Finite Group Databasebeta

← #29689333

Subgroups — #29689333

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D61C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21446750C212yesmaximal, complemented
5#21446748C212yesmaximal, complemented
6#21446631C211yesmaximal, characteristic, complemented
7#21446630C211yesmaximal, characteristic, complemented
8#21446628C211yesmaximal, characteristic, complemented
9#21445752C212yesmaximal, complemented
10#21445748C212yesmaximal
11C61×#1958C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C2×#14519766C2211yescharacteristic
15C2×#14519765C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519763—21no—
19#124×D61—24no—
20#124×D61C418yescomplemented
21C2×C22⋊C4×D61—21no—
22C2×C22⋊C4×D61—21no—
23C2×C42×D61C2211yescharacteristic
24#16839813—24no—
25C2×#14519706C2212yescomplemented
26C2×#14519702C2212yescomplemented
27C2×#14519701C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic, complemented
34#16839629—21no—
35#16839628C2211yescharacteristic, complemented
36#16839624—21no—
37#16839619C2212yes—
38C4⋊C4×Dic61C2212yescomplemented
39#16839609—21no—
40#16839607—21no—
41C2×#124×C61C2211yescharacteristic
42C2×#103×C61C2212yescomplemented
43C2×#102×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic, complemented
45C61×#382C2211yescharacteristic, complemented
46C24×D61—21no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yes—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C2×D4×D61—22no—
56C2×D4×D61—42no—
57C2×D4×D61—42no—
58C2×D4×D61C2×C414yes—
59C22×C4×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519766—22no—
65#14519766C2×C414yes—
66#14519765—22no—
67#14519765C2×C414yes—
68#14519764—22no—
69#14519764C2×C414yes—
70#14519763—28no—
71#14519763—28no—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C22⋊C4×D61—28no—
81C22⋊C4×D61—28no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88#14519706—24no—
89#14519702—42no—
90#14519701—42no—
91#14519700—42no—
92#14519699—42no—
93#14519698—42no—
94#14519697—42no—
95#14519689—24no—
96C4×#92418—22no—
97#14519684—21no—
98#14519684—21no—
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×C61C2311yescharacteristic
110#103×C61—42no—
111#102×C61—42no—
112C4×C8×C61—22no—
113#95×C61—21no—
114#95×C61—21no—
115C24×C61—21no—
116C23×D61—21no—
117C23×D61—24no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—44no—
121C23×D61—44no—
122C23×D61C2×D411yescharacteristic
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61C22×C412yes—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61C22×C412yes—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—48no—
145Q8×D61—48no—
146D4×D61—48no—
147D4×D61—48no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61C22⋊C414yes—
160C2×C4×D61C22⋊C414yes—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C2×D412yes—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—21no—
166C2×Dic122—24no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122C22×C412yes—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C42×C61C22×C412yes—
178C42×C61C2×D412yes—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181#11758198—24no—
182#11758198—24no—
183#11758198—24no—
184#11758198—24no—
185#11758198—44no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4×Dic61—44no—
189C4×Dic61C22⋊C414yes—
190C4×Dic61C22⋊C414yes—
191C4×Dic61C22×C412yes—
192C4×Dic61C2×D412yes—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C22×D411yescharacteristic
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—48no—
207C22×D61—48no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—48no—
213C22×D61—88no—
214C22×D61—88no—
215C22×D61C2×C22⋊C412yes—
216C22×D61#13514yes—
217Q8×C61—24no—
218Q8×C61—42no—
219D4×C61—24no—
220D4×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C412yes—
230C2×C4×C61C2×C22⋊C412yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61H(5,𝔽2)12yes—
234C2×C4×C61H(5,𝔽2)12yes—
235C22⋊D61—48no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—24no—
250D244—42no—
251C4×D61—28no—
252C4×D61—24no—
253C4×D61—24no—
254C4×D61—44no—
255C4×D61—48no—
256Dic122—24no—
257Dic122—48no—
258Dic122—42no—
259C8×C61—42no—
260#92418—42no—
261#21806—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61#53511yescharacteristic
269C22×C61#53511yescharacteristic
270C22×C61C2×#13512yes—
271C22×C61#57512yes—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—24no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—816no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61#41614yes—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—82no—
287C4×C61#51812yes—
288C4×C61#53512yes—
289Dic61—22no—
290Dic61—22no—
291Dic61—22no—
292Dic61—42no—
293Dic61—82no—
294C22×#124—611no—
295C2×#431—611no—
296#2283—611no—
297#1983—612no—
298#1983—612no—
299#1958—611no—
300#1958—611no—
301#1958—611no—
302#1958D6111yescharacteristic, complemented
303#1787—612no—
304#1786—612no—
305C2×C61—82no—
306C2×C61#230911yescharacteristic
307C2×C61C2×#41611yescharacteristic
308C2×C61#289611yescharacteristic
309C2×C61#321112yes—
310C2×C61#321112yes—
311D61—24no—
312D61—82no—
313C23×Q8—611no—
314C23×D4—611no—
315C2×#124—611no—
316C2×#124—611no—
317C2×#124—618no—
318C2×#124—611no—
319C2×#124—1221no—
320C2×#124—1221no—
321C2×#124—1224no—
322C2×#124C2×D6111yescharacteristic
323C22×C22⋊C4—1221no—
324C22×C22⋊C4—1221no—
325C22×C42—611no—
326#431—1224no—
327C2×C4⋊C8—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103C2×D6112yescomplemented
332C2×#102—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102C2×D6112yes—
336C2×C22⋊C8—612no—
337C2×C4×C8—611no—
338C2×C4×C8C2×D6111yescharacteristic, complemented
339#383—1221no—
340#382—611no—
341#382C2×D6111yescharacteristic, complemented
342#381—1221no—
343#380—612no—
344C4×C4⋊C4—612no—
345C4×C22⋊C4—1221no—
346#376—1221no—
347C61#2180611yescharacteristic, complemented, Sylow
348C25—1221no—
349C22×Q8—614no—
350C22×Q8—611no—
351C22×Q8—1221no—
352C22×Q8—1222no—
353C22×Q8—2442no—
354C22×Q8—2442no—
355C22×Q8C22×D6111yescharacteristic
356C22×D4—614no—
357C22×D4—611no—
358C22×D4—1222no—
359C22×D4—1221no—
360C22×D4—2442no—
361C22×D4—2442no—
362C22×D4C22×D6111yescharacteristic
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1221no—
367C22×C8—612no—
368C22×C8C22×D6112yes—
369#124—22no—
370#124—614no—
371#124—614no—
372#124—614no—
373#124—1228no—
374#124—1228no—
375#124—1222no—
376#124—1222no—
377#124—1222no—
378#124C4×D6114yes—
379C2×C4⋊C4—612no—
380C2×C4⋊C4C22×D6112yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1228no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1222no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—614no—
396C2×C42—614no—
397C2×C42—1222no—
398C2×C42C22×D6111yescharacteristic
399C4⋊C8—1224no—
400#103—42no—
401#103—2442no—
402#103—2442no—
403#103—2442no—
404#102—42no—
405#102—2442no—
406#102—2442no—
407#102—2442no—
408C22⋊C8—1224no—
409C4×C8—22no—
410C4×C8—1222no—
411#95—21no—
412#95—21no—
413#95—1222no—
414#95—1222no—
415#95—1221no—
416#95—1221no—
417C24—21no—
418C24—611no—
419C24—1224no—
420C24—1221no—
421C24—2444no—
422C24—2442no—
423C24—2442no—
424C24—2444no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—612no—
429C2×Q8—1224no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×Q8—2448no—
436C2×Q8—2448no—
437C2×Q8C2×C4×D6112yes—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—612no—
442C2×D4—1224no—
443C2×D4—1221no—
444C2×D4—1222no—
445C2×D4—1222no—
446C2×D4—2448no—
447C2×D4—2448no—
448C2×D4—2442no—
449C2×D4—2442no—
450C2×D4C2×C4×D6112yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—614no—
460C22×C4—1224no—
461C22×C4—1224no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—2444no—
491C42—612no—
492C42—612no—
493C42—614no—
494C42—614no—
495C42—2444no—
496C42C2×C4×D6112yes—
497C42D4×D6112yes—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—612no—
504C23—614no—
505C23—1222no—
506C23—1221no—
507C23—1221no—
508C23—1222no—
509C23—4888no—
510C23—4888no—
511C23—2448no—
512C23—2448no—
513C23—2448no—
514C23—2442no—
515C23—2442no—
516C23—2441no—
517C23—2441no—
518C23C2×D4×D6111yescharacteristic
519Q8—24no—
520Q8—42no—
521Q8—1224no—
522Q8—2448no—
523Q8—2442no—
524D4—24no—
525D4—42no—
526D4—1224no—
527D4—2448no—
528D4—2442no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1228no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—2444no—
549C2×C4—2448no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C22⋊C4×D6112yes—
556C2×C4C22⋊C4×D6112yes—
557C2×C4C22×C4×D6111yescharacteristic
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4#1451982212yes—
560C2×C4#1451982312yes—
561C8—42no—
562C8—2442no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—614no—
569C22—1224no—
570C22—1224no—
571C22—1222no—
572C22—1222no—
573C22—48816no—
574C22—4882no—
575C22—4882no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22C2×C22⋊C4×D6111yescharacteristic
580C22#1684032511yescharacteristic
581C22#1684033011yescharacteristic
582C22#135×D6112yes—
583C22#1684055912yes—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—82no—
588C4—1222no—
589C4—1222no—
590C4—1222no—
591C4—4882no—
592C4—2442no—
593C4#1684023412yes—
594C4#1684032812yes—
595C2—82no—
596C2—1224no—
597C2—4882no—
598C2#2144839811yescharacteristic
599C2#416×D6111yescharacteristic
600C2#2145433411yescharacteristic
601C2#2145677912yes—
602C2#2145678512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1958D61
hypocenterC61#21806
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #21448212C2semidirect
4. #21446750C2semidirect
5. #21446748C2semidirect
6. #21446631C2semidirect
7. #21446630C2semidirect
8. #21446628C2semidirect
9. #21445752C2semidirect
11. C61×#1958C2semidirect
20. #124×D61C4semidirect
25. C2×#14519706C22semidirect
26. C2×#14519702C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519700C22semidirect
33. C2×C4×#92418C22semidirect
35. #16839628C22semidirect
38. C4⋊C4×Dic61C22semidirect
42. C2×#103×C61C22semidirect
44. C2×C4×C8×C61C22semidirect
45. C61×#382C22semidirect
302. #1958D61semidirect
331. C2×#103C2×D61semidirect
338. C2×C4×C8C2×D61semidirect
341. #382C2×D61semidirect
347. C61#21806semidirect

Maximal subgroups

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

subgroupindex
C2×#124×D612
C2×#168398132
#214482122
#214467502
#214467482
#214466312
#214466302
#214466282
#214457522
#214457482
C61×#19582
#2180661