Finite Group Databasebeta

← #29681822

Subgroups — #29681822

Order 15616.

1718 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840334C211yesmaximal, characteristic
2C2×#16840239C211yesmaximal, characteristic, complemented
3C2×#16839734C212yesmaximal
4#21448404C211yesmaximal, characteristic, complemented
5#21446278C211yesmaximal, characteristic, complemented
6#21446277C211yesmaximal, characteristic, complemented
7#21446276C212yesmaximal, complemented
8#21446275C211yesmaximal, characteristic, complemented
9#21446274C212yesmaximal, complemented
10#21445550C212yesmaximal, complemented
11C61×#1899C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14#16840334—22no—
15#16840334—22no—
16#16840334—24no—
17C2×#14519776C412yescomplemented
18C2×#14519763C412yescomplemented
19C2×#14519761C2211yescharacteristic
20C2×#14519760C2211yescharacteristic
21C2×#14519758C2211yescharacteristic
22C2×#14519755C412yescomplemented
23C2×#14519754C412yescomplemented
24#16840239—28no—
25C2×C4×Dic122—22no—
26C2×#14519742—22no—
27C2×#14519736C2211yescharacteristic
28C2×#14519733C2212yes—
29C2×#14519733C2212yes—
30C2×#14519733C2212yes—
31C2×#14519733C2211yescharacteristic
32#16839734—28no—
33C2×#14519692C2212yes—
34C2×#14519691C2212yes—
35C2×#14519691C2212yes—
36#16839666—24no—
37#16839625C2211yescharacteristic
38#16839623C2212yes—
39#16839529C2212yes—
40#16839528C2212yes—
41#16839528C2212yes—
42C2×#123×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C2×#99×C61C2212yes—
45C61×#381C2211yescharacteristic
46C61×#342C2212yes—
47C2×#11758224—24no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—24no—
51C22×Dic122—24no—
52C22×Dic122—24no—
53C22×Dic122C2×C411yescharacteristic
54C22×Dic122C2×C412yes—
55C22×Dic122C2×C412yes—
56C22×Dic122D412yes—
57C22×Dic122D412yes—
58C22×Dic122D414yes—
59C22×C22⋊D61C2×C411yescharacteristic
60C22×C22⋊D61C2311yescharacteristic
61C23×Dic61C2×C411yescharacteristic
62C23×Dic61C2311yescharacteristic
63C23×Dic61C2311yescharacteristic
64C22×C4×D61C2×C411yescharacteristic
65#14519776—24no—
66#14519776—24no—
67#14519763—24no—
68#14519763—24no—
69#14519761—22no—
70#14519761—22no—
71#14519760—22no—
72#14519760—21no—
73#14519760—21no—
74#14519758—22no—
75#14519758—22no—
76#14519755—24no—
77#14519755—24no—
78#14519754—24no—
79#14519754—24no—
80C4×Dic122—216no—
81#14519742—216no—
82#14519736—24no—
83#14519733—24no—
84#14519733—22no—
85#14519733—22no—
86#14519733—24no—
87#14519733—24no—
88#14519733—24no—
89#14519733—24no—
90#14519733D418yes—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2×C412yes—
93C2×#11758196—22no—
94C2×#11758196—22no—
95C2×#11758196—22no—
96C2×#11758196C2×C412yes—
97C2×#11758196C2×C412yes—
98C2×#11758196C2×C412yes—
99C2×C22⋊Dic61C2×C412yes—
100C2×C22⋊Dic61C2312yes—
101C2×C22⋊Dic61C2311yescharacteristic
102C2×C22⋊Dic61C2312yes—
103C2×#11758198C2×C412yes—
104C2×#11758198C2×C412yes—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—22no—
107C2×C4×Dic61—22no—
108C2×C4×Dic61—22no—
109C2×C4×Dic61C2×C412yes—
110#14519692—24no—
111#14519691—24no—
112#14519691—24no—
113#14519684—22no—
114#14519684—24no—
115C22×D4×C61C2311yescharacteristic
116C23×C4×C61C2311yescharacteristic
117#123×C61—22no—
118#123×C61—21no—
119#123×C61—21no—
120#117×C61—24no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2×C412yes—
123C2×C22⋊C4×C61C2×C412yes—
124C2×C22⋊C4×C61C2312yes—
125C2×C22⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61C2312yes—
127C2×C42×C61—21no—
128#99×C61—24no—
129#95×C61—22no—
130C24×C61C22×C411yescharacteristic
131C24×C61C2411yescharacteristic
132C23×D61C22×C411yescharacteristic
133C2×D4×C61—22no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61C22×C411yescharacteristic
142C22×C4×C61C22×C412yes—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—22no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—24no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—24no—
158C22×Dic61—22no—
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C22×C412yes—
161C22×Dic61C22×C412yes—
162C22×Dic61C22×C411yescharacteristic
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D414yes—
169C22×Dic61#4114yes—
170#11758224—44no—
171#11758224—44no—
172#11758224—44no—
173#11758224—44no—
174C2×C4×D61—22no—
175C2×C4×D61—24no—
176C2×Dic122—28no—
177C2×Dic122—28no—
178C2×Dic122—24no—
179C2×Dic122—24no—
180C2×Dic122—28no—
181C2×Dic122—28no—
182C2×Dic122—24no—
183C2×Dic122—24no—
184C2×Dic122—22no—
185C2×Dic122—24no—
186C2×Dic122C22⋊C414yes—
187C2×Dic122C22⋊C414yes—
188C2×Dic122C22⋊C414yes—
189C2×Dic122C22⋊C414yes—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61C2×D414yes—
201C42×C61—24no—
202C22⋊Dic61—22no—
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C22⋊Dic61—24no—
206C22⋊Dic61—22no—
207C22⋊Dic61—22no—
208C22⋊Dic61C2×D414yes—
209#11758198—22no—
210#11758198—22no—
211#11758198—24no—
212C4⋊Dic61—24no—
213C4⋊Dic61—24no—
214#11758196—28no—
215#11758196—22no—
216#11758196—22no—
217#11758196—24no—
218#11758196—24no—
219#11758196—28no—
220#11758196—44no—
221C4×Dic61—28no—
222C4×Dic61—24no—
223C4×Dic61—24no—
224C4×Dic61—28no—
225C4×Dic61—44no—
226C23×C61—21no—
227C23×C61—22no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61C23×C411yescharacteristic
231C23×C61C22×D411yescharacteristic
232C23×C61C22×D411yescharacteristic
233C23×C61C22×D412yes—
234C23×C61C2×#4112yes—
235C22×D61—21no—
236C22×D61—22no—
237C22×D61—42no—
238D4×C61—41no—
239D4×C61—42no—
240D4×C61—41no—
241C2×C4×C61—24no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—24no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—44no—
252C2×C4×C61—42no—
253C2×C4×C61—44no—
254C2×C4×C61—42no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C412yes—
257C22⋊D61—41no—
258C22⋊D61—42no—
259C22⋊D61—41no—
260C22⋊D61—44no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—24no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—44no—
270C2×Dic61—44no—
271C2×Dic61—42no—
272C2×Dic61—44no—
273C2×Dic61—42no—
274C2×Dic61C2×C22⋊C414yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61C4×D414yes—
278C2×Dic61C4×D414yes—
279C2×Dic61#12014yes—
280C2×Dic61#12114yes—
281C2×Dic61#12114yes—
282C2×Dic61#12314yes—
283C4×D61—44no—
284Dic122—28no—
285Dic122—28no—
286Dic122—28no—
287Dic122—28no—
288Dic122—44no—
289#20986—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—41no—
297C22×C61—82no—
298C22×C61C22×C22⋊C411yescharacteristic
299C22×C61C2×C4×D412yes—
300C22×C61C2×#12011yescharacteristic
301C22×C61C2×#12112yes—
302C22×C61C2×#12311yescharacteristic
303C2×D61—42no—
304C2×D61—41no—
305C2×D61—82no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—44no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—82no—
312Dic61—24no—
313Dic61—44no—
314Dic61—42no—
315Dic61—41no—
316Dic61—82no—
317Dic61#38614yes—
318Dic61#38614yes—
319C2×#536—611no—
320C2×#519—611no—
321C2×#410—612no—
322#2310—611no—
323#1899—612no—
324#1899—611no—
325#1899—612no—
326#1899—611no—
327#1899—611no—
328#1899D6111yescharacteristic, complemented
329#1740—612no—
330C2×C61—21no—
331C2×C61—22no—
332C2×C61—22no—
333C2×C61—21no—
334C2×C61—81no—
335C2×C61C2×#38611yescharacteristic
336C2×C61#288412yes—
337D61—81no—
338C22×#41—611no—
339C23×Q8—611no—
340#536—1222no—
341#536—1222no—
342#536—1224no—
343C2×C4⋊Q8—612no—
344C2×#124—612no—
345C2×#123—611no—
346C2×#123—611no—
347C2×#123—611no—
348C2×#123C2×D6111yescharacteristic
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—612no—
351#519—1228no—
352C2×C4×Q8—1222no—
353C2×C4×Q8—1222no—
354C2×#117—611no—
355C2×#117—611no—
356C2×#117—612no—
357C2×#117—612no—
358C2×#117—612no—
359C2×#117C2×D6111yescharacteristic
360#410—1228no—
361C2×#99—612no—
362C2×#99—612no—
363C2×#99—612no—
364C2×#99C2×D6112yes—
365#391—1224no—
366#381—611no—
367#381—612no—
368#381C2×D6111yescharacteristic
369#342—612no—
370#342—612no—
371#342—612no—
372#342C2×D6112yes—
373C61#2098611yescharacteristic, complemented, Sylow
374C2×#41—1222no—
375C2×#41—1222no—
376C2×#41—1224no—
377C2×#41—1224no—
378C22×Q8—612no—
379C22×Q8—612no—
380C22×Q8—611no—
381C22×Q8—612no—
382C22×Q8—612no—
383C22×Q8—614no—
384C22×Q8—1224no—
385C22×Q8—1224no—
386C22×D4—611no—
387C22×D4—611no—
388C22×D4C22×D6111yescharacteristic
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4C22×D6111yescharacteristic
394C4⋊Q8—1224no—
395C4⋊Q8—1224no—
396#124—1224no—
397#124—1224no—
398#123—22no—
399#123—21no—
400#123—21no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405#123—1222no—
406#123—1221no—
407#123—1221no—
408C22⋊Q8—1224no—
409C22⋊Q8—1224no—
410C22⋊Q8—1224no—
411C22⋊Q8—1224no—
412C4×Q8—12216no—
413C4×Q8—12216no—
414#117—24no—
415#117—618no—
416#117—1224no—
417#117—1224no—
418#117—1222no—
419#117—1222no—
420#117—1224no—
421#117—1224no—
422#117—1224no—
423#117—1224no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4C4×D6112yes—
434C2×C22⋊C4—612no—
435C2×C22⋊C4—612no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—612no—
438C2×C22⋊C4—612no—
439C2×C22⋊C4—612no—
440C2×C22⋊C4C4×D6112yes—
441C2×C22⋊C4C22×D6112yes—
442C2×C22⋊C4C22×D6111yescharacteristic
443C2×C22⋊C4C22×D6112yes—
444C2×C42—21no—
445C2×C42—612no—
446C2×C42—1222no—
447C2×C42—1222no—
448C2×C42—1221no—
449C2×C42—1222no—
450#99—24no—
451#99—1224no—
452#99—1224no—
453#99—1224no—
454#95—22no—
455#95—1224no—
456#95—1222no—
457C24—611no—
458C24C2×C4×D6111yescharacteristic
459C24C23×D6111yescharacteristic
460#41—2444no—
461#41—2444no—
462#41—2444no—
463#41—2444no—
464C2×Q8—614no—
465C2×Q8—614no—
466C2×Q8—614no—
467C2×Q8—614no—
468C2×Q8—1224no—
469C2×Q8—1224no—
470C2×Q8—1228no—
471C2×Q8—1228no—
472C2×Q8—1228no—
473C2×Q8—1228no—
474C2×Q8—1224no—
475C2×Q8—1224no—
476C2×Q8—1222no—
477C2×Q8—1224no—
478C2×D4—22no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—1222no—
483C2×D4—1224no—
484C2×D4—1222no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1222no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—614no—
497C22×C4—614no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—611no—
502C22×C4—612no—
503C22×C4—1222no—
504C22×C4—1224no—
505C22×C4—1222no—
506C22×C4—1224no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4C2×C4×D6111yescharacteristic
512C22×C4C2×C4×D6112yes—
513C22×C4C2×C4×D6112yes—
514C22×C4D4×D6112yes—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—1224no—
521C4⋊C4—1224no—
522C4⋊C4—1224no—
523C4⋊C4—1228no—
524C4⋊C4—1228no—
525C4⋊C4—1224no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—2444no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—614no—
537C22⋊C4—1224no—
538C22⋊C4—1224no—
539C22⋊C4—1222no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4D4×D6114yes—
547C42—24no—
548C42—1224no—
549C42—1228no—
550C42—1224no—
551C42—1228no—
552C42—2444no—
553C23—21no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—1222no—
558C23—1221no—
559C23—2442no—
560C23C22×C4×D6111yescharacteristic
561C23C2×D4×D6111yescharacteristic
562C23C2×D4×D6111yescharacteristic
563C23C2×D4×D6112yes—
564C23#41×D6112yes—
565Q8—1228no—
566Q8—1228no—
567Q8—1228no—
568Q8—1228no—
569Q8—2444no—
570D4—41no—
571D4—42no—
572D4—41no—
573D4—2444no—
574D4—2441no—
575D4—2442no—
576D4—2441no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—614no—
594C2×C4—614no—
595C2×C4—614no—
596C2×C4—614no—
597C2×C4—614no—
598C2×C4—614no—
599C2×C4—614no—
600C2×C4—1224no—
601C2×C4—1222no—
602C2×C4—1224no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—2444no—
609C2×C4—2444no—
610C2×C4—2444no—
611C2×C4—2442no—
612C2×C4—2444no—
613C2×C4—2442no—
614C2×C4C22⋊C4×D6112yes—
615C2×C4C22⋊C4×D6112yes—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—42no—
623C22—82no—
624C22—4882no—
625C22—2441no—
626C22—2442no—
627C22C2×C22⋊C4×D6111yescharacteristic
628C22C4×D4×D6112yes—
629C22#120×D6111yescharacteristic
630C22#121×D6112yes—
631C22#123×D6111yescharacteristic
632C4—22no—
633C4—22no—
634C4—44no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—614no—
639C4—614no—
640C4—1224no—
641C4—4882no—
642C4—2444no—
643C4—2441no—
644C4—2442no—
645C2—21no—
646C2—22no—
647C2—22no—
648C2—21no—
649C2—81no—
650C2—4881no—
651C2#386×D6111yescharacteristic
652C2#2145428112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1899C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758224C22
cosocleC24×C61C24
hypercenter#1899D61
hypocenterC61#20986
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1899

Complements

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

NH ≅ G/Ntype
2. C2×#16840239C2semidirect
4. #21448404C2semidirect
5. #21446278C2semidirect
6. #21446277C2semidirect
7. #21446276C2semidirect
8. #21446275C2semidirect
9. #21446274C2semidirect
10. #21445550C2semidirect
11. C61×#1899C2semidirect
17. C2×#14519776C4semidirect
18. C2×#14519763C4semidirect
22. C2×#14519755C4semidirect
23. C2×#14519754C4semidirect
328. #1899D61semidirect
373. C61#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#168403342
C2×#168402392
C2×#168397342
#214484042
#214462782
#214462772
#214462762
#214462752
#214462742
#214455502
C61×#18992
#2098661