Finite Group Databasebeta

← #29713077

Subgroups — #29713077

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2C2×#16840097C211yesmaximal, characteristic, complemented
3C22×#14519700C211yesmaximal, characteristic, complemented
4#21447065C212yesmaximal, complemented
5#21447022C212yesmaximal, complemented
6#21446240C211yesmaximal, characteristic, complemented
7#21446112C211yesmaximal, characteristic
8#21446072C212yesmaximal, complemented
9#21445753C211yesmaximal, characteristic, complemented
10#21445753C212yesmaximal, complemented
11C61×#2487C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15D4×Dic122—24no—
16D4×Dic122—24no—
17C2×#14519777C2211yescharacteristic
18C2×#14519776—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519754—21no—
22C2×#14519754—21no—
23C2×C4×Dic122C2211yescharacteristic
24C2×D4×Dic61—21no—
25#16840097—24no—
26C2×#14519710C2212yes—
27C2×#14519700—24no—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519700C2212yes—
33C2×#14519700C2212yes—
34C2×C22⋊#92418C2211yescharacteristic
35C2×C22⋊#92418C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#400—21no—
41C61×#399—21no—
42C61×#389C2212yes—
43C61×#386—21no—
44C61×#385—21no—
45C61×#380C2211yescharacteristic
46C61×#380C2212yes—
47C22×Dic122—24no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122—22no—
52C22×Dic122—22no—
53C22×Dic122—24no—
54C22×Dic122—42no—
55C22×Dic122—42no—
56C22×Dic122D414yes—
57C22×Dic122C2311yescharacteristic
58C22×Dic122C2311yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418—22no—
63C22×#92418—42no—
64C22×#92418C2312yes—
65C22×#92418C2312yes—
66#14519777—22no—
67#14519777—22no—
68#14519776—28no—
69#14519756—22no—
70#14519756—22no—
71#14519756—22no—
72#14519756D414yes—
73#14519754—28no—
74#14519754—28no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77D4×Dic61—28no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87#14519710—24no—
88#14519700—24no—
89#14519700—24no—
90#14519700—24no—
91#14519700—24no—
92#14519700—42no—
93#14519700—42no—
94C22⋊#92418—22no—
95C22⋊#92418—42no—
96C22×D4×C61D411yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—22no—
115#95×C61—21no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61Q1614yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—42no—
131C22×C4×C61—82no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—24no—
141C22×Dic61—24no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—44no—
146C22×Dic61—44no—
147C2×Dic122—21no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—28no—
154C2×Dic122—24no—
155C2×Dic122—24no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—44no—
159C2×Dic122—48no—
160C2×Dic122—48no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122C2×D412yes—
166C2×Dic122C2×D412yes—
167C2×Dic122C2×D412yes—
168C2×Dic122#4114yes—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61—82no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—82no—
180C42×C61—21no—
181C42×C61—21no—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184C4⋊Dic61—24no—
185C4⋊Dic61—24no—
186C4⋊Dic61—21no—
187C4⋊Dic61C2×D412yes—
188#11758196—24no—
189#11758196—24no—
190#11758196—24no—
191C4×Dic61—24no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61#12012yes—
206C23×C61C2×Q1612yes—
207C23×C61C2×Q1612yes—
208C23×C61#13712yes—
209C23×C61C22×D411yescharacteristic
210D4×C61—28no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—84no—
231C2×C4×C61#12012yes—
232C2×C4×C61C2×Q1612yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—48no—
243C2×Dic61—48no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247Dic122—22no—
248Dic122—24no—
249Dic122—22no—
250Dic122—22no—
251Dic122—24no—
252Dic122—48no—
253Dic122—48no—
254Dic122—41no—
255Dic122—41no—
256Dic122—42no—
257Dic122#12314yes—
258#92418—42no—
259#92418—42no—
260#24042—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61#45114yes—
271C22×C61C2×#12011yescharacteristic
272C22×C61#54612yes—
273C22×C61C22×Q1611yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61#57712yes—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—82no—
281C4×C61—82no—
282C4×C61C2×#12311yescharacteristic
283C4×C61#53411yescharacteristic
284C4×C61#54612yes—
285Dic61—24no—
286Dic61—42no—
287Dic61—42no—
288Dic61—42no—
289C2×D4×Q8—611no—
290C2×#484—611no—
291C22×#103—611no—
292#2487D6111yescharacteristic
293#2043—612no—
294#2036—612no—
295#1892—611no—
296#1869—611no—
297#1863—612no—
298#1787—611no—
299#1787—612no—
300C2×C61—24no—
301C2×C61—42no—
302C2×C61#240511yescharacteristic
303C2×C61C2×#45111yescharacteristic
304C2×C61#301011yescharacteristic
305C2×C61#329912yes—
306C2×C61#332012yes—
307C23×Q8—611no—
308C23×Q8—611no—
309C23×C8—611no—
310D4×Q8—1224no—
311D4×Q8—1224no—
312C2×C4⋊Q8—611no—
313C2×C4⋊Q8—1221no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8—1221no—
317C2×C22⋊Q8—1221no—
318C2×C4×Q8—611no—
319C2×C4×D4—1221no—
320C2×C4×D4C2×D6111yescharacteristic
321C22×C4⋊C4C2×D6111yescharacteristic
322#484—1224no—
323C2×#107—612no—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—1224no—
331C2×C22⋊C8—611no—
332C2×C22⋊C8—612no—
333#400—21no—
334#399—21no—
335#389C2×D6112yes—
336#386—21no—
337#385—21no—
338#380C2×D6111yescharacteristic
339#380C2×D6112yes—
340#340—612no—
341#336—612no—
342C61#2404211yescharacteristic, complemented, Sylow
343C22×Q8—614no—
344C22×Q8—611no—
345C22×Q8—611no—
346C22×Q8—611no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—1224no—
350C22×Q8—1224no—
351C22×Q8—1222no—
352C22×Q8—1221no—
353C22×Q8—1221no—
354C22×Q8—2442no—
355C22×Q8—2442no—
356C22×D4C22⋊D6111yescharacteristic
357C23×C4—21no—
358C23×C4—1221no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C22×C8—612no—
363C22×C8—612no—
364C22×C8—1222no—
365C22×C8—2442no—
366C4⋊Q8—1222no—
367C4⋊Q8—1222no—
368C4⋊Q8—1228no—
369C22⋊Q8—614no—
370C22⋊Q8—1228no—
371C22⋊Q8—1228no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—1228no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22⋊D6111yescharacteristic
391C2×C4⋊C4C22×D6112yes—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4C22⋊D6111yescharacteristic
398C2×C22⋊C4C22⋊D6111yescharacteristic
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#107—1224no—
402#103—1224no—
403#103—1224no—
404#103—1224no—
405#103—1224no—
406#103—2442no—
407#103—2442no—
408C22⋊C8—1222no—
409C22⋊C8—2442no—
410#95—22no—
411#95—21no—
412#95—22no—
413C24C2×C22⋊D6111yescharacteristic
414C24C2×C22⋊D6111yescharacteristic
415C2×Q8—612no—
416C2×Q8—612no—
417C2×Q8—612no—
418C2×Q8—614no—
419C2×Q8—1224no—
420C2×Q8—1228no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1222no—
425C2×Q8—1224no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—2444no—
431C2×Q8—2444no—
432C2×Q8—2444no—
433C2×Q8—2448no—
434C2×Q8—2444no—
435C2×Q8—2448no—
436C2×Q8—2444no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#1175820214yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—1221no—
451C22×C4—1224no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1224no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4D4×D6112yes—
459C22×C4D4×D6112yes—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C23×D6111yescharacteristic
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2444no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—612no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4—1224no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1224no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—1224no—
491C22⋊C4—1224no—
492C42—21no—
493C42—21no—
494C42—1224no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23#1451974512yes—
502C23C2×#1175820212yes—
503C23C2×#1175820212yes—
504C23#1451980712yes—
505C23C22×C22⋊D6111yescharacteristic
506Q8—614no—
507Q8—1222no—
508Q8—1224no—
509Q8—1222no—
510Q8—1222no—
511Q8—1224no—
512Q8—2448no—
513Q8—2448no—
514Q8—2441no—
515Q8—2441no—
516Q8—2442no—
517D4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—1222no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—2448no—
544C2×C4—2448no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4#1451974512yes—
549C2×C4C2×#1175820212yes—
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4#41×D6112yes—
553C2×C4#1451982312yes—
554C8—2442no—
555C8—2442no—
556C22—24no—
557C22—24no—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22#1683991714yes—
566C22C2×#1451974511yescharacteristic
567C22#1684040012yes—
568C22C22×#1175820211yescharacteristic
569C22C2×#1451980711yescharacteristic
570C22#1684057312yes—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—1224no—
578C4—2442no—
579C4—2442no—
580C4—2442no—
581C4#123×D6111yescharacteristic
582C4#1684032311yescharacteristic
583C4#1684040412yes—
584C2—24no—
585C2—42no—
586C2#2144914311yescharacteristic
587C2C2×#1683991711yescharacteristic
588C2#2145510311yescharacteristic
589C2#2145768912yes—
590C2#2145798412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2487C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2487D61
hypocenterC61#24042
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16840097C2semidirect
3. C22×#14519700C2semidirect
4. #21447065C2semidirect
5. #21447022C2semidirect
6. #21446240C2semidirect
8. #21446072C2semidirect
9. #21445753C2semidirect
10. #21445753C2semidirect
342. C61#24042semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
C2×#168400972
C22×#145197002
#214470652
#214470222
#214462402
#214461122
#214460722
#214457532
C61×#24872
#2404261