Finite Group Databasebeta

← #29683727

Subgroups — #29683727

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×#16839764C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic, complemented
5C22×#14519699C211yesmaximal, characteristic, complemented
6#21448442C211yesmaximal, characteristic, complemented
7#21446643C212yesmaximal, complemented
8#21446635C212yesmaximal, complemented
9#21446630C212yesmaximal, complemented
10#21446625C212yesmaximal, complemented
11C2×C61×#405C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14#16840331—24no—
15#16840331C418yescomplemented
16C2×#14519774C2212yes—
17C2×#14519765C2212yes—
18C2×#14519752—22no—
19C2×#14519747—22no—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22#16839768—24no—
23#16839764—24no—
24#16839759—24no—
25C2×#14519702C2211yescharacteristic, complemented
26C2×#14519702C2212yescomplemented
27C2×#14519702C2211yescharacteristic, complemented
28C2×#14519699—24no—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31C2×#14519699C2212yes—
32C2×#14519699C2212yes—
33C2×#14519699C2211yescharacteristic, complemented
34C2×#14519699C2212yescomplemented
35C2×#14519699C2211yescharacteristic, complemented
36#16839661—22no—
37#16839637—22no—
38#16839632C2212yescomplemented
39#16839628C2212yescomplemented
40C22×M16×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C61×#405—24no—
44C2×#97×C61C2212yescomplemented
45C2×C4×C8×C61C2212yescomplemented
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—
65C23×Dic61—21no—
66C22×C4×D61—21no—
67#14519774—24no—
68#14519774C2×C418yescomplemented
69#14519765—24no—
70#14519765C2×C418yescomplemented
71#14519752—216no—
72#14519747—216no—
73#14519736—22no—
74#14519736—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—24no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×#11758198—24no—
84C2×C4×Dic61—21no—
85#14519702—22no—
86#14519702—22no—
87#14519702—24no—
88#14519699—24no—
89#14519699—24no—
90#14519699—24no—
91#14519699—24no—
92#14519699—22no—
93#14519699—22no—
94#14519699—24no—
95#14519684—24no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61D414yescomplemented
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61D414yescomplemented
103C22×C8×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106#117×C61C2×C414yes—
107#117×C61D414yes—
108C2×C4⋊C4×C61D412yes—
109C2×C22⋊C4×C61D412yes—
110C2×C42×C61C2312yes—
111#97×C61—24no—
112C4×C8×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×C61C22×C412yes—
122C22×C4×C61C2×D414yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61#4112yes—
128C22×C4×C61#4112yes—
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—42no—
138C22×Dic61—82no—
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—
174M16×C61—28no—
175C2×C8×C61—28no—
176C2×C8×C61—21no—
177C2×C8×C61—22no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—22no—
182C2×C8×C61C2×D412yes—
183C2×C8×C61C2×D412yes—
184C2×C8×C61C2×D414yes—
185C4⋊C4×C61C22⋊C418yes—
186C22⋊C4×C61C22⋊C418yes—
187C42×C61C22×C414yes—
188C42×C61C2×D414yes—
189C22⋊Dic61—42no—
190#11758198—216no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—84no—
199C4×Dic61—42no—
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×C61C2×C22⋊C418yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C4×D414yes—
226C2×C4×C61C4×D414yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12114yes—
229C2×C4×C61#12412yes—
230C2×C4×C61#12712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—48no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—44no—
246C2×Dic61—42no—
247C2×Dic61—84no—
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—
263C8×C61—22no—
264C8×C61—22no—
265C8×C61—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#39014yes—
290C4×C61C2×C4×D411yescharacteristic
291C4×C61C2×C4×D411yescharacteristic
292C4×C61C2×#12111yescharacteristic
293C4×C61C2×#12111yescharacteristic
294Dic61—42no—
295Dic61—81no—
296Dic61—81no—
297Dic61—82no—
298C2×#535—611no—
299C2×#417—611no—
300C2×#418—611no—
301C2×#416—611no—
302C22×#102—611no—
303C2×#405D6111yescharacteristic, complemented
304#2317—611no—
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×M16C2×D6111yescharacteristic
320C23×C8C2×D6111yescharacteristic
321#535—618no—
322#535—1224no—
323C2×#127—612no—
324C2×#124—612no—
325C2×#121—1222no—
326C2×#120—1222no—
327C2×#117—611no—
328C2×#117C2×D6111yescharacteristic
329C22×C4⋊C4—611no—
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—24no—
345C2×#97C2×D6112yescomplemented
346C2×C4×C8C2×D6112yescomplemented
347#389—1222no—
348#385—1222no—
349#384—612no—
350#382—612no—
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—1224no—
367C22×D4—1221no—
368C22×D4—1222no—
369C22×D4—1222no—
370C22×D4—1222no—
371C23×C4—1221no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16D24414yescomplemented
377C22×C8—22no—
378C22×C8—22no—
379C22×C8D24414yescomplemented
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6112yes—
383#127—618no—
384#127—1224no—
385#124—618no—
386#124—1224no—
387#121—12216no—
388#120—12216no—
389#117—1222no—
390#117—1222no—
391#117C4×D6114yes—
392#117C22⋊D6114yes—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1224no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4D24412yes—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4D24412yes—
404C2×C42—1221no—
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—24no—
417C4×C8—24no—
418#95—1224no—
419C24—1224no—
420C24—1222no—
421C24—1221no—
422C24—4882no—
423C24—4882no—
424C24—2442no—
425C24—2442no—
426C24C2×D24411yescharacteristic
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—1221no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1221no—
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—1221no—
463C22×C4—1224no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—4882no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×D24414yes—
473C22×C4C2×D24412yes—
474C22×C4C2×D24412yes—
475C22×C4C2×D24411yescharacteristic
476C22×C4#1175822212yes—
477C22×C4#1175822212yes—
478C22×C4C2×C22⋊D6112yes—
479C22×C4C23×D6111yescharacteristic
480M16—28no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—28no—
488C2×C8C2×D24412yes—
489C2×C8C2×D24412yes—
490C2×C8C2×D24414yes—
491C4⋊C4—1222no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—4884no—
498C4⋊C4—2442no—
499C4⋊C4#1175819818yes—
500C22⋊C4—12216no—
501C22⋊C4—2442no—
502C22⋊C4#1175819818yes—
503C42—2442no—
504C42C2×C4×D6114yes—
505C42C2×C22⋊D6114yes—
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#1451976512yes—
525C23#1451977412yes—
526C23C22×D24411yescharacteristic
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—1222no—
545C2×C4—4884no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2448no—
555C2×C4C2×#1175819818yes—
556C2×C4C2×#1175819812yes—
557C2×C4C4×D24414yes—
558C2×C4C4×D24414yes—
559C2×C4#1451975114yes—
560C2×C4#1451975114yes—
561C2×C4#1451976512yes—
562C2×C4#1451977412yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C22×D24411yescharacteristic
566C2×C4C22×D24411yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4C2×#1175822211yescharacteristic
569C8—22no—
570C8—22no—
571C8—24no—
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#1683966414yes—
588C22C22×#1175819811yescharacteristic
589C22C2×#1451976511yescharacteristic
590C22C2×#1451977411yescharacteristic
591C22#1684056712yes—
592C22#1684057412yes—
593C4—28no—
594C4—4881no—
595C4—4882no—
596C4—4881no—
597C4—2442no—
598C4#1683966414yes—
599C4C2×C4×D24411yescharacteristic
600C4C2×C4×D24411yescharacteristic
601C4C2×#1451975111yescharacteristic
602C4C2×#1451975111yescharacteristic
603C2—24no—
604C2—4884no—
605C2—4882no—
606C2C2×#1683966411yescharacteristic
607C2#2145434612yes—
608C2#2145617512yes—
609C2#2145629511yescharacteristic
610C2#2145630511yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#405C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
cosocleC22×C4×C61C24
hypercenterC2×#405D61
hypocenterC61#21206
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. C2×#16839764C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839759C2semidirect
5. C22×#14519699C2semidirect
6. #21448442C2semidirect
7. #21446643C2semidirect
8. #21446635C2semidirect
9. #21446630C2semidirect
10. #21446625C2semidirect
11. C2×C61×#405C2semidirect
15. #16840331C4semidirect
25. C2×#14519702C22semidirect
26. C2×#14519702C22semidirect
27. C2×#14519702C22semidirect
33. C2×#14519699C22semidirect
34. C2×#14519699C22semidirect
35. C2×#14519699C22semidirect
38. #16839632C22semidirect
39. #16839628C22semidirect
44. C2×#97×C61C22semidirect
45. C2×C4×C8×C61C22semidirect
68. #14519774C2×C4semidirect
70. #14519765C2×C4semidirect
99. C2×M16×C61D4semidirect
102. C22×C8×C61D4semidirect
303. C2×#405D61semidirect
345. C2×#97C2×D61semidirect
346. C2×C4×C8C2×D61semidirect
351. C61#21206semidirect
376. C2×M16D244semidirect
379. C22×C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168397642
C2×#168397682
C2×#168397592
C22×#145196992
#214484422
#214466432
#214466352
#214466302
#214466252
C2×C61×#4052
#2120661