Finite Group Databasebeta

← #29683723

Subgroups — #29683723

Order 15616.

1718 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
1C2×#16840331C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic, complemented
3C2×#16839766C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C22×#14519697C211yesmaximal, characteristic, complemented
6C2×#16839718C211yesmaximal, characteristic, complemented
7#21446642C212yesmaximal, complemented
8#21446633C212yesmaximal, complemented
9#21446628C212yesmaximal, complemented
10#21446624C212yesmaximal, complemented
11C61×#2317C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C23×#92418C2211yescharacteristic
16#16840331—24no—
17#16840331C418yescomplemented
18C2×#14519774C2212yes—
19C2×#14519765C2212yes—
20C2×#14519752—22no—
21C2×#14519747—22no—
22#16839766—24no—
23#16839761—24no—
24#16839756—24no—
25C2×#14519700C2211yescharacteristic, complemented
26C2×#14519700C2212yescomplemented
27C2×#14519700C2211yescharacteristic, complemented
28C2×#14519697—24no—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2211yescharacteristic, complemented
34C2×#14519697C2212yescomplemented
35C2×#14519697C2211yescharacteristic, complemented
36#16839718—24no—
37C2×#14519688C2212yescomplemented
38C2×C4×#92418C2212yescomplemented
39C2×#117×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#389—22no—
43C61×#385—22no—
44C61×#384C2212yescomplemented
45C61×#382C2212yescomplemented
46C24×D61—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222C2×C414yes—
53C22×Dic122C2311yescharacteristic
54C22×D244—22no—
55C22×D244—22no—
56C22×D244—22no—
57C22×D244—22no—
58C22×D244—21no—
59C22×D244—22no—
60C22×D244—24no—
61C22×D244C2×C414yes—
62C22×D244C2312yes—
63C22×D244C2311yescharacteristic
64C22×C22⋊D61—21no—
65C22×C4×D61—21no—
66C2×#11758194—22no—
67C2×#11758194—22no—
68C2×#11758194D414yescomplemented
69C22×#92418—22no—
70C22×#92418—22no—
71C22×#92418D414yescomplemented
72C22×#92418C2311yescharacteristic
73C22×#92418C2312yes—
74C22×#92418C2311yescharacteristic
75#14519774—24no—
76#14519774C2×C418yescomplemented
77#14519765—24no—
78#14519765C2×C418yescomplemented
79#14519752—216no—
80#14519747—216no—
81C2×#11758198—24no—
82#14519700—22no—
83#14519700—22no—
84#14519700—24no—
85#14519697—24no—
86#14519697—24no—
87#14519697—24no—
88#14519697—24no—
89#14519697—22no—
90#14519697—22no—
91#14519697—24no—
92#14519688—24no—
93C4×#92418—24no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96#117×C61—22no—
97#117×C61—22no—
98#117×C61C2×C414yes—
99#117×C61D414yes—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—24no—
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61D412yes—
110C2×C42×C61—21no—
111C2×C42×C61C2312yes—
112#95×C61—24no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C23×D61—22no—
116C23×D61—24no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—82no—
120C23×D61—82no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—82no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D414yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C22×Dic61—21no—
139#11758222—28no—
140#11758222—28no—
141#11758222—44no—
142#11758222—44no—
143C2×D244—28no—
144C2×D244—28no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—24no—
154C2×D244—22no—
155C2×D244—21no—
156C2×D244—216no—
157C2×D244—44no—
158C2×D244—44no—
159C2×D244—42no—
160C2×D244—42no—
161C2×D244—84no—
162C2×D244—84no—
163C2×D244C22×C414yes—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61—84no—
182C4⋊C4×C61C22⋊C418yes—
183C22⋊C4×C61—42no—
184C22⋊C4×C61C22⋊C418yes—
185C42×C61—42no—
186C42×C61C22×C414yes—
187C42×C61C2×D414yes—
188#11758198—216no—
189#11758194—28no—
190C2×#92418—28no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—22no—
197C2×#92418C2×D412yes—
198C2×#92418C2×D412yes—
199C2×#92418C2×D414yes—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61C2×C22⋊C412yes—
203C23×C61#12412yes—
204C23×C61#12712yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—24no—
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—42no—
210C22×D61—41no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—44no—
214C22×D61—42no—
215C22×D61—44no—
216C22×D61—48no—
217C22×D61—88no—
218C22×D61—88no—
219C22×D61—84no—
220C22×D61—84no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—44no—
228C2×C4×C61—42no—
229C2×C4×C61—84no—
230C2×C4×C61C2×C22⋊C418yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C4×D414yes—
233C2×C4×C61C4×D414yes—
234C2×C4×C61#12114yes—
235C2×C4×C61#12114yes—
236C2×C4×C61#12412yes—
237C2×C4×C61#12712yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C22⋊D61—48no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248D244—24no—
249D244—24no—
250D244—22no—
251D244—22no—
252D244—42no—
253D244—42no—
254D244—41no—
255D244—41no—
256D244—48no—
257D244—816no—
258C4×D61—48no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263#92418—22no—
264#92418—22no—
265#92418—24no—
266#21206—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61#39014yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61C2×#12411yescharacteristic
274C22×C61C2×#12711yescharacteristic
275C22×C61#57612yes—
276C22×C61#57712yes—
277C2×D61—44no—
278C2×D61—48no—
279C2×D61—42no—
280C2×D61—44no—
281C2×D61—816no—
282C2×D61—82no—
283C2×D61—84no—
284C2×D61—81no—
285C2×D61—82no—
286C2×D61—81no—
287C2×D61—82no—
288C4×C61—28no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61—82no—
292C4×C61#39014yes—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297Dic61—42no—
298C2×#535—611no—
299C2×#417—611no—
300C2×#418—611no—
301C2×#416—611no—
302C22×#102—611no—
303C2×#405—611no—
304#2317D6111yescharacteristic, complemented
305#1961—612no—
306#1959—612no—
307#1958—612no—
308#1957—612no—
309C2×C61—24no—
310C2×C61C2×#39011yescharacteristic
311C2×C61#289712yes—
312C2×C61#313612yes—
313C2×C61#315411yescharacteristic
314C2×C61#315511yescharacteristic
315D61—82no—
316D61—84no—
317C22×#41—611no—
318C23×D4—611no—
319C22×M16—611no—
320C23×C8—611no—
321#535—618no—
322#535—1224no—
323C2×#127—612no—
324C2×#124—612no—
325C2×#121—1222no—
326C2×#120—1222no—
327C2×#117C2×D6111yescharacteristic
328C2×#117C2×D6111yescharacteristic
329C22×C4⋊C4C2×D6111yescharacteristic
330#418—1224no—
331#417—1224no—
332#416—1224no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#102—612no—
343C2×#102—1224no—
344#405—1224no—
345C2×#97—612no—
346C2×C4×C8—612no—
347#389—22no—
348#385—22no—
349#384C2×D6112yescomplemented
350#382C2×D6112yescomplemented
351C61#2120611yescharacteristic, complemented, Sylow
352C25—1221no—
353C2×#41—614no—
354C2×#41—1222no—
355C2×#41—1222no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—1222no—
359C22×Q8—611no—
360C22×D4—614no—
361C22×D4—612no—
362C22×D4—611no—
363C22×D4—1221no—
364C22×D4—1222no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1222no—
369C22×D4—1224no—
370C22×D4—1221no—
371C23×C4—21no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C22×C8—614no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—612no—
381C22×C8—1222no—
382C22×C8—1222no—
383#127—618no—
384#127—1224no—
385#124—618no—
386#124—1224no—
387#121—12216no—
388#120—12216no—
389#117—22no—
390#117—22no—
391#117C4×D6114yes—
392#117D24414yes—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4C22⋊D6112yes—
398C2×C4⋊C4C22×D6112yes—
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4C22⋊D6112yes—
404C2×C42—21no—
405C2×C42C22×D6112yes—
406#103—1222no—
407#103—1222no—
408#103—1224no—
409#102—1224no—
410#102—1224no—
411#102—1222no—
412#102—1222no—
413#102—1224no—
414#102—1224no—
415#102—1224no—
416#97—1224no—
417C4×C8—1224no—
418#95—24no—
419C24—1224no—
420C24—1222no—
421C24—1221no—
422C24—4882no—
423C24—4882no—
424C24—2442no—
425C24—2442no—
426C24C2×C22⋊D6111yescharacteristic
427#41—1228no—
428#41—1228no—
429#41—2444no—
430#41—2444no—
431C2×Q8—612no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—614no—
438C2×D4—612no—
439C2×D4—1224no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1222no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—12216no—
451C2×D4—1224no—
452C2×D4—1228no—
453C2×D4—1228no—
454C2×D4—4884no—
455C2×D4—4884no—
456C2×D4—2442no—
457C2×D4—2442no—
458C2×D4—2444no—
459C2×D4—2444no—
460C2×D4—2442no—
461C2×D4—2442no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—82no—
467C22×C4—1221no—
468C22×C4—1224no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×D24412yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6112yes—
475C22×C4#1175822612yes—
476C22×C4#1175822612yes—
477C22×C4C2×C22⋊D6114yes—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480M16—1228no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—614no—
484C2×C8—1221no—
485C2×C8—1222no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1222no—
490C2×C8—1228no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—84no—
499C4⋊C4#1175819818yes—
500C22⋊C4—42no—
501C22⋊C4—12216no—
502C22⋊C4#1175819818yes—
503C42—42no—
504C42C2×C4×D6114yes—
505C42C2×D24414yes—
506C23—22no—
507C23—22no—
508C23—1224no—
509C23—1222no—
510C23—4888no—
511C23—4884no—
512C23—4884no—
513C23—4888no—
514C23—2441no—
515C23—2442no—
516C23—2441no—
517C23—2442no—
518C23—2442no—
519C23—2444no—
520C23—2442no—
521C23—2444no—
522C23—2448no—
523C23C2×#1175819812yes—
524C23#1451976612yes—
525C23#1451977312yes—
526C23C22×C22⋊D6111yescharacteristic
527Q8—1222no—
528Q8—1222no—
529Q8—2441no—
530Q8—2441no—
531D4—1224no—
532D4—1224no—
533D4—1222no—
534D4—1222no—
535D4—48816no—
536D4—2448no—
537D4—2442no—
538D4—2442no—
539D4—2441no—
540D4—2441no—
541D4—2448no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—1222no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2448no—
555C2×C4C2×#1175819818yes—
556C2×C4C2×#1175819812yes—
557C2×C4#1451973914yes—
558C2×C4#1451973914yes—
559C2×C4#1451975214yes—
560C2×C4#1451975214yes—
561C2×C4#1451976612yes—
562C2×C4#1451977312yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C22×D24411yescharacteristic
567C2×C4C2×#1175822611yescharacteristic
568C2×C4C2×#1175822611yescharacteristic
569C8—1222no—
570C8—1222no—
571C8—1224no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—48816no—
577C22—4884no—
578C22—4882no—
579C22—4882no—
580C22—4881no—
581C22—4882no—
582C22—4881no—
583C22—2444no—
584C22—2448no—
585C22—2442no—
586C22—2444no—
587C22#1683966314yes—
588C22C22×#1175819811yescharacteristic
589C22C2×#1451976611yescharacteristic
590C22C2×#1451977311yescharacteristic
591C22#1684056512yes—
592C22#1684057012yes—
593C4—28no—
594C4—81no—
595C4—81no—
596C4—82no—
597C4—2442no—
598C4#1683966314yes—
599C4C2×#1451973911yescharacteristic
600C4C2×#1451973911yescharacteristic
601C4C2×#1451975211yescharacteristic
602C4C2×#1451975211yescharacteristic
603C2—24no—
604C2—4884no—
605C2—4882no—
606C2C2×#1683966311yescharacteristic
607C2#2145434212yes—
608C2#2145617812yes—
609C2#2145629711yescharacteristic
610C2#2145630211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
cosocleC22×C4×C61C24
hypercenter#2317D61
hypocenterC61#21206
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#16839761C2semidirect
3. C2×#16839766C2semidirect
4. C2×#16839756C2semidirect
5. C22×#14519697C2semidirect
6. C2×#16839718C2semidirect
7. #21446642C2semidirect
8. #21446633C2semidirect
9. #21446628C2semidirect
10. #21446624C2semidirect
11. C61×#2317C2semidirect
17. #16840331C4semidirect
25. C2×#14519700C22semidirect
26. C2×#14519700C22semidirect
27. C2×#14519700C22semidirect
33. C2×#14519697C22semidirect
34. C2×#14519697C22semidirect
35. C2×#14519697C22semidirect
37. C2×#14519688C22semidirect
38. C2×C4×#92418C22semidirect
44. C61×#384C22semidirect
45. C61×#382C22semidirect
68. C2×#11758194D4semidirect
71. C22×#92418D4semidirect
76. #14519774C2×C4semidirect
78. #14519765C2×C4semidirect
304. #2317D61semidirect
349. #384C2×D61semidirect
350. #382C2×D61semidirect
351. C61#21206semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168397612
C2×#168397662
C2×#168397562
C22×#145196972
C2×#168397182
#214466422
#214466332
#214466282
#214466242
C61×#23172
#2120661