Finite Group Databasebeta

← #29677819

Subgroups — #29677819

Order 15616.

1598 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
1C2×#16840242C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic
4C2×#16839718C211yesmaximal, characteristic, complemented
5#21445652C212yesmaximal, complemented
6#21445647C212yesmaximal, complemented
7C4×#14519700C212yesmaximal, complemented
8C4×#14519697C212yesmaximal, complemented
9#21445540C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#2274C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840242—24no—
16#16840242C418yescomplemented
17C2×C4×Dic122C2212yes—
18C2×C4×D244C2212yes—
19C2×#14519737—22no—
20C2×#14519735—21no—
21C22×C4⋊Dic61C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23#16839761C418yescomplemented
24#16839756C418yescomplemented
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2212yes—
27C2×#14519700C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2212yes—
30C2×#14519697C2211yescharacteristic
31#16839718—24no—
32C2×#14519688C2212yes—
33C2×C4×#92418C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#117×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C4×C4⋊C4×C61—21no—
42C4×C4⋊C4×C61—21no—
43C4×C4⋊C4×C61C2212yes—
44C4×C4⋊C4×C61C2212yes—
45C4×C22⋊C4×C61—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222C2×C414yes—
50C2×#11758222C2×C414yes—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194C2×C414yescomplemented
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418C2×C414yescomplemented
64C22×#92418C2311yescharacteristic
65C22×#92418C2312yes—
66C22×#92418C2311yescharacteristic
67C4×Dic122—24no—
68C4×Dic122C2×C418yescomplemented
69C4×D244—24no—
70C4×D244C2×C418yescomplemented
71#14519737—216no—
72#14519735—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61C2×C414yes—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C4⋊C4×D61—28no—
79C2×#11758198—22no—
80C2×C4×Dic61—22no—
81#14519700C2×C414yescomplemented
82#14519700C2×C414yescomplemented
83#14519700C2×C418yescomplemented
84#14519697C2×C414yescomplemented
85#14519697C2×C414yescomplemented
86#14519697C2×C418yescomplemented
87#14519688—24no—
88C4×#92418—24no—
89C23×C4×C61—21no—
90C23×C4×C61C2311yescharacteristic
91#117×C61—22no—
92#117×C61C2×C414yes—
93#117×C61C2×C414yescomplemented
94#117×C61D414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61C2×C414yescomplemented
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61—21no—
106C2×C42×C61—22no—
107C2×C42×C61—22no—
108C2×C42×C61C2312yes—
109#95×C61—22no—
110#95×C61—22no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—81no—
118C22×C4×C61—81no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—81no—
136C22×Dic61—81no—
137#11758222—28no—
138#11758222—28no—
139#11758222—28no—
140#11758222C42116yescomplemented
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C22×C412yes—
145C2×D244C22×C412yes—
146C2×D244C22×C412yes—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—24no—
150C2×C4×D61—28no—
151C2×C4×D61—82no—
152C2×C4×D61—82no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C22×C412yes—
157C2×Dic122C22×C412yes—
158C2×Dic122C22×C412yes—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C22⋊C414yes—
164C4⋊C4×C61C22⋊C414yes—
165C4⋊C4×C61C22×C414yes—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C22×C412yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61C22⋊C418yes—
170C42×C61—24no—
171C42×C61—82no—
172C42×C61—82no—
173C42×C61C22×C414yes—
174C42×C61C2×D414yes—
175#11758198—28no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61C22×C414yes—
179#11758196—28no—
180C4×Dic61—28no—
181#11758194—28no—
182C2×#92418—28no—
183C2×#92418—21no—
184C2×#92418—22no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—22no—
189C2×#92418C22×C412yes—
190C2×#92418C22×C412yes—
191C2×#92418C22×C414yes—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61C2×C22⋊C412yes—
197C23×C61C4×D412yes—
198C23×C61C4×D412yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—84no—
221C2×C4×C61C2×C4214yes—
222C2×C4×C61C2×C22⋊C414yes—
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61#11714yes—
226C2×C4×C61C4×D414yes—
227C2×C4×C61C4×D414yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—216no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—84no—
244D244—22no—
245D244—22no—
246D244—22no—
247D244C2×C4214yes—
248C4×D61—216no—
249C4×D61—88no—
250Dic122—22no—
251Dic122—22no—
252Dic122—22no—
253Dic122C2×C4214yes—
254#92418—22no—
255#92418—22no—
256#92418—24no—
257#20415—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61C4×C22⋊C414yes—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×C4×D411yescharacteristic
267C22×C61C2×C4×D411yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57811yescharacteristic
272C2×D61—24no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C4×C61—24no—
280C4×C61—24no—
281C4×C61—44no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61C4×C22⋊C414yes—
285C4×C61C22×C4211yescharacteristic
286C4×C61C2×#11711yescharacteristic
287C4×C61C2×C4×D411yescharacteristic
288C4×C61C2×C4×D411yescharacteristic
289Dic61—24no—
290Dic61—44no—
291C2×#520—611no—
292C2×#417—611no—
293C2×#416—611no—
294C2×#405—611no—
295#2274D6111yescharacteristic, complemented
296#1763—612no—
297#1762—612no—
298C4×#103—612no—
299C4×#102—612no—
300#1736—611no—
301C2×#340—611no—
302C2×C61—24no—
303C2×C61C2×C4×C22⋊C411yescharacteristic
304C2×C61#289711yescharacteristic
305C2×C61#289811yescharacteristic
306C2×C61#294611yescharacteristic
307C2×C61#294711yescharacteristic
308C2×C61#295111yescharacteristic
309C2×C61#295211yescharacteristic
310D61—44no—
311C22×#41—611no—
312C22×M16—611no—
313C23×C8—611no—
314#520—618no—
315#520—1224no—
316C2×C4×Q8—612no—
317C2×C4×D4—612no—
318C2×C4×D4—1222no—
319C2×#117—1221no—
320C2×#117C2×D6111yescharacteristic
321C2×#117C2×D6111yescharacteristic
322C22×C4⋊C4—611no—
323C22×C4⋊C4—1221no—
324C22×C4⋊C4C2×D6111yescharacteristic
325#417—618no—
326#416—618no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—612no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—612no—
333#405—1224no—
334C2×#97—612no—
335C2×C4×C8—612no—
336C4×C4⋊C4—21no—
337C4×C4⋊C4—21no—
338C4×C4⋊C4C2×D6112yes—
339C4×C4⋊C4C2×D6112yes—
340C4×C22⋊C4—22no—
341#340—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345C61#2041511yescharacteristic, complemented, Sylow
346C2×#41—614no—
347C2×#41—614no—
348C2×#41—614no—
349C2×#41—1222no—
350C2×#41—1222no—
351C2×#41—1222no—
352C22×Q8—611no—
353C22×D4—611no—
354C22×D4—1221no—
355C23×C4—21no—
356C23×C4—1221no—
357C23×C4—1221no—
358C23×C4—1222no—
359C23×C4C22×D6111yescharacteristic
360C2×M16—614no—
361C2×M16—1222no—
362C2×M16—1222no—
363C22×C8—614no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—612no—
367C22×C8—1222no—
368C22×C8—1222no—
369C4×Q8—618no—
370C4×Q8—1224no—
371C4×D4—618no—
372C4×D4—12216no—
373C4×D4—1224no—
374#117—22no—
375#117—1228no—
376#117C4×D6114yes—
377#117C4×D6114yescomplemented
378#117D24414yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—614no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4C4×D6114yescomplemented
388C2×C4⋊C4C22⋊D6111yescharacteristic
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C4⋊C4C22×D6112yes—
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4C22⋊D6112yes—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—1222no—
400C2×C42C22×D6112yes—
401#103—614no—
402#103—614no—
403#103—618no—
404#102—614no—
405#102—614no—
406#102—618no—
407#97—1224no—
408C4×C8—1224no—
409#95—22no—
410#95—22no—
411C24—1221no—
412C24C2×C22⋊D6111yescharacteristic
413#41—6116no—
414#41—1228no—
415#41—1228no—
416#41—1228no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×D4—612no—
424C2×D4—612no—
425C2×D4—612no—
426C2×D4—1224no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—1221no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1222no—
444C22×C4—1228no—
445C22×C4—4882no—
446C22×C4—4882no—
447C22×C4—4881no—
448C22×C4—4881no—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6112yes—
451C22×C4C2×C4×D6112yes—
452C22×C4C2×D24412yes—
453C22×C4#1175822212yes—
454C22×C4#1175822212yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6112yes—
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459M16—1228no—
460C2×C8—612no—
461C2×C8—612no—
462C2×C8—614no—
463C2×C8—1221no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1228no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—614no—
475C4⋊C4—1222no—
476C4⋊C4—1224no—
477C4⋊C4—1228no—
478C4⋊C4#1175819814yes—
479C4⋊C4#1175819814yes—
480C4⋊C4C2×C4×D6114yes—
481C4⋊C4C2×C4×D6112yes—
482C4⋊C4C2×C4×D6112yes—
483C22⋊C4—24no—
484C22⋊C4—1228no—
485C22⋊C4#1175819818yes—
486C42—24no—
487C42—82no—
488C42—82no—
489C42—1228no—
490C42C2×C4×D6114yes—
491C42C2×D24414yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23C2×#1175819812yes—
504C23C4×C22⋊D6112yes—
505C23C4×C22⋊D6112yes—
506C23C22×C22⋊D6111yescharacteristic
507Q8—614no—
508Q8—1222no—
509Q8—1222no—
510Q8—1222no—
511D4—614no—
512D4—12216no—
513D4—1222no—
514D4—1222no—
515D4—1222no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—12216no—
534C2×C4—1224no—
535C2×C4—4888no—
536C2×C4—4884no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4C42×D6114yes—
540C2×C4C2×#1175819814yes—
541C2×C4C2×#1175819814yes—
542C2×C4C2×#1175819812yes—
543C2×C4#1451973414yes—
544C2×C4C4×C22⋊D6112yes—
545C2×C4C4×C22⋊D6112yes—
546C2×C4C4×D24414yes—
547C2×C4C4×D24414yes—
548C2×C4C22×C4×D6111yescharacteristic
549C2×C4C22×C4×D6111yescharacteristic
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C22×D24411yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C8—1222no—
555C8—1222no—
556C8—1224no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—1224no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C4×#1175819814yes—
571C22C22×#1175819811yescharacteristic
572C22C2×C4×C22⋊D6111yescharacteristic
573C22C2×C4×C22⋊D6111yescharacteristic
574C22#1684056511yescharacteristic
575C22#1684057011yescharacteristic
576C22#1684057311yescharacteristic
577C22#1684057711yescharacteristic
578C4—24no—
579C4—24no—
580C4—44no—
581C4—42no—
582C4—42no—
583C4—1224no—
584C4—2444no—
585C4C4×#1175819814yes—
586C4C2×C42×D6111yescharacteristic
587C4C2×#1451973411yescharacteristic
588C4C2×C4×D24411yescharacteristic
589C4C2×C4×D24411yescharacteristic
590C2—24no—
591C2—2444no—
592C2C2×C4×#1175819811yescharacteristic
593C2#2145434211yescharacteristic
594C2#2145434911yescharacteristic
595C2#2145465111yescharacteristic
596C2#2145466211yescharacteristic
597C2#2145469211yescharacteristic
598C2#2145470011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#2274C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2274D61
hypocenterC61#20415
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
4. C2×#16839718C2semidirect
5. #21445652C2semidirect
6. #21445647C2semidirect
7. C4×#14519700C2semidirect
8. C4×#14519697C2semidirect
9. #21445540C2semidirect
10. C2×#16839524C2semidirect
11. C61×#2274C2semidirect
16. #16840242C4semidirect
23. #16839761C4semidirect
24. #16839756C4semidirect
60. C2×#11758194C2×C4semidirect
63. C22×#92418C2×C4semidirect
68. C4×Dic122C2×C4semidirect
70. C4×D244C2×C4semidirect
81. #14519700C2×C4semidirect
82. #14519700C2×C4semidirect
83. #14519700C2×C4semidirect
84. #14519697C2×C4semidirect
85. #14519697C2×C4semidirect
86. #14519697C2×C4semidirect
93. #117×C61C2×C4semidirect
97. C2×C4⋊C4×C61C2×C4semidirect
140. #11758222C42semidirect
295. #2274D61semidirect
345. C61#20415semidirect
377. #117C4×D61semidirect
387. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402422
C2×#168397612
C2×#168397562
C2×#168397182
#214456522
#214456472
C4×#145197002
C4×#145196972
#214455402
C2×#168395242
C61×#22742
#2041561