Finite Group Databasebeta

← #29836492

Subgroups — #29836492

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840196C211yesmaximal, characteristic, complemented
2C2×SD32×D61C211yesmaximal, characteristic, complemented
3#21457559C211yesmaximal, characteristic, complemented
4D8×D244C211yesmaximal, characteristic, complemented
5#21448086C211yesmaximal, characteristic, complemented
6#21447935C211yesmaximal, characteristic, complemented
7#21447904C211yesmaximal, characteristic, complemented
8#21447860C211yesmaximal, characteristic, complemented
9#21447746C211yesmaximal, characteristic, complemented
10#21447743C211yesmaximal, characteristic, complemented
11#21447737C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14#21447643C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic, complemented
17C2×Q16×D61C2211yescharacteristic, complemented
18C2×D8×D61C2211yescharacteristic
19C2×D8×D61C2211yescharacteristic
20#16840420—21no—
21D4×D244—21no—
22#16840196—24no—
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519722C2211yescharacteristic, complemented
25C2×#14519721C2211yescharacteristic
26C2×#14519720C2211yescharacteristic, complemented
27C2×#14519720C2211yescharacteristic, complemented
28SD32×D61—24no—
29C2×#14519718C2211yescharacteristic, complemented
30C2×#14519718C2211yescharacteristic, complemented
31C2×#14519716C2211yescharacteristic, complemented
32C2×C16×D61C2211yescharacteristic, complemented
33#16840139C2211yescharacteristic
34#16840000C2211yescharacteristic, complemented
35#16839999C2211yescharacteristic
36#16839999C2211yescharacteristic
37#16839995C2211yescharacteristic, complemented
38#16839981—21no—
39#16839957—21no—
40#16839921—21no—
41#16839899—21no—
42#16839897—21no—
43#16839897—21no—
44C4×Dic244C2211yescharacteristic, complemented
45C8×D244C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839565C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839563C2211yescharacteristic
51#16839541C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic, complemented
53#16839538C2211yescharacteristic, complemented
54C2×SD32×C61C2211yescharacteristic
55C2×D16×C61C2211yescharacteristic
56C61×#495C2211yescharacteristic, complemented
57C4×D8×C61C2211yescharacteristic
58C61×C4⋊C16C2211yescharacteristic, complemented
59C61×#358C2211yescharacteristic, complemented
60C61×#357C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×Q8×D61—21no—
63C22×D244—41no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66#14519796—22no—
67#14519796—41no—
68C2×Dic244C2311yescharacteristic
69C2×#11758202—21no—
70Q16×D61—22no—
71Q16×D61—41no—
72C2×#11758200—21no—
73C2×D488—21no—
74C2×D488C2311yescharacteristic
75C2×D488C2311yescharacteristic
76C2×#11758199—21no—
77C2×#11758199—21no—
78D8×D61—22no—
79D8×D61—22no—
80D8×D61—41no—
81D8×D61—41no—
82C2×C8×D61C2311yescharacteristic
83C2×C8×D61C2311yescharacteristic
84#14519774—21no—
85#14519763—41no—
86#14519754—41no—
87#14519753—21no—
88#14519752—21no—
89#14519752—21no—
90#14519751—41no—
91#14519750—41no—
92#14519748—21no—
93#14519747—41no—
94#14519747—41no—
95C4×Dic122—21no—
96C4×D244D411yescharacteristic
97#14519737—41no—
98#14519722—22no—
99#14519722—22no—
100#14519721D414yescomplemented
101#14519720—22no—
102#14519720—41no—
103#14519718—22no—
104#14519718—41no—
105#14519716—22no—
106C2×#11758185C2311yescharacteristic
107C2×#11758185C2311yescharacteristic
108C16×D61—22no—
109#14519711C2311yescharacteristic
110C4⋊#92418D411yescharacteristic
111#14519702—21no—
112#14519701—21no—
113#14519698—21no—
114#14519697—21no—
115#14519697—21no—
116#14519689D411yescharacteristic
117#14519689D411yescharacteristic
118C2×Q16×C61C2311yescharacteristic
119C2×SD16×C61—21no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122#124×C61—21no—
123C4×D4×C61—21no—
124SD32×C61—22no—
125D16×C61—22no—
126C2×C16×C61C2311yescharacteristic
127C2×C16×C61C2311yescharacteristic
128#107×C61C2311yescharacteristic
129#102×C61—21no—
130C4×C8×C61C2311yescharacteristic
131C23×D61—41no—
132C23×D61—41no—
133C2×Q8×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C22×C4×C61—41no—
137C2×C22⋊D61—41no—
138C2×C22⋊D61—41no—
139C2×C22⋊D61—41no—
140C2×C22⋊D61—41no—
141C22×Dic61—41no—
142Q8×D61—42no—
143Q8×D61—81no—
144#11758224—42no—
145#11758224—81no—
146D4×D61—42no—
147D4×D61—42no—
148D4×D61—81no—
149D4×D61—81no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—41no—
153C2×D244—41no—
154C2×D244—41no—
155C2×D244—41no—
156C2×D244—84no—
157C2×D244D811yescharacteristic
158C2×C4×D61—41no—
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—41no—
163Q16×C61—21no—
164Q16×C61—21no—
165SD16×C61—81no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61C2×D412yes—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C4⋊C4×C61—21no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C42×C61C2×D411yescharacteristic
177C22⋊Dic61—41no—
178#11758202—42no—
179#11758200—42no—
180#11758199—42no—
181#11758199—42no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198D811yescharacteristic
186#11758198D811yescharacteristic
187C4⋊Dic61—21no—
188C4⋊Dic61D811yescharacteristic
189#11758196—41no—
190#11758196—41no—
191C4×Dic61—41no—
192C2×#92418C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194Dic244—21no—
195Dic244C2×D412yescomplemented
196D488—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—81no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C16×C61—21no—
206C16×C61—21no—
207#11758185—21no—
208#11758185C2×D412yescomplemented
209C23×C61—41no—
210C23×C61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—161no—
218C22×D61—161no—
219C22×D61C2×D811yescharacteristic
220C22×D61C2×D811yescharacteristic
221Q8×C61—41no—
222Q8×C61—41no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61C2×D811yescharacteristic
231C2×C4×C61C2×D811yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C22⋊D61—82no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—82no—
240C2×Dic61C2×D811yescharacteristic
241C2×Dic61C2×D811yescharacteristic
242D244—22no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—82no—
248D244—82no—
249D244—82no—
250D244—161no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—82no—
256Dic122—22no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61C22×D411yescharacteristic
261C8×C61H(5,𝔽2)11yescharacteristic
262#92418—21no—
263#92418—21no—
264#33831—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C22×D811yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—164no—
284C2×D61—164no—
285C2×D61#50912yescomplemented
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61D4211yescharacteristic
291C4×C61#57611yescharacteristic
292Dic61—21no—
293Dic61—21no—
294Dic61—42no—
295Dic61—81no—
296C2×#509—611no—
297C22×SD32—611no—
298#3285—611no—
299D4×D8—611no—
300#2250—611no—
301#2220—611no—
302#2215—611no—
303#2210D6111yescharacteristic, complemented
304#2207—611no—
305#2187—611no—
306#2186—611no—
307#2185—611no—
308#2184—611no—
309#2175—611no—
310#2168—611no—
311C2×C61—82no—
312C2×C61—161no—
313C2×C61D4×D811yescharacteristic
314C2×C61C2×#50911yescharacteristic
315C2×C61#341611yescharacteristic
316D61—22no—
317D61—41no—
318D61—161no—
319D61—161no—
320C2×#135—611no—
321C22×Q16—611no—
322C22×D8—611no—
323C22×D8—611no—
324#548—1221no—
325D42—1221no—
326#509—1224no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×SD32—611no—
330C2×SD32—611no—
331C2×SD32—611no—
332C2×SD32—1224no—
333C2×SD32C2×D6111yescharacteristic
334C2×D16—611no—
335C2×D16—611no—
336C2×D16C2×D6111yescharacteristic
337C2×M32—611no—
338C22×C16—611no—
339#495C2×D6111yescharacteristic, complemented
340#493—611no—
341#467—611no—
342#466—611no—
343#466—611no—
344#466—611no—
345#464—1221no—
346#459—1221no—
347#452—1221no—
348#448—1221no—
349#447—1221no—
350#447—1221no—
351C4×Q16—611no—
352C4×D8C2×D6111yescharacteristic
353C8×D4—611no—
354#366—611no—
355C4⋊C16C2×D6111yescharacteristic, complemented
356#358—611no—
357#358C2×D6111yescharacteristic, complemented
358#357—611no—
359#357—611no—
360#357—611no—
361#357C2×D6111yescharacteristic, complemented
362C22⋊C16—611no—
363C22⋊C16—611no—
364#346—611no—
365C61#3383111yescharacteristic, complemented, Sylow
366C2×#41—1221no—
367C22×Q8—1221no—
368C22×D4—1221no—
369C22×D4—1221no—
370C22×D4—2441no—
371#135—1222no—
372#135—2441no—
373C2×Q16—611no—
374C2×Q16—1221no—
375C2×Q16—1222no—
376C2×Q16—2441no—
377C2×Q16C22×D6111yescharacteristic
378C2×SD16—21no—
379C2×SD16—1221no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—1221no—
383C2×D8—1221no—
384C2×D8—1222no—
385C2×D8—1222no—
386C2×D8—1221no—
387C2×D8—2441no—
388C2×D8—2441no—
389C2×D8C22×D6111yescharacteristic
390C2×D8C22×D6111yescharacteristic
391C22×C8—611no—
392C22×C8—611no—
393#127—1221no—
394#124—21no—
395#124—2441no—
396C22⋊Q8—1221no—
397C22⋊Q8—2441no—
398#121—1221no—
399#121—1221no—
400#121—1221no—
401#121—2441no—
402#121—2441no—
403#120—2441no—
404#120—2441no—
405C4×Q8—1221no—
406C4×D4—21no—
407C4×D4—611no—
408C4×D4—2441no—
409SD32—22no—
410SD32—614no—
411SD32—1222no—
412SD32—1222no—
413SD32—1222no—
414SD32—2441no—
415D16—22no—
416D16—1222no—
417D16—2441no—
418M32—1222no—
419C2×C16—611no—
420C2×C16—611no—
421C2×C16—1222no—
422C2×C16C22×D6111yescharacteristic
423C2×C16C22×D6111yescharacteristic
424#107—611no—
425#107C22×D6111yescharacteristic
426C4⋊C8—611no—
427#103—1221no—
428#103—1221no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—611no—
434C22⋊C8—611no—
435C4×C8C22×D6111yescharacteristic
436C24—2441no—
437C24—2441no—
438#41—4881no—
439#41—2442no—
440C2×Q8—21no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—2441no—
444C2×Q8—2442no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—611no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—4884no—
451C2×D4—4881no—
452C2×D4—4881no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4—2441no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—41no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—2441no—
467C22×C4—2441no—
468Q16—21no—
469Q16—21no—
470Q16—612no—
471Q16—1221no—
472Q16—2442no—
473SD16—81no—
474SD16—2442no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—1221no—
479D8—1221no—
480D8—1221no—
481D8—1221no—
482D8—4881no—
483D8—2442no—
484D8—2442no—
485D8D4×D6112yes—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8D4×D6111yescharacteristic
493C2×C8C23×D6111yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—611no—
503C22⋊C4—611no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—2441no—
509C42D4×D6111yescharacteristic
510C16—21no—
511C16—21no—
512C16—612no—
513C16—1221no—
514C23—41no—
515C23—41no—
516C23—611no—
517C23—611no—
518C23—9761no—
519C23—9761no—
520C23—4882no—
521C23—4882no—
522C23—4882no—
523C23—4882no—
524C23—2441no—
525C23—2441no—
526Q8—41no—
527Q8—41no—
528Q8—1222no—
529Q8—4882no—
530Q8—2441no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—1222no—
536D4—9761no—
537D4—4882no—
538D4—4882no—
539D4—4882no—
540D4—4882no—
541D4—4882no—
542D4—4882no—
543D4—4882no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—82no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4D8×D6111yescharacteristic
562C2×C4D8×D6111yescharacteristic
563C2×C4C2×D4×D6111yescharacteristic
564C8—21no—
565C8—1221no—
566C8—1221no—
567C8C2×D4×D6111yescharacteristic
568C8#1451982311yescharacteristic
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—612no—
576C22—1221no—
577C22—1221no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—9764no—
582C22—9764no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22C2×D8×D6111yescharacteristic
590C4—21no—
591C4—21no—
592C4—81no—
593C4—81no—
594C4—1221no—
595C4—1221no—
596C4—4881no—
597C4—2442no—
598C4#1684039611yescharacteristic
599C4#1684056611yescharacteristic
600C2—82no—
601C2—161no—
602C2—1222no—
603C2—9761no—
604C2—9761no—
605C2—2441no—
606C2#2145749111yescharacteristic
607C2#509×D6111yescharacteristic
608C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
hypercenter#2210D61
hypocenterC61#33831
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. C2×SD32×D61C2semidirect
3. #21457559C2semidirect
4. D8×D244C2semidirect
5. #21448086C2semidirect
6. #21447935C2semidirect
7. #21447904C2semidirect
8. #21447860C2semidirect
9. #21447746C2semidirect
10. #21447743C2semidirect
11. #21447737C2semidirect
12. #21447729C2semidirect
13. #21447683C2semidirect
14. #21447643C2semidirect
15. C61×#2210C2semidirect
16. C2×#14519796C22semidirect
17. C2×Q16×D61C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519722C22semidirect
26. C2×#14519720C22semidirect
27. C2×#14519720C22semidirect
29. C2×#14519718C22semidirect
30. C2×#14519718C22semidirect
31. C2×#14519716C22semidirect
32. C2×C16×D61C22semidirect
34. #16840000C22semidirect
37. #16839995C22semidirect
44. C4×Dic244C22semidirect
46. #16839587C22semidirect
47. #16839565C22semidirect
48. #16839564C22semidirect
49. #16839564C22semidirect
51. #16839541C22semidirect
52. #16839541C22semidirect
53. #16839538C22semidirect
56. C61×#495C22semidirect
58. C61×C4⋊C16C22semidirect
59. C61×#358C22semidirect
60. C61×#357C22semidirect
100. #14519721D4semidirect
195. Dic244C2×D4semidirect
208. #11758185C2×D4semidirect
285. C2×D61#509semidirect
303. #2210D61semidirect
339. #495C2×D61semidirect
355. C4⋊C16C2×D61semidirect
357. #358C2×D61semidirect
361. #357C2×D61semidirect
365. C61#33831semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
C2×SD32×D612
#214575592
D8×D2442
#214480862
#214479352
#214479042
#214478602
#214477462
#214477432
#214477372
#214477292
#214476832
#214476432
C61×#22102
#3383161