Finite Group Databasebeta

← #29677811

Subgroups — #29677811

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×#16840243C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4C2×#16839717C211yesmaximal, characteristic, complemented
5#21448176C211yesmaximal, characteristic, complemented
6#21445648C212yesmaximal, complemented
7#21445643C212yesmaximal, complemented
8#21445638C212yesmaximal, complemented
9#21445633C212yesmaximal, complemented
10#21445538C211yesmaximal, characteristic, complemented
11C2×C61×#340C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840243—24no—
15#16840243C418yescomplemented
16C2×#14519742C2212yes—
17C2×#14519739C2212yes—
18C2×C4×C22⋊D61—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C2×#14519735—21no—
22C2×C4⋊C4×D61—21no—
23#16839761C418yescomplemented
24#16839759C418yescomplemented
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519700C2212yes—
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2212yes—
31#16839717—24no—
32C2×#14519687C2212yes—
33C2×C8×Dic61C2212yes—
34C4⋊C4×Dic61—21no—
35C4⋊C4×Dic61—21no—
36C4⋊C4×Dic61C2212yes—
37C4⋊C4×Dic61C2212yes—
38C4×C22⋊Dic61—22no—
39#16839524C2212yes—
40#16839524C2212yes—
41C23×C8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
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—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194C2×C414yescomplemented
60C22×#92418C2312yes—
61#14519742—24no—
62#14519742C2×C418yescomplemented
63#14519739—24no—
64#14519739C2×C418yescomplemented
65C4×C22⋊D61—216no—
66#14519736—22no—
67#14519736C2×C414yes—
68#14519736C2×C414yescomplemented
69#14519736D414yes—
70#14519735—28no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—22no—
76C4⋊C4×D61—28no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61D412yes—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61C2312yes—
85#14519702C2×C414yescomplemented
86#14519702C2×C414yescomplemented
87#14519700C2×C418yescomplemented
88#14519699C2×C414yescomplemented
89#14519699C2×C414yescomplemented
90#14519697C2×C418yescomplemented
91#14519687—24no—
92C8×Dic61—24no—
93#14519684—22no—
94#14519684—22no—
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61—22no—
99C22×C8×C61—22no—
100C22×C8×C61C2×C414yescomplemented
101C22×C8×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61C2×C414yes—
108C2×C4⋊C4×C61C2×C414yescomplemented
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2312yes—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—24no—
116C22×C4×C61—24no—
117C22×C4×C61—81no—
118C22×C4×C61—81no—
119C22×C4×C61—81no—
120C22×C4×C61—81no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
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—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D412yes—
136C22×Dic61C2×D412yes—
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—
159C2×C8×C61—28no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C22×C412yes—
165C2×C8×C61C22×C412yes—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C414yes—
171C4⋊C4×C61C22×C414yes—
172C22⋊Dic61—24no—
173C22⋊Dic61C22⋊C418yes—
174#11758198—28no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61C22⋊C414yes—
178C4⋊Dic61C22⋊C414yes—
179C4⋊Dic61C22×C412yes—
180C4⋊Dic61C22×C412yes—
181#11758196—28no—
182C4×Dic61—24no—
183C4×Dic61—28no—
184C4×Dic61—82no—
185C4×Dic61—82no—
186C4×Dic61C22×C414yes—
187C4×Dic61C2×D414yes—
188#11758194—28no—
189C2×#92418—22no—
190C2×#92418—22no—
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—24no—
208C2×C4×C61—24no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—84no—
214C2×C4×C61—84no—
215C2×C4×C61C2×C4214yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61#11714yes—
218C2×C4×C61C4×D414yes—
219C2×C4×C61C4×D414yes—
220C2×C4×C61C4×D412yes—
221C2×C4×C61C4×D412yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—216no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61C2×C22⋊C414yes—
243C2×Dic61C2×C22⋊C414yes—
244D244—22no—
245D244—22no—
246D244—22no—
247D244C2×C4214yes—
248C4×D61—216no—
249C4×D61—88no—
250Dic122—22no—
251Dic122—22no—
252Dic122—22no—
253Dic122C2×C4214yes—
254C8×C61—22no—
255C8×C61—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—44no—
280C4×C61—44no—
281C4×C61C4×C22⋊C414yes—
282C4×C61C22×C4211yescharacteristic
283C4×C61C2×#11711yescharacteristic
284C4×C61C2×C4×D411yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286Dic61—24no—
287Dic61—24no—
288Dic61—24no—
289Dic61—42no—
290Dic61—42no—
291C2×#520—611no—
292C2×#417—611no—
293C2×#416—611no—
294C2×#405—611no—
295#2274—611no—
296#1763—612no—
297#1762—612no—
298C4×#103—612no—
299C4×#102—612no—
300#1736—611no—
301C2×#340D6111yescharacteristic, complemented
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×C8C2×D6111yescharacteristic
314#520—618no—
315#520—1224no—
316C2×C4×Q8—612no—
317C2×C4×D4—612no—
318C2×C4×D4—1222no—
319C2×#117—611no—
320C2×#117—611no—
321C2×#117—1221no—
322C22×C4⋊C4—1221no—
323C22×C4⋊C4C2×D6111yescharacteristic
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—612no—
337C4×C4⋊C4—612no—
338C4×C4⋊C4—1221no—
339C4×C4⋊C4—1221no—
340C4×C22⋊C4—1222no—
341#340—612no—
342#340—612no—
343#340C2×D6112yes—
344#340C2×D6112yes—
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—21no—
357C23×C4—1222no—
358C23×C4—1221no—
359C23×C4C22×D6111yescharacteristic
360C2×M16—614no—
361C2×M16—1222no—
362C2×M16—1222no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—612no—
366C22×C8C4×D6114yescomplemented
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369C4×Q8—618no—
370C4×Q8—1224no—
371C4×D4—618no—
372C4×D4—1224no—
373C4×D4—12216no—
374#117—614no—
375#117—614no—
376#117—614no—
377#117—1228no—
378#117—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1228no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4C4×D6114yes—
390C2×C4⋊C4C4×D6114yescomplemented
391C2×C4⋊C4C22×D6112yes—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—612no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1221no—
396C2×C42—612no—
397C2×C42—1222no—
398C2×C42—1221no—
399C2×C42—1222no—
400C2×C42—1222no—
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—1222no—
410#95—1222no—
411C24—1221no—
412C24D4×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—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1228no—
443C22×C4—1221no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—4882no—
450C22×C4—4882no—
451C22×C4C2×C4×D6112yes—
452C22×C4C2×C4×D6112yes—
453C22×C4C2×C4×D6112yes—
454C22×C4D4×D6112yes—
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822412yes—
457C22×C4#1175822612yes—
458C22×C4C23×D6111yescharacteristic
459M16—1228no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—28no—
465C2×C8—614no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8C2×C4×D6112yes—
469C2×C8C2×C4×D6112yes—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—614no—
475C4⋊C4—614no—
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—1228no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4C2×C4×D6114yes—
482C4⋊C4C2×C4×D6114yes—
483C22⋊C4—618no—
484C22⋊C4—1228no—
485C22⋊C4—1224no—
486C42—614no—
487C42—614no—
488C42—1228no—
489C42—1224no—
490C42—4882no—
491C42—4882no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23C22⋊C4×D6112yes—
504C23#1451973712yes—
505C23#1451973912yes—
506C23C2×D4×D6111yescharacteristic
507Q8—614no—
508Q8—1222no—
509Q8—1222no—
510Q8—1222no—
511D4—614no—
512D4—12216no—
513D4—1222no—
514D4—1222no—
515D4—1222no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—12216no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—4888no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4C42×D6114yes—
542C2×C4C22⋊C4×D6112yes—
543C2×C4#1451973514yes—
544C2×C4#1451973714yes—
545C2×C4#1451973712yes—
546C2×C4#1451973914yes—
547C2×C4#1451973912yes—
548C2×C4C22×C4×D6111yescharacteristic
549C2×C4C22×C4×D6111yescharacteristic
550C2×C4C22×C4×D6111yescharacteristic
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822611yescharacteristic
554C8—22no—
555C8—22no—
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—
570C22#1683960914yes—
571C22C2×C22⋊C4×D6111yescharacteristic
572C22C2×#1451973711yescharacteristic
573C22C2×#1451973911yescharacteristic
574C22#1684056311yescharacteristic
575C22#1684056811yescharacteristic
576C22#1684056811yescharacteristic
577C22#1684057511yescharacteristic
578C4—44no—
579C4—44no—
580C4—1224no—
581C4—1224no—
582C4—1224no—
583C4—2442no—
584C4—2442no—
585C4#1683960914yes—
586C4C2×C42×D6111yescharacteristic
587C4C2×#1451973511yescharacteristic
588C4C2×#1451973711yescharacteristic
589C4C2×#1451973911yescharacteristic
590C2—24no—
591C2—2444no—
592C2C2×#1683960911yescharacteristic
593C2#2145434011yescharacteristic
594C2#2145434711yescharacteristic
595C2#2145464511yescharacteristic
596C2#2145465611yescharacteristic
597C2#2145469311yescharacteristic
598C2#2145470211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#340D61
hypocenterC61#20415
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. C2×#16839717C2semidirect
5. #21448176C2semidirect
6. #21445648C2semidirect
7. #21445643C2semidirect
8. #21445638C2semidirect
9. #21445633C2semidirect
10. #21445538C2semidirect
11. C2×C61×#340C2semidirect
15. #16840243C4semidirect
23. #16839761C4semidirect
24. #16839759C4semidirect
59. C2×#11758194C2×C4semidirect
62. #14519742C2×C4semidirect
64. #14519739C2×C4semidirect
68. #14519736C2×C4semidirect
85. #14519702C2×C4semidirect
86. #14519702C2×C4semidirect
87. #14519700C2×C4semidirect
88. #14519699C2×C4semidirect
89. #14519699C2×C4semidirect
90. #14519697C2×C4semidirect
100. C22×C8×C61C2×C4semidirect
108. C2×C4⋊C4×C61C2×C4semidirect
140. #11758222C42semidirect
301. C2×#340D61semidirect
345. C61#20415semidirect
366. C22×C8C4×D61semidirect
390. 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×#168402432
C2×#168397612
C2×#168397592
C2×#168397172
#214481762
#214456482
#214456432
#214456382
#214456332
#214455382
C2×C61×#3402
#2041561