Finite Group Databasebeta

← #30616688

Subgroups — #30616688

Order 15616.

1164 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758213C211yesmaximal, characteristic, complemented
2#21462266C211yesmaximal, characteristic, complemented
3#21461994C212yesmaximal, complemented
4#21461983C212yesmaximal, complemented
5#21461802C214yesmaximal, complemented
6#21461582C214yesmaximal, complemented
7#21459114C211yesmaximal, characteristic
8C22×#11758213C2212yes—
9C2×#14519841C2218yes—
10C4×#11758213—21no—
11C4×#11758213—21no—
12C4×#11758213C2212yes—
13C4×#11758213C2212yes—
14C2×#14519832—21no—
15C2×C4×#92420—21no—
16#16840707—21no—
17#16840706—21no—
18#16840705—21no—
19#16840704—21no—
20C4×C4⋊#6923C2212yes—
21C4×C22⋊#6923C2212yes—
22#16840698C2212yes—
23#16840632C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840317C411yescharacteristic, complemented
26#16840317C2211yescharacteristic
27#16840310C411yescharacteristic, complemented
28#16840310C2211yescharacteristic
29#16840294—21no—
30#16840294C412yescomplemented
31#16840293—21no—
32#16840293C412yescomplemented
33#16840291—21no—
34#16840291—21no—
35#16840272—21no—
36#16840272C2212yes—
37#16840269—21no—
38#41×Dic61—21no—
39C4×#11758224—21no—
40C4×#11758224C2212yes—
41C2×C42×D61C2211yescharacteristic
42C61×#532C411yescharacteristic, complemented
43C2×#11758213—22no—
44C2×#11758213—28no—
45C2×#11758213C2314yes—
46C22×#92420—22no—
47C2×C4⋊#6923—22no—
48C2×C22⋊#6923—22no—
49C2×C4×#6923C2314yes—
50#14519841—216no—
51#14519832—22no—
52#14519832—22no—
53C4×#92420—22no—
54C4×#92420—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4—22no—
59C2×#11758224—21no—
60C2×#11758224C2312yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519777C2×C411yescharacteristic
66#14519770—22no—
67#14519770—22no—
68#14519769—21no—
69#14519768—21no—
70#14519768C2×C412yes—
71#14519764—21no—
72#14519764C2×C412yes—
73#14519759—22no—
74#14519759—22no—
75#14519758—22no—
76#14519757—21no—
77#14519757C2×C412yes—
78#14519756—21no—
79#14519756C2×C412yes—
80#14519753—21no—
81#14519753—21no—
82#14519748—21no—
83#14519748—21no—
84#14519748—21no—
85#14519748C2×C412yes—
86#14519745—22no—
87Q8×Dic61—21no—
88Q8×Dic61—21no—
89C4×Dic122—21no—
90C4×Dic122C2×C412yes—
91#14519742—21no—
92#14519742C2×C412yes—
93D4×Dic61—22no—
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61—21no—
98D4×Dic61C2×C412yes—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101#14519737—22no—
102#14519737—22no—
103#14519736—21no—
104#14519736—22no—
105#14519735—21no—
106#14519735C2312yes—
107#14519733—22no—
108#14519733—22no—
109C2×C4×Dic61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C2×#41×C61—21no—
122C4⋊Q8×C61C2×C411yescharacteristic
123#127×C61C2×C411yescharacteristic
124#124×C61—21no—
125#121×C61—21no—
126#121×C61C2×C412yes—
127C4×D4×C61—21no—
128C4×D4×C61C2×C412yes—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C22×#6923—24no—
132#41×C61—44no—
133C2×Q8×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—22no—
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C2×C22⋊D61—22no—
143C2×C22⋊D61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149#11758224—28no—
150#11758224—44no—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122C22×C412yes—
168C22⋊#6923—44no—
169C2×#92420—28no—
170C2×#92420—24no—
171C4⋊#6923—44no—
172C4×#6923—28no—
173#11758213—28no—
174#11758213—24no—
175#11758213#4118yescomplemented
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22×C411yescharacteristic
211C4×Dic61C22×C411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C23×C61—22no—
217C23×C61C2×C22⋊C411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—41no—
221C22×D61C2×C22⋊C411yescharacteristic
222C22×D61C22×D411yescharacteristic
223C2×#6923—44no—
224C2×#6923—44no—
225C2×#6923—44no—
226Q8×C61—42no—
227D4×C61—24no—
228D4×C61—42no—
229D4×C61—44no—
230D4×C61—42no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—44no—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C23×C411yescharacteristic
242C22⋊D61—44no—
243C22⋊D61—44no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—44no—
250C2×Dic61—44no—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C22×D411yescharacteristic
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61C2×#4114yes—
269Dic122—24no—
270Dic122—42no—
271#92420—28no—
272#21383—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61#42112yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—44no—
286C2×D61#53212yes—
287#6923—84no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—42no—
293C4×C61#42112yes—
294C4×C61C4×#4112yes—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—21no—
300Dic61—42no—
301Dic61—42no—
302Dic61#52512yes—
303#3444—611no—
304C2×C4×M16—611no—
305#2283—611no—
306#1986—612no—
307#1980—612no—
308#1871—614no—
309#1759—614no—
310C2×C61—22no—
311C2×C61—82no—
312C2×C61#229711yescharacteristic
313C2×C61C2×#42111yescharacteristic
314C2×C61C2×#42111yescharacteristic
315D61—22no—
316D61—41no—
317C22×M16—612no—
318#533—611no—
319#533—611no—
320#533—611no—
321#532—611no—
322#532—611no—
323#532#692311yescharacteristic, complemented
324#529—612no—
325#529—612no—
326#529—1221no—
327#529—1221no—
328#529—1221no—
329#529—1221no—
330#525—612no—
331#525—1221no—
332#525—1221no—
333C4×#41—612no—
334C4×#41—1221no—
335C4×#41—1221no—
336C22×C42—611no—
337C2×#104—618no—
338C4×M16—612no—
339C4×M16—612no—
340C4×M16—1221no—
341C4×M16—1221no—
342C2×#97—1221no—
343C2×C4×C8—1221no—
344#383—1221no—
345#382—1221no—
346#381—1221no—
347#380—1221no—
348C4×C4⋊C4—612no—
349C4×C22⋊C4—612no—
350#376—612no—
351#339—618no—
352C61#2138311yescharacteristic, complemented, Sylow
353C2×#41—21no—
354C2×#41—612no—
355C2×#41—1221no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C2×M16—614no—
360C2×M16—1228no—
361C2×M16—1222no—
362C22×C8—1222no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8C2×#692311yescharacteristic
366#127C2×#692311yescharacteristic
367#126—1222no—
368#126—1222no—
369#125—612no—
370#125—1221no—
371#125—1221no—
372#124—21no—
373#124—612no—
374#124—1221no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384#121—21no—
385#121—612no—
386#121—1221no—
387#121—1221no—
388#121—1221no—
389#121C2×#692312yes—
390#120—1222no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×D4—21no—
398C4×D4—612no—
399C4×D4—1221no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4C2×#692312yes—
409#117—612no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413#117—1221no—
414#117—1221no—
415C2×C4⋊C4—1222no—
416C2×C22⋊C4—1222no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—614no—
429C2×C42—1222no—
430C2×C42C2×#692311yescharacteristic
431#104—12216no—
432#97—1222no—
433#97—1222no—
434C4×C8—1222no—
435C4×C8—1222no—
436#95—1222no—
437#95—1222no—
438#95—1222no—
439#95—1222no—
440C24—611no—
441#41—44no—
442#41—1228no—
443#41—2444no—
444C2×Q8—21no—
445C2×Q8—612no—
446C2×Q8—1221no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4C22×#692312yes—
453C22×C4—22no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—1224no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4C22⋊#692311yescharacteristic
476C22×C4C22⋊#692311yescharacteristic
477C22×C4C22×#692311yescharacteristic
478M16—618no—
479M16—1224no—
480M16—1228no—
481C2×C8—1224no—
482C2×C8—1228no—
483C4⋊C4—21no—
484C4⋊C4—612no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—2444no—
495C4⋊C4C22×#692312yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—2444no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—1222no—
519C42—1222no—
520C42—1228no—
521C42—1222no—
522C42C22⋊#692311yescharacteristic
523C42C22⋊#692311yescharacteristic
524C42C22×#692311yescharacteristic
525C42C22×#692311yescharacteristic
526C23—22no—
527C23—611no—
528C23—611no—
529C23—1222no—
530C23—1222no—
531C23—2441no—
532C23C2×C22⋊#692311yescharacteristic
533Q8—42no—
534Q8—1224no—
535Q8—2442no—
536D4—24no—
537D4—42no—
538D4—44no—
539D4—42no—
540D4—2444no—
541D4—2444no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—611no—
548C2×C4—614no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2444no—
570C2×C4—2441no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4C2×C22⋊#692311yescharacteristic
576C2×C4C2×C22⋊#692311yescharacteristic
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C2×C22⋊#692311yescharacteristic
579C2×C4C2×C22⋊#692311yescharacteristic
580C2×C4C23×#692311yescharacteristic
581C8—1228no—
582C22—21no—
583C22—21no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—612no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—2444no—
594C22#1684076012yes—
595C22C22×C22⋊#692311yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—612no—
602C4—1221no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—1221no—
607C4—4884no—
608C4—2442no—
609C4—2442no—
610C4#1684075912yes—
611C4#41×#692312yes—
612C2—22no—
613C2—82no—
614C2—1222no—
615C2—2441no—
616C2#2146228011yescharacteristic
617C2C2×#1684076011yescharacteristic
618C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC2×C4×D61C24
hypercenter#532#6923
hypocenterC61#21383
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×#11758213C2semidirect
2. #21462266C2semidirect
3. #21461994C2semidirect
4. #21461983C2semidirect
5. #21461802C2semidirect
6. #21461582C2semidirect
24. #16840319C4semidirect
25. #16840317C4semidirect
27. #16840310C4semidirect
30. #16840294C4semidirect
32. #16840293C4semidirect
42. C61×#532C4semidirect
175. #11758213#41semidirect
323. #532#6923semidirect
352. C61#21383semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582132
#214622662
#214619942
#214619832
#214618022
#214615822
#214591142
#2138361