Finite Group Databasebeta

← #29685312

Subgroups — #29685312

Order 15616.

1430 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459111C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3C2×C42×Dic61C211yesmaximal, characteristic, complemented
4#21446794C212yesmaximal, complemented
5#21446645C212yesmaximal, complemented
6#21446263C211yesmaximal, characteristic, complemented
7#21446262C211yesmaximal, characteristic, complemented
8#21446260C211yesmaximal, characteristic, complemented
9#104×Dic61C212yesmaximal, complemented
10#21445623C212yesmaximal, 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×Dic61—22no—
26C22×C4×Dic61C2211yescharacteristic
27C22×C4×Dic61C2211yescharacteristic
28#16839816—22no—
29#16839812—22no—
30#16839785C2212yes—
31C2×#14519705C2212yes—
32C2×#14519704C2212yes—
33C2×#14519703C2212yes—
34#16839726C2211yescharacteristic
35M16×Dic61C2212yes—
36C42×Dic61—22no—
37C42×Dic61C2212yes—
38C42×Dic61C2212yes—
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—216no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—24no—
94C2×C4×Dic61—24no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—24no—
97C2×C4×Dic61C2×C412yes—
98C2×C4×Dic61C2×C412yes—
99C2×C4×Dic61D412yes—
100C2×C4×Dic61D412yes—
101C2×C4×Dic61D412yes—
102C2×C4×Dic61C2312yes—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105C42×D61—21no—
106#14519706—24no—
107#14519705—24no—
108#14519704—24no—
109#14519703—24no—
110C22⋊#92418—22no—
111#14519687—22no—
112C8×Dic61—22no—
113C2×#41×C61C2311yescharacteristic
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C2×M16×C61C2312yes—
117#123×C61—21no—
118C22⋊Q8×C61C2×C412yes—
119#121×C61C2×C412yes—
120#120×C61—21no—
121C4×D4×C61—22no—
122#117×C61C2311yescharacteristic
123C2×C42×C61—21no—
124C2×C42×C61—24no—
125C2×C42×C61C2312yes—
126#104×C61—24no—
127C22⋊C8×C61—22no—
128C24×C61C2×D411yescharacteristic
129#41×C61—22no—
130#41×C61—22no—
131C2×Q8×C61C22×C411yescharacteristic
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61C22×C411yescharacteristic
135C22×C4×C61—24no—
136C22×C4×C61—28no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—22no—
146C2×C22⋊D61—21no—
147C2×C22⋊D61—22no—
148C22×Dic61—28no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—28no—
152C22×Dic61—22no—
153C22×Dic61C22⋊C412yes—
154C22×Dic61C22⋊C412yes—
155C22×Dic61C22⋊C412yes—
156C22×Dic61C22⋊C412yes—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D412yes—
159#11758224—48no—
160#11758222—22no—
161#11758222—22no—
162C2×D244C22×C411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—22no—
165C2×Dic122—22no—
166C2×Dic122C22×C411yescharacteristic
167M16×C61—22no—
168M16×C61—22no—
169C2×C8×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61C22×C412yes—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—22no—
175C42×C61—24no—
176C42×C61—21no—
177C42×C61—22no—
178C42×C61#4114yescomplemented
179C22⋊Dic61—22no—
180C22⋊Dic61—21no—
181C22⋊Dic61—22no—
182C22⋊Dic61—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—216no—
195C4×Dic61—216no—
196C4×Dic61—24no—
197C4×Dic61—21no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22⋊C412yes—
202C4×Dic61C22⋊C412yes—
203C4×Dic61C22⋊C412yes—
204C4×Dic61C22⋊C412yes—
205C4×Dic61C22×C412yes—
206C4×Dic61C22×C411yescharacteristic
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D412yes—
211C4×Dic61#4114yescomplemented
212#11758194—22no—
213#11758194—22no—
214C2×#92418—22no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—24no—
218C23×C61—21no—
219C23×C61C2×C22⋊C412yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222Q8×C61—22no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—44no—
227C2×C4×C61—28no—
228C2×C4×C61—216no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C412yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4112yes—
241C2×C4×C61C2×#4112yes—
242C22⋊D61—41no—
243C22⋊D61—41no—
244C22⋊D61—44no—
245C22⋊D61—44no—
246C2×Dic61—28no—
247C2×Dic61—216no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61#10418yescomplemented
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262D244—22no—
263C4×D61—42no—
264C4×D61—44no—
265Dic122—22no—
266Dic122—44no—
267C8×C61—42no—
268#92418—42no—
269#21415—611nomaximal, Sylow
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—42no—
276C22×C61—41no—
277C22×C61C2×#10414yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61#52511yescharacteristic
280C22×C61#52511yescharacteristic
281C2×D61—42no—
282C2×D61—41no—
283C4×C61—28no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61C4×#4112yes—
288C4×C61#53212yes—
289Dic61—28no—
290Dic61—22no—
291Dic61—22no—
292Dic61—42no—
293Dic61—41no—
294Dic61C2×#10414yes—
295Dic61C2×#10414yes—
296#3444—611no—
297#2920—611no—
298C2×C43—611no—
299#1994—612no—
300#1962—612no—
301#1896—611no—
302#1896—611no—
303#1896—611no—
304#1896D6111yescharacteristic, complemented
305C4×#104—612no—
306C4×#104—612no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—24no—
310C2×C61—81no—
311C2×C61#229711yescharacteristic
312C2×C61C22×#10412yes—
313D61—81no—
314#533—611no—
315#533—612no—
316#532—612no—
317#532—611no—
318#529—612no—
319#529—612no—
320#529—1222no—
321#529—1222no—
322#525—611no—
323#525—1222no—
324#525C2×D6111yescharacteristic
325C4×#41—611no—
326C4×#41—611no—
327C4×#41—1222no—
328C22×C42—611no—
329C22×C42—611no—
330C22×C42—1222no—
331C22×C42C2×D6111yescharacteristic
332#432—1222no—
333#431—1222no—
334#423—612no—
335C2×#104—612no—
336C2×#104—612no—
337C2×#104—612no—
338C2×#104C2×D6112yes—
339#407—611no—
340#407C2×D6111yescharacteristic
341C4×M16—612no—
342C43—612no—
343C43—612no—
344C43—1222no—
345#339—612no—
346#339—612no—
347#339—612no—
348#339C2×D6112yes—
349C61#2141511yescharacteristic, complemented, Sylow
350C2×#41—611no—
351C2×#41—1222no—
352C2×#41C22×D6111yescharacteristic
353C23×C4—21no—
354C23×C4—612no—
355C23×C4—1221no—
356C23×C4C22×D6111yescharacteristic
357C2×M16—612no—
358C2×M16C22×D6112yes—
359C4⋊Q8—612no—
360C4⋊Q8—611no—
361#127—611no—
362#126—1222no—
363#126—1222no—
364#125—612no—
365#125—1222no—
366#124—611no—
367#124—611no—
368#124—1222no—
369#123—21no—
370#123—1222no—
371#123—1222no—
372#123—1221no—
373C22⋊Q8—612no—
374C22⋊Q8—1222no—
375C22⋊Q8—1222no—
376C22⋊Q8C4×D6112yes—
377#121—612no—
378#121—1222no—
379#121—1222no—
380#121C4×D6112yes—
381#120—21no—
382#120—1221no—
383C4×Q8—612no—
384C4×Q8—612no—
385C4×Q8—1222no—
386C4×Q8—1222no—
387C4×D4—22no—
388C4×D4—612no—
389C4×D4—612no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—1222no—
397#117—611no—
398#117—1221no—
399#117—1222no—
400#117—1222no—
401#117—1221no—
402#117—1222no—
403#117C22×D6111yescharacteristic
404C2×C42—21no—
405C2×C42—24no—
406C2×C42—612no—
407C2×C42—612no—
408C2×C42—612no—
409C2×C42—612no—
410C2×C42—612no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—612no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1224no—
417C2×C42—1224no—
418C2×C42—1224no—
419C2×C42—12216no—
420C2×C42—1221no—
421C2×C42C22×D6112yes—
422C4⋊C8—1224no—
423#104—24no—
424#104—1224no—
425#104—1224no—
426#104—1224no—
427C22⋊C8—22no—
428C22⋊C8—1222no—
429#97—1222no—
430C4×C8—1222no—
431C24D4×D6111yescharacteristic
432#41—22no—
433#41—22no—
434#41—1222no—
435#41—1222no—
436#41—2448no—
437C2×Q8—611no—
438C2×Q8—1222no—
439C2×Q8C2×C4×D6111yescharacteristic
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—611no—
443C2×D4—1222no—
444C2×D4—1221no—
445C2×D4—1222no—
446C2×D4C2×C4×D6111yescharacteristic
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—1222no—
460C22×C4—1228no—
461C22×C4—1222no—
462C22×C4—1228no—
463C22×C4—1224no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4C2×C4×D6112yes—
467C22×C4D4×D6112yes—
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470M16—22no—
471M16—22no—
472M16—1222no—
473M16—1222no—
474C2×C8—22no—
475C2×C8—1222no—
476C4⋊C4—22no—
477C4⋊C4—612no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4C2×C4×D6112yes—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—1222no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1221no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C42—24no—
497C42—21no—
498C42—22no—
499C42—612no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—612no—
505C42—612no—
506C42—612no—
507C42—612no—
508C42—612no—
509C42—614no—
510C42—1222no—
511C42—1222no—
512C42—12216no—
513C42—12216no—
514C42—1224no—
515C42—1221no—
516C42—1222no—
517C42#1175822414yescomplemented
518C23—21no—
519C23—21no—
520C23—24no—
521C23—21no—
522C23—1221no—
523C23C22⋊C4×D6112yes—
524C23C2×D4×D6111yescharacteristic
525Q8—22no—
526Q8—1222no—
527Q8—2444no—
528D4—22no—
529D4—41no—
530D4—41no—
531D4—44no—
532D4—1222no—
533D4—2444no—
534D4—2441no—
535D4—2441no—
536D4—2444no—
537C2×C4—28no—
538C2×C4—216no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—42no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—618no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1224no—
556C2×C4—1228no—
557C2×C4—12216no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1222no—
561C2×C4—1221no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2442no—
565C2×C4C22⋊C4×D6112yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4#41×D6112yes—
569C2×C4C2×#1175822412yes—
570C8—42no—
571C8—2442no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—41no—
579C22—2442no—
580C22—2441no—
581C22#104×D6114yes—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22#1684026811yescharacteristic
584C22#1684027011yescharacteristic
585C4—28no—
586C4—22no—
587C4—42no—
588C4—41no—
589C4—614no—
590C4—614no—
591C4—1228no—
592C4—1222no—
593C4—1222no—
594C4—2442no—
595C4—2441no—
596C4C4×#1175822412yes—
597C4#1684030712yes—
598C2—21no—
599C2—21no—
600C2—24no—
601C2—81no—
602C2—4881no—
603C2#2144830411yescharacteristic
604C2C2×#104×D6112yes—
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#21415
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×C42×Dic61C2semidirect
4. #21446794C2semidirect
5. #21446645C2semidirect
6. #21446263C2semidirect
7. #21446262C2semidirect
8. #21446260C2semidirect
9. #104×Dic61C2semidirect
10. #21445623C2semidirect
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
178. C42×C61#41semidirect
211. C4×Dic61#41semidirect
257. C2×Dic61#104semidirect
304. #1896D61semidirect
349. C61#21415semidirect
517. 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×C42×Dic612
#214467942
#214466452
#214462632
#214462622
#214462602
#104×Dic612
#214456232
C61×#18962
#2141561