Finite Group Databasebeta

← #29685673

Subgroups — #29685673

Order 15616.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic61C211yesmaximal, characteristic, complemented
2C2×#16839901C211yesmaximal, characteristic
3C2×#16839865C212yesmaximal, complemented
4#21448325C211yesmaximal, characteristic
5#21446172C212yesmaximal, complemented
6#21446134C211yesmaximal, characteristic
7#21446071C212yesmaximal, complemented
8#21446006C211yesmaximal, characteristic, complemented
9#21445651C212yesmaximal, complemented
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519751C2211yescharacteristic
14C2×Q8×Dic61—28no—
15C2×Q8×Dic61—21no—
16C2×Q8×Dic61—21no—
17C2×Q8×Dic61C2212yes—
18C2×Q8×Dic61C2212yes—
19C2×#14519739C2211yescharacteristic
20C2×#14519737—21no—
21C22×C4⋊Dic61—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23C22×C4×Dic61—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839901C418yescomplemented
26#16839865—28no—
27C2×#14519710C2212yes—
28C2×#14519701C2212yes—
29C2×#14519699C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519687C2212yes—
32#16839663C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839651—21no—
35#16839639—21no—
36C4⋊C4×Dic61C2211yescharacteristic
37#16839609—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#380C2211yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61—21no—
49C23×Dic61D411yescharacteristic
50C23×Dic61D411yescharacteristic
51C22×C4×D61—21no—
52C2×#11758201—24no—
53C2×#11758201—24no—
54C2×#11758201C2×C418yescomplemented
55C22×#92418C2312yes—
56#14519751—22no—
57#14519751C2×C414yes—
58Q8×Dic61—28no—
59Q8×Dic61—28no—
60Q8×Dic61—24no—
61Q8×Dic61—24no—
62Q8×Dic61—24no—
63Q8×Dic61D418yescomplemented
64#14519739—24no—
65#14519737—44no—
66C2×C4⋊Dic61—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61—28no—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—24no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519710—24no—
88#14519701C2×C418yescomplemented
89#14519699—24no—
90C22⋊#92418C2×C414yescomplemented
91#14519687—24no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—24no—
95C22×Q8×C61—24no—
96C22×Q8×C61—21no—
97C22×Q8×C61—22no—
98C22×Q8×C61C2×C414yescomplemented
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106#103×C61—24no—
107#103×C61—24no—
108C22⋊C8×C61—22no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—21no—
112C2×Q8×C61—216no—
113C2×Q8×C61—24no—
114C2×Q8×C61—24no—
115C2×Q8×C61—216no—
116C2×Q8×C61—24no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61C22×C414yes—
121C2×Q8×C61C2×D414yes—
122C2×Q8×C61#4114yes—
123C22×C4×C61—24no—
124C22×C4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—24no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—41no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—21no—
156C2×D244C22×C412yes—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C22⋊C4×C61—42no—
166#11758201—216no—
167#11758198—24no—
168#11758198—42no—
169#11758198—42no—
170#11758198—42no—
171C4⋊Dic61—28no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—28no—
176C4⋊Dic61C22×C412yes—
177C4⋊Dic61C2×D414yes—
178#11758196—42no—
179C4×Dic61—28no—
180C4×Dic61—22no—
181C4×Dic61—28no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—42no—
185C4×Dic61C2×D414yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418C22×C414yes—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61C4×D412yes—
194C23×C61AGL(1,ℤ/8)12yes—
195C23×C61#13712yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204Q8×C61—216no—
205Q8×C61—24no—
206Q8×C61—24no—
207Q8×C61—24no—
208Q8×C61C4×D418yescomplemented
209C2×C4×C61—216no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—21no—
213C2×C4×C61—24no—
214C2×C4×C61—21no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C4×D412yes—
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D412yes—
226C2×C4×C61C2×#4112yes—
227C2×C4×C61C2×#4111yescharacteristic
228C22⋊D61—48no—
229C2×Dic61—28no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—24no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—44no—
242C2×Dic61—48no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61#12012yes—
248C2×Dic61#12012yes—
249C2×Dic61#13712yes—
250D244—24no—
251D244—42no—
252C4×D61—84no—
253C8×C61—42no—
254#92418—24no—
255#21454—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61#44014yescomplemented
263C22×C61C2×C4×D411yescharacteristic
264C22×C61C2×#12011yescharacteristic
265C22×C61#52511yescharacteristic
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C22×C61C2×#13711yescharacteristic
269C22×C61#57511yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C4×C61—28no—
278C4×C61—22no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61C2×C4×D412yes—
282C4×C61#52512yes—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—42no—
287Dic61—41no—
288Dic61—44no—
289C22×C4×Q8—611no—
290C2×#448—611no—
291C2×#440—612no—
292C4×#121—611no—
293#1880—612no—
294#1874—611no—
295#1863—612no—
296#1852—611no—
297#1787D6111yescharacteristic, complemented
298#1763—612no—
299#1749—611no—
300C2×C61—24no—
301C2×C61C4×#12011yescharacteristic
302C2×C61C2×#44011yescharacteristic
303C2×C61C4×#13711yescharacteristic
304C2×C61#300411yescharacteristic
305C2×C61#300611yescharacteristic
306C2×C61#311211yescharacteristic
307C2×C61#311711yescharacteristic
308D61—82no—
309C23×Q8C2×D6111yescharacteristic
310C22×SD16—612no—
311C2×#121—611no—
312C2×C4×Q8—612no—
313C2×C4×Q8—612no—
314C2×C4×Q8—1221no—
315C2×C4×Q8—1221no—
316C2×C4×Q8—1228no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—1221no—
321C22×C42—611no—
322C22×C42—1221no—
323#448—618no—
324#440—1228no—
325C2×#107—612no—
326C2×#103—612no—
327C2×#103C2×D6112yes—
328C2×#103C2×D6112yes—
329C2×#102—612no—
330C2×C22⋊C8—611no—
331C2×C22⋊C8C2×D6111yescharacteristic
332C2×#97—612no—
333#390—611no—
334#389—611no—
335#388—1221no—
336#386—1221no—
337#380C2×D6111yescharacteristic
338C4×C4⋊C4—611no—
339C4×C22⋊C4—1221no—
340#340—612no—
341#340—612no—
342#336—612no—
343C61#2145411yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C4×D6114yescomplemented
349C22×Q8C22×D6112yes—
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4C22×D6111yescharacteristic
358C2×SD16—618no—
359C2×SD16—1224no—
360C2×SD16—1224no—
361C22×C8—612no—
362C22×C8C22×D6112yes—
363#121—614no—
364#121—1222no—
365C4×Q8—618no—
366C4×Q8—1224no—
367C4×Q8—1224no—
368C4×Q8—1228no—
369C4×Q8—1228no—
370C4×Q8—1224no—
371C4×D4—1224no—
372C4×D4—2444no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1228no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—1224no—
393C2×C42—1222no—
394C2×C42—1222no—
395C2×C42—1228no—
396C2×C42—1221no—
397#107—1224no—
398#103—24no—
399#103—24no—
400#103—618no—
401#102—1224no—
402C22⋊C8—22no—
403C22⋊C8—614no—
404#97—1224no—
405#95—21no—
406#95—1221no—
407#95—1221no—
408C24—1221no—
409C24D4×D6111yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8C2×C4×D6114yes—
419C2×Q8D4×D6114yes—
420C2×Q8#1175822414yes—
421C2×D4—612no—
422C2×D4—1224no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1221no—
426C2×D4—2442no—
427C2×D4—2442no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1224no—
445C22×C4—1221no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2441no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4C2×C4×D6112yes—
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822412yes—
457C22×C4C23×D6111yescharacteristic
458SD16—12216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—614no—
463C2×C8—1222no—
464C2×C8—1222no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—612no—
468C4⋊C4—614no—
469C4⋊C4—1228no—
470C4⋊C4—1228no—
471C4⋊C4—1224no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—2442no—
475C22⋊C4—42no—
476C22⋊C4—1224no—
477C22⋊C4—2442no—
478C22⋊C4—2442no—
479C22⋊C4—2442no—
480C42—614no—
481C42—1222no—
482C42—1222no—
483C42—1228no—
484C42—1228no—
485C42—1222no—
486C42—2442no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—1222no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23#1451973712yes—
499C23#1451980412yes—
500C23#1451980912yes—
501C23C2×D4×D6111yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8#1451973718yescomplemented
507D4—1224no—
508D4—2448no—
509D4—2442no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—1228no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—4884no—
540C2×C4—2444no—
541C2×C4—2448no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4#1451973712yes—
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6112yes—
549C2×C4C2×#1175822412yes—
550C2×C4C2×#1175822411yescharacteristic
551C8—42no—
552C8—1224no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—4881no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22#1683986514yescomplemented
567C22C2×#1451973711yescharacteristic
568C22#120×D6111yescharacteristic
569C22#1684026811yescharacteristic
570C22C2×#1451980411yescharacteristic
571C22#137×D6111yescharacteristic
572C22C2×#1451980911yescharacteristic
573C22#1684056111yescharacteristic
574C4—28no—
575C4—22no—
576C4—81no—
577C4—81no—
578C4—1222no—
579C4—1222no—
580C4—1222no—
581C4—2442no—
582C4—2441no—
583C4—2444no—
584C4C2×#1451973712yes—
585C4#1684026812yes—
586C2—24no—
587C2—4882no—
588C2#2144831811yescharacteristic
589C2C2×#1683986511yescharacteristic
590C2#2145463411yescharacteristic
591C2#2145505411yescharacteristic
592C2#2145507011yescharacteristic
593C2#2145593611yescharacteristic
594C2#2145599111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1787D61
hypocenterC61#21454
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic61C2semidirect
3. C2×#16839865C2semidirect
5. #21446172C2semidirect
7. #21446071C2semidirect
8. #21446006C2semidirect
9. #21445651C2semidirect
10. #21445583C2semidirect
11. C61×#1787C2semidirect
25. #16839901C4semidirect
54. C2×#11758201C2×C4semidirect
63. Q8×Dic61D4semidirect
88. #14519701C2×C4semidirect
90. C22⋊#92418C2×C4semidirect
98. C22×Q8×C61C2×C4semidirect
208. Q8×C61C4×D4semidirect
262. C22×C61#440semidirect
297. #1787D61semidirect
343. C61#21454semidirect
348. C22×Q8C4×D61semidirect
506. Q8#14519737semidirect
566. C22#16839865semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic612
C2×#168399012
C2×#168398652
#214483252
#214461722
#214461342
#214460712
#214460062
#214456512
#214455832
C61×#17872
#2145461