Finite Group Databasebeta

← #29680891

Subgroups — #29680891

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840243C211yesmaximal, characteristic
2#21454408C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic
4#21454336C211yesmaximal, characteristic
5#21446051C214yesmaximal, complemented
6#21446044C214yesmaximal, complemented
7C2×#16839531C211yesmaximal, characteristic, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C61×#2317C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840243—22no—
14#16840243—22no—
15#16840243—22no—
16#16840243C2214yes—
17C2×#14519742—21no—
18C2×#14519739—21no—
19C2×C4×C22⋊D61—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519735—21no—
22C2×C4⋊C4×D61—21no—
23#16839805—21no—
24#16839805—21no—
25#16839802C2212yescomplemented
26#16839802C2212yescomplemented
27#16839798—22no—
28#16839780C418yescomplemented
29C2×#14519705C2214yes—
30#16839770—21no—
31#16839766—21no—
32#16839761C2212yes—
33#16839761C2212yes—
34#16839756—22no—
35#16839531C2214yescomplemented
36#16839522C2214yescomplemented
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#389—22no—
41C61×#385—22no—
42C61×#384C2212yescomplemented
43C61×#382C2212yescomplemented
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222D412yes—
50C2×#11758222Q814yes—
51C2×#11758222C2312yes—
52C22×Dic122D411yescharacteristic
53C22×D244D411yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57C2×#11758194—21no—
58C2×#11758194—22no—
59C2×#11758194—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yescomplemented
63C2×#11758194C2312yes—
64C2×#11758194C2312yes—
65C22×#92418—21no—
66C22×#92418—21no—
67#14519742—24no—
68#14519742—24no—
69#14519739—24no—
70#14519739—24no—
71C4×C22⋊D61—28no—
72C4×C22⋊D61—28no—
73#14519736—21no—
74#14519736—21no—
75#14519736D412yes—
76#14519736C2312yes—
77#14519735—24no—
78#14519735—24no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61D411yescharacteristic
81C2×#11758196—22no—
82C4⋊C4×D61—24no—
83C4⋊C4×D61—24no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61—21no—
88#14519710—22no—
89#14519710—22no—
90#14519708—22no—
91#14519708—22no—
92#14519705C2×C4116yescomplemented
93#14519700—22no—
94#14519700—22no—
95#14519697—22no—
96#14519697—22no—
97C23×C4×C61D412yes—
98C23×C4×C61C2311yescharacteristic
99#117×C61—22no—
100#117×C61—21no—
101#117×C61—21no—
102#117×C61—21no—
103#117×C61—21no—
104#117×C61C2×C414yescomplemented
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—24no—
111C2×C4⋊C4×C61D412yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2312yes—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2312yes—
119#95×C61—24no—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—82no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D414yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×Q812yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—22no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146#11758222—24no—
147#11758222—24no—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—22no—
152#11758222—22no—
153#11758222C4⋊C418yescomplemented
154#11758222C22×C414yes—
155#11758222C2×Q814yes—
156C2×D244—21no—
157C2×D244—22no—
158C2×D244C4⋊C414yescomplemented
159C2×D244C2×D412yes—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—28no—
165C2×C4×D61—84no—
166C2×Dic122—21no—
167C2×Dic122—22no—
168C2×Dic122C4⋊C414yescomplemented
169C2×Dic122C2×D412yes—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—84no—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—22no—
184C42×C61—41no—
185C42×C61—41no—
186C42×C61C22×C414yescomplemented
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—28no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C2×D412yes—
193C4⋊Dic61C2×D412yes—
194#11758196—28no—
195C4×Dic61—28no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198#11758194—24no—
199#11758194—22no—
200#11758194—22no—
201#11758194—22no—
202#11758194—42no—
203#11758194C22×C414yescomplemented
204C2×#92418—24no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C4×D412yes—
211C23×C61#12011yescharacteristic
212C23×C61#12011yescharacteristic
213C23×C61C22⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—42no—
218C22×D61—42no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—82no—
230C2×C4×C61C2×C4⋊C412yes—
231C2×C4×C61C2×C4⋊C412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C4×D412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C22⋊Q812yes—
238C2×C4×C61C22⋊Q812yes—
239C2×C4×C61C22⋊Q812yes—
240C2×C4×C61#12312yes—
241C2×C4×C61#12312yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—24no—
249C22⋊D61—24no—
250C22⋊D61—28no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257D244—22no—
258D244—24no—
259D244C2×C4⋊C414yes—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—28no—
263C4×D61—88no—
264Dic122—22no—
265Dic122—24no—
266Dic122C2×C4⋊C414yes—
267#92418—24no—
268#92418—42no—
269#20864—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61#38512yes—
279C22×C61#38512yes—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C2×#12011yescharacteristic
282C22×C61C2×C22⋊Q811yescharacteristic
283C2×D61—24no—
284C2×D61—48no—
285C2×D61—42no—
286C2×D61—42no—
287C4×C61—44no—
288C4×C61—44no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#38512yes—
292C4×C61#38512yes—
293C4×C61C22×C4⋊C411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×#12311yescharacteristic
297Dic61—24no—
298Dic61—42no—
299Dic61—42no—
300C2×#520—611no—
301#2911—611no—
302C2×#421—611no—
303#2896—611no—
304#2317D6111yescharacteristic, complemented
305#1860—614no—
306#1859—614no—
307C2×#343—611no—
308#1733—611no—
309C2×C61—21no—
310C2×C61—22no—
311C2×C61—21no—
312C2×C61—22no—
313C2×C61C2×#38511yescharacteristic
314C2×C61#301911yescharacteristic
315C2×C61#302011yescharacteristic
316D61—44no—
317C22×#41—611no—
318C22×M16—611no—
319C22×M16—611no—
320#520—614no—
321#520—1222no—
322#520—1222no—
323#520—1222no—
324C2×C4×Q8—1221no—
325C2×C4×D4—1221no—
326C2×C4×D4—1222no—
327C2×#117—611no—
328C2×#117—1221no—
329C2×#117C2×D6111yescharacteristic
330C2×#117C2×D6111yescharacteristic
331C22×C4⋊C4—1221no—
332C22×C4⋊C4C2×D6111yescharacteristic
333#428—612no—
334#428—612no—
335#428—1221no—
336#428—1221no—
337#427—1222no—
338#421—618no—
339C2×#104—614no—
340#419—1221no—
341#418—1221no—
342#417—612no—
343#417—612no—
344#416—1222no—
345#389—22no—
346#385—22no—
347#384C2×D6112yescomplemented
348#382C2×D6112yescomplemented
349#343—614no—
350#339—614no—
351C61#2086411yescharacteristic, complemented, Sylow
352C2×#41—614no—
353C2×#41—612no—
354C2×#41—614no—
355C2×#41—612no—
356C2×#41—1222no—
357C2×#41—1221no—
358C2×#41—1222no—
359C2×#41—1221no—
360C22×Q8—611no—
361C22×D4—611no—
362C22×D4—1221no—
363C23×C4—1222no—
364C23×C4—1221no—
365C23×C4C22⋊D6112yes—
366C23×C4C22×D6111yescharacteristic
367C2×M16—614no—
368C2×M16—612no—
369C2×M16—612no—
370C2×M16—1221no—
371C2×M16—1222no—
372C2×M16—1221no—
373C2×M16—1222no—
374C2×M16—1222no—
375C22×C8—1221no—
376C22×C8—1221no—
377C4×Q8—1224no—
378C4×Q8—1224no—
379C4×D4—1224no—
380C4×D4—1224no—
381C4×D4—1228no—
382C4×D4—1228no—
383#117—22no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—612no—
389#117—612no—
390#117—1224no—
391#117—1224no—
392#117—1221no—
393#117—1221no—
394#117C4×D6114yescomplemented
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—24no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4C22⋊D6112yes—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6112yes—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1221no—
414C2×C42—21no—
415C2×C42—1222no—
416C2×C42—1221no—
417C2×C42C22×D6112yes—
418#107—1222no—
419#107—1222no—
420#106—1222no—
421#106—1222no—
422#104—6116no—
423#103—1222no—
424#103—1222no—
425#102—1222no—
426#102—1222no—
427#95—24no—
428C24—1221no—
429C24C2×C22⋊D6111yescharacteristic
430#41—614no—
431#41—614no—
432#41—618no—
433#41—1222no—
434#41—1222no—
435#41—1224no—
436#41—1224no—
437#41—1224no—
438#41—1224no—
439#41—1224no—
440C2×Q8—612no—
441C2×Q8—614no—
442C2×Q8—1221no—
443C2×Q8—1222no—
444C2×D4—612no—
445C2×D4—614no—
446C2×D4—1222no—
447C2×D4—1224no—
448C2×D4—1222no—
449C2×D4—1224no—
450C2×D4—1221no—
451C2×D4—1222no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—82no—
458C22×C4—1228no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—1222no—
463C22×C4—1224no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—4884no—
467C22×C4C2×C4×D6112yes—
468C22×C4#1175822212yes—
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4Q8×D6112yes—
474C22×C4#1175822612yes—
475C22×C4C2×C22⋊D6114yes—
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478M16—614no—
479M16—1222no—
480M16—1224no—
481M16—1222no—
482M16—1222no—
483M16—2442no—
484C2×C8—1222no—
485C2×C8—1224no—
486C2×C8—1222no—
487C2×C8—2442no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—84no—
498C4⋊C4—612no—
499C4⋊C4—612no—
500C4⋊C4—1228no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C22⋊C4—24no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—1228no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C42—22no—
510C42—41no—
511C42—41no—
512C42—1228no—
513C42—1222no—
514C42—1222no—
515C42C2×C4×D6114yescomplemented
516C23—41no—
517C23—41no—
518C23—1222no—
519C23—1224no—
520C23—2442no—
521C23—2442no—
522C23C4×C22⋊D6112yes—
523C23#1451974511yescharacteristic
524C23#1451974511yescharacteristic
525C23#1451975312yes—
526C23C22×C22⋊D6111yescharacteristic
527Q8—614no—
528Q8—1222no—
529Q8—1224no—
530D4—614no—
531D4—1224no—
532D4—1224no—
533D4—1228no—
534D4—1222no—
535D4—1224no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1228no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—4888no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4C4⋊C4×D6112yes—
558C2×C4C4⋊C4×D6112yes—
559C2×C4C4×C22⋊D6112yes—
560C2×C4#1451973912yes—
561C2×C4#1451973912yes—
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974511yescharacteristic
564C2×C4#1451975312yes—
565C2×C4#1451975512yes—
566C2×C4#1451975512yes—
567C2×C4#1451976212yes—
568C2×C4#1451976212yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×Q8×D6111yescharacteristic
573C2×C4C2×#1175822211yescharacteristic
574C2×C4C2×#1175822611yescharacteristic
575C8—1224no—
576C8—2442no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—42no—
584C22—42no—
585C22—1224no—
586C22—2448no—
587C22—2442no—
588C22—2442no—
589C22#1683963312yes—
590C22#1683963312yes—
591C22C2×C4×C22⋊D6111yescharacteristic
592C22C2×#1451974511yescharacteristic
593C22C2×#1451975311yescharacteristic
594C4—44no—
595C4—44no—
596C4—81no—
597C4—81no—
598C4—1224no—
599C4—2442no—
600C4—2442no—
601C4#1683963312yes—
602C4#1683963312yes—
603C4C2×C4⋊C4×D6111yescharacteristic
604C4C2×#1451973911yescharacteristic
605C4C2×#1451975511yescharacteristic
606C4C2×#1451976211yescharacteristic
607C2—21no—
608C2—22no—
609C2—21no—
610C2—22no—
611C2—2444no—
612C2C2×#1683963311yescharacteristic
613C2#2145517511yescharacteristic
614C2#2145518211yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2317D61
hypocenterC61#20864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #21454408C2semidirect
5. #21446051C2semidirect
6. #21446044C2semidirect
7. C2×#16839531C2semidirect
8. #21445529C2semidirect
9. C61×#2317C2semidirect
25. #16839802C22semidirect
26. #16839802C22semidirect
28. #16839780C4semidirect
35. #16839531C22semidirect
36. #16839522C22semidirect
42. C61×#384C22semidirect
43. C61×#382C22semidirect
62. C2×#11758194C2×C4semidirect
92. #14519705C2×C4semidirect
104. #117×C61C2×C4semidirect
153. #11758222C4⋊C4semidirect
158. C2×D244C4⋊C4semidirect
168. C2×Dic122C4⋊C4semidirect
186. C42×C61C22×C4semidirect
203. #11758194C22×C4semidirect
304. #2317D61semidirect
347. #384C2×D61semidirect
348. #382C2×D61semidirect
351. C61#20864semidirect
394. #117C4×D61semidirect
515. C42C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402432
#214544082
C2×#168397802
#214543362
#214460512
#214460442
C2×#168395312
#214455292
C61×#23172
#2086461