Finite Group Databasebeta

← #30616694

Subgroups — #30616694

Order 15616.

1258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic
2C2×C4×C4⋊#6923C211yesmaximal, characteristic
3#21461997C212yesmaximal, complemented
4#21461985C212yesmaximal, complemented
5#21461804C214yesmaximal
6C4×#14519840C214yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8C22×C4⋊#6923C2212yes—
9C22×C4×#6923—21no—
10#16840781—21no—
11#16840780—21no—
12#16840779—21no—
13#16840778—21no—
14D61⋊(C4⋊C8)C2212yes—
15C2×#14519840C2218yes—
16D61⋊(C22⋊C8)C2212yes—
17C4×D61⋊C8C2212yes—
18C4×C4⋊#6923—21no—
19C4×C4⋊#6923—21no—
20C4×C4⋊#6923C2212yes—
21C4×C4⋊#6923C2212yes—
22C2×(C2×#6923)⋊C4—21no—
23#16840635C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318—21no—
26#16840317C411yescharacteristic, complemented
27#16840317C411yescharacteristic, complemented
28#16840316—21no—
29#16840316C412yescomplemented
30#16840315—21no—
31#16840314—21no—
32#16840314C412yescomplemented
33C4⋊Q8×D61C2211yescharacteristic
34C4⋊Q8×D61C2211yescharacteristic
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C2212yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C2212yes—
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×D61⋊C8C2314yes—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—28no—
47C2×C4⋊#6923C2314yes—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—44no—
51#14519844—22no—
52C4⋊#92420—22no—
53#14519840—216no—
54#14519834—22no—
55C22⋊#92420—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58(C2×#6923)⋊C4—22no—
59(C2×#6923)⋊C4—22no—
60C2×Q8×D61—21no—
61C2×Q8×D61C2312yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65#14519777C2×C411yescharacteristic, complemented
66#14519777C2×C411yescharacteristic, complemented
67#14519776—21no—
68#14519776—21no—
69#14519776—21no—
70#14519776C2×C412yescomplemented
71#14519775—21no—
72#14519775C2×C412yescomplemented
73#14519769—21no—
74#14519768—21no—
75#14519768—21no—
76#14519767—22no—
77#14519767—22no—
78#14519757—21no—
79#14519757C2×C412yes—
80#14519756—21no—
81#14519756C2×C412yes—
82#14519755—22no—
83#14519755—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753C2×C412yes—
88Q8×Dic61—21no—
89Q8×Dic61C2×C412yescomplemented
90C4×Dic122—21no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122C2×C412yescomplemented
94#14519742—21no—
95#14519742—22no—
96#14519742—22no—
97#14519742C2×C412yescomplemented
98#14519736—21no—
99#14519735—21no—
100#14519735—21no—
101#14519734—22no—
102#14519733—22no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61C2312yes—
109C22⋊C4×D61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊Q8×C61C2×C411yescharacteristic, complemented
123C4⋊Q8×C61C2×C411yescharacteristic, complemented
124#125×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61C2×C412yes—
127C4×Q8×C61—21no—
128C4×Q8×C61C2×C412yescomplemented
129#117×C61—21no—
130C2×C42×C61C2×C411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C22×#6923—28no—
133C22×#6923—24no—
134C22×#6923—44no—
135C22×#6923—44no—
136C2×Q8×C61—21no—
137C2×Q8×C61C22×C412yes—
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—28no—
145Q8×D61—44no—
146C2×C4×D61—24no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—42no—
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61C22⋊C411yescharacteristic
154C2×C4×D61SD1614yes—
155C2×C4×D61Q1614yes—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122C22×C412yes—
168C2×#92420—24no—
169C4⋊#6923—28no—
170C4⋊#6923—24no—
171C4⋊#6923#4118yes—
172C4×#6923—44no—
173C4×#6923—44no—
174D61⋊C8—28no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—22no—
182C42×C61—22no—
183C42×C61C22⋊C411yescharacteristic
184C42×C61C22⋊C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C22⋊Dic61—22no—
188#11758198—22no—
189#11758198—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22×C411yescharacteristic
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C23×C61C2×C22⋊C411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61C2×C22⋊C411yescharacteristic
223C22×D61C2×SD1612yes—
224C22×D61C2×Q1612yes—
225C22×D61C22×D411yescharacteristic
226C2×#6923—28no—
227C2×#6923—416no—
228C2×#6923—44no—
229C2×#6923—44no—
230Q8×C61—24no—
231Q8×C61—42no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61#10314yescomplemented
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C23×C411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—41no—
247C2×Dic61#10314yescomplemented
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—44no—
262C4×D61—44no—
263C4×D61#10314yescomplemented
264C4×D61#10314yescomplemented
265C4×D61C2×SD1612yes—
266C4×D61C2×SD1612yes—
267C4×D61C2×Q1612yes—
268C4×D61C2×Q1612yes—
269C4×D61C2×#4114yes—
270Dic122—24no—
271Dic122—44no—
272Dic122—42no—
273Dic122—44no—
274#92420—44no—
275#21385—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61C2×#10312yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61C2×#10312yes—
285C2×D61#52512yes—
286C2×D61C22×SD1611yescharacteristic
287C2×D61C22×Q1611yescharacteristic
288C2×D61#57512yes—
289#6923—48no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61C2×#10312yes—
296C4×C61C2×#10312yes—
297C4×C61C4×#4112yes—
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—42no—
302Dic61—42no—
303Dic61C2×#10312yes—
304Dic61C2×#10312yes—
305Dic61#53212yes—
306C2×#533—611no—
307C2×#431—611no—
308C2×C4×C4⋊C4—611no—
309#1987—612no—
310#1981—612no—
311#1872—614no—
312C4×#103—614no—
313C2×C61—22no—
314C2×C61#229711yescharacteristic
315C2×C61C22×#10311yescharacteristic
316C2×C61#289611yescharacteristic
317D61—22no—
318D61#310112yes—
319D61#310412yes—
320#533—611no—
321#533—611no—
322#533—612no—
323#533—612no—
324#533—611no—
325#533—1221no—
326#533—1221no—
327#533—1221no—
328#533—1221no—
329#533#692311yescharacteristic, complemented
330C2×C4⋊Q8—611no—
331C2×C4⋊Q8—611no—
332C2×#125—1221no—
333C2×C22⋊Q8—612no—
334C2×C22⋊Q8—1221no—
335C2×C4×Q8—612no—
336C2×C4×Q8—1221no—
337C2×#117—1221no—
338C22×C4⋊C4—612no—
339C22×C42—611no—
340C22×C42—1221no—
341#431—1221no—
342#431—1221no—
343#431—1221no—
344#431—1221no—
345C2×C4⋊C8—612no—
346C2×#103—618no—
347C2×C22⋊C8—612no—
348C2×C4×C8—612no—
349C4×C4⋊C4—612no—
350C4×C4⋊C4—612no—
351C4×C4⋊C4—1221no—
352C4×C4⋊C4—1221no—
353C2×#95—1221no—
354#340—618no—
355C61#2138511yescharacteristic, complemented, Sylow
356C22×Q8—612no—
357C22×Q8—1221no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—1222no—
362C22×C8—614no—
363C4⋊Q8—612no—
364C4⋊Q8—612no—
365C4⋊Q8—611no—
366C4⋊Q8—611no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8C2×#692311yescharacteristic, complemented
372C4⋊Q8C2×#692311yescharacteristic, complemented
373#125—21no—
374#125—1222no—
375#125—1222no—
376#125—1221no—
377#125—1221no—
378#125—1221no—
379C22⋊Q8—21no—
380C22⋊Q8—612no—
381C22⋊Q8—612no—
382C22⋊Q8—612no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390C22⋊Q8C2×#692312yes—
391C4×Q8—21no—
392C4×Q8—612no—
393C4×Q8—612no—
394C4×Q8—612no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C2×#692312yescomplemented
403#117—21no—
404#117—1222no—
405#117—1222no—
406#117—1221no—
407#117—1221no—
408#117—1221no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1228no—
417C2×C4⋊C4—1222no—
418C2×C22⋊C4—1222no—
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—611no—
429C2×C42—611no—
430C2×C42—1222no—
431C2×C42—1222no—
432C2×C42—1222no—
433C2×C42—2444no—
434C2×C42C2×#692311yescharacteristic
435C4⋊C8—1222no—
436C4⋊C8—1222no—
437#103—12216no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C4×C8—1222no—
441C4×C8—1222no—
442#95—1222no—
443#95—1222no—
444C24—611no—
445C2×Q8—21no—
446C2×Q8—612no—
447C2×Q8—1222no—
448C2×Q8—1222no—
449C2×Q8—1228no—
450C2×Q8—1221no—
451C2×Q8—2444no—
452C2×Q8C22×#692312yes—
453C22×C4—611no—
454C22×C4—614no—
455C22×C4—614no—
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—611no—
470C22×C4—1224no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1224no—
474C22×C4—1228no—
475C22×C4—2442no—
476C22×C4—2444no—
477C22×C4—2444no—
478C22×C4C22⋊#692311yescharacteristic
479C22×C4C22⋊#692311yescharacteristic
480C22×C4C22×#692311yescharacteristic
481C2×C8—1224no—
482C2×C8—1228no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—618no—
489C4⋊C4—612no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1224no—
499C4⋊C4—1228no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4C22×#692312yes—
506C22⋊C4—22no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—22no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42—2444no—
527C42—2444no—
528C42C22⋊#692311yescharacteristic
529C42C22⋊#692311yescharacteristic
530C42C22×#692311yescharacteristic
531C42C22×#692311yescharacteristic
532C23—611no—
533C23—612no—
534C23—612no—
535C23—611no—
536C23—1222no—
537C23—1222no—
538C23C2×C22⋊#692311yescharacteristic
539Q8—24no—
540Q8—42no—
541Q8—1224no—
542Q8—2442no—
543Q8—2444no—
544Q8—2444no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—614no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—614no—
554C2×C4—614no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—612no—
559C2×C4—614no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1228no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2444no—
575C2×C4—2444no—
576C2×C4—2441no—
577C2×C4—2444no—
578C2×C4—2444no—
579C2×C4—24416no—
580C2×C4#1451984014yescomplemented
581C2×C4C2×C22⋊#692311yescharacteristic
582C2×C4C2×C22⋊#692311yescharacteristic
583C2×C4C2×C22⋊#692311yescharacteristic
584C2×C4C2×C22⋊#692311yescharacteristic
585C2×C4C2×C22⋊#692311yescharacteristic
586C2×C4C23×#692311yescharacteristic
587C8—2444no—
588C22—21no—
589C22—21no—
590C22—612no—
591C22—611no—
592C22—611no—
593C22—612no—
594C22—612no—
595C22—1224no—
596C22—1224no—
597C22—1221no—
598C22—1221no—
599C22C2×#1451984012yes—
600C22C22×C22⋊#692311yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—42no—
605C4—42no—
606C4—612no—
607C4—612no—
608C4—612no—
609C4—1221no—
610C4—1221no—
611C4—1221no—
612C4—2448no—
613C4—2442no—
614C4—2442no—
615C4C2×#1451984012yes—
616C4C2×#1451984012yes—
617C4#41×#692312yes—
618C2—22no—
619C2—612no—
620C2—612no—
621C2—1222no—
622C2#2146228011yescharacteristic
623C2C22×#1451984011yescharacteristic
624C2#2146243911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#21385
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461997C2semidirect
4. #21461985C2semidirect
6. C4×#14519840C2semidirect
24. #16840319C4semidirect
26. #16840317C4semidirect
27. #16840317C4semidirect
29. #16840316C4semidirect
32. #16840314C4semidirect
42. C61×#533C4semidirect
65. #14519777C2×C4semidirect
66. #14519777C2×C4semidirect
70. #14519776C2×C4semidirect
72. #14519775C2×C4semidirect
89. Q8×Dic61C2×C4semidirect
93. C4×Dic122C2×C4semidirect
97. #14519742C2×C4semidirect
122. C4⋊Q8×C61C2×C4semidirect
123. C4⋊Q8×C61C2×C4semidirect
128. C4×Q8×C61C2×C4semidirect
236. C2×C4×C61#103semidirect
247. C2×Dic61#103semidirect
263. C4×D61#103semidirect
264. C4×D61#103semidirect
329. #533#6923semidirect
355. C61#21385semidirect
371. C4⋊Q8C2×#6923semidirect
372. C4⋊Q8C2×#6923semidirect
402. C4×Q8C2×#6923semidirect
580. C2×C4#14519840semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
C2×C4×C4⋊#69232
#214619972
#214619852
#214618042
C4×#145198402
#533×D612
#2138561