Finite Group Databasebeta

← #29697355

Subgroups — #29697355

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454477C211yesmaximal, characteristic, complemented
2C2×#16839756C211yesmaximal, characteristic
3#21454269C211yesmaximal, characteristic
4#21446531C212yesmaximal, complemented
5#21446465C212yesmaximal, complemented
6#21446095C212yesmaximal, complemented
7#21445855C211yesmaximal, characteristic, complemented
8#21445764C211yesmaximal, characteristic
9#21445660C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic, complemented
11C61×#1826C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519785C2212yes—
14C23×#92418C2211yescharacteristic
15#16839823—22no—
16#16839819—21no—
17#16839815—21no—
18C2×#14519712C2212yescomplemented
19#16839789C2212yescomplemented
20C2×#14519705C2212yes—
21C2×#14519705C2212yes—
22#16839756C411yescharacteristic, complemented
23#16839756C411yescharacteristic, complemented
24#16839756C411yescharacteristic, complemented
25#16839756C411yescharacteristic, complemented
26#16839756C2211yescharacteristic
27#16839756C2211yescharacteristic
28#16839756C2211yescharacteristic
29#16839756C2211yescharacteristic
30C2×#14519700C411yescharacteristic, complemented
31C2×#14519700C411yescharacteristic, complemented
32C2×#14519697C411yescharacteristic, complemented
33C2×#14519697C411yescharacteristic, complemented
34#16839730C2212yes—
35#16839727—22no—
36#16839726C2211yescharacteristic
37#16839724—21no—
38C2×C22⋊#92418C2211yescharacteristic
39C22⋊C8×D61—21no—
40#16839718C2212yescomplemented
41C2×#14519688C2212yescomplemented
42#16839526C2212yescomplemented
43#16839517C2212yescomplemented
44C2×#117×C61C2211yescharacteristic
45C61×#407C2211yescharacteristic
46C4×C22⋊C4×C61C2211yescharacteristic
47C61×#339C2212yescomplemented
48C61×#339C2212yescomplemented
49C2×#11758222—21no—
50C2×#11758222—21no—
51C2×#11758222—24no—
52C2×#11758222—42no—
53C2×#11758222C2×C411yescharacteristic
54C2×#11758222C2×C411yescharacteristic
55C2×#11758222C2311yescharacteristic
56C2×#11758222C2311yescharacteristic
57C22×Dic122C2×C411yescharacteristic
58C22×D244C2×C411yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61#14519785—24no—
62#14519785—42no—
63C2×#11758194C2312yes—
64C2×#11758189—22no—
65C22×#92418—42no—
66C22×#92418C2×C411yescharacteristic
67C22×#92418C2×C411yescharacteristic
68C22×#92418C2×C411yescharacteristic
69C22×#92418C2×C411yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2312yes—
73C2×C8×D61—22no—
74#14519712—42no—
75C4⋊#92418—22no—
76C4⋊#92418—22no—
77#14519705—24no—
78#14519705—42no—
79#14519700C2×C411yescharacteristic
80#14519700C2×C411yescharacteristic
81#14519700C2×C411yescharacteristic
82#14519700C2×C411yescharacteristic
83#14519700C2×C411yescharacteristic
84#14519700C2×C411yescharacteristic
85#14519700C2×C411yescharacteristic
86#14519700C2×C411yescharacteristic
87#14519697C2×C411yescharacteristic
88#14519697C2×C411yescharacteristic
89#14519697C2×C411yescharacteristic
90#14519697C2×C411yescharacteristic
91#14519697C2×C411yescharacteristic
92#14519697C2×C411yescharacteristic
93#14519697C2×C411yescharacteristic
94#14519697C2×C411yescharacteristic
95C22⋊#92418—21no—
96C22⋊#92418—21no—
97C22⋊#92418D411yescharacteristic
98C22⋊#92418D411yescharacteristic
99C22⋊#92418D412yes—
100#14519689—22no—
101#14519689—22no—
102#14519688—22no—
103#14519688—42no—
104C4×#92418—22no—
105C23×C4×C61C2311yescharacteristic
106C2×M16×C61C2312yes—
107#117×C61—21no—
108#117×C61—21no—
109#117×C61C2×C411yescharacteristic
110#117×C61C2×C411yescharacteristic
111#117×C61C2311yescharacteristic
112#117×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2×C411yescharacteristic
114C2×C4⋊C4×C61C2×C411yescharacteristic
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2312yes—
119C22⋊C8×C61—21no—
120C22⋊C8×C61—21no—
121#95×C61—21no—
122C24×C61#4111yescharacteristic
123C23×D61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—21no—
138C2×C22⋊D61—21no—
139C2×C22⋊D61—44no—
140C22×Dic61—21no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—41no—
147#11758222—41no—
148#11758222—44no—
149#11758222—44no—
150#11758222#4114yes—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—41no—
154C2×D244C22×C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—44no—
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122—41no—
164C2×Dic122C22×C411yescharacteristic
165C2×Dic122C22×C411yescharacteristic
166M16×C61—22no—
167M16×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—42no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—22no—
185C42×C61—22no—
186#11758194—22no—
187#11758194—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—41no—
194C2×#92418—41no—
195C2×#92418—41no—
196C2×#92418—41no—
197C2×#92418C22×C411yescharacteristic
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C2×#92418C22×C411yescharacteristic
201C2×#92418C2×D412yes—
202C2×#92418C2×D412yes—
203#11758189—44no—
204C8×D61—44no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61C4×D411yescharacteristic
209C23×C61C4×D411yescharacteristic
210C23×C61C2×#4111yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—81no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—81no—
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C4×D411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4112yes—
237C2×C4×C61H(5,𝔽2)12yes—
238C22⋊D61—24no—
239C22⋊D61—42no—
240C22⋊D61—42no—
241C22⋊D61—42no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—42no—
245D244—21no—
246D244—21no—
247D244—21no—
248D244—42no—
249D244#11712yes—
250C4×D61—24no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254Dic122—21no—
255Dic122—21no—
256Dic122—21no—
257Dic122—42no—
258Dic122#11712yes—
259C8×C61—42no—
260#92418—21no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—22no—
265#92418—22no—
266#92418—42no—
267#22586—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—44no—
276C22×C61#44412yes—
277C22×C61C2×C4×D411yescharacteristic
278C22×C61#55112yes—
279C2×D61—22no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—82no—
293C4×C61C2×#11711yescharacteristic
294C4×C61#51811yescharacteristic
295C4×C61#54612yes—
296Dic61—22no—
297Dic61—41no—
298#2924—611no—
299C2×#416—611no—
300#2880—611no—
301#1938—612no—
302#1929—612no—
303#1866—612no—
304#1826D6111yescharacteristic, complemented
305#1810—611no—
306#1789—611no—
307#1765—612no—
308#1752—611no—
309C2×C61—22no—
310C2×C61—22no—
311C2×C61—41no—
312C2×C61#234211yescharacteristic
313C2×C61C2×#44411yescharacteristic
314C2×C61#295711yescharacteristic
315D61—42no—
316D61—81no—
317C22×#41—611no—
318C2×#131—612no—
319C23×C8—611no—
320C2×#117C2×D6111yescharacteristic
321#433—1222no—
322#432—1221no—
323#431—1221no—
324C2×#108—612no—
325#424—612no—
326C2×#104—612no—
327C2×#104—612no—
328#416—611no—
329#416—611no—
330#416—611no—
331#416—611no—
332#416—611no—
333#416—611no—
334#416—611no—
335#416—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#102—611no—
339C2×#102—611no—
340#408—612no—
341#408—1222no—
342#407—611no—
343#407—1221no—
344#407C2×D6111yescharacteristic
345C2×C22⋊C8—611no—
346C2×C22⋊C8—1221no—
347#405—612no—
348C2×#97—612no—
349C4×C22⋊C4C2×D6111yescharacteristic
350#341—612no—
351#339C2×D6112yescomplemented
352#339C2×D6112yescomplemented
353#336—612no—
354C61#2258611yescharacteristic, complemented, Sylow
355C2×#41—611no—
356C2×#41—611no—
357C2×#41—611no—
358C2×#41—611no—
359C2×#41—1224no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—2442no—
363C22×Q8—611no—
364C22×D4—611no—
365C22×D4—1221no—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368#131—1224no—
369#131—2442no—
370C2×M16—612no—
371C2×M16—1222no—
372C2×M16C22×D6112yes—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—612no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—1222no—
381C22×C8—2442no—
382#117—21no—
383#117—21no—
384#117C4×D6111yescharacteristic
385#117C4×D6111yescharacteristic
386#117C22×D6111yescharacteristic
387#117C22×D6111yescharacteristic
388C2×C4⋊C4C4×D6111yescharacteristic
389C2×C4⋊C4C4×D6111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42C22×D6112yes—
394#108—2442no—
395C4⋊C8—1222no—
396C4⋊C8—1222no—
397#104—1224no—
398#104—2442no—
399#103—611no—
400#103—611no—
401#103—611no—
402#103—611no—
403#103—611no—
404#103—611no—
405#103—611no—
406#103—611no—
407#102—611no—
408#102—611no—
409#102—611no—
410#102—611no—
411#102—611no—
412#102—611no—
413#102—611no—
414#102—611no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—611no—
418C22⋊C8—611no—
419C22⋊C8—612no—
420C22⋊C8—1222no—
421C22⋊C8—1222no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1222no—
425#97—2442no—
426C4×C8—1222no—
427#95—21no—
428C24—1221no—
429C24#1175822411yescharacteristic
430#41—614no—
431#41—1222no—
432#41—1222no—
433#41—1222no—
434#41—1222no—
435#41—1222no—
436#41—2444no—
437#41—2441no—
438#41—2441no—
439#41—2444no—
440C2×Q8—611no—
441C2×Q8—611no—
442C2×Q8—1222no—
443C2×Q8—1221no—
444C2×Q8—2441no—
445C2×D4—611no—
446C2×D4—611no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—2441no—
454C2×D4—2444no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2444no—
466C22×C4C2×C4×D6111yescharacteristic
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4#1175822411yescharacteristic
471C22×C4C23×D6111yescharacteristic
472M16—22no—
473M16—22no—
474M16—1222no—
475M16—1222no—
476M16—2444no—
477C2×C8—22no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—612no—
483C2×C8—612no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—2444no—
490C2×C8—2441no—
491C2×C8—2441no—
492C2×C8—2441no—
493C2×C8—2441no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C2×C4×D6111yescharacteristic
497C4⋊C4C2×C4×D6111yescharacteristic
498C4⋊C4C2×C4×D6111yescharacteristic
499C4⋊C4C2×C4×D6111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—42no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—1221no—
515C23—1221no—
516C23—4881no—
517C23—2442no—
518C23—2442no—
519C23#1451973711yescharacteristic
520C23#1451973711yescharacteristic
521C23C2×#1175822411yescharacteristic
522Q8—612no—
523Q8—1221no—
524Q8—1221no—
525Q8—1221no—
526Q8—2442no—
527D4—612no—
528D4—1224no—
529D4—1221no—
530D4—1221no—
531D4—1221no—
532D4—2442no—
533D4—2442no—
534D4—2442no—
535D4—2442no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—1221no—
553C2×C4—1224no—
554C2×C4—1221no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4#1451973711yescharacteristic
560C2×C4#1451973711yescharacteristic
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#41×D6112yes—
564C2×C4#1451982312yes—
565C8—42no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—1222no—
571C8—1222no—
572C8—2442no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—1222no—
582C22—4884no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22#1683988112yes—
589C22C2×#1451973711yescharacteristic
590C22#1684043512yes—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—1222no—
599C4—2441no—
600C4#117×D6111yescharacteristic
601C4#1684023611yescharacteristic
602C4#1684040412yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—4881no—
607C2—2442no—
608C2#2144866111yescharacteristic
609C2C2×#1683988111yescharacteristic
610C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1826C2
socleC22×C61C2×C4×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1826D61
hypocenterC61#22586
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
1. #21454477C2semidirect
4. #21446531C2semidirect
5. #21446465C2semidirect
6. #21446095C2semidirect
7. #21445855C2semidirect
9. #21445660C2semidirect
10. #21445597C2semidirect
11. C61×#1826C2semidirect
18. C2×#14519712C22semidirect
19. #16839789C22semidirect
22. #16839756C4semidirect
23. #16839756C4semidirect
24. #16839756C4semidirect
25. #16839756C4semidirect
30. C2×#14519700C4semidirect
31. C2×#14519700C4semidirect
32. C2×#14519697C4semidirect
33. C2×#14519697C4semidirect
40. #16839718C22semidirect
41. C2×#14519688C22semidirect
42. #16839526C22semidirect
43. #16839517C22semidirect
47. C61×#339C22semidirect
48. C61×#339C22semidirect
304. #1826D61semidirect
351. #339C2×D61semidirect
352. #339C2×D61semidirect
354. C61#22586semidirect

Maximal subgroups

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

subgroupindex
#214544772
C2×#168397562
#214542692
#214465312
#214464652
#214460952
#214458552
#214457642
#214456602
#214455972
C61×#18262
#2258661