Finite Group Databasebeta

← #29685326

Subgroups — #29685326

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459111C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3C2×#16839608C211yesmaximal, characteristic, complemented
4#21446798C212yesmaximal, complemented
5#21446649C212yesmaximal, complemented
6#21446263C211yesmaximal, characteristic, complemented
7#21446262C211yesmaximal, characteristic, complemented
8#21446260C211yesmaximal, characteristic, complemented
9#21445631C212yesmaximal, complemented
10#21445628C212yesmaximal, complemented
11C61×#1896C211yesmaximal, characteristic, complemented
12#16840315C2211yescharacteristic
13#16840314C412yescomplemented
14#16840308C2211yescharacteristic
15#16840307C412yescomplemented
16#16840298C412yescomplemented
17#16840293C412yescomplemented
18#16840292—22no—
19#16840289—22no—
20#16840273C2211yescharacteristic
21#16840268—22no—
22#41×Dic61C2211yescharacteristic
23#16840230C2211yescharacteristic
24C4×#11758224—22no—
25C22×C4×Dic61C2211yescharacteristic
26C22×C4×Dic61C2211yescharacteristic
27#16839816—22no—
28#16839812—22no—
29#16839785C2212yes—
30C2×#14519705C2212yes—
31C2×#14519704C2212yes—
32C2×#14519703C2212yes—
33#16839726C2211yescharacteristic
34M16×Dic61C2212yes—
35#16839608—22no—
36#16839608C2212yes—
37#16839608C2212yes—
38C2×#14519684—22no—
39#16839523C2212yes—
40#16839522C2212yes—
41#16839521C2212yes—
42C61×#525C2211yescharacteristic
43C22×C42×C61C2211yescharacteristic
44C2×#104×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#339C2212yes—
47C2×#11758222C2311yescharacteristic
48C2×#11758224—22no—
49C23×Dic61—21no—
50C23×Dic61D412yes—
51C2×#11758194C2312yes—
52#14519776C2×C412yescomplemented
53#14519775C2×C411yescharacteristic, complemented
54#14519773C2×C411yescharacteristic, complemented
55#14519772—22no—
56#14519770—22no—
57#14519768C2×C412yescomplemented
58#14519767—22no—
59#14519766C2×C411yescharacteristic, complemented
60#14519764C2×C411yescharacteristic, complemented
61#14519763—22no—
62#14519761—22no—
63#14519760—21no—
64#14519758—22no—
65#14519756C2×C412yes—
66#14519754—22no—
67#14519754—22no—
68#14519751C2×C412yes—
69#14519749—22no—
70#14519749—22no—
71#14519746—21no—
72Q8×Dic61C2×C412yescomplemented
73C4×Dic122—22no—
74#14519742—22no—
75#14519742C2×C412yescomplemented
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61C2×C412yescomplemented
79#14519739C2×C412yescomplemented
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82#14519737—22no—
83#14519737—22no—
84#14519737—22no—
85#14519736—21no—
86#14519736C2311yescharacteristic
87#14519735—22no—
88#14519734—21no—
89#14519733—22no—
90#14519733—22no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—24no—
94C2×C4×Dic61C2×C412yes—
95C2×C4×Dic61C2×C412yes—
96C2×C4×Dic61D412yes—
97C2×C4×Dic61D412yes—
98C2×C4×Dic61D412yes—
99C2×C4×Dic61C2312yes—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C42×D61—21no—
103#14519706—24no—
104#14519705—24no—
105#14519704—24no—
106#14519703—24no—
107C22⋊#92418—22no—
108#14519687—22no—
109C8×Dic61—22no—
110#14519684—24no—
111#14519684—24no—
112C2×#41×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61C2311yescharacteristic
115C2×M16×C61C2312yes—
116#123×C61—21no—
117C22⋊Q8×C61C2×C412yes—
118#121×C61C2×C412yes—
119#120×C61—21no—
120C4×D4×C61—22no—
121#117×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61—24no—
124C2×C42×C61C2312yes—
125#104×C61—24no—
126C22⋊C8×C61—22no—
127C24×C61C2×D411yescharacteristic
128#41×C61—22no—
129#41×C61—22no—
130C2×Q8×C61C22×C411yescharacteristic
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61C22×C411yescharacteristic
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—44no—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—22no—
145C2×C22⋊D61—21no—
146C2×C22⋊D61—22no—
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—44no—
151C22×Dic61—44no—
152C22×Dic61C22⋊C412yes—
153C22×Dic61C22⋊C412yes—
154C22×Dic61C22⋊C412yes—
155C22×Dic61C22⋊C412yes—
156C22×Dic61C2×D412yes—
157C22×Dic61C2×D412yes—
158#11758224—48no—
159#11758222—22no—
160#11758222—22no—
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×Dic122—22no—
165C2×Dic122C22×C411yescharacteristic
166M16×C61—22no—
167M16×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—22no—
174C42×C61—24no—
175C42×C61—21no—
176C42×C61—22no—
177C42×C61#4114yescomplemented
178C22⋊Dic61—22no—
179C22⋊Dic61—21no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61C22×C412yes—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193C4×Dic61—24no—
194C4×Dic61—21no—
195C4×Dic61—22no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61C22⋊C412yes—
199C4×Dic61C22⋊C412yes—
200C4×Dic61C22⋊C412yes—
201C4×Dic61C22⋊C412yes—
202C4×Dic61C22×C412yes—
203C4×Dic61C22×C411yescharacteristic
204C4×Dic61C22×C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D412yes—
208C4×Dic61#4114yescomplemented
209#11758194—22no—
210#11758194—22no—
211C2×#92418—22no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—24no—
215C23×C61—21no—
216C23×C61C2×C22⋊C412yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219Q8×C61—22no—
220D4×C61—22no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—44no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—48no—
232C2×C4×C61—44no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4112yes—
238C2×C4×C61C2×#4112yes—
239C22⋊D61—44no—
240C22⋊D61—41no—
241C22⋊D61—41no—
242C22⋊D61—44no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—44no—
253C2×Dic61—48no—
254C2×Dic61—44no—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259D244—22no—
260C4×D61—44no—
261C4×D61—42no—
262Dic122—22no—
263Dic122—44no—
264C8×C61—42no—
265#92418—42no—
266#21417—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—42no—
273C22×C61—41no—
274C22×C61#42114yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61#52511yescharacteristic
277C22×C61#52511yescharacteristic
278C2×D61—42no—
279C2×D61—41no—
280C4×C61—22no—
281C4×C61—44no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61C4×#4112yes—
285C4×C61#53212yes—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—22no—
290Dic61—44no—
291Dic61—42no—
292Dic61—41no—
293#3444—611no—
294#2920—611no—
295C2×#376—611no—
296#1995—612no—
297#1963—612no—
298#1896—611no—
299#1896—611no—
300#1896—611no—
301#1896D6111yescharacteristic, complemented
302#1759—612no—
303#1759—612no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—24no—
307C2×C61—81no—
308C2×C61#229711yescharacteristic
309C2×C61C2×#42112yes—
310D61—81no—
311#533—611no—
312#533—612no—
313#532—612no—
314#532—611no—
315#529—612no—
316#529—612no—
317#529—1222no—
318#529—1222no—
319#525—611no—
320#525—1222no—
321#525C2×D6111yescharacteristic
322C4×#41—611no—
323C4×#41—611no—
324C4×#41—1222no—
325C22×C42—611no—
326C22×C42—611no—
327C22×C42C2×D6111yescharacteristic
328#432—1222no—
329#431—1222no—
330#423—612no—
331C2×#104—612no—
332C2×#104—612no—
333C2×#104—612no—
334C2×#104C2×D6112yes—
335#407—611no—
336#407C2×D6111yescharacteristic
337C4×M16—612no—
338#376—612no—
339#376—612no—
340#376—1222no—
341C2×#95—1222no—
342#339—612no—
343#339—612no—
344#339—612no—
345#339C2×D6112yes—
346C61#2141711yescharacteristic, complemented, Sylow
347C2×#41—611no—
348C2×#41—1222no—
349C2×#41C22×D6111yescharacteristic
350C23×C4—21no—
351C23×C4—612no—
352C23×C4—1221no—
353C23×C4C22×D6111yescharacteristic
354C2×M16—612no—
355C2×M16C22×D6112yes—
356C4⋊Q8—612no—
357C4⋊Q8—611no—
358#127—611no—
359#126—1222no—
360#126—1222no—
361#125—612no—
362#125—1222no—
363#124—611no—
364#124—611no—
365#124—1222no—
366#123—21no—
367#123—1222no—
368#123—1222no—
369#123—1221no—
370C22⋊Q8—612no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8C4×D6112yes—
374#121—612no—
375#121—1222no—
376#121—1222no—
377#121C4×D6112yes—
378#120—21no—
379#120—1221no—
380C4×Q8—612no—
381C4×Q8—612no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×D4—22no—
385C4×D4—612no—
386C4×D4—612no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394#117—611no—
395#117—1221no—
396#117—1222no—
397#117—1222no—
398#117—1221no—
399#117—1222no—
400#117C22×D6111yescharacteristic
401C2×C42—21no—
402C2×C42—24no—
403C2×C42—612no—
404C2×C42—612no—
405C2×C42—612no—
406C2×C42—612no—
407C2×C42—612no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—612no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1224no—
414C2×C42—1221no—
415C2×C42C22×D6112yes—
416C4⋊C8—1224no—
417#104—24no—
418#104—1224no—
419#104—1224no—
420#104—1224no—
421C22⋊C8—22no—
422C22⋊C8—1222no—
423#97—1222no—
424C4×C8—1222no—
425#95—1224no—
426#95—1224no—
427C24D4×D6111yescharacteristic
428#41—22no—
429#41—22no—
430#41—1222no—
431#41—1222no—
432#41—2448no—
433C2×Q8—611no—
434C2×Q8—1222no—
435C2×Q8C2×C4×D6111yescharacteristic
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—611no—
439C2×D4—1222no—
440C2×D4—1221no—
441C2×D4—1222no—
442C2×D4C2×C4×D6111yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—44no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—2444no—
461C22×C4—2444no—
462C22×C4C2×C4×D6112yes—
463C22×C4D4×D6112yes—
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—22no—
467M16—22no—
468M16—1222no—
469M16—1222no—
470C2×C8—22no—
471C2×C8—1222no—
472C4⋊C4—22no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4C2×C4×D6112yes—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1221no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C42—24no—
493C42—21no—
494C42—22no—
495C42—612no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—612no—
501C42—612no—
502C42—612no—
503C42—612no—
504C42—612no—
505C42—614no—
506C42—1222no—
507C42—1222no—
508C42—1224no—
509C42—1221no—
510C42—1222no—
511C42#1175822414yescomplemented
512C23—21no—
513C23—21no—
514C23—24no—
515C23—21no—
516C23—1221no—
517C23C22⋊C4×D6112yes—
518C23C2×D4×D6111yescharacteristic
519Q8—22no—
520Q8—1222no—
521Q8—2444no—
522D4—22no—
523D4—41no—
524D4—41no—
525D4—44no—
526D4—1222no—
527D4—2444no—
528D4—2441no—
529D4—2441no—
530D4—2444no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1224no—
550C2×C4—2444no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2444no—
556C2×C4—2448no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C22⋊C4×D6112yes—
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4#41×D6112yes—
563C2×C4C2×#1175822412yes—
564C8—42no—
565C8—2442no—
566C22—24no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—41no—
573C22—2442no—
574C22—2441no—
575C22#1683977714yes—
576C22C2×C22⋊C4×D6111yescharacteristic
577C22#1684026811yescharacteristic
578C22#1684027011yescharacteristic
579C4—22no—
580C4—44no—
581C4—42no—
582C4—41no—
583C4—1222no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—2444no—
588C4—2442no—
589C4—2441no—
590C4C4×#1175822412yes—
591C4#1684030712yes—
592C2—21no—
593C2—21no—
594C2—24no—
595C2—81no—
596C2—4881no—
597C2#2144830411yescharacteristic
598C2C2×#1683977712yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1896C2
socleC22×C61C22×C22⋊C4
prime radical#21459111C2
cosocleC22×C4×C61C24
hypercenter#1896D61
hypocenterC61#21417
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1896

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. C2×#16839608C2semidirect
4. #21446798C2semidirect
5. #21446649C2semidirect
6. #21446263C2semidirect
7. #21446262C2semidirect
8. #21446260C2semidirect
9. #21445631C2semidirect
10. #21445628C2semidirect
11. C61×#1896C2semidirect
13. #16840314C4semidirect
15. #16840307C4semidirect
16. #16840298C4semidirect
17. #16840293C4semidirect
52. #14519776C2×C4semidirect
53. #14519775C2×C4semidirect
54. #14519773C2×C4semidirect
57. #14519768C2×C4semidirect
59. #14519766C2×C4semidirect
60. #14519764C2×C4semidirect
72. Q8×Dic61C2×C4semidirect
75. #14519742C2×C4semidirect
78. D4×Dic61C2×C4semidirect
79. #14519739C2×C4semidirect
177. C42×C61#41semidirect
208. C4×Dic61#41semidirect
301. #1896D61semidirect
346. C61#21417semidirect
511. C42#11758224semidirect

Maximal subgroups

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

subgroupindex
#214591112
#214544482
C2×#168396082
#214467982
#214466492
#214462632
#214462622
#214462602
#214456312
#214456282
C61×#18962
#2141761