Finite Group Databasebeta

← #29691228

Subgroups — #29691228

Order 15616.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839903C211yesmaximal, characteristic
2#21448325C211yesmaximal, characteristic
3C2×#16839667C211yesmaximal, characteristic, complemented
4#21446698C212yesmaximal, complemented
5#21446507C212yesmaximal, complemented
6#21446459C212yesmaximal, complemented
7#21446154C212yesmaximal, complemented
8#21446136C211yesmaximal, characteristic
9#21445753C211yesmaximal, characteristic, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1852C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×#11758190C2212yes—
14C2×#14519751C2211yescharacteristic
15C2×#14519739C2211yescharacteristic
16C2×#14519737—21no—
17C22×C4×Dic61C2211yescharacteristic
18#16839903C418yescomplemented
19C2×#14519702C2212yes—
20C2×#14519700C2212yes—
21C2×#14519700C2212yes—
22C2×#14519697C2212yes—
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519687C2212yes—
25#16839667—21no—
26#16839667—21no—
27#16839667C2212yes—
28#16839667C2212yes—
29#16839663C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839651—21no—
32#16839639—21no—
33C4⋊C4×Dic61C2211yescharacteristic
34#16839609—21no—
35C2×#14519684—21no—
36C2×#14519684—21no—
37#16839524C2212yes—
38#16839516C2212yes—
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#107×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122—42no—
47C22×Dic122—42no—
48C22×Dic122C2×C414yescomplemented
49C22×Dic122C2312yes—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C2×#11758190—42no—
57C2×#11758190—42no—
58C2×#11758190C2×C418yescomplemented
59C22×#92418C2312yes—
60#14519751—22no—
61#14519751C2×C414yes—
62#14519739—42no—
63#14519737—44no—
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61—42no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C2×C4×Dic61C2311yescharacteristic
76#14519702C2×C418yescomplemented
77#14519700—42no—
78#14519700—42no—
79#14519697—42no—
80C22⋊#92418—22no—
81#14519687—42no—
82#14519684—22no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—21no—
87#14519684—21no—
88C23×C4×C61—21no—
89C23×C4×C61C2311yescharacteristic
90C22×C8×C61C2312yes—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—42no—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2312yes—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—21no—
97#107×C61—42no—
98C22⋊C8×C61C2×C414yescomplemented
99#95×C61—21no—
100C24×C61C2×D411yescharacteristic
101C23×D61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61C22×C412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4112yes—
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—24no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C22×Dic61—24no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—21no—
135C2×D244C22×C412yes—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—24no—
144C2×Dic122—22no—
145C2×Dic122—48no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122—48no—
149C2×Dic122C22×C414yes—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61C22×C414yes—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—42no—
159#11758198—24no—
160#11758198—42no—
161#11758198—42no—
162#11758198—42no—
163C4⋊Dic61—21no—
164C4⋊Dic61C22×C412yes—
165#11758196—42no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174#11758190—48no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61C4×D412yes—
180C23×C61AGL(1,ℤ/8)12yes—
181C23×C61#13712yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—84no—
204C2×C4×C61C4×D412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×C4×C61H(5,𝔽2)12yes—
209C22⋊D61—48no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—44no—
224C2×Dic61—48no—
225C2×Dic61—44no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61#13712yes—
231D244—24no—
232D244—42no—
233C4×D61—84no—
234Dic122—24no—
235Dic122—24no—
236Dic122—48no—
237Dic122—42no—
238Dic122—42no—
239C8×C61—24no—
240#92418—42no—
241#22047—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61#44014yes—
249C22×C61C2×C4×D411yescharacteristic
250C22×C61#53411yescharacteristic
251C22×C61#53511yescharacteristic
252C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C61C2×#13711yescharacteristic
254C22×C61C2×#13711yescharacteristic
255C22×C61#57511yescharacteristic
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C4×C61—22no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61—82no—
267C4×C61#51812yes—
268C4×C61#53812yes—
269Dic61—22no—
270Dic61—44no—
271Dic61—41no—
272Dic61—42no—
273Dic61—41no—
274Dic61—41no—
275C2×#448—611no—
276C4×#121—611no—
277C2×#391—611no—
278#1973—612no—
279#1935—612no—
280#1928—612no—
281#1877—612no—
282#1874—611no—
283#1852D6111yescharacteristic, complemented
284#1787—611no—
285#1749—611no—
286C2×C61—24no—
287C2×C61#232111yescharacteristic
288C2×C61C2×#44011yescharacteristic
289C2×C61C4×#13711yescharacteristic
290C2×C61#319311yescharacteristic
291C2×C61#319611yescharacteristic
292C2×C61#321311yescharacteristic
293C2×C61#321811yescharacteristic
294D61—82no—
295C23×Q8—611no—
296C22×SD16—612no—
297C2×#121—611no—
298C2×C4×D4—611no—
299C2×C4×D4—1221no—
300C22×C4⋊C4C2×D6111yescharacteristic
301C22×C42—611no—
302#448—618no—
303C2×#107C2×D6112yes—
304C2×#103—612no—
305C2×#103—612no—
306C2×#103—612no—
307C2×#102—612no—
308C2×C22⋊C8—611no—
309C2×C22⋊C8C2×D6111yescharacteristic
310C2×#97—612no—
311#391—612no—
312#391—612no—
313#391—1221no—
314#391—1221no—
315#390—611no—
316#389—611no—
317#388—1221no—
318#386—1221no—
319#380C2×D6111yescharacteristic
320C4×C4⋊C4—611no—
321C4×C22⋊C4—1221no—
322C2×#95—1221no—
323C2×#95—1221no—
324#340—612no—
325#340C2×D6112yes—
326#336—612no—
327C61#2204711yescharacteristic, complemented, Sylow
328C22×Q8—614no—
329C22×Q8—612no—
330C22×Q8—1221no—
331C22×Q8—1222no—
332C22×Q8—2442no—
333C22×Q8—2442no—
334C22×D4—611no—
335C22×D4—1221no—
336C23×C4—21no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—1221no—
340C23×C4—1221no—
341C23×C4C22×D6111yescharacteristic
342C2×SD16—618no—
343C2×SD16—2442no—
344C2×SD16—2442no—
345C22×C8—612no—
346C22×C8C22×D6112yes—
347#121—614no—
348#121—1222no—
349C4×D4—2442no—
350C4×D4—2444no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4C22×D6112yes—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C42—611no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—1221no—
368C2×C42—2442no—
369#107—42no—
370#103—618no—
371#103—2442no—
372#103—2442no—
373#102—2442no—
374C22⋊C8—1222no—
375C22⋊C8C4×D6114yescomplemented
376#97—2442no—
377#95—21no—
378#95—1222no—
379#95—1222no—
380#95—1222no—
381#95—1222no—
382#95—1221no—
383#95—1221no—
384C24—1221no—
385C24D4×D6111yescharacteristic
386C2×Q8—614no—
387C2×Q8—1222no—
388C2×Q8—1224no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—1222no—
393C2×Q8—2448no—
394C2×Q8—2442no—
395C2×Q8—2442no—
396C2×Q8—2448no—
397C2×D4—612no—
398C2×D4—1224no—
399C2×D4—1221no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—2442no—
403C2×D4—2442no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1221no—
420C22×C4—1224no—
421C22×C4—1222no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4—2441no—
426C22×C4—2441no—
427C22×C4—2441no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4C2×C4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822612yes—
433C22×C4C23×D6111yescharacteristic
434SD16—2448no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8C2×C4×D6114yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—612no—
447C4⋊C4—1221no—
448C4⋊C4—2442no—
449C22⋊C4—42no—
450C22⋊C4—1224no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C22⋊C4—2442no—
454C42—1222no—
455C42—1222no—
456C42—2441no—
457C42—2441no—
458C42—2442no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1222no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23#1451973912yes—
471C23#1451980112yes—
472C23#1451980912yes—
473C23C2×D4×D6111yescharacteristic
474Q8—1224no—
475Q8—1224no—
476Q8—2448no—
477Q8—2442no—
478Q8—2442no—
479D4—1224no—
480D4—2448no—
481D4—2442no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—84no—
496C2×C4—612no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1222no—
500C2×C4—1224no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—4884no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4—2448no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451973912yes—
519C2×C4C22×C4×D6111yescharacteristic
520C2×C4C2×#1175822611yescharacteristic
521C2×C4#1451982212yes—
522C2×C4#1451982312yes—
523C8—24no—
524C8—2442no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—2442no—
536C22—2442no—
537C22—2442no—
538C22#1683986114yes—
539C22C2×#1451973911yescharacteristic
540C22#1684032311yescharacteristic
541C22#1684032711yescharacteristic
542C22C2×#1451980111yescharacteristic
543C22#137×D6111yescharacteristic
544C22C2×#1451980911yescharacteristic
545C22#1684055811yescharacteristic
546C4—22no—
547C4—81no—
548C4—82no—
549C4—81no—
550C4—1222no—
551C4—2442no—
552C4—2441no—
553C4—2441no—
554C4—2444no—
555C4—2441no—
556C4#1684023412yes—
557C4#1684035212yes—
558C2—24no—
559C2—4882no—
560C2#2144847611yescharacteristic
561C2C2×#1683986111yescharacteristic
562C2#2145463711yescharacteristic
563C2#2145662711yescharacteristic
564C2#2145665411yescharacteristic
565C2#2145680311yescharacteristic
566C2#2145684911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1852D61
hypocenterC61#22047
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
3. C2×#16839667C2semidirect
4. #21446698C2semidirect
5. #21446507C2semidirect
6. #21446459C2semidirect
7. #21446154C2semidirect
9. #21445753C2semidirect
10. #21445583C2semidirect
11. C61×#1852C2semidirect
18. #16839903C4semidirect
48. C22×Dic122C2×C4semidirect
58. C2×#11758190C2×C4semidirect
76. #14519702C2×C4semidirect
98. C22⋊C8×C61C2×C4semidirect
283. #1852D61semidirect
327. C61#22047semidirect
375. C22⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399032
#214483252
C2×#168396672
#214466982
#214465072
#214464592
#214461542
#214461362
#214457532
#214455832
C61×#18522
#2204761