Finite Group Databasebeta

← #29679419

Subgroups — #29679419

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840241C211yesmaximal, characteristic
2C2×#16839734C212yesmaximal
3#21448436C212yesmaximal, complemented
4#21445807C211yesmaximal, characteristic, complemented
5#21445806C212yesmaximal, complemented
6#21445805C212yesmaximal, complemented
7#21445804C212yesmaximal, complemented
8#21445550C212yesmaximal, complemented
9C61×#1799C211yesmaximal, characteristic, complemented
10C22×#11758224C2211yescharacteristic
11#16840241—24no—
12#16840241—24no—
13C2×C4×Dic122C411yescharacteristic, complemented
14C2×#14519742C411yescharacteristic, complemented
15C2×D4×Dic61C2211yescharacteristic
16C2×C4×C22⋊D61C411yescharacteristic, complemented
17C2×C4×C22⋊D61C2211yescharacteristic
18C2×#14519737C411yescharacteristic, complemented
19C2×#14519737C2211yescharacteristic
20C2×#14519734C411yescharacteristic, complemented
21C2×#14519733C411yescharacteristic, complemented
22C2×#14519733C2214yes—
23C2×#14519733C2211yescharacteristic
24C22×#11758196C411yescharacteristic, complemented
25C22×#11758196C2211yescharacteristic
26C2×C4⋊C4×D61C411yescharacteristic, complemented
27#16839734—28no—
28C2×#14519692C2212yes—
29C2×#14519691C2214yes—
30#16839659—24no—
31#16839657—24no—
32#16839622C2212yes—
33#16839620C2214yes—
34#16839529C2212yes—
35#16839528C2214yes—
36C2×C4×D4×C61C2211yescharacteristic
37C2×#99×C61C2212yes—
38C61×#380C2212yes—
39C61×#342C2212yes—
40C2×#11758224—24no—
41C2×#11758224—24no—
42C2×#11758224D414yes—
43C2×#11758224Q814yes—
44C22×Dic122C2×C411yescharacteristic
45C22×C22⋊D61C2×C411yescharacteristic
46C22×C22⋊D61C2311yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2311yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C4×Dic122—22no—
55C4×Dic122—22no—
56#14519742—24no—
57D4×Dic61—24no—
58C4×C22⋊D61—22no—
59C4×C22⋊D61—22no—
60C4×C22⋊D61—22no—
61C4×C22⋊D61—22no—
62#14519737—24no—
63#14519737—24no—
64#14519734—22no—
65#14519734—22no—
66#14519733—24no—
67#14519733—24no—
68#14519733—28no—
69#14519733—28no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×#11758196—24no—
75C2×#11758196—24no—
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C411yescharacteristic
84C2×#11758196D414yes—
85C2×#11758196D414yes—
86C4⋊C4×D61—24no—
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×C22⋊Dic61C2314yes—
89C2×C22⋊Dic61C2311yescharacteristic
90C2×#11758198C2×C411yescharacteristic
91C2×#11758198C2×C411yescharacteristic
92C2×C4×Dic61—24no—
93C2×C4×Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98#14519692—24no—
99#14519691—28no—
100#14519684—22no—
101#14519684—24no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61C2×C411yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61C2×C411yescharacteristic
110C2×C22⋊C4×C61C2314yes—
111C2×C22⋊C4×C61C2311yescharacteristic
112C2×C42×C61C2×C411yescharacteristic
113#99×C61—24no—
114#95×C61—22no—
115C24×C61C22×C411yescharacteristic
116C24×C61C2411yescharacteristic
117C23×D61C22×C411yescharacteristic
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61C2×D412yes—
121C2×D4×C61C2×Q812yes—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—24no—
125C22×C4×C61—81no—
126C22×C4×C61—81no—
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61C4⋊C414yescomplemented
139C2×C22⋊D61C2×D412yes—
140C2×C22⋊D61C2×Q812yes—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—81no—
146C22×Dic61—81no—
147C22×Dic61C4⋊C414yescomplemented
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C2×D414yes—
156C22×Dic61C2×D414yes—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×Q812yes—
159C22×Dic61#4114yes—
160#11758224—216no—
161#11758224—216no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61C4⋊C414yescomplemented
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122C4⋊C414yescomplemented
172C4⋊C4×C61—22no—
173C4⋊C4×C61—82no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C42×C61—21no—
179C42×C61—21no—
180C22⋊Dic61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184#11758198—21no—
185#11758198—21no—
186#11758198—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—82no—
189#11758196—21no—
190#11758196—21no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—48no—
195#11758196—84no—
196#11758196C22⋊C414yes—
197#11758196C22⋊C412yes—
198#11758196C22⋊C412yes—
199#11758196C22⋊C414yes—
200#11758196C22⋊C412yes—
201#11758196C22⋊C412yes—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—48no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61C2×C4⋊C412yes—
215C23×C61C23×C411yescharacteristic
216C23×C61C22×D412yes—
217C23×C61C22×D411yescharacteristic
218C23×C61C22×Q811yescharacteristic
219C23×C61C2×#4112yes—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61C2×C4⋊C412yes—
225D4×C61—24no—
226D4×C61—24no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61—81no—
241C2×C4×C61—81no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C4⋊C412yes—
246C22⋊D61—24no—
247C22⋊D61—24no—
248C22⋊D61—28no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—28no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—44no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—44no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—82no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C4⋊C412yes—
268C2×Dic61C2×C4⋊C412yes—
269C2×Dic61C2×C4⋊C412yes—
270C2×Dic61#11712yes—
271C2×Dic61#11712yes—
272C2×Dic61#12118yes—
273C2×Dic61C22⋊Q818yes—
274C4×D61—28no—
275C4×D61—82no—
276C4×D61—82no—
277Dic122—28no—
278#20634—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—21no—
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61C22×C4⋊C411yescharacteristic
293C22×C61C2×#11711yescharacteristic
294C22×C61C2×#12112yes—
295C22×C61C2×C22⋊Q812yes—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—42no—
301C4×C61—22no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—84no—
308Dic61—22no—
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—84no—
314Dic61#38014yes—
315Dic61#38014yes—
316C2×#520—611no—
317C2×#410—612no—
318#2316—612no—
319#1799—612no—
320#1799—612no—
321#1799—612no—
322#1799—611no—
323#1799D6111yescharacteristic, complemented
324#1740—612no—
325C2×C61—22no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—22no—
329C2×C61—42no—
330C2×C61C2×#38011yescharacteristic
331C2×C61#288311yescharacteristic
332C2×C61#288411yescharacteristic
333D61—42no—
334C22×#41—611no—
335#520—1224no—
336#520—1224no—
337C2×C4×Q8—611no—
338C2×C4×Q8—611no—
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4C2×D6111yescharacteristic
345C2×#117—611no—
346C2×#117—611no—
347C2×#117—611no—
348C2×#117—614no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4—611no—
351C22×C4⋊C4—611no—
352#410—1228no—
353C2×#99—612no—
354C2×#99—614no—
355C2×#99C2×D6112yes—
356#389—1224no—
357#389—1224no—
358#380—612no—
359#380—614no—
360#380C2×D6112yes—
361#342—614no—
362#342—612no—
363#342C2×D6112yes—
364C61#2063411yescharacteristic, complemented, Sylow
365C2×#41—614no—
366C2×#41—614no—
367C2×#41—1224no—
368C2×#41—1224no—
369C22×Q8—611no—
370C22×D4—611no—
371C22×D4—611no—
372C22×D4C22×D6111yescharacteristic
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4C4×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1224no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—1224no—
388C4×D4—1224no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1224no—
392C4×D4—1222no—
393C4×D4—1222no—
394#117—1222no—
395#117—1222no—
396#117—1224no—
397#117—1224no—
398#117—1228no—
399#117—1228no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—614no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1224no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1224no—
418C2×C4⋊C4C4×D6111yescharacteristic
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—614no—
424C2×C22⋊C4C4×D6111yescharacteristic
425C2×C22⋊C4C22×D6114yes—
426C2×C22⋊C4C22×D6111yescharacteristic
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—1224no—
433C2×C42C4×D6111yescharacteristic
434#99—24no—
435#99—1228no—
436#99—1224no—
437#95—22no—
438#95—1224no—
439#95—1222no—
440C24—611no—
441C24C2×C4×D6111yescharacteristic
442C24C23×D6111yescharacteristic
443#41—12216no—
444#41—12216no—
445C2×Q8—614no—
446C2×Q8—1222no—
447C2×Q8—1222no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—612no—
451C2×D4—612no—
452C2×D4—614no—
453C2×D4—1222no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4D4×D6112yes—
458C2×D4Q8×D6112yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—614no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—614no—
470C22×C4—611no—
471C22×C4—614no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1224no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4D4×D6112yes—
496C22×C4D4×D6112yes—
497C4⋊C4—22no—
498C4⋊C4—82no—
499C4⋊C4—612no—
500C4⋊C4—612no—
501C4⋊C4—614no—
502C4⋊C4—612no—
503C4⋊C4—612no—
504C4⋊C4—614no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4884no—
512C4⋊C4—4882no—
513C4⋊C4—2448no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C42—21no—
526C42—21no—
527C42—1222no—
528C42—1222no—
529C42—1222no—
530C42—1222no—
531C42—1222no—
532C42—2448no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—612no—
540C23—1221no—
541C23—1221no—
542C23—2441no—
543C23—2441no—
544C23C4⋊C4×D6112yes—
545C23C22×C4×D6111yescharacteristic
546C23C2×D4×D6112yes—
547C23C2×D4×D6111yescharacteristic
548C23C2×Q8×D6111yescharacteristic
549C23#41×D6112yes—
550Q8—1228no—
551D4—24no—
552D4—24no—
553D4—1228no—
554D4—1224no—
555D4—1224no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—618no—
578C2×C4—618no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1228no—
583C2×C4—1228no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—4882no—
589C2×C4—4882no—
590C2×C4—4882no—
591C2×C4—2444no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2444no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C22⋊C4×D6112yes—
602C2×C4C4⋊C4×D6112yes—
603C22—22no—
604C22—21no—
605C22—22no—
606C22—21no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22—1222no—
616C22—2442no—
617C22—2442no—
618C22—2441no—
619C22—2441no—
620C22C2×C22⋊C4×D6111yescharacteristic
621C22C2×C4⋊C4×D6111yescharacteristic
622C22#117×D6111yescharacteristic
623C22#121×D6112yes—
624C22C22⋊Q8×D6112yes—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—84no—
632C4—614no—
633C4—614no—
634C4—1222no—
635C4—1222no—
636C4—1222no—
637C4—4884no—
638C4—2441no—
639C4—2441no—
640C2—22no—
641C2—21no—
642C2—21no—
643C2—22no—
644C2—42no—
645C2—2442no—
646C2#380×D6111yescharacteristic
647C2#2145427711yescharacteristic
648C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1799C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758224C22
cosocleC24×C61C24
hypercenter#1799D61
hypocenterC61#20634
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21448436C2semidirect
4. #21445807C2semidirect
5. #21445806C2semidirect
6. #21445805C2semidirect
7. #21445804C2semidirect
8. #21445550C2semidirect
9. C61×#1799C2semidirect
13. C2×C4×Dic122C4semidirect
14. C2×#14519742C4semidirect
16. C2×C4×C22⋊D61C4semidirect
18. C2×#14519737C4semidirect
20. C2×#14519734C4semidirect
21. C2×#14519733C4semidirect
24. C22×#11758196C4semidirect
26. C2×C4⋊C4×D61C4semidirect
138. C2×C22⋊D61C4⋊C4semidirect
147. C22×Dic61C4⋊C4semidirect
168. C2×C4×D61C4⋊C4semidirect
171. C2×Dic122C4⋊C4semidirect
323. #1799D61semidirect
364. C61#20634semidirect

Maximal subgroups

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

subgroupindex
C2×#168402412
C2×#168397342
#214484362
#214458072
#214458062
#214458052
#214458042
#214455502
C61×#17992
#2063461