Finite Group Databasebeta

← #29713079

Subgroups — #29713079

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×#16840098C211yesmaximal, characteristic, complemented
3C22×#14519702C211yesmaximal, characteristic, complemented
4#21450136C211yesmaximal, characteristic
5#21447066C212yesmaximal, complemented
6#21447023C212yesmaximal, complemented
7#21446242C211yesmaximal, characteristic, complemented
8#21446076C212yesmaximal, complemented
9#21445755C212yesmaximal, complemented
10#21445755C211yesmaximal, characteristic, complemented
11C61×#1869C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14D4×Dic122—24no—
15D4×Dic122—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519776—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519754—21no—
21C2×#14519754—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×D4×Dic61—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25#16840098—24no—
26C2×#14519711C2212yes—
27C2×#14519702—24no—
28C2×#14519702C2212yes—
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33C2×#14519702C2212yes—
34#16839701—21no—
35#16839698—21no—
36#16839660C2212yes—
37#16839642—21no—
38#16839634—21no—
39#16839622C2211yescharacteristic
40#16839622C2212yes—
41#16839525C2212yes—
42C23×C8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×C22⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
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—
62C23×Dic61—21no—
63#14519777—22no—
64#14519777—22no—
65#14519776—28no—
66#14519756—22no—
67#14519756—22no—
68#14519756—22no—
69#14519756D414yes—
70#14519754—28no—
71#14519754—28no—
72C4×Dic122—22no—
73C4×Dic122—22no—
74D4×Dic61—28no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—22no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—22no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—22no—
91C2×C4×Dic61—21no—
92#14519711—24no—
93#14519702—24no—
94#14519702—24no—
95#14519702—24no—
96#14519702—24no—
97#14519702—42no—
98#14519702—42no—
99#14519684—21no—
100#14519684—22no—
101#14519684—22no—
102C22×D4×C61D411yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61—22no—
106C22×C8×C61—42no—
107C22×C8×C61C2312yes—
108C22×C8×C61C2312yes—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C22⋊C8×C61—22no—
116C22⋊C8×C61—42no—
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×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—24no—
137C22×Dic61—24no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—44no—
144C22×Dic61—44no—
145C22×Dic61—42no—
146C22×Dic61—82no—
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—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—44no—
176C4⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61—21no—
180C42×C61—21no—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—42no—
184C22⋊Dic61—82no—
185C4⋊Dic61—24no—
186C4⋊Dic61—24no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—42no—
193C4⋊Dic61C2×D412yes—
194#11758196—24no—
195#11758196—24no—
196#11758196—24no—
197#11758196—82no—
198C4×Dic61—24no—
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#12012yes—
225C2×C4×C61C2×Q1612yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4112yes—
229C2×C4×C61H(5,𝔽2)12yes—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—48no—
236C2×Dic61—48no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—84no—
247Dic122—22no—
248Dic122—24no—
249Dic122—22no—
250Dic122—22no—
251Dic122—24no—
252Dic122—48no—
253Dic122—48no—
254Dic122—41no—
255Dic122—41no—
256Dic122—42no—
257Dic122#12314yes—
258C8×C61—42no—
259C8×C61—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×C61C2×#12311yescharacteristic
281C4×C61#53411yescharacteristic
282C4×C61#54612yes—
283Dic61—24no—
284Dic61—42no—
285Dic61—42no—
286Dic61—42no—
287Dic61—82no—
288Dic61—82no—
289C2×D4×Q8—611no—
290C2×#484—611no—
291C22×#103—611no—
292#2487—611no—
293#2043—612no—
294#2036—612no—
295#1892—611no—
296#1869D6111yescharacteristic
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×C8C2×D6111yescharacteristic
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⋊C4—611no—
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⋊C8C2×D6111yescharacteristic
332C2×C22⋊C8C2×D6112yes—
333#400—1221no—
334#399—1221no—
335#389—612no—
336#386—1221no—
337#385—1221no—
338#380—611no—
339#380—612no—
340#340—612no—
341#336C2×D6112yes—
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—1221no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—2442no—
355C22×Q8—2442no—
356C22×D4D24411yescharacteristic
357C23×C4—1221no—
358C23×C4—1221no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C22×C8—22no—
363C22×C8—42no—
364C22×C8C22×D6112yes—
365C22×C8C22×D6112yes—
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—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4D24411yescharacteristic
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4D24411yescharacteristic
398C2×C22⋊C4D24411yescharacteristic
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—22no—
409C22⋊C8—42no—
410#95—1222no—
411#95—1221no—
412#95—1222no—
413C24C2×D24411yescharacteristic
414C24C2×D24411yescharacteristic
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×D4Dic24414yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—1221no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—4882no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2442no—
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4D4×D6112yes—
463C22×C4D4×D6112yes—
464C22×C4C23×D6111yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—44no—
472C4⋊C4—22no—
473C4⋊C4—612no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—1221no—
479C4⋊C4—1224no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—4882no—
485C4⋊C4—2442no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—4882no—
491C22⋊C4—2442no—
492C42—21no—
493C42—21no—
494C42—1224no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23#1451974712yes—
502C23C2×Dic24412yes—
503C23C2×Dic24412yes—
504C23#1451981012yes—
505C23C22×D24411yescharacteristic
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—1222no—
532C2×C4—1224no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—4884no—
537C2×C4—2448no—
538C2×C4—2448no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4#1451974712yes—
549C2×C4C2×Dic24412yes—
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4#41×D6112yes—
553C2×C4#1451982212yes—
554C8—42no—
555C8—42no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—42no—
564C22—41no—
565C22#1683992014yes—
566C22C2×#1451974711yescharacteristic
567C22#1684040712yes—
568C22C22×Dic24411yescharacteristic
569C22C2×#1451981011yescharacteristic
570C22#1684057412yes—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—1224no—
576C4—4882no—
577C4—4882no—
578C4—2442no—
579C4—2442no—
580C4—2442no—
581C4#123×D6111yescharacteristic
582C4#1684032211yescharacteristic
583C4#1684040112yes—
584C2—24no—
585C2—42no—
586C2#2144914511yescharacteristic
587C2C2×#1683992011yescharacteristic
588C2#2145510711yescharacteristic
589C2#2145769612yes—
590C2#2145799212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1869C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1869D61
hypocenterC61#24042
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C2×#16840098C2semidirect
3. C22×#14519702C2semidirect
5. #21447066C2semidirect
6. #21447023C2semidirect
7. #21446242C2semidirect
8. #21446076C2semidirect
9. #21445755C2semidirect
10. #21445755C2semidirect
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×#168400982
C22×#145197022
#214501362
#214470662
#214470232
#214462422
#214460762
#214457552
C61×#18692
#2404261