Finite Group Databasebeta

← #29685672

Subgroups — #29685672

Order 15616.

1868 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic
2C2×#16839912C211yesmaximal, characteristic
3C2×#16839865C212yesmaximal, complemented
4C22⋊Q8×Dic61C211yesmaximal, characteristic, complemented
5#21446150C212yesmaximal, complemented
6#21446067C212yesmaximal, complemented
7#21446007C211yesmaximal, characteristic, complemented
8#21445751C211yesmaximal, characteristic
9#21445643C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1872C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×#11758201C2212yes—
15C2×Q8×Dic61C2211yescharacteristic, complemented
16C2×#14519739—28no—
17C2×#14519739—21no—
18C2×#14519739—21no—
19C2×#14519739C2212yes—
20C2×#14519739C2212yes—
21C22×#11758198—21no—
22C22×C4×Dic61C2211yescharacteristic
23#16839912C418yescomplemented
24#16839865—28no—
25C2×#14519710C2212yescomplemented
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519687C2212yescomplemented
31#16839669C2211yescharacteristic, complemented
32#16839661C2211yescharacteristic, complemented
33#16839648—21no—
34#16839637—21no—
35#16839622C2211yescharacteristic
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2211yescharacteristic, complemented
38C22⋊C4×Dic61—21no—
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—24no—
51C22×D244C2×C414yes—
52C22×D244C2312yes—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×C4×D61—28no—
56C22×C4×D61—22no—
57C22×C4×D61—22no—
58C22×C4×D61—44no—
59C22×C4×D61—44no—
60C2×#11758201—24no—
61C2×#11758201—24no—
62C2×#11758201C2×C418yescomplemented
63C22×#92418C2312yes—
64Q8×Dic61—24no—
65#14519739—28no—
66#14519739—28no—
67#14519739—24no—
68#14519739—24no—
69#14519739—24no—
70#14519739D418yes—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×#11758198—28no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×C4×Dic61—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—24no—
87#14519699—24no—
88#14519699—24no—
89#14519697C2×C418yescomplemented
90C22⋊#92418C2×C414yescomplemented
91#14519687—24no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2×C414yescomplemented
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—24no—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109#103×C61—24no—
110C22⋊C8×C61—22no—
111C24×C61C2×D411yescharacteristic
112C23×D61—24no—
113C23×D61—24no—
114C23×D61—24no—
115C23×D61—22no—
116C23×D61—42no—
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—42no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C22×C412yescomplemented
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—41no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×D244—24no—
149C2×D244—216no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—216no—
156C2×D244C22×C414yes—
157C2×D244C2×D414yes—
158C2×D244#4114yes—
159C2×C4×D61—28no—
160C2×C4×D61—28no—
161C2×C4×D61—432no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C412yescomplemented
174C4⋊C4×C61C2×D414yes—
175C22⋊C4×C61—24no—
176C22⋊Dic61—42no—
177#11758201—216no—
178#11758198—28no—
179#11758198—28no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—42no—
183C4⋊Dic61—42no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—42no—
188C4×Dic61—42no—
189C4×Dic61C2×D414yes—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yescomplemented
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61C4×D412yes—
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61#13712yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—216no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—48no—
206C22×D61—48no—
207C22×D61—48no—
208C22×D61—48no—
209C22×D61—48no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—48no—
215Q8×C61—24no—
216Q8×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—48no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—41no—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D412yes—
234C2×C4×C61C2×#4112yes—
235C2×C4×C61C2×#4111yescharacteristic
236C2×Dic61—28no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—44no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61#12012yes—
254C2×Dic61#12012yes—
255C2×Dic61AGL(1,ℤ/8)12yes—
256D244—216no—
257D244—24no—
258D244—24no—
259D244—24no—
260D244C4×D418yes—
261C4×D61—416no—
262C4×D61—416no—
263C8×C61—42no—
264#92418—24no—
265#21453—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61#44014yescomplemented
273C22×C61C2×C4×D411yescharacteristic
274C22×C61C2×#12011yescharacteristic
275C22×C61#52511yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C2×D61—28no—
281C2×D61—432no—
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—44no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—44no—
291C4×C61—42no—
292C4×C61C2×C4×D412yes—
293C4×C61#52512yes—
294Dic61—22no—
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300C22×C4×D4—611no—
301C2×#450—611no—
302C2×#440—612no—
303C4×C22⋊Q8—611no—
304#1877—612no—
305#1872D6111yescharacteristic, complemented
306#1862—612no—
307#1852—611no—
308#1786—611no—
309#1762—612no—
310#1749—611no—
311C2×C61—24no—
312C2×C61C4×#12011yescharacteristic
313C2×C61C2×#44011yescharacteristic
314C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C61#300311yescharacteristic
316C2×C61#300711yescharacteristic
317C2×C61#311011yescharacteristic
318C2×C61#311411yescharacteristic
319D61—48no—
320C23×D4—611no—
321C24×C4—1221no—
322C22×SD16—612no—
323C2×C22⋊Q8C2×D6111yescharacteristic
324C2×C4×Q8—611no—
325C2×C4×D4—612no—
326C2×C4×D4—612no—
327C2×C4×D4—1221no—
328C2×C4×D4—1221no—
329C2×C4×D4—1228no—
330C22×C4⋊C4C2×D6111yescharacteristic
331C22×C22⋊C4—1221no—
332C22×C42—611no—
333#450—618no—
334#440—1228no—
335C2×#107—612no—
336C2×#103C2×D6112yescomplemented
337C2×#102—612no—
338C2×#102—612no—
339C2×#102—612no—
340C2×C22⋊C8—611no—
341C2×C22⋊C8C2×D6111yescharacteristic
342C2×#97—612no—
343#391—611no—
344#389—611no—
345#387—1221no—
346#385—1221no—
347#380—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—1221no—
350C4×C22⋊C4—1221no—
351#340—612no—
352#340C2×D6112yescomplemented
353#336—612no—
354C61#2145311yescharacteristic, complemented, Sylow
355C25—1221no—
356C22×Q8C22×D6111yescharacteristic
357C22×D4—614no—
358C22×D4—612no—
359C22×D4—1224no—
360C22×D4—1224no—
361C22×D4—1221no—
362C22×D4—1222no—
363C23×C4—21no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—1228no—
369C23×C4—2444no—
370C23×C4—2444no—
371C23×C4C22×D6111yescharacteristic
372C2×SD16—618no—
373C2×SD16—1224no—
374C2×SD16—1224no—
375C22×C8—612no—
376C22×C8C22×D6112yes—
377C22⋊Q8—22no—
378C22⋊Q8C4×D6114yescomplemented
379C4×Q8—1224no—
380C4×D4—618no—
381C4×D4—1224no—
382C4×D4—1224no—
383C4×D4—1228no—
384C4×D4—1228no—
385C4×D4—1224no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1228no—
399C2×C22⋊C4—1221no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1224no—
405C2×C42—1221no—
406C2×C42—1221no—
407#107—1224no—
408#103—24no—
409#102—618no—
410#102—1224no—
411#102—1224no—
412C22⋊C8—22no—
413C22⋊C8—614no—
414#97—1224no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—1224no—
421C24—1224no—
422C24—1222no—
423C24—1224no—
424C24—2442no—
425C24—2442no—
426C24—2442no—
427C24—2442no—
428C24D4×D6111yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C4×D6112yescomplemented
433C2×D4—614no—
434C2×D4—614no—
435C2×D4—614no—
436C2×D4—12216no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—12216no—
440C2×D4—1222no—
441C2×D4—1222no—
442C2×D4—1222no—
443C2×D4—1224no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1228no—
457C22×C4—1228no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2444no—
466C22×C4—24432no—
467C22×C4—2444no—
468C22×C4—2444no—
469C22×C4—2444no—
470C22×C4—2441no—
471C22×C4C2×C4×D6112yes—
472C22×C4D4×D6111yescharacteristic
473C22×C4#1175822412yes—
474C22×C4C23×D6111yescharacteristic
475SD16—12216no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—614no—
480C2×C8—1222no—
481C2×C8—1222no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C4⋊C4C2×C4×D6112yescomplemented
491C4⋊C4D4×D6114yes—
492C22⋊C4—24no—
493C22⋊C4—1228no—
494C22⋊C4—1228no—
495C22⋊C4—2442no—
496C42—614no—
497C42—1222no—
498C42—1222no—
499C42—1222no—
500C42—2442no—
501C42—2442no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—12216no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—2448no—
511C23—2448no—
512C23—2448no—
513C23—2448no—
514C23—2448no—
515C23—2442no—
516C23—2442no—
517C23—2442no—
518C23—2442no—
519C23—2448no—
520C23#1451973712yes—
521C23#1451980412yes—
522C23#1451980912yes—
523C23C2×D4×D6111yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—618no—
527D4—12216no—
528D4—1224no—
529D4—1224no—
530D4—1224no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1228no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—24416no—
559C2×C4—24416no—
560C2×C4—2444no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4#1451973712yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×D4×D6112yes—
570C2×C4#41×D6112yes—
571C2×C4C2×#1175822411yescharacteristic
572C8—42no—
573C8—1224no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—1228no—
581C22—24432no—
582C22—2444no—
583C22—2444no—
584C22—2444no—
585C22—2444no—
586C22—2444no—
587C22—2444no—
588C22#1683986514yescomplemented
589C22C2×#1451973711yescharacteristic
590C22#120×D6111yescharacteristic
591C22#1684027011yescharacteristic
592C22AGL(1,ℤ/8)×D6111yescharacteristic
593C22C2×#1451980411yescharacteristic
594C22C2×#1451980911yescharacteristic
595C22#1684056111yescharacteristic
596C4—22no—
597C4—42no—
598C4—42no—
599C4—44no—
600C4—1222no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4C4×D4×D6112yes—
607C4#1684027212yes—
608C2—24no—
609C2—2448no—
610C2#2144832411yescharacteristic
611C2C2×#1683986511yescharacteristic
612C2#2145463211yescharacteristic
613C2#2145504711yescharacteristic
614C2#2145507811yescharacteristic
615C2#2145591511yescharacteristic
616C2#2145596011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. C2×#16839865C2semidirect
4. C22⋊Q8×Dic61C2semidirect
5. #21446150C2semidirect
6. #21446067C2semidirect
7. #21446007C2semidirect
9. #21445643C2semidirect
10. #21445583C2semidirect
11. C61×#1872C2semidirect
15. C2×Q8×Dic61C22semidirect
23. #16839912C4semidirect
25. C2×#14519710C22semidirect
30. C2×#14519687C22semidirect
31. #16839669C22semidirect
32. #16839661C22semidirect
37. C4⋊C4×Dic61C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
62. C2×#11758201C2×C4semidirect
89. #14519697C2×C4semidirect
90. C22⋊#92418C2×C4semidirect
102. C22⋊Q8×C61C2×C4semidirect
123. C2×Q8×C61C22×C4semidirect
173. C4⋊C4×C61C22×C4semidirect
192. C2×#92418C22×C4semidirect
272. C22×C61#440semidirect
305. #1872D61semidirect
336. C2×#103C2×D61semidirect
352. #340C2×D61semidirect
354. C61#21453semidirect
378. C22⋊Q8C4×D61semidirect
432. C2×Q8C2×C4×D61semidirect
490. C4⋊C4C2×C4×D61semidirect
588. C22#16839865semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
C2×#168399122
C2×#168398652
C22⋊Q8×Dic612
#214461502
#214460672
#214460072
#214457512
#214456432
#214455832
C61×#18722
#2145361