Finite Group Databasebeta

← #29689377

Subgroups — #29689377

Order 15616.

1354 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840311C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446752C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446641C212yesmaximal, complemented
8#21446637C212yesmaximal, complemented
9#21445758C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14#16840311—24no—
15#16840311C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519750—21no—
21C2×#14519737—21no—
22C2×#14519737—21no—
23C2×C42×D61C2211yescharacteristic
24#16839818—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×#14519687C2212yes—
34#16839629—21no—
35#16839627C2212yes—
36#16839624—21no—
37#16839619C2212yes—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×#11758198—21no—
40#16839608—21no—
41C2×#124×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#102×C61C2212yes—
45C4×C4⋊C4×C61C2211yescharacteristic
46C2×#11758226—22no—
47C2×#11758226—42no—
48C2×#11758226—42no—
49C2×#11758226C2×C414yes—
50C2×#11758224—22no—
51C2×#11758224—42no—
52C2×#11758224—42no—
53C2×#11758224C2×C414yes—
54C22×Dic122C2311yescharacteristic
55C22×D244—21no—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C23×Dic61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2312yes—
66#14519775—22no—
67#14519775C2×C414yes—
68#14519773—22no—
69#14519773C2×C414yes—
70#14519765—22no—
71#14519765C2×C414yes—
72#14519750—28no—
73#14519750—28no—
74#14519737—28no—
75#14519737—28no—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519702—42no—
90#14519701—42no—
91#14519700—42no—
92#14519699—42no—
93#14519698—42no—
94#14519697—42no—
95#14519689—24no—
96#14519687—24no—
97#14519684—21no—
98#14519684—22no—
99#14519684—22no—
100#14519684—22no—
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C22×C8×C61C2312yes—
104#124×C61—22no—
105#124×C61C2×C414yes—
106C2×C4⋊C4×C61C2312yes—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C42×C61—21no—
110C2×C42×C61C2311yescharacteristic
111C4⋊C8×C61—22no—
112#103×C61—42no—
113#102×C61—42no—
114#95×C61—21no—
115C24×C61—21no—
116C23×D61—21no—
117C23×D61C2×D411yescharacteristic
118C2×Q8×C61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61C22×C412yes—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C22×C412yes—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—44no—
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145#11758226—48no—
146#11758226—44no—
147#11758226—44no—
148#11758224—48no—
149#11758224—44no—
150#11758224—44no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—24no—
154C2×D244—21no—
155C2×D244—44no—
156C2×D244C22×C412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122C22×C412yes—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C42×C61—42no—
180C42×C61C22×C412yes—
181C42×C61C2×D412yes—
182#11758198—24no—
183#11758198—24no—
184#11758198—44no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187#11758196—24no—
188#11758196—24no—
189C4×Dic61—42no—
190C4×Dic61C22⋊C414yes—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22×C412yes—
193C4×Dic61C2×D412yes—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—22no—
208C22×D61—42no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—41no—
212C22×D61C2×C22⋊C412yes—
213Q8×C61—24no—
214Q8×C61—41no—
215Q8×C61—41no—
216D4×C61—24no—
217D4×C61—41no—
218D4×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61H(5,𝔽2)12yes—
232C2×C4×C61H(5,𝔽2)12yes—
233C22⋊D61—48no—
234C22⋊D61—48no—
235C22⋊D61—48no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—48no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—24no—
250D244—48no—
251D244—41no—
252D244—41no—
253C4×D61—24no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61—48no—
259Dic122—24no—
260Dic122—41no—
261Dic122—41no—
262C8×C61—42no—
263#92418—42no—
264#21812—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—82no—
269C22×C61—82no—
270C22×C61C22×C22⋊C411yescharacteristic
271C22×C61#53512yes—
272C22×C61#57511yescharacteristic
273C22×C61#57512yes—
274C22×C61#57511yescharacteristic
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—82no—
284C2×D61—82no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—82no—
289C4×C61#51812yes—
290C4×C61#53511yescharacteristic
291C4×C61#53511yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—22no—
295Dic61—42no—
296Dic61—82no—
297C2×#532—611no—
298C2×#432—611no—
299#2283—611no—
300#1983—611no—
301#1983—611no—
302#1983—611no—
303#1983D6111yescharacteristic, complemented
304#1961—612no—
305#1960—612no—
306#1788—612no—
307#1788—612no—
308C2×C61—82no—
309C2×C61#230911yescharacteristic
310C2×C61#289611yescharacteristic
311C2×C61#289611yescharacteristic
312C2×C61#321912yes—
313C2×C61#321912yes—
314D61—42no—
315D61—82no—
316C22×#41—611no—
317C22×#41—611no—
318#532—618no—
319#532—1224no—
320C2×C4⋊Q8—611no—
321C2×#127—611no—
322C2×#124—611no—
323C2×#124C2×D6111yescharacteristic
324C2×#121—1221no—
325C2×#121—1221no—
326C2×C4×D4—1221no—
327C2×C4×D4—1221no—
328C22×C42—611no—
329#432—1224no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8C2×D6111yescharacteristic
332C2×#103—612no—
333C2×#103—612no—
334C2×#103—612no—
335C2×#103C2×D6112yes—
336C2×#102—612no—
337C2×#102—612no—
338C2×#102—612no—
339C2×#102C2×D6112yes—
340C2×C22⋊C8—612no—
341C2×#97—612no—
342#383—1221no—
343#382—612no—
344#381—1221no—
345#380—612no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4C2×D6111yescharacteristic
348C4×C22⋊C4—1221no—
349#376—1221no—
350C61#2181211yescharacteristic, complemented, Sylow
351C2×#41—614no—
352C2×#41—614no—
353C2×#41—1222no—
354C2×#41—1222no—
355C2×#41—2442no—
356C2×#41—2442no—
357C2×#41—2442no—
358C2×#41—2442no—
359C22×Q8—611no—
360C22×Q8C22×D6111yescharacteristic
361C22×D4—611no—
362C22×D4—1221no—
363C22×D4—1221no—
364C22×D4—1221no—
365C22×D4—1221no—
366C22×D4C22×D6111yescharacteristic
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—1221no—
371C23×C4—1221no—
372C22×C8—612no—
373C22×C8C22×D6112yes—
374C4⋊Q8—614no—
375C4⋊Q8—1222no—
376#127—614no—
377#127—1222no—
378#124—22no—
379#124—614no—
380#124—1222no—
381#124C4×D6114yes—
382#121—1228no—
383#121—1228no—
384C4×D4—1228no—
385C4×D4—1228no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6112yes—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1222no—
395C2×C42—21no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—614no—
400C2×C42—614no—
401C2×C42—1221no—
402C2×C42C22×D6111yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—1222no—
405#103—42no—
406#103—2442no—
407#103—2442no—
408#103—2442no—
409#102—42no—
410#102—2442no—
411#102—2442no—
412#102—2442no—
413C22⋊C8—1224no—
414#97—1224no—
415#95—21no—
416#95—1222no—
417#95—1222no—
418#95—1222no—
419#95—1221no—
420C24—21no—
421C24—611no—
422C24—1221no—
423#41—2448no—
424#41—2444no—
425#41—2444no—
426#41—2448no—
427#41—2444no—
428#41—2444no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—612no—
433C2×Q8—1221no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8C2×C4×D6112yes—
437C2×D4—21no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—612no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—2444no—
449C2×D4—2442no—
450C2×D4—2444no—
451C2×D4—2442no—
452C2×D4—2444no—
453C2×D4C2×C4×D6112yes—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—614no—
462C22×C4—614no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—2441no—
470C22×C4—2442no—
471C22×C4—2441no—
472C22×C4—2444no—
473C22×C4—2444no—
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—1224no—
492C22⋊C4—1224no—
493C22⋊C4—2444no—
494C42—42no—
495C42—612no—
496C42—612no—
497C42—614no—
498C42—614no—
499C42—2442no—
500C42C2×C4×D6112yes—
501C42D4×D6112yes—
502C23—22no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—612no—
508C23—1221no—
509C23—1222no—
510C23—1221no—
511C23—1222no—
512C23—1222no—
513C23—2441no—
514C23—2442no—
515C23—2441no—
516C23—2442no—
517C23C2×D4×D6111yescharacteristic
518Q8—24no—
519Q8—41no—
520Q8—41no—
521Q8—1224no—
522Q8—2441no—
523Q8—2441no—
524D4—24no—
525D4—41no—
526D4—41no—
527D4—1224no—
528D4—2448no—
529D4—2448no—
530D4—2448no—
531D4—2448no—
532D4—2441no—
533D4—2441no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2448no—
556C2×C4—2448no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4C22⋊C4×D6112yes—
562C2×C4C22⋊C4×D6112yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#1451982212yes—
566C2×C4#1451982312yes—
567C8—42no—
568C8—2442no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—82no—
573C22—82no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—4882no—
578C22—4882no—
579C22—2442no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22—2442no—
584C22C2×C22⋊C4×D6111yescharacteristic
585C22#1684032812yes—
586C22#1684055811yescharacteristic
587C22#1684055912yes—
588C22#1684056111yescharacteristic
589C4—22no—
590C4—22no—
591C4—42no—
592C4—82no—
593C4—1222no—
594C4—1222no—
595C4—1222no—
596C4—4882no—
597C4—2442no—
598C4#1684023412yes—
599C4#1684032511yescharacteristic
600C4#1684033011yescharacteristic
601C2—82no—
602C2—4882no—
603C2—2442no—
604C2#2144839811yescharacteristic
605C2#2145433411yescharacteristic
606C2#2145433511yescharacteristic
607C2#2145685512yes—
608C2#2145686112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC2×#16840311C2
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21812
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839818C2semidirect
3. #21448215C2semidirect
4. #21446752C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
7. #21446641C2semidirect
8. #21446637C2semidirect
9. #21445758C2semidirect
10. #21445756C2semidirect
11. C61×#1983C2semidirect
15. #16840311C4semidirect
303. #1983D61semidirect
350. C61#21812semidirect

Maximal subgroups

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

subgroupindex
C2×#168403112
C2×#168398182
#214482152
#214467522
#214467512
#214467492
#214466412
#214466372
#214457582
#214457562
C61×#19832
#2181261