Finite Group Databasebeta

← #29685327

Subgroups — #29685327

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459112C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3#376×D61C211yesmaximal, characteristic, complemented
4#21446798C212yesmaximal, complemented
5#21446649C212yesmaximal, complemented
6#21446261C212yesmaximal, complemented
7#21446259C212yesmaximal, complemented
8#21445632C211yesmaximal, characteristic, complemented
9#21445630C211yesmaximal, characteristic, complemented
10#21445629C211yesmaximal, characteristic, complemented
11C61×#1759C211yesmaximal, characteristic, complemented
12#16840316C412yescomplemented
13#16840315C2211yescharacteristic
14#16840309C412yescomplemented
15#16840308C2211yescharacteristic
16#16840299C412yescomplemented
17#16840294C412yescomplemented
18#16840292—22no—
19#16840290—22no—
20#16840274C2211yescharacteristic
21#16840271—22no—
22#16840269C2211yescharacteristic
23C4×#11758222—22no—
24C4×#11758222C2211yescharacteristic
25C2×C42×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839817—22no—
28#16839813—22no—
29#16839785C2212yes—
30C2×#14519705C2212yes—
31C2×#14519704C2212yes—
32C2×#14519703C2212yes—
33#16839725C2212yes—
34C4×#11758194C2211yescharacteristic
35#16839608C2211yescharacteristic
36#16839608C2212yes—
37#16839607—22no—
38#95×D61—22no—
39#16839523C2212yes—
40#16839522C2212yes—
41#16839521C2212yes—
42C4×#41×C61C2211yescharacteristic
43C2×#104×C61C2212yes—
44C4×M16×C61C2211yescharacteristic
45C61×#376C2211yescharacteristic
46C61×#339C2212yes—
47C2×#11758222—22no—
48C2×#11758222C2311yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61D412yes—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194C2312yes—
53#14519776C2×C412yescomplemented
54#14519775C2×C411yescharacteristic, complemented
55#14519773C2×C411yescharacteristic, complemented
56#14519772—22no—
57#14519770—22no—
58#14519768C2×C412yescomplemented
59#14519767—22no—
60#14519766C2×C411yescharacteristic, complemented
61#14519764C2×C411yescharacteristic, complemented
62#14519763—22no—
63#14519762—22no—
64#14519761—21no—
65#14519759—22no—
66#14519758—21no—
67#14519757C2×C412yes—
68#14519755—22no—
69#14519754—22no—
70#14519753C2×C412yes—
71#14519752C2×C412yes—
72#14519750—22no—
73#14519749—22no—
74#14519748C2×C412yes—
75#14519747—21no—
76#14519745—21no—
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122C2×C412yescomplemented
80C4×D244—22no—
81C4×D244—22no—
82C4×D244C2×C412yescomplemented
83C4×C22⋊D61—22no—
84C4×C22⋊D61—22no—
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88#14519736—22no—
89#14519736C2311yescharacteristic
90#14519734—22no—
91#14519734—21no—
92#14519734—22no—
93C2×C4×Dic61—22no—
94C2×C4×Dic61D412yes—
95C2×C4×Dic61C2312yes—
96C2×C4×Dic61C2311yescharacteristic
97C42×D61—21no—
98C42×D61—24no—
99C42×D61—24no—
100C42×D61C2×C412yes—
101C42×D61C2×C412yes—
102C42×D61D412yes—
103C42×D61D412yes—
104#14519706—24no—
105#14519705—24no—
106#14519704—24no—
107#14519703—24no—
108#14519689—24no—
109#14519688—21no—
110C4×#92418—21no—
111#14519684—22no—
112#14519684—24no—
113C2×#41×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C4×Q8×C61C2×C412yescomplemented
116C4×D4×C61—22no—
117C4×D4×C61C2×C412yescomplemented
118#117×C61—21no—
119#117×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2312yes—
122C2×C42×C61C2311yescharacteristic
123#104×C61—24no—
124#97×C61—21no—
125C4×C8×C61—21no—
126#95×C61—22no—
127C23×D61C2×D411yescharacteristic
128#41×C61—22no—
129#41×C61—22no—
130C2×Q8×C61C22×C411yescharacteristic
131C2×D4×C61—21no—
132C2×D4×C61C22×C411yescharacteristic
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61—22no—
141C22×Dic61—22no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D412yes—
144#11758222—22no—
145#11758222—22no—
146#11758222—48no—
147C2×D244—22no—
148C2×D244C22×C411yescharacteristic
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—28no—
152C2×C4×D61—22no—
153C2×C4×D61—21no—
154C2×C4×D61—44no—
155C2×C4×D61—48no—
156C2×C4×D61C22⋊C412yes—
157C2×C4×D61C22⋊C412yes—
158C2×C4×D61C22⋊C412yes—
159C2×C4×D61C22⋊C412yes—
160C2×C4×D61C22×C412yes—
161C2×C4×D61C2×D412yes—
162C2×Dic122—22no—
163C2×Dic122C22×C411yescharacteristic
164M16×C61—22no—
165M16×C61—22no—
166C2×C8×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61—22no—
174C42×C61C22×C412yes—
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61#4114yescomplemented
178C22⋊Dic61—21no—
179C22⋊Dic61—22no—
180C22⋊Dic61—22no—
181#11758198—22no—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C22×C412yes—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194C4×Dic61—28no—
195C4×Dic61—22no—
196C4×Dic61—21no—
197C4×Dic61—22no—
198C4×Dic61C22⋊C412yes—
199C4×Dic61C22⋊C412yes—
200C4×Dic61C22⋊C412yes—
201C4×Dic61C22⋊C412yes—
202C4×Dic61C22×C411yescharacteristic
203C4×Dic61C22×C411yescharacteristic
204C4×Dic61C2×D412yes—
205C4×Dic61#4114yescomplemented
206#11758194—22no—
207#11758194—22no—
208C2×#92418—22no—
209C23×C61—21no—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—24no—
213C22×D61—21no—
214C22×D61C2×C22⋊C412yes—
215Q8×C61—21no—
216Q8×C61—21no—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—42no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4112yes—
232C2×C4×C61C2×#4112yes—
233C22⋊D61—44no—
234C22⋊D61—42no—
235C22⋊D61—44no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—24no—
240C2×Dic61—42no—
241C2×Dic61—44no—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C412yes—
245D244—21no—
246D244—21no—
247D244—44no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—24no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—44no—
256C4×D61—416no—
257C4×D61—42no—
258Dic122—21no—
259Dic122—21no—
260Dic122—44no—
261C8×C61—42no—
262#92418—42no—
263#21417—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61#52512yes—
269C2×D61—28no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—42no—
273C2×D61—41no—
274C2×D61#42114yes—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—44no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61C4×#4111yescharacteristic
281C4×C61C4×#4111yescharacteristic
282C4×C61#53212yes—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—44no—
287Dic61—42no—
288Dic61—41no—
289#3444—611no—
290#2920—611no—
291C2×#376—611no—
292#1995—612no—
293#1963—612no—
294#1896—612no—
295#1896—612no—
296#1759—611no—
297#1759—611no—
298#1759—611no—
299#1759D6111yescharacteristic, complemented
300C2×C61—22no—
301C2×C61—81no—
302C2×C61#229711yescharacteristic
303C2×C61C2×#42112yes—
304D61—24no—
305D61—81no—
306#533—611no—
307#533—612no—
308#532—612no—
309#532—611no—
310#529—612no—
311#529—612no—
312#529—1222no—
313#529—1222no—
314#525—611no—
315#525—611no—
316#525—1222no—
317C4×#41—611no—
318C4×#41—1222no—
319C4×#41C2×D6111yescharacteristic
320C22×C42—611no—
321C22×C42—612no—
322#432—1222no—
323#431—1222no—
324#423—612no—
325C2×#104—612no—
326C2×#104—612no—
327C2×#104—612no—
328C2×#104C2×D6112yes—
329#407—612no—
330C4×M16—611no—
331C4×M16C2×D6111yescharacteristic
332#376—612no—
333#376—611no—
334#376—1222no—
335#376C2×D6111yescharacteristic
336C2×#95—1222no—
337#339—612no—
338#339—612no—
339#339—612no—
340#339C2×D6112yes—
341C61#2141711yescharacteristic, complemented, Sylow
342C2×#41—611no—
343C2×#41—1222no—
344C2×#41C22×D6111yescharacteristic
345C23×C4—611no—
346C23×C4—612no—
347C23×C4—1222no—
348C2×M16—612no—
349C2×M16C22×D6112yes—
350C4⋊Q8—612no—
351C4⋊Q8—611no—
352#127—611no—
353#126—1222no—
354#126—1222no—
355#125—612no—
356#125—1222no—
357#124—611no—
358#124—611no—
359#124—1222no—
360#123—1221no—
361#123—1222no—
362#123—1222no—
363#123—1221no—
364C22⋊Q8—612no—
365C22⋊Q8—612no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368#121—612no—
369#121—612no—
370#121—1222no—
371#121—1222no—
372#120—1221no—
373#120—1221no—
374C4×Q8—612no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×Q8C4×D6112yescomplemented
378C4×D4—22no—
379C4×D4—612no—
380C4×D4—1222no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
385C4×D4—1222no—
386C4×D4—1222no—
387C4×D4C4×D6112yescomplemented
388#117—21no—
389#117—611no—
390#117—1222no—
391#117—1222no—
392#117—1222no—
393#117—1221no—
394#117C22×D6111yescharacteristic
395C2×C42—21no—
396C2×C42—612no—
397C2×C42—611no—
398C2×C42—612no—
399C2×C42—612no—
400C2×C42—612no—
401C2×C42—612no—
402C2×C42—612no—
403C2×C42—1224no—
404C2×C42—1222no—
405C2×C42—1224no—
406C2×C42—1221no—
407C2×C42C22×D6112yes—
408C2×C42C22×D6111yescharacteristic
409C4⋊C8—1224no—
410#104—24no—
411#104—1224no—
412#104—1224no—
413#104—1224no—
414C22⋊C8—1224no—
415#97—21no—
416#97—1221no—
417C4×C8—21no—
418C4×C8—1221no—
419#95—22no—
420#95—1224no—
421#95—1222no—
422C24—611no—
423#41—22no—
424#41—22no—
425#41—1222no—
426#41—1222no—
427#41—2448no—
428C2×Q8—611no—
429C2×Q8—1222no—
430C2×Q8C2×C4×D6111yescharacteristic
431C2×D4—21no—
432C2×D4—611no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—1221no—
437C2×D4C2×C4×D6111yescharacteristic
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—1228no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—2444no—
456C22×C4—2448no—
457C22×C4D4×D6112yes—
458C22×C4C23×D6111yescharacteristic
459M16—22no—
460M16—22no—
461M16—1222no—
462M16—1222no—
463C2×C8—22no—
464C2×C8—1222no—
465C4⋊C4—22no—
466C4⋊C4—612no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4C2×C4×D6112yes—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C22⋊C4—1222no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C22⋊C4—1221no—
484C22⋊C4—1222no—
485C42—21no—
486C42—22no—
487C42—22no—
488C42—611no—
489C42—611no—
490C42—612no—
491C42—612no—
492C42—612no—
493C42—612no—
494C42—612no—
495C42—614no—
496C42—1228no—
497C42—1221no—
498C42—1222no—
499C42—1222no—
500C42C2×C4×D6112yes—
501C42#1175822214yescomplemented
502C42D4×D6111yescharacteristic
503C42D4×D6111yescharacteristic
504C23—21no—
505C23—612no—
506C23—1224no—
507C23—1222no—
508C23—1221no—
509C23C2×D4×D6111yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—1221no—
513Q8—1221no—
514Q8—2444no—
515D4—21no—
516D4—21no—
517D4—42no—
518D4—1221no—
519D4—1221no—
520D4—2444no—
521D4—2444no—
522D4—2444no—
523D4—2442no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1224no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2444no—
546C2×C4—24416no—
547C2×C4—2442no—
548C2×C4—2444no—
549C2×C4—2442no—
550C2×C4C22⋊C4×D6112yes—
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22×C4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4#41×D6112yes—
555C2×C4C2×#1175822212yes—
556C8—42no—
557C8—2442no—
558C22—22no—
559C22—42no—
560C22—41no—
561C22—614no—
562C22—1228no—
563C22—1222no—
564C22—1222no—
565C22—2442no—
566C22—2441no—
567C22C2×C22⋊C4×D6111yescharacteristic
568C22#1684027112yes—
569C4—22no—
570C4—22no—
571C4—44no—
572C4—42no—
573C4—41no—
574C4—1222no—
575C4—1222no—
576C4—1222no—
577C4—2444no—
578C4—2442no—
579C4—2441no—
580C4#1684023011yescharacteristic
581C4C4×#1175822211yescharacteristic
582C4#1684030912yes—
583C2—22no—
584C2—81no—
585C2—1224no—
586C2—4881no—
587C2#2144830511yescharacteristic
588C2#421×D6112yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1759C2
socleC22×C61C22×C22⋊C4
prime radical#21459112C2
cosocleC22×C4×C61C24
hypercenter#1759D61
hypocenterC61#21417
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #376×D61C2semidirect
4. #21446798C2semidirect
5. #21446649C2semidirect
6. #21446261C2semidirect
7. #21446259C2semidirect
8. #21445632C2semidirect
9. #21445630C2semidirect
10. #21445629C2semidirect
11. C61×#1759C2semidirect
12. #16840316C4semidirect
14. #16840309C4semidirect
16. #16840299C4semidirect
17. #16840294C4semidirect
53. #14519776C2×C4semidirect
54. #14519775C2×C4semidirect
55. #14519773C2×C4semidirect
58. #14519768C2×C4semidirect
60. #14519766C2×C4semidirect
61. #14519764C2×C4semidirect
79. C4×Dic122C2×C4semidirect
82. C4×D244C2×C4semidirect
115. C4×Q8×C61C2×C4semidirect
117. C4×D4×C61C2×C4semidirect
177. C42×C61#41semidirect
205. C4×Dic61#41semidirect
299. #1759D61semidirect
341. C61#21417semidirect
377. C4×Q8C4×D61semidirect
387. C4×D4C4×D61semidirect
501. C42#11758222semidirect

Maximal subgroups

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

subgroupindex
#214591122
#214544502
#376×D612
#214467982
#214466492
#214462612
#214462592
#214456322
#214456302
#214456292
C61×#17592
#2141761