Finite Group Databasebeta

← #29834332

Subgroups — #29834332

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C211yesmaximal, characteristic
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21456307C211yesmaximal, characteristic, complemented
4#21454666C211yesmaximal, characteristic, complemented
5#21447877C211yesmaximal, characteristic, complemented
6#21447876C211yesmaximal, characteristic, complemented
7#21447875C211yesmaximal, characteristic, complemented
8#21447874C211yesmaximal, characteristic, complemented
9#21447863C211yesmaximal, characteristic, complemented
10#21447862C211yesmaximal, characteristic, complemented
11#21447861C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447537C211yesmaximal, characteristic, complemented
14#21447525C211yesmaximal, characteristic, complemented
15C61×#2208C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17C2×#14519800C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840337—21no—
21#16840234—21no—
22#16840199—24no—
23C2×#14519722C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25C2×#14519720C2211yescharacteristic
26C2×#14519720C2211yescharacteristic
27SD32×D61—24no—
28C2×D976C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519716C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840155—21no—
33#16840154—21no—
34#16840147C2211yescharacteristic, complemented
35#16840146C2211yescharacteristic, complemented
36#16840140C2211yescharacteristic, complemented
37#16839872—21no—
38#16839871—21no—
39C4×D488C2211yescharacteristic
40D8×Dic61C2211yescharacteristic
41#16839844C2211yescharacteristic
42#16839799C2211yescharacteristic
43#16839765—21no—
44#16839716C2211yescharacteristic
45C4⋊#11758185C2211yescharacteristic, complemented
46#16839580C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839566C2211yescharacteristic, complemented
50#16839566C2211yescharacteristic, complemented
51#16839565C2211yescharacteristic, complemented
52#16839564C2211yescharacteristic, complemented
53#16839562C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×SD32×C61C2211yescharacteristic
56C61×#494C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#358C2211yescharacteristic, complemented
61C2×#11758226—21no—
62C2×Q8×D61—21no—
63C2×D4×D61—21no—
64#14519809—44no—
65#14519800—22no—
66#14519800—22no—
67C2×Dic244C2311yescharacteristic
68C2×#11758202—21no—
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—22no—
72C2×#11758190—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×D488C2311yescharacteristic
76C2×D488C2311yescharacteristic
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61D414yes—
80C2×#11758189D411yescharacteristic
81C2×C8×D61C2311yescharacteristic
82#14519775—21no—
83#14519766—21no—
84#14519765—21no—
85#14519762—41no—
86#14519762—41no—
87#14519755—41no—
88#14519755—41no—
89C4×D244—21no—
90D4×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519735—21no—
95#14519734D411yescharacteristic
96C2×#11758198—41no—
97C22⋊C4×D61—41no—
98#14519722—22no—
99#14519721—22no—
100#14519720—22no—
101#14519720—22no—
102D976—22no—
103#14519718—22no—
104#14519716—22no—
105C2×#11758185C2311yescharacteristic
106C2×#11758185C2311yescharacteristic
107C16×D61—22no—
108#14519711C2311yescharacteristic
109#14519708—21no—
110#14519699—21no—
111#14519698—21no—
112#14519697—21no—
113#14519688D411yescharacteristic
114#14519687D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×Q16×C61C2311yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×D8×C61C2311yescharacteristic
120C4⋊Q8×C61—21no—
121C4×D4×C61—21no—
122SD32×C61—22no—
123SD32×C61—22no—
124C2×C16×C61C2311yescharacteristic
125C2×C16×C61C2311yescharacteristic
126#107×C61C2311yescharacteristic
127#102×C61—21no—
128C4×C8×C61C2311yescharacteristic
129C23×D61—41no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×D4×C61—21no—
133C22×C4×C61—41no—
134C2×C22⋊D61—41no—
135C22×Dic61—41no—
136#11758226—42no—
137#11758226—42no—
138Q8×D61—42no—
139Q8×D61—42no—
140D4×D61—24no—
141D4×D61—42no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—41no—
152Q16×C61—21no—
153Q16×C61—21no—
154Q16×C61—21no—
155Q16×C61—21no—
156Q16×C61—42no—
157D8×C61—21no—
158D8×C61C2×D412yes—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C42×C61C2×D411yescharacteristic
166C22⋊Dic61—41no—
167#11758202—42no—
168#11758202—42no—
169#11758201—42no—
170#11758201—42no—
171#11758199—24no—
172#11758198—41no—
173#11758198—41no—
174#11758198—41no—
175#11758198—41no—
176#11758198—41no—
177#11758198—41no—
178#11758198—41no—
179#11758198D811yescharacteristic
180#11758198D811yescharacteristic
181C4⋊Dic61—21no—
182#11758196—41no—
183#11758196—41no—
184#11758196D811yescharacteristic
185#11758196D811yescharacteristic
186C4×Dic61—41no—
187C4×Dic61C2×D411yescharacteristic
188C2×#92418C2×D411yescharacteristic
189C2×#92418C2×D411yescharacteristic
190Dic244—21no—
191Dic244—21no—
192D488—21no—
193D488—21no—
194D488—21no—
195D488C2×D412yes—
196#11758190—42no—
197#11758189—22no—
198C8×D61C2×D412yes—
199C8×D61#4112yes—
200C16×C61—21no—
201C16×C61—21no—
202#11758185—21no—
203#11758185—21no—
204C23×C61—41no—
205C22×D61—41no—
206C22×D61—44no—
207C22×D61—41no—
208C22×D61—161no—
209C22×D61—161no—
210C22×D61C2×D811yescharacteristic
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—22no—
216D4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—161no—
221C2×C4×C61C2×D811yescharacteristic
222C2×C4×C61C2×D811yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C22⋊D61—44no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—161no—
228C2×Dic61C2×D811yescharacteristic
229C2×Dic61C2×D811yescharacteristic
230C2×Dic61C2×D811yescharacteristic
231D244—22no—
232D244—41no—
233D244—41no—
234D244—41no—
235D244—82no—
236C4×D61—82no—
237C4×D61—82no—
238C4×D61—161no—
239C4×D61—161no—
240C4×D61#12112yes—
241C4×D61AGL(1,ℤ/8)12yes—
242Dic122—41no—
243Dic122—41no—
244Dic122—82no—
245C8×C61—21no—
246C8×C61C22×D411yescharacteristic
247C8×C61C2×#4111yescharacteristic
248#92418—21no—
249#92418#12112yes—
250#33586—611nomaximal, Sylow
251C22×C61—42no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C22×D811yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—164no—
264C2×D61#45912yes—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—81no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
272Dic61—21no—
273Dic61—21no—
274Dic61—81no—
275Dic61—81no—
276Dic61#45912yes—
277C2×#509—611no—
278C22×SD32—611no—
279#3156—611no—
280#2948—611no—
281#2210—611no—
282#2210—611no—
283#2210—611no—
284#2210—611no—
285#2208—611no—
286#2208—611no—
287#2208—611no—
288#2208D6111yescharacteristic, complemented
289#2151—611no—
290#2141—611no—
291#2138—611no—
292C2×C61—82no—
293C2×C61C2×#45911yescharacteristic
294C2×C61#341611yescharacteristic
295C2×C61#341611yescharacteristic
296D61—22no—
297D61—82no—
298D61—161no—
299C2×#137—1221no—
300C2×#135—611no—
301C22×Q16—611no—
302C22×D8—611no—
303#536—1221no—
304#518—1221no—
305#509—1224no—
306C2×SD32—611no—
307C2×SD32—611no—
308C2×SD32—611no—
309C2×SD32—611no—
310C2×SD32—1224no—
311C2×SD32C2×D6111yescharacteristic
312C2×SD32C2×D6111yescharacteristic
313C2×D16—611no—
314C2×D16—611no—
315C2×M32—611no—
316C22×C16—611no—
317#497—1221no—
318#497—1221no—
319#495—611no—
320#495—611no—
321#494—611no—
322#494C2×D6111yescharacteristic, complemented
323#442—1221no—
324#442—1221no—
325C4×D8—611no—
326C4×D8—611no—
327C4×D8—611no—
328C4×D8C2×D6111yescharacteristic
329#427—611no—
330#418—1221no—
331#405—611no—
332C4⋊C16—611no—
333C4⋊C16—611no—
334C4⋊C16—611no—
335C4⋊C16C2×D6111yescharacteristic, complemented
336#358—611no—
337#358—611no—
338#358—611no—
339#358—611no—
340#358C2×D6111yescharacteristic, complemented
341#358C2×D6111yescharacteristic, complemented
342#357—611no—
343#357—611no—
344C61#3358611yescharacteristic, complemented, Sylow
345C2×#41—1221no—
346C22×Q8—1221no—
347C22×D4—1221no—
348#137—2444no—
349#135—1222no—
350#135—1222no—
351C2×Q16—21no—
352C2×Q16—611no—
353C2×Q16—1222no—
354C2×Q16—1222no—
355C2×Q16—1221no—
356C2×Q16—1221no—
357C2×Q16C22×D6111yescharacteristic
358C2×Q16C22×D6111yescharacteristic
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—614no—
363C2×D8—611no—
364C2×D8—611no—
365C2×D8—1222no—
366C2×D8—1221no—
367C2×D8C22×D6111yescharacteristic
368C2×M16—611no—
369C22×C8—611no—
370C4⋊Q8—21no—
371C4⋊Q8—1221no—
372#124—1221no—
373#124—1221no—
374#123—2441no—
375#123—2441no—
376C22⋊Q8—2441no—
377C22⋊Q8—2441no—
378C4×D4—21no—
379C4×D4—1221no—
380C4×D4—1221no—
381C4×D4—1221no—
382C4×D4—2441no—
383C4×D4—2441no—
384#117—611no—
385#117—1221no—
386C2×C22⋊C4—2441no—
387C2×C22⋊C4—2441no—
388SD32—22no—
389SD32—22no—
390SD32—1222no—
391SD32—1222no—
392SD32—1222no—
393SD32—1222no—
394D16—1222no—
395D16—1222no—
396M32—1222no—
397C2×C16—611no—
398C2×C16—611no—
399C2×C16—1222no—
400C2×C16C22×D6111yescharacteristic
401C2×C16C22×D6111yescharacteristic
402#107—611no—
403#107C22×D6111yescharacteristic
404#106—1221no—
405#102—21no—
406#102—1221no—
407#102—1221no—
408#102—1221no—
409#97—611no—
410#97—611no—
411C4×C8—611no—
412C4×C8C22×D6111yescharacteristic
413C24—2441no—
414#41—2442no—
415#41—2442no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—1221no—
419C2×Q8—2441no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×D4—21no—
423C2×D4—1224no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—2441no—
427C2×D4—2442no—
428C2×D4—2441no—
429C22×C4—41no—
430C22×C4—611no—
431C22×C4—2441no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4—2441no—
436Q16—21no—
437Q16—21no—
438Q16—21no—
439Q16—21no—
440Q16—42no—
441Q16—1221no—
442Q16—1221no—
443Q16—2442no—
444Q16—2442no—
445SD16—2442no—
446SD16—2442no—
447SD16—2442no—
448D8—21no—
449D8—612no—
450D8—1224no—
451D8—1221no—
452D8—1221no—
453D8—1221no—
454D8D4×D6112yes—
455M16—1222no—
456C2×C8—611no—
457C2×C8—612no—
458C2×C8—612no—
459C2×C8—611no—
460C2×C8D4×D6111yescharacteristic
461C2×C8C23×D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—611no—
466C4⋊C4—611no—
467C4⋊C4—1221no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C22⋊C4—41no—
471C22⋊C4—611no—
472C22⋊C4—611no—
473C22⋊C4—2441no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C22⋊C4—2441no—
478C22⋊C4—2441no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C42—611no—
482C42—2441no—
483C42D4×D6111yescharacteristic
484C16—21no—
485C16—21no—
486C16—1221no—
487C16—1221no—
488C23—41no—
489C23—611no—
490C23—9761no—
491C23—9761no—
492C23—2444no—
493C23—2441no—
494C23—2441no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—4882no—
500Q8—2441no—
501Q8—2441no—
502D4—22no—
503D4—41no—
504D4—1222no—
505D4—4882no—
506D4—2444no—
507D4—2441no—
508D4—2441no—
509D4—2441no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—161no—
514C2×C4—611no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—9761no—
520C2×C4—9761no—
521C2×C4—9761no—
522C2×C4—4882no—
523C2×C4—4882no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4D8×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C8—21no—
530C8—612no—
531C8—1221no—
532C8C2×D4×D6111yescharacteristic
533C8#41×D6111yescharacteristic
534C22—42no—
535C22—81no—
536C22—81no—
537C22—612no—
538C22—1221no—
539C22—1221no—
540C22—9764no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22—2442no—
547C22C2×D8×D6111yescharacteristic
548C4—21no—
549C4—21no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—612no—
554C4—1221no—
555C4—1221no—
556C4—4881no—
557C4—4881no—
558C4#121×D6111yescharacteristic
559C4AGL(1,ℤ/8)×D6111yescharacteristic
560C2—82no—
561C2—1222no—
562C2—9761no—
563C2—4882no—
564C2#459×D6111yescharacteristic
565C2#2145896111yescharacteristic
566C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
prime radicalC2×#16840199C2
hypercenter#2208D61
hypocenterC61#33586
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
2. C2×SD32×D61C2semidirect
3. #21456307C2semidirect
4. #21454666C2semidirect
5. #21447877C2semidirect
6. #21447876C2semidirect
7. #21447875C2semidirect
8. #21447874C2semidirect
9. #21447863C2semidirect
10. #21447862C2semidirect
11. #21447861C2semidirect
12. #21447576C2semidirect
13. #21447537C2semidirect
14. #21447525C2semidirect
15. C61×#2208C2semidirect
34. #16840147C22semidirect
35. #16840146C22semidirect
36. #16840140C22semidirect
45. C4⋊#11758185C22semidirect
46. #16839580C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839566C22semidirect
50. #16839566C22semidirect
51. #16839565C22semidirect
52. #16839564C22semidirect
53. #16839562C22semidirect
56. C61×#494C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#358C22semidirect
288. #2208D61semidirect
322. #494C2×D61semidirect
335. C4⋊C16C2×D61semidirect
340. #358C2×D61semidirect
341. #358C2×D61semidirect
344. C61#33586semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
C2×SD32×D612
#214563072
#214546662
#214478772
#214478762
#214478752
#214478742
#214478632
#214478622
#214478612
#214475762
#214475372
#214475252
C61×#22082
#3358661