Finite Group Databasebeta

← #29587334

Subgroups — #29587334

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454408C211yesmaximal, characteristic, complemented
2#21454339C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C2×#16839728C211yesmaximal, characteristic
5C2×#16839520C211yesmaximal, characteristic, complemented
6#21445515C211yesmaximal, characteristic, complemented
7C61×#1729C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C22×#11758194C2211yescharacteristic
11C2×#14519736C2211yescharacteristic
12#16839805—22no—
13#16839802—22no—
14#16839798—22no—
15#16839772—21no—
16#16839770—21no—
17#16839768—21no—
18#16839766—21no—
19#16839764—22no—
20#16839761—22no—
21#16839759—22no—
22#16839756—22no—
23#16839728—22no—
24#16839728C414yescomplemented
25C2×#14519689—22no—
26#16839520—22no—
27#16839516—22no—
28C22×M16×C61C2211yescharacteristic
29C23×C8×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C61×#336—22no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222Q812yes—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61—21no—
47C22×C4×D61—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758194C2×C412yes—
59C2×#11758194D412yes—
60C22×#92418D412yes—
61C22×#92418D412yes—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66C2×C4⋊Dic61D412yes—
67C2×C22⋊Dic61—21no—
68C2×C4×Dic61—21no—
69#14519710—24no—
70#14519708—24no—
71#14519702—24no—
72#14519700—24no—
73#14519699—24no—
74#14519697—24no—
75#14519689—28no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C22×C8×C61—21no—
84C22×C8×C61—21no—
85C22×C8×C61—21no—
86C22×C8×C61—21no—
87C22×C8×C61—21no—
88C22×C8×C61—21no—
89C22×C8×C61C2×C412yes—
90C22×C8×C61D412yes—
91#117×C61—21no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95C2×C4⋊C4×C61D412yes—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61—21no—
98C24×C61#4111yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×Q811yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—22no—
116C2×C22⋊D61—22no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C4⋊C412yes—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C4⋊C412yes—
157M16×C61—41no—
158M16×C61—41no—
159M16×C61—41no—
160M16×C61—41no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—41no—
168C2×C8×C61—41no—
169C2×C8×C61—41no—
170C2×C8×C61—41no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C42×C61—41no—
176C42×C61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4×Dic61—41no—
182C4×Dic61—41no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—41no—
188#11758194—41no—
189#11758194—41no—
190#11758194—41no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418C22⋊C414yes—
197C23×C61—42no—
198C23×C61#11711yescharacteristic
199C23×C61#12111yescharacteristic
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61C22⋊Q811yescharacteristic
203C23×C61#12611yescharacteristic
204C23×C61#12611yescharacteristic
205C22×D61—22no—
206C22×D61—44no—
207C22×D61—41no—
208C22×D61—41no—
209C2×C4×C61—44no—
210C2×C4×C61—42no—
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×C4⋊C411yescharacteristic
213C2×C4×C61C4×D411yescharacteristic
214C2×C4×C61C4×D411yescharacteristic
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12011yescharacteristic
219C2×C4×C61#12011yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61C22⋊Q811yescharacteristic
226C2×C4×C61C22⋊Q811yescharacteristic
227C2×C4×C61C22⋊Q811yescharacteristic
228C2×C4×C61C22⋊Q811yescharacteristic
229C2×C4×C61C22⋊Q811yescharacteristic
230C2×C4×C61#12311yescharacteristic
231C2×C4×C61#12311yescharacteristic
232C2×C4×C61#12311yescharacteristic
233C2×C4×C61#12311yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12411yescharacteristic
236C2×C4×C61#12411yescharacteristic
237C2×C4×C61#12411yescharacteristic
238C2×C4×C61#12711yescharacteristic
239C22⋊D61—42no—
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—44no—
247C2×Dic61—42no—
248D244—22no—
249D244—22no—
250D244—41no—
251D244—41no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—42no—
261C8×C61—42no—
262#92418—24no—
263#92418—42no—
264#11111—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61#38011yescharacteristic
273C22×C61#38811yescharacteristic
274C22×C61#38811yescharacteristic
275C22×C61#39211yescharacteristic
276C22×C61#39611yescharacteristic
277C22×C61#39611yescharacteristic
278C22×C61#39711yescharacteristic
279C2×D61—44no—
280C2×D61—42no—
281C2×D61—82no—
282C2×D61—82no—
283C4×C61—82no—
284C4×C61#38511yescharacteristic
285C4×C61#38511yescharacteristic
286C4×C61#38611yescharacteristic
287C4×C61#39011yescharacteristic
288C4×C61#39311yescharacteristic
289C4×C61#39411yescharacteristic
290C4×C61#39411yescharacteristic
291C4×C61#39911yescharacteristic
292Dic61—42no—
293Dic61—82no—
294#2911—611no—
295#2896—611no—
296#2896—611no—
297C2×#408—611no—
298C2×#338—611no—
299#1729—611no—
300#1729D6111yescharacteristic, complemented
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61#143911yescharacteristic
308C2×C61#201712yes—
309D61—82no—
310C22×#41—611no—
311C22×M16—611no—
312C22×M16—611no—
313C22×M16C2×D6111yescharacteristic
314C23×C8C2×D6111yescharacteristic
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317#428—1222no—
318#428—1222no—
319#427—1222no—
320#419—1221no—
321#419—1221no—
322#418—1221no—
323#418—1221no—
324#417—1222no—
325#417—1222no—
326#416—1222no—
327#416—1222no—
328#408—614no—
329#408—1222no—
330C2×C22⋊C8—1222no—
331#338—1222no—
332#336—22no—
333#336—1222no—
334C61#1111111yescharacteristic, complemented, Sylow
335C2×#41—612no—
336C2×#41—612no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C2×#41—1221no—
346C2×#41—1221no—
347C22×Q8—611no—
348C22×D4—611no—
349C22×D4—1221no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—612no—
359C2×M16—612no—
360C2×M16—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C2×M16—1221no—
370C22×C8—21no—
371C22×C8—21no—
372C22×C8—21no—
373C22×C8—21no—
374C22×C8—21no—
375C22×C8—21no—
376C22×C8—612no—
377C22×C8—612no—
378C22×C8C4×D6112yes—
379C22×C8C22⋊D6112yes—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—1221no—
385#117—1221no—
386#117—1221no—
387#117—1221no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4D24412yes—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—1221no—
394#107—1224no—
395#106—1224no—
396#103—1224no—
397#103—1224no—
398#102—1224no—
399#102—1224no—
400C22⋊C8—1228no—
401C24—1221no—
402C24#1175822211yescharacteristic
403#41—1222no—
404#41—1222no—
405#41—1222no—
406#41—1222no—
407#41—1222no—
408#41—1222no—
409#41—1222no—
410#41—1222no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416#41—2441no—
417#41—2441no—
418#41—2441no—
419C2×Q8—612no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×D4—612no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1221no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2441no—
437C22×C4—22no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×D24411yescharacteristic
448C22×C4#1175822211yescharacteristic
449C22×C4#1175822211yescharacteristic
450C22×C4#1175822211yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822411yescharacteristic
457C22×C4Q8×D6111yescharacteristic
458C22×C4#1175822611yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—1222no—
465M16—1222no—
466M16—1222no—
467M16—1222no—
468M16—2441no—
469M16—2441no—
470M16—2441no—
471M16—2441no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—614no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—2441no—
495C22⋊C4—2441no—
496C42—41no—
497C42—41no—
498C42—2441no—
499C42—2441no—
500C23—42no—
501C23—1222no—
502C23—2444no—
503C23—2441no—
504C23—2441no—
505C23#1451973411yescharacteristic
506C23#1451975011yescharacteristic
507C23#1451975011yescharacteristic
508C23#1451975111yescharacteristic
509C23#1451975511yescharacteristic
510C23#1451977111yescharacteristic
511C23#1451977211yescharacteristic
512Q8—1222no—
513Q8—1222no—
514Q8—2441no—
515Q8—2441no—
516D4—1222no—
517D4—1222no—
518D4—2442no—
519D4—2442no—
520D4—2442no—
521D4—2442no—
522D4—2441no—
523D4—2441no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—1222no—
527C2×C4—2444no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C22⋊C4×D6111yescharacteristic
536C2×C4C4⋊C4×D6111yescharacteristic
537C2×C4#1451973711yescharacteristic
538C2×C4C4×C22⋊D6111yescharacteristic
539C2×C4#1451973911yescharacteristic
540C2×C4C4×D24411yescharacteristic
541C2×C4#1451974511yescharacteristic
542C2×C4#1451974711yescharacteristic
543C2×C4#1451974711yescharacteristic
544C2×C4#1451974911yescharacteristic
545C2×C4#1451975011yescharacteristic
546C2×C4#1451975011yescharacteristic
547C2×C4#1451975011yescharacteristic
548C2×C4#1451975211yescharacteristic
549C2×C4#1451975311yescharacteristic
550C2×C4#1451975511yescharacteristic
551C2×C4#1451975511yescharacteristic
552C2×C4#1451975511yescharacteristic
553C2×C4#1451975711yescharacteristic
554C2×C4#1451975911yescharacteristic
555C2×C4#1451976011yescharacteristic
556C2×C4#1451976211yescharacteristic
557C2×C4#1451976211yescharacteristic
558C2×C4#1451976211yescharacteristic
559C2×C4#1451976311yescharacteristic
560C2×C4#1451976311yescharacteristic
561C2×C4#1451976511yescharacteristic
562C2×C4#1451977311yescharacteristic
563C8—42no—
564C8—42no—
565C8—1224no—
566C8—2442no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4882no—
575C22—4882no—
576C22—2444no—
577C22—2442no—
578C22#1683962111yescharacteristic
579C22#1683965111yescharacteristic
580C22#1683965311yescharacteristic
581C22#1683967111yescharacteristic
582C22#1683968411yescharacteristic
583C22#1683968611yescharacteristic
584C22#1683969111yescharacteristic
585C4—82no—
586C4—4882no—
587C4—2442no—
588C4#1683963311yescharacteristic
589C4#1683963611yescharacteristic
590C4#1683964011yescharacteristic
591C4#1683966211yescharacteristic
592C4#1683967511yescharacteristic
593C4#1683967711yescharacteristic
594C4#1683968011yescharacteristic
595C4#1683969711yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—4882no—
603C2#2144406311yescharacteristic
604C2#2144692312yes—
subgroupG/H
centerC2×C4#14519773
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1729C2
socleC22×C61#392
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1729D61
hypocenterC61#11111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. #21454408C2semidirect
5. C2×#16839520C2semidirect
6. #21445515C2semidirect
7. C61×#1729C2semidirect
24. #16839728C4semidirect
300. #1729D61semidirect
334. C61#11111semidirect

Maximal subgroups

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

subgroupindex
#214544082
#214543392
#214543362
C2×#168397282
C2×#168395202
#214455152
C61×#17292
#1111161