Finite Group Databasebeta

← #29679964

Subgroups — #29679964

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840241C211yesmaximal, characteristic
2#21448421C212yesmaximal, complemented
3#21445895C212yesmaximal, complemented
4#21445807C211yesmaximal, characteristic, complemented
5#21445804C212yesmaximal, complemented
6#21445735C212yesmaximal
7#99×Dic61C212yesmaximal, complemented
8#21445606C212yesmaximal, 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×#14519733C2211yescharacteristic
23C22×#11758196C411yescharacteristic, complemented
24C22×#11758196C2211yescharacteristic
25C2×C4⋊C4×D61C411yescharacteristic, complemented
26C2×#14519692C2212yes—
27C2×#14519691C2214yes—
28#16839649—24no—
29#16839643—24no—
30#16839623C2214yes—
31#16839622C2212yes—
32C22⋊C4×Dic61C2214yes—
33#16839529C2212yes—
34#16839528C2214yes—
35C2×C4×D4×C61C2211yescharacteristic
36C2×#99×C61C2212yes—
37C61×#380C2212yes—
38C61×#342C2212yes—
39C2×#11758224—22no—
40C2×#11758224—22no—
41C2×#11758224—24no—
42C2×#11758224—24no—
43C22×Dic122C2×C411yescharacteristic
44C22×C22⋊D61C2×C411yescharacteristic
45C22×C22⋊D61C2311yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C4×Dic122—22no—
54C4×Dic122—22no—
55#14519742—24no—
56D4×Dic61—24no—
57C4×C22⋊D61—22no—
58C4×C22⋊D61—22no—
59C4×C22⋊D61—22no—
60C4×C22⋊D61—22no—
61#14519737—24no—
62#14519737—24no—
63#14519734—22no—
64#14519734—22no—
65#14519733—24no—
66#14519733—24no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196C2×C411yescharacteristic
72C2×#11758196C2×C411yescharacteristic
73C2×#11758196C2×C411yescharacteristic
74C2×#11758196C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196D414yes—
80C2×#11758196D414yes—
81C4⋊C4×D61—24no—
82C2×C22⋊Dic61C2×C411yescharacteristic
83C2×C22⋊Dic61C2314yes—
84C2×C22⋊Dic61C2311yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2×C411yescharacteristic
87C2×C4×Dic61—24no—
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93#14519692—24no—
94#14519691—28no—
95#14519684—22no—
96#14519684—24no—
97#14519684—28no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61C2×C411yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61C2×C411yescharacteristic
105C2×C22⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2314yes—
107C2×C22⋊C4×C61C2311yescharacteristic
108C2×C42×C61C2×C411yescharacteristic
109#99×C61—24no—
110#95×C61—22no—
111C24×C61C22×C411yescharacteristic
112C24×C61C2411yescharacteristic
113C23×D61C22×C411yescharacteristic
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61#4112yes—
117C2×D4×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—24no—
121C22×C4×C61—81no—
122C22×C4×C61—81no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—21no—
133C2×C22⋊D61—21no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61#4114yes—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—81no—
146C22×Dic61—81no—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C2×D414yes—
155C22×Dic61C2×D414yes—
156#11758224—216no—
157#11758224—216no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—82no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C22⋊Dic61—22no—
177C22⋊Dic61—22no—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180#11758198—21no—
181#11758198—21no—
182#11758198—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61—82no—
185#11758196—21no—
186#11758196—21no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196C22⋊C414yes—
191#11758196C22⋊C412yes—
192#11758196C22⋊C412yes—
193#11758196C22⋊C414yes—
194#11758196C22⋊C412yes—
195#11758196C22⋊C412yes—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—84no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61#11712yes—
209C23×C61C23×C411yescharacteristic
210C23×C61C22×D412yes—
211C23×C61C2×#4112yes—
212C23×C61C2×#4111yescharacteristic
213C23×C61C2×#4111yescharacteristic
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61#11712yes—
219D4×C61—24no—
220D4×C61—24no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—44no—
232C2×C4×C61—44no—
233C2×C4×C61—44no—
234C2×C4×C61—81no—
235C2×C4×C61—81no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61#11712yes—
240C22⋊D61—28no—
241C22⋊D61—24no—
242C22⋊D61—24no—
243C2×Dic61—28no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—24no—
251C2×Dic61—24no—
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×Dic61#11712yes—
268C4×D61—28no—
269C4×D61—82no—
270C4×D61—82no—
271Dic122—28no—
272#20725—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—21no—
275C22×C61—22no—
276C22×C61—21no—
277C22×C61—22no—
278C22×C61—42no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61C22×C22⋊C411yescharacteristic
286C22×C61C2×#11711yescharacteristic
287C22×C61C2×#11711yescharacteristic
288C22×C61#52512yes—
289C22×C61#52512yes—
290C2×D61—22no—
291C2×D61—42no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—42no—
295C4×C61—22no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—22no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—84no—
302Dic61—22no—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—22no—
307Dic61—41no—
308Dic61—41no—
309Dic61—84no—
310C2×#520—611no—
311#2313—612no—
312#1820—612no—
313#1799—612no—
314#1799—611no—
315#1799D6111yescharacteristic, complemented
316#1782—612no—
317C4×#99—612no—
318C4×#99—612no—
319C2×C61—22no—
320C2×C61—22no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—42no—
324C2×C61#228011yescharacteristic
325C2×C61#288311yescharacteristic
326C2×C61#288411yescharacteristic
327D61—42no—
328C22×#41—611no—
329#520—1224no—
330#520—1224no—
331C2×C4×Q8—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×C4×D4—611no—
335C2×C4×D4—611no—
336C2×C4×D4—611no—
337C2×C4×D4—611no—
338C2×C4×D4C2×D6111yescharacteristic
339C2×#117—611no—
340C2×#117—611no—
341C2×#117—611no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4—611no—
344C22×C4⋊C4—611no—
345C2×#99—612no—
346C2×#99—614no—
347C2×#99C2×D6112yes—
348#387—1224no—
349#387—1224no—
350#381—614no—
351#380—612no—
352#380C2×D6112yes—
353C4×C22⋊C4—614no—
354#342—614no—
355#342—612no—
356#342C2×D6112yes—
357C61#2072511yescharacteristic, complemented, Sylow
358C2×#41—1224no—
359C2×#41—1224no—
360C2×#41—1222no—
361C2×#41—1222no—
362C22×Q8—611no—
363C22×D4—611no—
364C22×D4—611no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4C4×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×Q8—1224no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—1224no—
381C4×D4—1224no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1224no—
385C4×D4—1222no—
386C4×D4—1222no—
387#117—1222no—
388#117—1222no—
389#117—1224no—
390#117—1224no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—614no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4C4×D6111yescharacteristic
408C2×C22⋊C4—611no—
409C2×C22⋊C4—611no—
410C2×C22⋊C4—611no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—614no—
413C2×C22⋊C4C4×D6111yescharacteristic
414C2×C22⋊C4C22×D6114yes—
415C2×C22⋊C4C22×D6111yescharacteristic
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—1224no—
422C2×C42C4×D6111yescharacteristic
423#99—24no—
424#99—1228no—
425#99—1224no—
426#95—22no—
427#95—1224no—
428#95—1228no—
429#95—1222no—
430C24—611no—
431C24C2×C4×D6111yescharacteristic
432C24C23×D6111yescharacteristic
433#41—12216no—
434#41—12216no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—614no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1222no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1222no—
447C2×D4#1175822412yes—
448C2×D4#1175822612yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—614no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—614no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1224no—
476C22×C4—4881no—
477C22×C4—4881no—
478C22×C4—4881no—
479C22×C4—4881no—
480C22×C4C2×C4×D6111yescharacteristic
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4D4×D6112yes—
486C22×C4D4×D6112yes—
487C4⋊C4—22no—
488C4⋊C4—82no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—614no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—614no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—4882no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1224no—
512C22⋊C4—1224no—
513C42—21no—
514C42—21no—
515C42—1222no—
516C42—1222no—
517C42—1222no—
518C42—1222no—
519C42—1222no—
520C42—4884no—
521C23—21no—
522C23—21no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—612no—
528C23—1221no—
529C23—1221no—
530C23—2441no—
531C23—2441no—
532C23#1451973512yes—
533C23C22×C4×D6111yescharacteristic
534C23C2×D4×D6112yes—
535C23#41×D6112yes—
536C23C2×#1175822411yescharacteristic
537C23C2×#1175822611yescharacteristic
538Q8—1228no—
539D4—24no—
540D4—24no—
541D4—1228no—
542D4—1224no—
543D4—1224no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—1228no—
565C2×C4—1228no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—4882no—
577C2×C4—4882no—
578C2×C4—4882no—
579C2×C4—2444no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4C22⋊C4×D6111yescharacteristic
588C2×C4C22⋊C4×D6111yescharacteristic
589C2×C4C22⋊C4×D6112yes—
590C2×C4#1451973512yes—
591C22—22no—
592C22—21no—
593C22—22no—
594C22—21no—
595C22—22no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—1222no—
604C22—2442no—
605C22—2442no—
606C22—2441no—
607C22—2441no—
608C22C2×C22⋊C4×D6111yescharacteristic
609C22#117×D6111yescharacteristic
610C22C2×#1451973511yescharacteristic
611C22#1684027212yes—
612C22#1684027512yes—
613C4—22no—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—84no—
620C4—1222no—
621C4—1222no—
622C4—1222no—
623C4—1222no—
624C4—1222no—
625C4—4884no—
626C4—2441no—
627C4—2441no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—2442no—
634C2#2144820111yescharacteristic
635C2#2145427711yescharacteristic
636C2#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#20725
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
2. #21448421C2semidirect
3. #21445895C2semidirect
4. #21445807C2semidirect
5. #21445804C2semidirect
7. #99×Dic61C2semidirect
8. #21445606C2semidirect
9. C61×#1799C2semidirect
13. C2×C4×Dic122C4semidirect
14. C2×#14519742C4semidirect
16. C2×C4×C22⋊D61C4semidirect
18. C2×#14519737C4semidirect
20. C2×#14519734C4semidirect
21. C2×#14519733C4semidirect
23. C22×#11758196C4semidirect
25. C2×C4⋊C4×D61C4semidirect
315. #1799D61semidirect
357. C61#20725semidirect

Maximal subgroups

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

subgroupindex
C2×#168402412
#214484212
#214458952
#214458072
#214458042
#214457352
#99×Dic612
#214456062
C61×#17992
#2072561