Finite Group Databasebeta

← #29713933

Subgroups — #29713933

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840039C211yesmaximal, characteristic, complemented
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447019C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446614C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446018C212yesmaximal, complemented
10#21445824C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×#14519748—21no—
15C2×C4×D244C2211yescharacteristic
16#16840039—24no—
17C4×#11758201—24no—
18C2×#14519710C2212yes—
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×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61—21no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×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#14519710—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×C61C2312yes—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61D414yescomplemented
83#126×C61—216no—
84#124×C61—28no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—24no—
87C4×Q8×C61—28no—
88C4×Q8×C61—22no—
89C4×Q8×C61D414yescomplemented
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92#117×C61—24no—
93#117×C61—24no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D412yes—
103C2×C42×C61—22no—
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—42no—
115C2×Q8×C61—42no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C2×C22⋊D61—82no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244#4112yes—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C4⋊C4×C61—28no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—28no—
157C22⋊C4×C61—24no—
158C42×C61—28no—
159C42×C61—24no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—44no—
163#11758201—28no—
164#11758198—44no—
165#11758198—42no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—42no—
171C4⋊Dic61—81no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418#4112yes—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—41no—
199Q8×C61—41no—
200Q8×C61—48no—
201D4×C61—44no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61#12012yes—
221C2×C4×C61#12012yes—
222C2×C4×C61#13512yes—
223C2×C4×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C22⋊D61—88no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—42no—
237D244—22no—
238D244—22no—
239D244—41no—
240D244—41no—
241C4×D61—84no—
242#92418—22no—
243#92418—41no—
244#92418—42no—
245#24119—611nomaximal, Sylow
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61C2×#13511yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57811yescharacteristic
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—44no—
271C4×C61—42no—
272C4×C61#44914yes—
273C4×C61#52511yescharacteristic
274C4×C61#53711yescharacteristic
275C4×C61#54811yescharacteristic
276C4×C61#55011yescharacteristic
277Dic61—81no—
278Dic61—82no—
279C2×#556D6111yescharacteristic, complemented
280C2×#474—611no—
281C2×C4×SD16—611no—
282#2537—611no—
283#2036—612no—
284#2000—612no—
285#1990—611no—
286#1954—611no—
287#1875—612no—
288#1854—612no—
289#1803—611no—
290C2×C61—82no—
291C2×C61#240711yescharacteristic
292C2×C61C2×#44911yescharacteristic
293C2×C61#301311yescharacteristic
294C2×C61#330211yescharacteristic
295C2×C61#332411yescharacteristic
296C2×C61#334011yescharacteristic
297C2×C61#335311yescharacteristic
298D61—82no—
299C22×SD16—611no—
300#556—24no—
301#556—24no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303C2×#126—22no—
304C2×#124—21no—
305C2×C22⋊Q8C2×D6112yes—
306C2×#121—612no—
307C2×#121—1221no—
308C2×C4×Q8—21no—
309C2×C4×Q8C2×D6111yescharacteristic
310C2×C4×D4—611no—
311C2×C4×D4C2×D6111yescharacteristic
312C2×#117C2×D6112yes—
313#474—1224no—
314C4×SD16—1224no—
315C2×#107—612no—
316C2×#106—611no—
317C2×C4⋊C8—611no—
318C2×#103—611no—
319C2×#103—612no—
320C2×#102—611no—
321C2×#102—612no—
322C2×C22⋊C8—612no—
323C2×C22⋊C8—612no—
324C2×C4×C8—611no—
325#394—1222no—
326#384—611no—
327#380—612no—
328#380—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332C61#2411911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8C22×D6111yescharacteristic
335C22×D4—611no—
336C22×D4—1222no—
337C22×D4C22⋊D6111yescharacteristic
338C23×C4—1221no—
339C23×C4C22×D6112yes—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8—612no—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D6114yescomplemented
348#126—216no—
349#124—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—1224no—
353#121—2442no—
354#121—2444no—
355C4×Q8—28no—
356C4×Q8—22no—
357C4×Q8C22⋊D6114yescomplemented
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—1222no—
361C4×D4—1222no—
362#117—24no—
363#117—24no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4C22⋊D6111yescharacteristic
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6112yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4C22⋊D6112yes—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42C22×D6111yescharacteristic
383#107—1224no—
384#106—1222no—
385C4⋊C8—1222no—
386#103—1222no—
387#103—2442no—
388#102—1222no—
389#102—1224no—
390C22⋊C8—1224no—
391C22⋊C8—2442no—
392C4×C8—1222no—
393#95—1222no—
394C24—1221no—
395C24C2×C22⋊D6112yes—
396C2×Q8—21no—
397C2×Q8—24no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8C2×C22⋊D6112yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—4882no—
415C2×D4—2444no—
416C2×D4—2444no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4C2×C22⋊D6112yes—
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435SD16—1228no—
436C2×C8—612no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1221no—
445C4⋊C4—28no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—4881no—
456C4⋊C4—2442no—
457C4⋊C4C2×C22⋊D6112yes—
458C4⋊C4C2×C22⋊D6114yes—
459C22⋊C4—28no—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2442no—
465C42—28no—
466C42—24no—
467C42C2×C22⋊D6112yes—
468C42C2×C22⋊D6112yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—1222no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23—2441no—
483C23C22×C22⋊D6111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—48no—
487Q8—41no—
488Q8—41no—
489D4—44no—
490D4—1222no—
491D4—1222no—
492D4—4888no—
493D4—2441no—
494D4—2441no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—4884no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974612yes—
521C2×C4#1451974612yes—
522C2×C4#1451979712yes—
523C2×C4#1451980212yes—
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4#41×D6111yescharacteristic
527C2×C4#41×D6111yescharacteristic
528C2×C4#1451982311yescharacteristic
529C2×C4#1451982411yescharacteristic
530C8—1222no—
531C8—2441no—
532C8—2442no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22C2×#1451974611yescharacteristic
548C22#1684040011yescharacteristic
549C22#1684041111yescharacteristic
550C22C2×#1451979711yescharacteristic
551C22C2×#1451980211yescharacteristic
552C22#1684057011yescharacteristic
553C22#1684057711yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—4882no—
560C4—4881no—
561C4#1683990714yes—
562C4#1684026911yescharacteristic
563C4#1684034011yescharacteristic
564C4#1684041911yescharacteristic
565C4#1684043111yescharacteristic
566C2—82no—
567C2—4882no—
568C2#2144915811yescharacteristic
569C2C2×#1683990711yescharacteristic
570C2#2145513411yescharacteristic
571C2#2145773011yescharacteristic
572C2#2145804411yescharacteristic
573C2#2145822611yescharacteristic
574C2#2145839711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×#556D61
hypocenterC61#24119
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16840039C2semidirect
2. C2×C4×#11758201C2semidirect
4. #21447019C2semidirect
5. #21446820C2semidirect
6. #21446781C2semidirect
7. #21446614C2semidirect
8. #21446141C2semidirect
9. #21446018C2semidirect
10. #21445824C2semidirect
11. C2×C61×#556C2semidirect
82. C4⋊Q8×C61D4semidirect
89. C4×Q8×C61D4semidirect
279. C2×#556D61semidirect
332. C61#24119semidirect
347. C4⋊Q8C22⋊D61semidirect
357. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400392
C2×C4×#117582012
#214507322
#214470192
#214468202
#214467812
#214466142
#214461412
#214460182
#214458242
C2×C61×#5562
#2411961