Finite Group Databasebeta

← #29696758

Subgroups — #29696758

Order 15616.

1412 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840421C211yesmaximal, characteristic
2#21448651C212yesmaximal, complemented
3#21446654C212yesmaximal, complemented
4#21446653C214yesmaximal, complemented
5#21445806C212yesmaximal, complemented
6C4×#14519692C211yesmaximal, characteristic, complemented
7#21445606C212yesmaximal, complemented
8C61×#1799C211yesmaximal, characteristic, complemented
9C22×#11758222C411yescharacteristic, complemented
10C23×Dic122C2211yescharacteristic
11#16840421—22no—
12#16840421—24no—
13#16840421—22no—
14C2×#14519765C411yescharacteristic, complemented
15C2×#14519763C412yescomplemented
16C2×#14519756C411yescharacteristic, complemented
17C2×#14519755C412yescomplemented
18C2×#14519751C2211yescharacteristic
19C2×#14519749C2212yes—
20C2×C4×Dic122C411yescharacteristic, complemented
21C2×#14519742—22no—
22C2×#14519737C2212yes—
23C2×#14519692C2212yes—
24C2×#14519691C2214yes—
25C2×#14519691C2214yes—
26#16839667—22no—
27#16839666—24no—
28#16839620C2214yes—
29C22⋊C4×Dic61C2214yes—
30C4×C22⋊Dic61C2212yes—
31#16839529C2211yescharacteristic
32#16839529C2211yescharacteristic
33C2×C4×D4×C61C2211yescharacteristic
34C2×#99×C61C2212yes—
35C61×#380C2212yes—
36C61×#342C2212yes—
37C2×#11758222—21no—
38C2×#11758222—22no—
39C2×#11758222—21no—
40C2×#11758222—44no—
41C22×Dic122—44no—
42C22×Dic122C2×C411yescharacteristic
43C22×Dic122C2×C412yes—
44C22×Dic122C2×C411yescharacteristic
45C22×Dic122D412yes—
46C22×Dic122D414yes—
47C22×Dic122D412yes—
48C22×D244C2×C411yescharacteristic
49C22×C22⋊D61C2×C412yes—
50C22×C22⋊D61C2312yes—
51C23×Dic61C2312yes—
52C22×C4×D61C2×C412yes—
53#14519765—22no—
54#14519765—22no—
55#14519763—24no—
56#14519763—24no—
57#14519756—22no—
58#14519756—22no—
59#14519755—24no—
60#14519755—24no—
61#14519751—22no—
62#14519751—22no—
63#14519749—24no—
64#14519749—24no—
65C4×Dic122—21no—
66C4×Dic122—22no—
67C4×Dic122—21no—
68#14519742—48no—
69#14519737—24no—
70#14519737—24no—
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196—24no—
73C2×#11758196C2×C412yes—
74C2×#11758196C2×C412yes—
75C2×C22⋊Dic61C2×C412yes—
76C2×C22⋊Dic61C2312yes—
77C2×C22⋊Dic61C2312yes—
78C2×#11758198C2×C412yes—
79C2×#11758198C2×C412yes—
80C2×#11758198C2×C412yes—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—24no—
83C2×C4×Dic61C2×C412yes—
84#14519692—24no—
85#14519691—28no—
86#14519691—44no—
87#14519684—22no—
88#14519684—24no—
89#14519684—24no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61C2×C411yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C22⋊C4×C61C2×C411yescharacteristic
98C2×C22⋊C4×C61C2314yes—
99C2×C22⋊C4×C61C2311yescharacteristic
100C2×C42×C61C2×C411yescharacteristic
101#99×C61—42no—
102#95×C61—22no—
103C24×C61C22×C411yescharacteristic
104C24×C61C2411yescharacteristic
105C23×D61C22×C412yes—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—24no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61—44no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—44no—
132C22×Dic61C22×C412yes—
133C22×Dic61C22×C412yes—
134C22×Dic61C22×C412yes—
135C22×Dic61C2×D414yes—
136C22×Dic61C2×D414yes—
137#11758222—44no—
138#11758222—48no—
139#11758222—44no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—42no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—44no—
146C2×Dic122—24no—
147C2×Dic122—28no—
148C2×Dic122—44no—
149C2×Dic122—48no—
150C2×Dic122—42no—
151C2×Dic122—44no—
152C2×Dic122C22⋊C411yescharacteristic
153C2×Dic122C22⋊C412yes—
154C2×Dic122C22⋊C414yes—
155C2×Dic122C22⋊C412yes—
156C2×Dic122C22⋊C414yes—
157C2×Dic122C22⋊C412yes—
158C2×Dic122C22⋊C411yescharacteristic
159C4⋊C4×C61—22no—
160C4⋊C4×C61—44no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C42×C61—21no—
166C42×C61—21no—
167C22⋊Dic61—22no—
168C22⋊Dic61—24no—
169C22⋊Dic61—22no—
170C22⋊Dic61—22no—
171C22⋊Dic61#4114yescomplemented
172#11758198—24no—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177#11758196—24no—
178#11758196—22no—
179#11758196—22no—
180#11758196—48no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—48no—
184C4×Dic61—44no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61C23×C411yescharacteristic
193C23×C61C22×D412yes—
194C23×C61C2×#4112yes—
195C23×C61H(5,𝔽2)11yescharacteristic
196C23×C612−1+411yescharacteristic
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—42no—
201C22×D61—42no—
202D4×C61—42no—
203D4×C61—42no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C22⋊D61—44no—
224C22⋊D61—44no—
225C22⋊D61—44no—
226C22⋊D61—44no—
227C2×Dic61—28no—
228C2×Dic61—24no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—42no—
234C2×Dic61—44no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C414yes—
242C2×Dic61C2×C22⋊C412yes—
243D244—42no—
244D244—42no—
245C4×D61—44no—
246C4×D61—44no—
247Dic122—24no—
248Dic122—22no—
249Dic122—28no—
250Dic122—24no—
251Dic122—44no—
252Dic122—42no—
253Dic122#9914yescomplemented
254#22534—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—21no—
257C22×C61—22no—
258C22×C61—21no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C4×#4111yescharacteristic
269C22×C61#52011yescharacteristic
270C22×C61#54812yes—
271C22×C61#55012yes—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—82no—
277C4×C61—21no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—84no—
283C4×C61C2×#9912yes—
284Dic61—22no—
285Dic61—24no—
286Dic61—44no—
287Dic61—42no—
288Dic61—82no—
289Dic61C2×#9914yes—
290C2×#548—611no—
291Q8×C22⋊C4—612no—
292#1964—614no—
293#1964—612no—
294#1799—612no—
295#1799D6111yescharacteristic, complemented
296C4×#99—612no—
297C4×#99—611no—
298C2×C61—22no—
299C2×C61—22no—
300C2×C61—21no—
301C2×C61—21no—
302C2×C61—81no—
303C2×C61#234111yescharacteristic
304C2×C61C22×#9911yescharacteristic
305C2×C61#288411yescharacteristic
306D61—82no—
307C22×#41—611no—
308C23×Q8—611no—
309#548—1222no—
310#548—1222no—
311#548—1224no—
312C2×#124—612no—
313C2×#124—611no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—612no—
316C2×#121—612no—
317C2×#121—611no—
318C2×C4×Q8—611no—
319C2×C4×Q8—1222no—
320C2×C4×D4—612no—
321C2×C4×D4C2×D6111yescharacteristic
322C2×#99—612no—
323C2×#99—614no—
324C2×#99—614no—
325C2×#99C2×D6112yes—
326#391—1224no—
327#391—1222no—
328#380—614no—
329#380C2×D6112yes—
330C4×C22⋊C4—612no—
331C4×C22⋊C4—614no—
332#342—611no—
333#342—611no—
334#342C2×D6112yes—
335C61#2253411yescharacteristic, complemented, Sylow
336C2×#41—1221no—
337C2×#41—1222no—
338C2×#41—1221no—
339C2×#41—2444no—
340C22×Q8—611no—
341C22×Q8—612no—
342C22×Q8—611no—
343C22×Q8—612no—
344C22×Q8—614no—
345C22×Q8—612no—
346C22×Q8—2444no—
347C22×D4—612no—
348C22×D4—612no—
349C22×D4—611no—
350C22×D4C22×D6111yescharacteristic
351C23×C4—612no—
352C23×C4—612no—
353C23×C4C4×D6111yescharacteristic
354C23×C4C22×D6111yescharacteristic
355#124—1222no—
356#124—1222no—
357#124—1224no—
358#124—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1224no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363#121—1224no—
364#121—1224no—
365#121—1222no—
366#121—1222no—
367C4×Q8—1221no—
368C4×Q8—1222no—
369C4×Q8—1221no—
370C4×Q8—2448no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—1224no—
374C4×D4—1224no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4C4×D6111yescharacteristic
381C2×C22⋊C4—612no—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4C4×D6111yescharacteristic
388C2×C22⋊C4C22×D6114yes—
389C2×C22⋊C4C22×D6111yescharacteristic
390C2×C42—612no—
391C2×C42—1224no—
392C2×C42—1222no—
393C2×C42C4×D6111yescharacteristic
394#99—42no—
395#99—1228no—
396#99—1224no—
397#99—2444no—
398#95—22no—
399#95—1224no—
400#95—1224no—
401#95—1222no—
402C24—612no—
403C24C2×C4×D6111yescharacteristic
404C24C23×D6111yescharacteristic
405#41—2444no—
406#41—2448no—
407#41—2444no—
408C2×Q8—611no—
409C2×Q8—612no—
410C2×Q8—611no—
411C2×Q8—614no—
412C2×Q8—612no—
413C2×Q8—614no—
414C2×Q8—612no—
415C2×Q8—1228no—
416C2×Q8—1224no—
417C2×Q8—2444no—
418C2×Q8—2442no—
419C2×Q8—2444no—
420C2×Q8—2448no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—2442no—
434C2×D4—2444no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—612no—
443C22×C4—614no—
444C22×C4—612no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4C2×C4×D6111yescharacteristic
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4C2×C4×D6111yescharacteristic
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6111yescharacteristic
456C22×C4D4×D6112yes—
457C22×C4D4×D6112yes—
458C4⋊C4—22no—
459C4⋊C4—44no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—2448no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—614no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1222no—
475C22⋊C4—1224no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C42—21no—
479C42—21no—
480C42—1222no—
481C42—1222no—
482C42—2444no—
483C42—2448no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—2442no—
495C23—2442no—
496C23C22×C4×D6111yescharacteristic
497C23C2×D4×D6112yes—
498C23#41×D6112yes—
499C23#1451982211yescharacteristic
500C23#1451982511yescharacteristic
501Q8—614no—
502Q8—1228no—
503Q8—1224no—
504Q8—1224no—
505Q8—1222no—
506Q8—2442no—
507Q8—2444no—
508D4—42no—
509D4—42no—
510D4—2442no—
511D4—2444no—
512D4—2444no—
513D4—2444no—
514D4—2444no—
515D4—2442no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—614no—
535C2×C4—1228no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1224no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22⋊C4×D6111yescharacteristic
553C22—22no—
554C22—21no—
555C22—22no—
556C22—21no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—81no—
564C22—81no—
565C22—4882no—
566C22—4882no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C2×C22⋊C4×D6111yescharacteristic
571C22#1684023011yescharacteristic
572C22#1684024311yescharacteristic
573C22#1684042212yes—
574C22#1684043012yes—
575C4—21no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—84no—
581C4—614no—
582C4—1222no—
583C4—1224no—
584C4—4882no—
585C4—2444no—
586C4—2442no—
587C4#99×D6112yes—
588C2—22no—
589C2—22no—
590C2—21no—
591C2—21no—
592C2—81no—
593C2—4882no—
594C2#2144865611yescharacteristic
595C2C2×#99×D6111yescharacteristic
596C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1799C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840421C2
cosocleC24×C61C24
hypercenter#1799D61
hypocenterC61#22534
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21448651C2semidirect
3. #21446654C2semidirect
4. #21446653C2semidirect
5. #21445806C2semidirect
6. C4×#14519692C2semidirect
7. #21445606C2semidirect
8. C61×#1799C2semidirect
9. C22×#11758222C4semidirect
14. C2×#14519765C4semidirect
15. C2×#14519763C4semidirect
16. C2×#14519756C4semidirect
17. C2×#14519755C4semidirect
20. C2×C4×Dic122C4semidirect
171. C22⋊Dic61#41semidirect
253. Dic122#99semidirect
295. #1799D61semidirect
335. C61#22534semidirect

Maximal subgroups

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

subgroupindex
C2×#168404212
#214486512
#214466542
#214466532
#214458062
C4×#145196922
#214456062
C61×#17992
#2253461