Finite Group Databasebeta

← #29718919

Subgroups — #29718919

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839895C211yesmaximal, characteristic
2#21454477C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4#21447176C212yesmaximal, complemented
5#21447074C212yesmaximal, complemented
6#21446728C212yesmaximal, complemented
7#21446480C212yesmaximal, complemented
8#21446283C211yesmaximal, characteristic, complemented
9#21446023C211yesmaximal, characteristic, complemented
10#21445739C211yesmaximal, characteristic, complemented
11C61×#3094C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758199C2212yes—
14C22×#11758194C2211yescharacteristic
15#16839895—21no—
16#16839895—21no—
17#16839895—22no—
18#16839823—22no—
19#16839819—21no—
20#16839815—21no—
21C2×#14519712C2212yescomplemented
22#16839789C2212yescomplemented
23#16839766—21no—
24#16839766—21no—
25#16839766C2212yes—
26#16839766C2212yes—
27C2×#14519697—21no—
28C2×#14519697—21no—
29C2×#14519697C2212yes—
30C2×#14519694C2212yescomplemented
31C2×#14519693C2212yes—
32C2×#14519693C2212yes—
33#16839726C2211yescharacteristic
34C2×C22⋊#92418C2211yescharacteristic
35#16839718C2212yescomplemented
36#16839531C2212yescomplemented
37#16839526C2212yescomplemented
38C2×#121×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C61×#482—21no—
41C61×#482—21no—
42C61×#481—21no—
43C61×#480—21no—
44C61×#428C2211yescharacteristic, complemented
45C61×#428C2211yescharacteristic, complemented
46C61×#418C2211yescharacteristic, complemented
47C61×#418C2211yescharacteristic, complemented
48C61×#407C2211yescharacteristic
49C24×D61—21no—
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—24no—
53C22×D244—21no—
54C22×D244—21no—
55C22×D244—21no—
56C22×D244—42no—
57C22×D244C2312yes—
58C2×#11758199—24no—
59C2×#11758199—22no—
60C2×#11758199—22no—
61C2×#11758199—42no—
62C2×#11758194—21no—
63C2×#11758194—42no—
64C2×#11758194C2312yes—
65C2×#11758194C2312yes—
66C22×#92418—21no—
67C22×#92418C2312yes—
68#14519712—42no—
69C4⋊#92418—22no—
70C4⋊#92418—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519697—22no—
76#14519697—22no—
77#14519694—42no—
78#14519693—24no—
79#14519693—42no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C22⋊#92418D411yescharacteristic
83C22⋊#92418D411yescharacteristic
84C22⋊#92418D412yes—
85#14519688—22no—
86C4×#92418—22no—
87C22×D4×C61—21no—
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C2×M16×C61C2311yescharacteristic
91C2×M16×C61C2311yescharacteristic
92#121×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95#121×C61—21no—
96#117×C61—21no—
97#117×C61—21no—
98#117×C61D411yescharacteristic
99#117×C61D411yescharacteristic
100#117×C61C2311yescharacteristic
101#117×C61C2311yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108#107×C61—21no—
109#107×C61—21no—
110#106×C61—21no—
111#106×C61—21no—
112#102×C61—21no—
113#102×C61—21no—
114#102×C61—21no—
115#102×C61—21no—
116C22⋊C8×C61—21no—
117C22⋊C8×C61—21no—
118C24×C61—21no—
119C24×C61#4111yescharacteristic
120C23×D61—22no—
121C23×D61—44no—
122C23×D61—41no—
123C23×D61—41no—
124C23×D61—82no—
125C23×D61—82no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—41no—
131C2×D4×C61—41no—
132C2×D4×C61—82no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61#4111yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×D244—24no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—44no—
152C2×D244—44no—
153C2×D244—44no—
154C2×D244—42no—
155C2×D244—42no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244#4114yes—
159M16×C61—22no—
160M16×C61—22no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C4⋊C4×C61C2×D411yescharacteristic
170C4⋊C4×C61C2×D411yescharacteristic
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181#11758199—44no—
182#11758199—44no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—42no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—42no—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—81no—
201C23×C61—81no—
202C23×C61#12111yescharacteristic
203C23×C61#12111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C22×D61—24no—
206C22×D61—44no—
207C22×D61—44no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—82no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—816no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—42no—
220D4×C61—82no—
221D4×C61—82no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—81no—
236C2×C4×C61—81no—
237C2×C4×C61#12111yescharacteristic
238C2×C4×C61#12111yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4112yes—
242C2×C4×C61H(5,𝔽2)12yes—
243D244—24no—
244D244—24no—
245D244—44no—
246D244—44no—
247D244—42no—
248D244—42no—
249C8×C61—41no—
250C8×C61—41no—
251#92418—22no—
252#92418—22no—
253#92418—42no—
254#92418—42no—
255#24520—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—44no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—162no—
268C22×C61#47212yes—
269C22×C61C2×#12111yescharacteristic
270C22×C61#55112yes—
271C2×D61—44no—
272C2×D61—84no—
273C2×D61—88no—
274C2×D61—88no—
275C2×D61—82no—
276C2×D61—82no—
277C2×D61—84no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61#52511yescharacteristic
286C4×C61#53511yescharacteristic
287C4×C61#54612yes—
288#3094D6111yescharacteristic, complemented
289C2×#447—611no—
290#2924—611no—
291C2×#418—611no—
292#2063—612no—
293#2045—612no—
294#1978—612no—
295#1931—612no—
296#1900—611no—
297#1855—611no—
298#1783—611no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—41no—
302C2×C61—161no—
303C2×C61#241111yescharacteristic
304C2×C61C2×#47211yescharacteristic
305C2×C61#306811yescharacteristic
306D61—84no—
307C23×D4—611no—
308C22×D8—612no—
309C22×M16—611no—
310C2×#121C2×D6111yescharacteristic
311C2×#117C2×D6111yescharacteristic
312#482—21no—
313#482—21no—
314#481—21no—
315#480—21no—
316#447—1221no—
317#447—1221no—
318#447—1222no—
319#433—1222no—
320#432—1221no—
321#431—1221no—
322C2×#108—612no—
323#428C2×D6111yescharacteristic, complemented
324#428C2×D6111yescharacteristic, complemented
325#424—612no—
326#418—612no—
327#418—612no—
328#418—1221no—
329#418—1221no—
330#418C2×D6111yescharacteristic, complemented
331#418C2×D6111yescharacteristic, complemented
332C2×#102—612no—
333C2×#102—1221no—
334C2×#102—1221no—
335C2×#100—612no—
336C2×#100—612no—
337C2×#100—612no—
338#407—611no—
339#407C2×D6111yescharacteristic
340C2×C22⋊C8—611no—
341#405—612no—
342#343—612no—
343#341—612no—
344C61#2452011yescharacteristic, complemented, Sylow
345C25—1221no—
346C22×D4—21no—
347C22×D4—612no—
348C22×D4—1224no—
349C22×D4—1221no—
350C22×D4—1221no—
351C22×D4—1221no—
352C22×D4—1221no—
353C22×D4—1221no—
354C22×D4—2442no—
355C22×D4C22×D6111yescharacteristic
356C23×C4C22×D6111yescharacteristic
357C2×D8—1222no—
358C2×D8—1222no—
359C2×D8—1224no—
360C2×D8—2442no—
361C2×M16—612no—
362C2×M16—612no—
363C2×M16—1221no—
364C2×M16—2442no—
365C2×M16C22×D6111yescharacteristic
366C2×M16C22×D6111yescharacteristic
367C22×C8—612no—
368C22×C8—1221no—
369#121—21no—
370#121—21no—
371#121—21no—
372#121—21no—
373#117—21no—
374#117—21no—
375#117C22⋊D6111yescharacteristic
376#117C22⋊D6111yescharacteristic
377#117C22×D6111yescharacteristic
378#117C22×D6111yescharacteristic
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C4⋊C4C22⋊D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42—21no—
385#108—2442no—
386#107—21no—
387#107—21no—
388#106—21no—
389#106—21no—
390C4⋊C8—1222no—
391C4⋊C8—1222no—
392#102—21no—
393#102—21no—
394#102—21no—
395#102—21no—
396#102—1222no—
397#102—1222no—
398#102—1222no—
399#102—1222no—
400#102—1222no—
401#102—1222no—
402#100—1224no—
403#100—2442no—
404#100—2442no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—611no—
408C22⋊C8—611no—
409C22⋊C8—612no—
410C22⋊C8—1221no—
411C22⋊C8—1221no—
412#97—1222no—
413C4×C8—1222no—
414C24—21no—
415C24—1222no—
416C24—4882no—
417C24—4882no—
418C24—2444no—
419C24—2441no—
420C24—2441no—
421C24#1175822411yescharacteristic
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—82no—
429C2×D4—614no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1222no—
437C2×D4—2444no—
438C2×D4—2444no—
439C2×D4—2444no—
440C2×D4—2442no—
441C2×D4—2442no—
442C2×D4—2441no—
443C2×D4—2441no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822411yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455D8—2444no—
456D8—2444no—
457M16—22no—
458M16—22no—
459M16—1222no—
460M16—1222no—
461M16—1222no—
462M16—1222no—
463M16—2442no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—2442no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C4⋊C4C2×C22⋊D6111yescharacteristic
480C4⋊C4C2×C22⋊D6111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—81no—
497C23—81no—
498C23—1224no—
499C23—4882no—
500C23—4882no—
501C23—4882no—
502C23—4882no—
503C23—4882no—
504C23—48816no—
505C23—4882no—
506C23—2444no—
507C23—2444no—
508C23—2442no—
509C23—2442no—
510C23#1451974911yescharacteristic
511C23#1451974911yescharacteristic
512C23C2×#1175822411yescharacteristic
513D4—41no—
514D4—41no—
515D4—42no—
516D4—82no—
517D4—82no—
518D4—1224no—
519D4—1224no—
520D4—2444no—
521D4—2444no—
522D4—2442no—
523D4—2442no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4#1451974911yescharacteristic
540C2×C4#1451974911yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4#41×D6112yes—
544C2×C4#1451982312yes—
545C8—41no—
546C8—41no—
547C8—1222no—
548C8—1222no—
549C8—2442no—
550C8—2442no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—44no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—162no—
563C22—4884no—
564C22—4884no—
565C22—4888no—
566C22—4888no—
567C22—4882no—
568C22—4882no—
569C22—2444no—
570C22#1684003012yes—
571C22C2×#1451974911yescharacteristic
572C22#1684043512yes—
573C4—21no—
574C4—21no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4#1684026911yescharacteristic
581C4#1684032611yescharacteristic
582C4#1684040412yes—
583C2—22no—
584C2—22no—
585C2—41no—
586C2—161no—
587C2—4884no—
588C2#2144920511yescharacteristic
589C2C2×#1684003011yescharacteristic
590C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3094C2
socleC22×C61C2×#121
prime radicalC2×#16839895C2
hypercenter#3094D61
hypocenterC61#24520
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
2. #21454477C2semidirect
3. C2×#16839766C2semidirect
4. #21447176C2semidirect
5. #21447074C2semidirect
6. #21446728C2semidirect
7. #21446480C2semidirect
8. #21446283C2semidirect
9. #21446023C2semidirect
10. #21445739C2semidirect
11. C61×#3094C2semidirect
21. C2×#14519712C22semidirect
22. #16839789C22semidirect
30. C2×#14519694C22semidirect
35. #16839718C22semidirect
36. #16839531C22semidirect
37. #16839526C22semidirect
44. C61×#428C22semidirect
45. C61×#428C22semidirect
46. C61×#418C22semidirect
47. C61×#418C22semidirect
288. #3094D61semidirect
323. #428C2×D61semidirect
324. #428C2×D61semidirect
330. #418C2×D61semidirect
331. #418C2×D61semidirect
344. C61#24520semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
#214544772
C2×#168397662
#214471762
#214470742
#214467282
#214464802
#214462832
#214460232
#214457392
C61×#30942
#2452061