Finite Group Databasebeta

← #29713922

Subgroups — #29713922

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic, complemented
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447012C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446136C212yesmaximal, complemented
9#21446012C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C61×#554C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×#14519748—21no—
15C2×C4×D244C2211yescharacteristic
16#16840049—24no—
17C4×#11758201—24no—
18C2×#14519708C2212yes—
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839678—22no—
29#16839632C2211yescharacteristic
30#16839622C2212yes—
31#16839619—21no—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#554—24no—
36C61×#554—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#125×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×D4×C61—21no—
43C2×C4×D4×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56#14519748—44no—
57C4×D244—22no—
58C4×D244—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×C22⋊Dic61—22no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66#14519708—24no—
67#14519708—22no—
68C4⋊#92418—22no—
69#14519701—22no—
70#14519701—42no—
71#14519697—24no—
72#14519697—22no—
73C22⋊#92418—24no—
74C22⋊#92418—42no—
75C4×#92418—22no—
76#14519684—22no—
77C22×Q8×C61—21no—
78C22×Q8×C61C2311yescharacteristic
79C22×D4×C61D411yescharacteristic
80C23×C4×C61—22no—
81C23×C4×C61C2312yes—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61D414yescomplemented
84#125×C61—28no—
85C22⋊Q8×C61—216no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88C4×Q8×C61—22no—
89C4×Q8×C61D414yescomplemented
90C4×D4×C61—28no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—24no—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×Q8×C61—24no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—81no—
115C2×Q8×C61—81no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61SD1614yescomplemented
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—82no—
131C22×C4×C61—82no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—44no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—82no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244#4112yes—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C4⋊C4×C61—28no—
151C4⋊C4×C61—28no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C2×D412yes—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—28no—
160C22⋊C4×C61—24no—
161C42×C61—24no—
162C42×C61C2×D412yes—
163C42×C61C2×D412yes—
164C22⋊Dic61—44no—
165#11758201—28no—
166#11758198—44no—
167#11758198—42no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61—81no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418—22no—
181C2×#92418—21no—
182C2×#92418#4112yes—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61C2×SD1614yes—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198Q8×C61—22no—
199Q8×C61—22no—
200Q8×C61—41no—
201Q8×C61—41no—
202Q8×C61—84no—
203D4×C61—28no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—88no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12012yes—
225C2×C4×C61C2×SD1612yes—
226C2×C4×C61#13712yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C22⋊D61—88no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240D244—22no—
241D244—22no—
242D244—41no—
243D244—41no—
244C4×D61—84no—
245#92418—22no—
246#92418—41no—
247#92418—42no—
248#24118—611nomaximal, Sylow
249C22×C61—24no—
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#54611yescharacteristic
258C22×C61#54711yescharacteristic
259C22×C61C22×SD1611yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—44no—
275C4×C61#44814yes—
276C4×C61#52511yescharacteristic
277C4×C61#53711yescharacteristic
278C4×C61#54811yescharacteristic
279C4×C61#55011yescharacteristic
280Dic61—81no—
281Dic61—82no—
282C2×#554D6111yescharacteristic, complemented
283C2×#476—611no—
284C2×C4×SD16—611no—
285#2537—611no—
286#2035—612no—
287#2000—612no—
288#1990—611no—
289#1954—611no—
290#1874—612no—
291#1853—612no—
292#1803—611no—
293C2×C61—44no—
294C2×C61#240711yescharacteristic
295C2×C61C2×#44811yescharacteristic
296C2×C61#301211yescharacteristic
297C2×C61#329511yescharacteristic
298C2×C61#333111yescharacteristic
299C2×C61#333511yescharacteristic
300C2×C61#335711yescharacteristic
301D61—82no—
302C22×SD16—611no—
303#554—24no—
304#554—24no—
305C2×C4⋊Q8C2×D6111yescharacteristic
306C2×#125—21no—
307C2×C22⋊Q8—22no—
308C2×C22⋊Q8C2×D6112yes—
309C2×#121—612no—
310C2×#121—1221no—
311C2×C4×Q8C2×D6111yescharacteristic
312C2×C4×D4—21no—
313C2×C4×D4—611no—
314C2×C4×D4C2×D6111yescharacteristic
315C22×C4⋊C4C2×D6112yes—
316#476—1224no—
317C4×SD16—1224no—
318C2×#106—611no—
319C2×#106—612no—
320C2×C4⋊C8—611no—
321C2×#103—611no—
322C2×#103—612no—
323C2×#102—611no—
324C2×#102—612no—
325C2×C22⋊C8—612no—
326C2×C22⋊C8—612no—
327C2×C4×C8—611no—
328#394—1222no—
329#384—611no—
330#380—612no—
331#380—1221no—
332#340—612no—
333#340—612no—
334#340—612no—
335C61#2411811yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—611no—
339C22×D4—1222no—
340C22×D4C22⋊D6111yescharacteristic
341C23×C4—22no—
342C23×C4—1221no—
343C23×C4C22×D6112yes—
344C2×SD16—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8—612no—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D6114yescomplemented
352#125—28no—
353C22⋊Q8—216no—
354C22⋊Q8—24no—
355C22⋊Q8—24no—
356#121—1224no—
357#121—2442no—
358#121—2444no—
359C4×Q8—22no—
360C4×Q8C22⋊D6114yescomplemented
361C4×D4—28no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—1222no—
365C4×D4—1222no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4C22⋊D6112yes—
384C2×C42—21no—
385C2×C42C22×D6111yescharacteristic
386#106—1222no—
387#106—1224no—
388C4⋊C8—1222no—
389#103—1222no—
390#103—2442no—
391#102—1222no—
392#102—1224no—
393C22⋊C8—1224no—
394C22⋊C8—2442no—
395C4×C8—1222no—
396#95—1222no—
397C24—1221no—
398C24C2×C22⋊D6112yes—
399C2×Q8—21no—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C2×C22⋊D6112yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—612no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—4882no—
417C2×D4—2444no—
418C2×D4—2444no—
419C2×D4#1175820114yescomplemented
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C23×D6111yescharacteristic
441SD16—1228no—
442C2×C8—612no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1221no—
449C2×C8—1222no—
450C2×C8—1221no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—4881no—
463C4⋊C4—2442no—
464C4⋊C4C2×C22⋊D6112yes—
465C4⋊C4C2×C22⋊D6114yes—
466C22⋊C4—28no—
467C22⋊C4—24no—
468C22⋊C4—2444no—
469C22⋊C4—2444no—
470C22⋊C4—2442no—
471C42—24no—
472C42C2×C22⋊D6112yes—
473C42C2×C22⋊D6112yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—1222no—
481C23—2441no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23C2×#1175820114yes—
488C23C22×C22⋊D6111yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494D4—28no—
495D4—1222no—
496D4—1222no—
497D4—4888no—
498D4—2441no—
499D4—2441no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—4884no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974612yes—
527C2×C4#1451974612yes—
528C2×C4C2×#1175820112yes—
529C2×C4#1451980812yes—
530C2×C4C22×C22⋊D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4#41×D6111yescharacteristic
534C2×C4#1451982311yescharacteristic
535C2×C4#1451982411yescharacteristic
536C8—1222no—
537C8—2441no—
538C8—2442no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22C2×#1451974611yescharacteristic
554C22#1684040011yescharacteristic
555C22#1684041111yescharacteristic
556C22C22×#1175820111yescharacteristic
557C22C2×#1451980811yescharacteristic
558C22#1684056511yescharacteristic
559C22#1684057311yescharacteristic
560C4—22no—
561C4—42no—
562C4—44no—
563C4—42no—
564C4—42no—
565C4—4881no—
566C4—4882no—
567C4#1683990114yes—
568C4#1684026911yescharacteristic
569C4#1684034011yescharacteristic
570C4#1684041911yescharacteristic
571C4#1684043111yescharacteristic
572C2—44no—
573C2—4882no—
574C2#2144915811yescharacteristic
575C2C2×#1683990111yescharacteristic
576C2#2145512311yescharacteristic
577C2#2145763711yescharacteristic
578C2#2145812911yescharacteristic
579C2#2145817011yescharacteristic
580C2#2145844511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×#554D61
hypocenterC61#24118
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
2. C2×C4×#11758201C2semidirect
4. #21447012C2semidirect
5. #21446820C2semidirect
6. #21446781C2semidirect
7. #21446614C2semidirect
8. #21446136C2semidirect
9. #21446012C2semidirect
10. #21445824C2semidirect
11. C2×C61×#554C2semidirect
83. C4⋊Q8×C61D4semidirect
89. C4×Q8×C61D4semidirect
119. C2×D4×C61SD16semidirect
282. C2×#554D61semidirect
335. C61#24118semidirect
351. C4⋊Q8C22⋊D61semidirect
360. C4×Q8C22⋊D61semidirect
419. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×C4×#117582012
#214507322
#214470122
#214468202
#214467812
#214466142
#214461362
#214460122
#214458242
C2×C61×#5542
#2411861