285D61—44no—
286C22×#41—1221no—
287C22×SD16—611no—
288#547—1224no—
289#547—1224no—
290C2×C4⋊Q8—611no—
291C2×#123—1222no—
292C2×C22⋊Q8—612no—
293C2×C22⋊Q8—1221no—
294C2×#121—612no—
295C2×C4×Q8—611no—
296C2×C4×D4—611no—
297C2×C4×D4C2×D6111yescharacteristic
298C22×C4⋊C4—612no—
299#477—1224no—
300#440—1224no—
301C2×#107—612no—
302C2×#107—611no—
303C2×C4⋊C8C2×D6111yescharacteristic
304C2×#103—612no—
305C2×#103—611no—
306C2×#102—611no—
307C2×#102C2×D6112yes—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—612no—
310C2×#97—611no—
311#398—1221no—
312#394—1221no—
313#388—1222no—
314#382C2×D6111yescharacteristic
315#380—612no—
316#340—612no—
317#340—612no—
318#340C2×D6112yes—
319C61#2413211yescharacteristic, complemented, Sylow
320C2×#41—1228no—
321C2×#41—2448no—
322C22×Q8—611no—
323C22×Q8—1221no—
324C22×D4—611no—
325C22×D4—1222no—
326C22×D4C22×D6111yescharacteristic
327C23×C4—21no—
328C23×C4—612no—
329C23×C4—1222no—
330C2×SD16—1222no—
331C2×SD16—1222no—
332C2×SD16—1222no—
333C22×C8—611no—
334C22×C8—611no—
335C22×C8C22×D6112yes—
336C4⋊Q8—614no—
337C4⋊Q8—1222no—
338#123—12216no—
339C22⋊Q8—1228no—
340C22⋊Q8—1224no—
341C22⋊Q8—2442no—
342#121—1224no—
343#121—1224no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×D4—22no—
347C4×D4—1222no—
348C4×D4—1222no—
349C4×D4D24414yes—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—1221no—
367C2×C42C22×D6111yescharacteristic
368#107—1224no—
369#107—1222no—
370C4⋊C8—22no—
371#103—1224no—
372#103—1222no—
373#102—42no—
374#102—1222no—
375C22⋊C8—1224no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—21no—
379#95—21no—
380#95—1222no—
381C24—21no—
382C24—612no—
383#41—24432no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1224no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—2444no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—1224no—
398C2×D4—1228no—
399C2×D4—1222no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4C2×D24412yes—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—612no—
408C22×C4—1228no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—2444no—
419C22×C4—2444no—
420C22×C4D4×D6111yescharacteristic
421C22×C4D4×D6111yescharacteristic
422C22×C4C23×D6111yescharacteristic
423SD16—1228no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—612no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—1221no—
432C2×C8—1221no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—614no—
437C4⋊C4—1228no—
438C4⋊C4—1222no—
439C4⋊C4—1222no—
440C4⋊C4—1224no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2444no—
444C4⋊C4—2442no—
445C4⋊C4C2×D24412yes—
446C22⋊C4—24no—
447C22⋊C4—1228no—
448C22⋊C4—1224no—
449C22⋊C4—2444no—
450C22⋊C4—2444no—
451C42—2442no—
452C42C2×D24412yes—
453C42D4×D6112yes—
454C23—22no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—614no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—1222no—
464C23—1222no—
465C23—1222no—
466C23C2×D4×D6111yescharacteristic
467Q8—1222no—
468Q8—1222no—
469Q8—2448no—
470Q8—2441no—
471Q8—2441no—
472D4—22no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—1228no—
477D4—24416no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—1222no—
487C2×C4—1224no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—24416no—
492C2×C4—2441no—
493C2×C4—2442no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974712yes—
503C2×C4#1451974712yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C22×D24411yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4#41×D6111yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982511yescharacteristic
510C8—42no—
511C8—1222no—
512C8—2441no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—1224no—
519C22—1224no—
520C22—1224no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22C2×#1451974711yescharacteristic
526C22#1684040911yescharacteristic
527C22#1684041411yescharacteristic
528C22AGL(1,ℤ/8)×D6112yes—
529C22#1684057612yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—82no—
534C4—4881no—
535C4—2444no—
536C4—2442no—
537C4—2442no—
538C4#1684027411yescharacteristic
539C4#1684034311yescharacteristic
540C4#1684042211yescharacteristic
541C4#1684043011yescharacteristic
542C2—82no—
543C2—2444no—
544C2#2144916511yescharacteristic
545C2#2145504811yescharacteristic
546C2#2145511711yescharacteristic
547C2#2145781712yes—
548C2#2145826612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC2×#16840417C2
hypercenter#1804D61
hypocenterC61#24132
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×#16839862C2semidirect
4. #21450729C2semidirect
5. #21447000C2semidirect
6. #21446823C2semidirect
7. #21446791C2semidirect
8. #21446616C2semidirect
9. #21446125C2semidirect
10. #21446008C2semidirect
11. C61×#1804C2semidirect
147. C2×Dic122#41semidirect
178. C2×#92418#41semidirect
278. #1804D61semidirect
319. C61#24132semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
C2×#168400582
C2×#168398622
#214507292
#214470002
#214468232
#214467912
#214466162
#214461252
#214460082
C61×#18042
#2413261