Finite Group Databasebeta

← #29715692

Subgroups — #29715692

Order 15616.

1478 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519796C211yesmaximal, characteristic, complemented
2C2×#16839990C212yesmaximal, complemented
3C2×#16839827C211yesmaximal, characteristic, complemented
4#21449162C212yesmaximal, complemented
5#21447142C211yesmaximal, characteristic, complemented
6#21446830C211yesmaximal, characteristic, complemented
7#21446814C211yesmaximal, characteristic, complemented
8#21446672C212yesmaximal, complemented
9#21445843C211yesmaximal, characteristic, complemented
10#21445756C212yesmaximal, complemented
11C61×#2054C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C2×#14519796—28no—
14C2×#14519796—24no—
15C22×Dic244C2211yescharacteristic
16C22×#11758200C2212yes—
17C22×C8×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519757C2212yes—
20C2×#14519748C2212yes—
21C2×C4×D244C2211yescharacteristic
22#16839990—28no—
23#16839827—24no—
24C2×#14519708C2212yes—
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839678—22no—
32#16839671—22no—
33#16839668C2212yes—
34#16839639—22no—
35#16839637—22no—
36#16839619C2212yes—
37#16839525C2211yescharacteristic
38C22×D8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#102×C61C2212yes—
41C61×#390C2212yes—
42C61×#340C2211yescharacteristic
43C2×#11758224—24no—
44C2×#11758224—24no—
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—42no—
48C2×#11758224—42no—
49C22×Dic122C2312yes—
50C22×D244D411yescharacteristic
51C22×C22⋊D61—22no—
52C22×C22⋊D61—22no—
53C23×Dic61—22no—
54C22×C4×D61C2311yescharacteristic
55C22×C4×D61C2311yescharacteristic
56#14519796—232no—
57C2×Dic244—24no—
58C2×Dic244—22no—
59C2×#11758200—24no—
60C2×#11758200—24no—
61C2×#11758200D418yescomplemented
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C2×C8×D61—24no—
65C2×C8×D61—24no—
66C2×C8×D61—22no—
67C2×C8×D61D414yescomplemented
68#14519757—24no—
69#14519757—42no—
70#14519748—24no—
71#14519748—42no—
72C4×D244—22no—
73C4×D244—22no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×C22⋊Dic61—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D411yescharacteristic
81C2×#11758198D411yescharacteristic
82#14519708—24no—
83#14519702—24no—
84#14519700—42no—
85#14519698—42no—
86#14519689—22no—
87#14519689—22no—
88C4×#92418—22no—
89#14519684—22no—
90#14519684—22no—
91#14519684—22no—
92C22×D4×C61C2312yes—
93C2×D8×C61—24no—
94C2×D8×C61—22no—
95C22×C8×C61C2311yescharacteristic
96C22×C8×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102#102×C61—24no—
103C24×C61—22no—
104C23×D61C2×D411yescharacteristic
105C23×D61C2×D411yescharacteristic
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61C2×D414yes—
112C22×C4×C61—22no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—44no—
117C2×C22⋊D61—44no—
118C2×C22⋊D61—44no—
119C2×C22⋊D61—44no—
120C2×C22⋊D61—44no—
121C2×C22⋊D61—82no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—44no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129#11758224—216no—
130#11758224—48no—
131#11758224—48no—
132#11758224—48no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—21no—
141C2×C4×D61—22no—
142C2×C4×D61C2×D412yes—
143C2×C4×D61#4112yes—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122C2×D414yes—
150D8×C61—28no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—21no—
154C2×C8×C61C2×D412yes—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C22⋊C4×C61—44no—
158C22⋊C4×C61—82no—
159C42×C61—21no—
160C42×C61—21no—
161C22⋊Dic61—44no—
162#11758200—216no—
163#11758198—22no—
164#11758198—22no—
165#11758198—44no—
166#11758198—82no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—44no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418C2×D412yes—
174C2×#92418C2×D414yes—
175Dic244—28no—
176C8×D61—28no—
177C8×D61—44no—
178C23×C61—24no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61C22×D411yescharacteristic
186C22×D61—21no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61#12112yes—
195C22×D61#13512yescomplemented
196D4×C61—24no—
197D4×C61—24no—
198D4×C61—42no—
199D4×C61—42no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61#13512yescomplemented
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D412yes—
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)12yes—
212C22⋊D61—416no—
213C22⋊D61—88no—
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—416no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61#12112yes—
225D244—28no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—24no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61—42no—
232Dic122—24no—
233Dic122—24no—
234Dic122—42no—
235Dic122—42no—
236C8×C61—22no—
237C8×C61—41no—
238#92418—41no—
239#92418#12014yes—
240#24265—611nomaximal, Sylow
241C22×C61—44no—
242C22×C61—44no—
243C22×C61—44no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61C2×#12111yescharacteristic
249C22×C61D4212yes—
250C22×C61C2×#13511yescharacteristic
251C22×C61C2×#13511yescharacteristic
252C22×C61#57712yes—
253C2×D61—22no—
254C2×D61—24no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—41no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—41no—
263C4×C61—22no—
264C4×C61—41no—
265C4×C61—82no—
266C4×C61C2×#12011yescharacteristic
267C4×C61#53511yescharacteristic
268C4×C61#54612yes—
269Dic61—22no—
270Dic61—44no—
271Dic61—41no—
272Dic61—82no—
273C22×#135—611no—
274C2×#465—612no—
275C2×C8×D4—611no—
276#2407—612no—
277#2056—611no—
278#2054D6111yescharacteristic, complemented
279#2001—611no—
280#1999—611no—
281#1968—612no—
282#1807—611no—
283#1788—612no—
284C2×C61—84no—
285C2×C61#240811yescharacteristic
286C2×C61#305011yescharacteristic
287C2×C61#306011yescharacteristic
288C2×C61#328212yes—
289C2×C61#328412yes—
290D61—42no—
291D61—42no—
292C22×#41—612no—
293C2×#135—1224no—
294C2×#135—1228no—
295C22×Q16—611no—
296C22×SD16—612no—
297C22×D8C2×D6111yescharacteristic
298C23×C8—611no—
299C23×C8—611no—
300C2×C22⋊Q8—612no—
301C2×#121—612no—
302C2×C4×D4—611no—
303#465—1228no—
304C8×D4—1224no—
305C2×#106—612no—
306C2×C4⋊C8C2×D6111yescharacteristic
307C2×#103—612no—
308C2×#103—612no—
309C2×#102—612no—
310C2×#102C2×D6112yes—
311C2×C22⋊C8—611no—
312C2×C22⋊C8—611no—
313C2×C4×C8—611no—
314#394—1222no—
315#392—1222no—
316#391—612no—
317#390C2×D6112yes—
318#386—1222no—
319#385—1222no—
320#380—612no—
321#340—611no—
322#340C2×D6111yescharacteristic
323C61#2426511yescharacteristic, complemented, Sylow
324C2×#41—1224no—
325C2×#41—1224no—
326C2×#41—1224no—
327C2×#41—1224no—
328C2×#41—2442no—
329C2×#41—2442no—
330C22×Q8—612no—
331C22×D4—611no—
332C22×D4—1222no—
333C22×D4—1222no—
334C22×D4C22×D6112yes—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—1222no—
338#135—12232no—
339C2×Q16—1224no—
340C2×Q16—1222no—
341C2×SD16—618no—
342C2×SD16—1224no—
343C2×SD16—1224no—
344C2×D8—24no—
345C2×D8—22no—
346C22×C8—611no—
347C22×C8—614no—
348C22×C8—611no—
349C22×C8—1224no—
350C22×C8—1222no—
351C22×C8—1224no—
352C22×C8C22×D6111yescharacteristic
353C22×C8C22×D6111yescharacteristic
354C22⋊Q8—1224no—
355C22⋊Q8—2442no—
356#121—1224no—
357#121—2442no—
358C4×D4—1222no—
359C4×D4—1222no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—611no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4—1222no—
371C2×C42C22×D6111yescharacteristic
372#106—1224no—
373C4⋊C8—22no—
374#103—1224no—
375#103—2442no—
376#102—24no—
377#102—2442no—
378C22⋊C8—1222no—
379C22⋊C8—1222no—
380C4×C8—1222no—
381#95—1222no—
382#95—1222no—
383#95—1222no—
384C24—22no—
385C24—611no—
386C24—611no—
387#41—12216no—
388#41—2448no—
389#41—2448no—
390#41—2448no—
391C2×Q8—614no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—1222no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—4882no—
406C2×D4—2444no—
407C2×D4—2444no—
408C2×D4—2444no—
409C2×D4—2444no—
410C2×D4—2444no—
411C2×D4D4×D6114yes—
412C22×C4—22no—
413C22×C4—611no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—611no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—2444no—
426C22×C4—2444no—
427C22×C4—2444no—
428C22×C4D4×D6111yescharacteristic
429C22×C4D4×D6111yescharacteristic
430C22×C4C23×D6111yescharacteristic
431Q16—1228no—
432SD16—12216no—
433D8—28no—
434C2×C8—22no—
435C2×C8—21no—
436C2×C8—22no—
437C2×C8—612no—
438C2×C8—614no—
439C2×C8—1228no—
440C2×C8—1221no—
441C2×C8—1222no—
442C2×C8—2444no—
443C2×C8D4×D6112yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—1222no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—2444no—
450C22⋊C4—44no—
451C22⋊C4—82no—
452C22⋊C4—1222no—
453C22⋊C4—1222no—
454C22⋊C4—4882no—
455C22⋊C4—2444no—
456C22⋊C4—2444no—
457C42—21no—
458C42—21no—
459C23—24no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—612no—
467C23—612no—
468C23—1221no—
469C23—1221no—
470C23—1221no—
471C23—1221no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23C2×D4×D6111yescharacteristic
477Q8—1224no—
478Q8—1224no—
479Q8—2442no—
480Q8—2442no—
481D4—24no—
482D4—24no—
483D4—42no—
484D4—42no—
485D4—1228no—
486D4—4888no—
487D4—24416no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—612no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1224no—
500C2×C4—1221no—
501C2×C4—1222no—
502C2×C4—1224no—
503C2×C4—24416no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4#1451979612yescomplemented
513C2×C4C2×D4×D6111yescharacteristic
514C2×C4C2×D4×D6112yes—
515C2×C4#41×D6111yescharacteristic
516C2×C4#1451982312yes—
517C8—22no—
518C8—41no—
519C8—614no—
520C8—2441no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—1222no—
529C22—1222no—
530C22—1222no—
531C22—1224no—
532C22—1222no—
533C22—2441no—
534C22—2442no—
535C22—2442no—
536C22—2442no—
537C22—2441no—
538C22#121×D6111yescharacteristic
539C22#1684039612yes—
540C22#135×D6111yescharacteristic
541C22C2×#1451979611yescharacteristic
542C22#1684057212yes—
543C4—22no—
544C4—41no—
545C4—82no—
546C4—1222no—
547C4—4882no—
548C4—2441no—
549C4—2444no—
550C4#120×D6111yescharacteristic
551C4#1684033011yescharacteristic
552C4#1684040412yes—
553C2—84no—
554C2—2442no—
555C2—2442no—
556C2#2144917011yescharacteristic
557C2#2145539811yescharacteristic
558C2#2145547911yescharacteristic
559C2#2145753212yes—
560C2#2145755212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2054C2
socleC23×C61C22×D4
hypercenter#2054D61
hypocenterC61#24265
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. C22×#14519796C2semidirect
2. C2×#16839990C2semidirect
3. C2×#16839827C2semidirect
4. #21449162C2semidirect
5. #21447142C2semidirect
6. #21446830C2semidirect
7. #21446814C2semidirect
8. #21446672C2semidirect
9. #21445843C2semidirect
10. #21445756C2semidirect
11. C61×#2054C2semidirect
61. C2×#11758200D4semidirect
67. C2×C8×D61D4semidirect
195. C22×D61#135semidirect
207. C2×C4×C61#135semidirect
278. #2054D61semidirect
323. C61#24265semidirect
512. C2×C4#14519796semidirect

Maximal subgroups

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

subgroupindex
C22×#145197962
C2×#168399902
C2×#168398272
#214491622
#214471422
#214468302
#214468142
#214466722
#214458432
#214457562
C61×#20542
#2426561