Finite Group Databasebeta

← #29679539

Subgroups — #29679539

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840243C211yesmaximal, characteristic
2C2×#16839790C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic
4C2×#16839756C211yesmaximal, characteristic
5#21448439C211yesmaximal, characteristic, complemented
6#21445833C212yesmaximal, complemented
7#21445828C212yesmaximal, complemented
8#21445823C212yesmaximal, complemented
9#21445818C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840243—24no—
15#16840243C418yescomplemented
16C2×#14519742C2212yes—
17C2×#14519739C2212yes—
18C2×C4×C22⋊D61—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519735—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23#16839790—24no—
24C2×#14519706C2212yes—
25C2×#14519706C2212yes—
26#16839764—24no—
27#16839756—24no—
28C2×#14519702C2212yes—
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519699C2212yes—
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34#16839660—21no—
35#16839660—21no—
36#16839634—22no—
37#16839631C2212yes—
38#16839627C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×M16×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222C2×C414yes—
50C2×#11758222D414yes—
51C2×#11758222Q814yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—22no—
58C22×#92418—22no—
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519742—24no—
64#14519742C2×C418yescomplemented
65#14519739—24no—
66#14519739C2×C418yescomplemented
67C4×C22⋊D61—216no—
68#14519736C2×C414yes—
69#14519736D414yes—
70#14519735—28no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C4⋊C4×D61—28no—
81C2×C22⋊Dic61D412yes—
82C2×#11758198—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2312yes—
85#14519706—24no—
86#14519706—24no—
87#14519702—24no—
88#14519700—22no—
89#14519700—22no—
90#14519699—24no—
91#14519697—22no—
92#14519697—22no—
93#14519684—22no—
94#14519684—22no—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C22×C8×C61C2312yes—
101#117×C61—22no—
102#117×C61—22no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61C2×C414yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—24no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—81no—
118C22×C4×C61—81no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×Q812yes—
124C22×C4×C61#4112yes—
125C22×C4×C61#4112yes—
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—24no—
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—42no—
133C22×Dic61—81no—
134C22×Dic61—81no—
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
137#11758222—28no—
138#11758222—28no—
139#11758222—28no—
140#11758222C4⋊C4116yescomplemented
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C22×C412yes—
145C2×D244C2×D412yes—
146C2×D244C2×Q812yes—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—28no—
151C2×C4×D61—82no—
152C2×C4×D61—82no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C22×C412yes—
157C2×Dic122C2×D412yes—
158C2×Dic122C2×Q812yes—
159M16×C61—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C414yes—
170C22⋊C4×C61—42no—
171C42×C61—42no—
172C22⋊Dic61C22⋊C418yes—
173#11758198—28no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—42no—
177C4⋊Dic61C22⋊C414yes—
178C4⋊Dic61C22⋊C414yes—
179#11758196—28no—
180#11758196—82no—
181#11758196—82no—
182C4×Dic61—28no—
183C4×Dic61C22×C414yes—
184C4×Dic61C2×D414yes—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—44no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61C2×C22⋊C412yes—
197C23×C61#12112yes—
198C23×C61C22⋊Q812yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C2×C4×C61—24no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—84no—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C4⋊C414yes—
219C2×C4×C61#11714yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12112yes—
222C2×C4×C61C22⋊Q814yes—
223C2×C4×C61C22⋊Q812yes—
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—216no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—84no—
242C2×Dic61C2×C22⋊C414yes—
243C2×Dic61C2×C22⋊C414yes—
244D244—22no—
245D244—22no—
246D244—22no—
247D244C2×C4⋊C414yes—
248C4×D61—216no—
249C4×D61—88no—
250Dic122—22no—
251Dic122—22no—
252Dic122—22no—
253Dic122C2×C4⋊C414yes—
254C8×C61—42no—
255#92418—41no—
256#92418—41no—
257#20652—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61#38014yes—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×C22⋊Q811yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57811yescharacteristic
272C2×D61—24no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C4×C61—44no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61#38014yes—
283C4×C61C22×C4⋊C411yescharacteristic
284C4×C61C2×#11711yescharacteristic
285C4×C61C2×#12111yescharacteristic
286C4×C61C2×C22⋊Q811yescharacteristic
287Dic61—24no—
288Dic61—24no—
289Dic61—24no—
290Dic61—82no—
291C2×#520—611no—
292C2×#424—611no—
293C2×#417—611no—
294C2×#416—611no—
295#2317—611no—
296#1805—612no—
297#1804—612no—
298#1803—612no—
299#1802—612no—
300#1736D6111yescharacteristic, complemented
301C2×#340—611no—
302C2×C61—24no—
303C2×C61C2×#38011yescharacteristic
304C2×C61#289711yescharacteristic
305C2×C61#289811yescharacteristic
306C2×C61#304511yescharacteristic
307C2×C61#304611yescharacteristic
308C2×C61#308311yescharacteristic
309C2×C61#308411yescharacteristic
310D61—44no—
311C22×#41—611no—
312C22×M16C2×D6111yescharacteristic
313C23×C8—611no—
314#520—618no—
315#520—1224no—
316C2×C4×Q8—612no—
317C2×C4×D4—612no—
318C2×C4×D4—1222no—
319C2×#117—611no—
320C2×#117—1221no—
321C2×#117C2×D6111yescharacteristic
322C22×C4⋊C4—611no—
323C22×C4⋊C4—1221no—
324C22×C4⋊C4C2×D6111yescharacteristic
325#424—1224no—
326C2×C4⋊C8—612no—
327C2×C4⋊C8—612no—
328#417—1224no—
329#416—1224no—
330C2×#103—612no—
331C2×#103—611no—
332C2×#103—611no—
333C2×#102—612no—
334C2×#102—611no—
335C2×#102—611no—
336#389—1221no—
337#389—1221no—
338#385—1222no—
339#384—612no—
340#382—612no—
341#340—612no—
342#340—612no—
343#340C2×D6112yes—
344#340C2×D6112yes—
345C61#2065211yescharacteristic, complemented, Sylow
346C2×#41—614no—
347C2×#41—614no—
348C2×#41—614no—
349C2×#41—1222no—
350C2×#41—1222no—
351C2×#41—1222no—
352C22×Q8—611no—
353C22×D4—611no—
354C22×D4—1221no—
355C23×C4—21no—
356C23×C4—1222no—
357C23×C4—1221no—
358C23×C4—1221no—
359C23×C4C22×D6111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—1222no—
366C22×C8—1222no—
367C22×C8—1222no—
368C22×C8C22×D6112yes—
369C4×Q8—618no—
370C4×Q8—1224no—
371C4×D4—618no—
372C4×D4—1224no—
373C4×D4—12216no—
374#117—22no—
375#117—22no—
376#117—614no—
377#117—614no—
378#117—1228no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1228no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4C4×D6114yes—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—612no—
397C2×C22⋊C4—1222no—
398C2×C42—21no—
399C2×C42—612no—
400C2×C42—1222no—
401C4⋊C8—1224no—
402C4⋊C8—1224no—
403#103—1222no—
404#103—1222no—
405#103—1224no—
406#102—1222no—
407#102—1222no—
408#102—1224no—
409#95—1222no—
410#95—1222no—
411C24—1221no—
412C24D4×D6111yescharacteristic
413#41—6116no—
414#41—1228no—
415#41—1228no—
416#41—1228no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×D4—612no—
424C2×D4—612no—
425C2×D4—612no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—1228no—
441C22×C4—1221no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1222no—
446C22×C4—4882no—
447C22×C4—4882no—
448C22×C4—4881no—
449C22×C4—4881no—
450C22×C4—2442no—
451C22×C4C2×C4×D6112yes—
452C22×C4#1175822212yes—
453C22×C4#1175822212yes—
454C22×C4D4×D6112yes—
455C22×C4D4×D6112yes—
456C22×C4D4×D6111yescharacteristic
457C22×C4Q8×D6112yes—
458C22×C4C23×D6111yescharacteristic
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—2444no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—614no—
477C4⋊C4—614no—
478C4⋊C4—1228no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—4882no—
482C4⋊C4—4882no—
483C4⋊C4—2442no—
484C4⋊C4C2×C4×D6114yes—
485C22⋊C4—42no—
486C22⋊C4—618no—
487C22⋊C4—1228no—
488C42—42no—
489C42—614no—
490C42—614no—
491C42—1228no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23C22⋊C4×D6112yes—
504C23#1451975012yes—
505C23#1451975512yes—
506C23C2×D4×D6111yescharacteristic
507Q8—614no—
508Q8—1222no—
509Q8—1222no—
510Q8—1222no—
511D4—614no—
512D4—12216no—
513D4—1222no—
514D4—1222no—
515D4—1222no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—614no—
527C2×C4—614no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—12216no—
536C2×C4—4888no—
537C2×C4—4884no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4C22⋊C4×D6112yes—
542C2×C4C4⋊C4×D6114yes—
543C2×C4#1451973414yes—
544C2×C4#1451975014yes—
545C2×C4#1451975012yes—
546C2×C4#1451975514yes—
547C2×C4#1451975512yes—
548C2×C4C22×C4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×Q8×D6111yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C8—42no—
555C8—2441no—
556C8—2441no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—1224no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22#1683962114yes—
571C22C2×C22⋊C4×D6111yescharacteristic
572C22C2×#1451975011yescharacteristic
573C22C2×#1451975511yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684057111yescharacteristic
576C22#1684057211yescharacteristic
577C22#1684057811yescharacteristic
578C4—44no—
579C4—81no—
580C4—81no—
581C4—1224no—
582C4—1224no—
583C4—1224no—
584C4—4882no—
585C4#1683962114yes—
586C4C2×C4⋊C4×D6111yescharacteristic
587C4C2×#1451973411yescharacteristic
588C4C2×#1451975011yescharacteristic
589C4C2×#1451975511yescharacteristic
590C2—24no—
591C2—2444no—
592C2C2×#1683962111yescharacteristic
593C2#2145434411yescharacteristic
594C2#2145435111yescharacteristic
595C2#2145535711yescharacteristic
596C2#2145536811yescharacteristic
597C2#2145566911yescharacteristic
598C2#2145568011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20652
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#16839790C2semidirect
5. #21448439C2semidirect
6. #21445833C2semidirect
7. #21445828C2semidirect
8. #21445823C2semidirect
9. #21445818C2semidirect
10. C2×#16839524C2semidirect
11. C61×#1736C2semidirect
15. #16840243C4semidirect
64. #14519742C2×C4semidirect
66. #14519739C2×C4semidirect
140. #11758222C4⋊C4semidirect
300. #1736D61semidirect
345. C61#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#168402432
C2×#168397902
C2×#168397642
C2×#168397562
#214484392
#214458332
#214458282
#214458232
#214458182
C2×#168395242
C61×#17362
#2065261