Finite Group Databasebeta

← #30615176

Subgroups — #30615176

Order 15616.

1378 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462266C212yesmaximal, complemented
2#21461716C212yesmaximal, complemented
3#21461682C212yesmaximal, complemented
4#21461681C212yesmaximal, complemented
5#21461648C212yesmaximal
6#21461566C212yesmaximal, complemented
7#21461564C212yesmaximal, complemented
8C2×#16840230C211yesmaximal, characteristic
9C2×#14519836C2214yes—
10C2×#14519835C2214yes—
11#16840707—22no—
12#16840706—22no—
13#16840705C2214yes—
14#16840704C2214yes—
15C4×C4⋊#6923—22no—
16C4×C22⋊#6923C2214yes—
17#16840698—22no—
18#16840640C2214yes—
19#16840640C2214yes—
20C22×#11758222C2211yescharacteristic
21#16840230—22no—
22#16840230—22no—
23#16840230—22no—
24#16840230—22no—
25C2×#14519742C411yescharacteristic, complemented
26C2×#14519742C411yescharacteristic, complemented
27C2×#14519739C2211yescharacteristic
28C2×#14519739C2211yescharacteristic
29C2×#14519737C411yescharacteristic, complemented
30C2×#14519737C411yescharacteristic, complemented
31C2×#14519737C2211yescharacteristic
32C2×#14519737C2211yescharacteristic
33C2×#14519734C411yescharacteristic, complemented
34C2×#14519734C2211yescharacteristic
35C22×C4×Dic61C411yescharacteristic, complemented
36C2×C42×D61C411yescharacteristic, complemented
37C2×C42×D61C2211yescharacteristic
38C2×#117×C61C411yescharacteristic, complemented
39C2×C4⋊#6923—24no—
40C2×C22⋊#6923C2318yes—
41C2×C4×#6923—24no—
42#14519836—28no—
43#14519835—28no—
44(C2×#6923)⋊C4—24no—
45(C2×#6923)⋊C4—24no—
46(C2×#6923)⋊C4—24no—
47(C2×#6923)⋊C4—24no—
48C2×#11758222—24no—
49C2×#11758222—24no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2311yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519742—22no—
65#14519742—22no—
66#14519742—22no—
67#14519742—22no—
68#14519739—22no—
69#14519739—22no—
70#14519739—22no—
71#14519739—22no—
72#14519737—22no—
73#14519737—22no—
74#14519737—22no—
75#14519737—22no—
76#14519737—22no—
77#14519737—22no—
78#14519737—22no—
79#14519737—22no—
80#14519734—22no—
81#14519734—22no—
82#14519734—22no—
83#14519734—22no—
84C2×#11758196C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×#11758196C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758198C2×C411yescharacteristic
89C2×#11758198C2×C411yescharacteristic
90C2×#11758198C2311yescharacteristic
91C2×#11758198C2311yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98C2×C4×Dic61C2×C411yescharacteristic
99C2×C4×Dic61C2×C411yescharacteristic
100C2×C4×Dic61C2×C411yescharacteristic
101C2×C4×Dic61C2×C411yescharacteristic
102C2×C4×Dic61C2×C411yescharacteristic
103C2×C4×Dic61C2×C411yescharacteristic
104C42×D61—21no—
105C42×D61—21no—
106C42×D61—21no—
107C42×D61—21no—
108C42×D61D412yes—
109C42×D61D412yes—
110C42×D61D412yes—
111C42×D61D412yes—
112C23×C4×C61C2×C411yescharacteristic
113#117×C61—22no—
114#117×C61—22no—
115C2×C4⋊C4×C61C2×C411yescharacteristic
116C2×C4⋊C4×C61C2×C411yescharacteristic
117C2×C22⋊C4×C61C2×C411yescharacteristic
118C2×C22⋊C4×C61C2×C411yescharacteristic
119C2×C42×C61C2×C411yescharacteristic
120C2×C42×C61C2×C411yescharacteristic
121C24×C61C22×C411yescharacteristic
122C23×D61C22×C411yescharacteristic
123C23×D61C2411yescharacteristic
124C22×#6923—28no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C2×C22⋊D61—22no—
136C2×C22⋊D61—22no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61#4114yes—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153#11758222—216no—
154#11758222—216no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244#4114yes—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×C4×D61C2×D412yes—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C22⋊#6923—28no—
182C22⋊#6923—28no—
183C4⋊#6923—48no—
184C4×#6923—48no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C22⋊C412yes—
192C42×C61C22⋊C412yes—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C412yes—
216C4×Dic61C22⋊C412yes—
217C4×Dic61C22⋊C412yes—
218C4×Dic61C22⋊C412yes—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61#11712yes—
224C23×C61C23×C411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61#11712yes—
232C22×D61C22×D412yes—
233C22×D61C2×#4112yes—
234C22×D61C2×#4112yes—
235C2×#6923—48no—
236C2×#6923—48no—
237C2×#6923—48no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61#11712yes—
255C22⋊D61—28no—
256C22⋊D61—28no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61#11712yes—
277D244—28no—
278C4×D61—24no—
279C4×D61—24no—
280C4×D61—24no—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—24no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—42no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—42no—
290Dic122—28no—
291#20723—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—22no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61C22×C22⋊C411yescharacteristic
300C22×C61C2×#11711yescharacteristic
301C22×C61C2×#11711yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—42no—
311C2×D61#52514yes—
312#6923—88no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—41no—
319C4×C61—41no—
320Dic61—21no—
321Dic61—21no—
322Dic61—22no—
323Dic61—22no—
324Dic61—41no—
325Dic61—41no—
326Dic61#41012yes—
327Dic61#41012yes—
328C2×C4×#41—611no—
329#2283—612no—
330#1820—612no—
331#1799—612no—
332#1799—612no—
333#1782—612no—
334C4×#99—612no—
335C4×#99—612no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61—42no—
339C2×C61#228011yescharacteristic
340C2×C61C2×#41011yescharacteristic
341C2×C61C2×#41011yescharacteristic
342D61—24no—
343D61—42no—
344C22×#41—611no—
345C4×#41—1222no—
346C4×#41—1222no—
347C4×#41—1222no—
348C4×#41—1222no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×#117—611no—
358C2×#117—611no—
359C2×#117#692311yescharacteristic, complemented
360C22×C42—611no—
361C22×C42—611no—
362C22×C42—611no—
363C2×#99—614no—
364C2×#99—614no—
365#383—1222no—
366#382—1222no—
367#381—614no—
368#380—614no—
369C4×C4⋊C4—1222no—
370C4×C22⋊C4—614no—
371#376—1222no—
372#342—614no—
373#342—614no—
374C61#2072311yescharacteristic, complemented, Sylow
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1224no—
378C2×#41—1224no—
379C22×Q8—611no—
380C22×D4—611no—
381C22×D4—611no—
382C22×D4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4C2×#692311yescharacteristic
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408#117—22no—
409#117—22no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413#117—1222no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4C2×#692311yescharacteristic
420C2×C4⋊C4C2×#692311yescharacteristic
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—618no—
426C2×C22⋊C4C2×#692311yescharacteristic
427C2×C22⋊C4C2×#692311yescharacteristic
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—612no—
432C2×C42—612no—
433C2×C42—612no—
434C2×C42—612no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1224no—
449C2×C42C2×#692311yescharacteristic
450C2×C42C2×#692311yescharacteristic
451#99—1228no—
452#99—1228no—
453#95—1224no—
454#95—1224no—
455#95—1224no—
456#95—1224no—
457C24—611no—
458C24—611no—
459C24C22×#692311yescharacteristic
460#41—12216no—
461#41—12216no—
462C2×Q8—1222no—
463C2×Q8—1222no—
464C2×Q8—1222no—
465C2×D4—614no—
466C2×D4—614no—
467C2×D4—1222no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4—1222no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1222no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1228no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4C22×#692311yescharacteristic
512C22×C4C22×#692311yescharacteristic
513C22×C4C22×#692311yescharacteristic
514C22×C4C22×#692311yescharacteristic
515C22×C4C22×#692311yescharacteristic
516C22×C4C22×#692311yescharacteristic
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—2448no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1228no—
531C22⋊C4—1228no—
532C42—21no—
533C42—21no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—612no—
539C42—612no—
540C42—612no—
541C42—612no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—2448no—
553C42C22⋊#692312yes—
554C42C22⋊#692312yes—
555C23—21no—
556C23—21no—
557C23—41no—
558C23—41no—
559C23—612no—
560C23—612no—
561C23—612no—
562C23—612no—
563C23—1221no—
564C23—1221no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23#1451985612yes—
570C23C23×#692311yescharacteristic
571Q8—1228no—
572D4—1228no—
573D4—1228no—
574D4—1228no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2448no—
618C2×C4—2448no—
619C2×C4—2448no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4C2×C22⋊#692311yescharacteristic
623C2×C4C2×C22⋊#692311yescharacteristic
624C2×C4C2×C22⋊#692312yes—
625C2×C4C2×C22⋊#692312yes—
626C2×C4#1451985612yes—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—614no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—2442no—
639C22—2442no—
640C22—2442no—
641C22—2442no—
642C22—2441no—
643C22—2441no—
644C22C22×C22⋊#692311yescharacteristic
645C22C2×#1451985611yescharacteristic
646C22C2×#1451985611yescharacteristic
647C4—22no—
648C4—22no—
649C4—21no—
650C4—21no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—612no—
655C4—612no—
656C4—1221no—
657C4—1221no—
658C4—1222no—
659C4—1222no—
660C4—4888no—
661C4—2441no—
662C4—2441no—
663C2—21no—
664C2—21no—
665C2—42no—
666C2—1224no—
667C2—2442no—
668C2#2146226311yescharacteristic
669C2D61⋊#41011yescharacteristic
670C2D61⋊#41011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C22
cosocleC23×D61C24
hypercenterC2×#117#6923
hypocenterC61#20723
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21462266C2semidirect
2. #21461716C2semidirect
3. #21461682C2semidirect
4. #21461681C2semidirect
6. #21461566C2semidirect
7. #21461564C2semidirect
25. C2×#14519742C4semidirect
26. C2×#14519742C4semidirect
29. C2×#14519737C4semidirect
30. C2×#14519737C4semidirect
33. C2×#14519734C4semidirect
35. C22×C4×Dic61C4semidirect
36. C2×C42×D61C4semidirect
38. C2×#117×C61C4semidirect
359. C2×#117#6923semidirect
374. C61#20723semidirect

Maximal subgroups

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

subgroupindex
#214622662
#214617162
#214616822
#214616812
#214616482
#214615662
#214615642
C2×#168402302
#2072361