Finite Group Databasebeta

← #29716071

Subgroups — #29716071

Order 15616.

1378 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840041C212yesmaximal, complemented
3#21448327C211yesmaximal, characteristic
4#21447147C211yesmaximal, characteristic, complemented
5#21446956C211yesmaximal, characteristic, complemented
6#21446952C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446135C212yesmaximal
9#21446005C212yesmaximal, complemented
10#21445798C211yesmaximal, characteristic, complemented
11C61×#2057C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—28no—
18C2×#14519752C2211yescharacteristic
19C2×#14519739—21no—
20C2×#14519739C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840041—28no—
25C2×#14519710C2212yes—
26C2×#14519706C2212yes—
27C2×#14519699C2212yes—
28C2×#14519697C2212yes—
29C2×#14519689C2212yes—
30#16839664C2211yescharacteristic
31#16839652—21no—
32#16839640—21no—
33#16839619C2211yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#107×C61C2212yes—
42C61×#389C2211yescharacteristic
43C61×#340C2212yes—
44C61×#336C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—22no—
47C2×Q8×D61—28no—
48C2×Q8×D61—44no—
49C2×Q8×D61—44no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C4×D61—22no—
53C22×C4×D61—21no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—24no—
59#14519776—28no—
60#14519776—24no—
61#14519776D418yescomplemented
62#14519775—28no—
63#14519752—22no—
64#14519752Q814yescomplemented
65#14519739—24no—
66#14519739—44no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C4⋊C4×D61—28no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61—24no—
85#14519710—24no—
86#14519706—24no—
87#14519699—24no—
88#14519697—24no—
89#14519689—24no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C42×C61C2311yescharacteristic
101#107×C61—24no—
102#95×C61—21no—
103C23×D61—21no—
104C23×D61C2×D411yescharacteristic
105C2×Q8×C61—24no—
106C2×Q8×C61—42no—
107C2×Q8×C61—42no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D411yescharacteristic
119Q8×D61—432no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—24no—
123C2×D244—21no—
124C2×D244—42no—
125C2×D244—42no—
126C2×D244C2×Q812yescomplemented
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—28no—
135C2×C4×D61—44no—
136C2×C4×D61—44no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61D814yes—
141C2×C4×D61C2×Q812yes—
142C2×C4×D61#4112yes—
143C2×Dic122—24no—
144C2×Dic122—28no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×Dic122—44no—
148C2×Dic122—44no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61C2×D414yes—
158C4⋊C4×C61C2×Q812yescomplemented
159C42×C61—22no—
160#11758198—24no—
161#11758198—42no—
162#11758198—42no—
163#11758198—42no—
164#11758198—42no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D414yes—
169#11758196—28no—
170#11758196—28no—
171C4×Dic61—22no—
172C4×Dic61—22no—
173C4×Dic61—42no—
174C4×Dic61C2×D414yes—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61C22×D411yescharacteristic
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61C22⋊Q812yes—
191C22×D61C2×D812yes—
192C22×D61#13712yes—
193Q8×C61—48no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61C2×D812yes—
208C2×C4×C61#13712yes—
209C2×C4×C61C22×D412yes—
210C2×C4×C61C22×Q811yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C612−1+412yes—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—24no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61#12012yes—
224C2×Dic61#12012yes—
225C2×Dic61C22⋊Q812yes—
226C2×Dic61C2×D812yes—
227C2×Dic61AGL(1,ℤ/8)12yes—
228D244—24no—
229D244—48no—
230D244—42no—
231C4×D61—28no—
232C4×D61—24no—
233C4×D61—24no—
234C4×D61—28no—
235C4×D61—416no—
236C4×D61—84no—
237Dic122—48no—
238Dic122—416no—
239C8×C61—42no—
240#92418—42no—
241#24298—611nomaximal, Sylow
242C22×C61C2×#12011yescharacteristic
243C22×C61C2×C22⋊Q811yescharacteristic
244C22×C61#53611yescharacteristic
245C22×C61C22×D811yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C61C2×#13711yescharacteristic
248C22×C61C2×#13711yescharacteristic
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61#47414yescomplemented
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—44no—
266C4×C61—42no—
267C4×C61—81no—
268C4×C61#54812yes—
269C4×C61D4×Q812yes—
270Dic61—22no—
271Dic61—22no—
272Dic61—44no—
273Dic61—81no—
274Dic61#44714yes—
275C22×C4⋊Q8—611no—
276C2×#474—612no—
277C4×#121—611no—
278#2057—611no—
279#2057D6111yescharacteristic, complemented
280#2023—611no—
281#2023—611no—
282#1988—612no—
283#1874—612no—
284#1852—612no—
285#1797—611no—
286C2×C61#240911yescharacteristic
287C2×C61C2×#44711yescharacteristic
288C2×C61#300411yescharacteristic
289C2×C61C2×#47411yescharacteristic
290C2×C61#307911yescharacteristic
291C2×C61#322611yescharacteristic
292C2×C61#323911yescharacteristic
293D61—24no—
294D61—82no—
295C23×Q8—1221no—
296C2×C4⋊Q8—612no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8—1221no—
299C2×C4⋊Q8—1221no—
300C2×C4⋊Q8—1228no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×#121—611no—
303C2×C4×D4—611no—
304C2×C4×D4—1221no—
305C22×C4⋊C4—612no—
306C22×C4⋊C4—1221no—
307C22×C42—611no—
308#474—1228no—
309C2×#107—612no—
310C2×#107C2×D6112yes—
311C2×C4⋊C8—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×C22⋊C8—612no—
315#390—611no—
316#389C2×D6111yescharacteristic
317#388—1221no—
318#386—1221no—
319#380—611no—
320C4×C4⋊C4—611no—
321C4×C22⋊C4—1221no—
322#340—612no—
323#340—612no—
324#340—612no—
325#340C2×D6112yes—
326#336—611no—
327#336C2×D6111yescharacteristic
328C61#2429811yescharacteristic, complemented, Sylow
329C22×Q8—21no—
330C22×Q8—1222no—
331C22×Q8—1228no—
332C22×Q8—1221no—
333C22×Q8—2444no—
334C22×Q8—2444no—
335C22×D4—611no—
336C22×D4—1221no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C23×C4—1222no—
342C22×C8—612no—
343C22×C8C22×D6112yes—
344C4⋊Q8—24no—
345C4⋊Q8—618no—
346C4⋊Q8—1224no—
347C4⋊Q8—1224no—
348C4⋊Q8—1228no—
349C4⋊Q8—1228no—
350#121—614no—
351#121—1222no—
352C4×D4—1224no—
353C4×D4—2444no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4—1228no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1224no—
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C4⋊C4C22×D6112yes—
366C2×C4⋊C4C22×D6111yescharacteristic
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—1224no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#107—24no—
379#107—1224no—
380C4⋊C8—1224no—
381#102—1224no—
382#102—1224no—
383C22⋊C8—1224no—
384#95—21no—
385#95—1221no—
386#95—1221no—
387C24—611no—
388C24—1221no—
389C2×Q8—24no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—1228no—
393C2×Q8—1224no—
394C2×Q8—2442no—
395C2×Q8—2444no—
396C2×Q8—2444no—
397C2×Q8—24432no—
398C2×Q8—2442no—
399C2×D4—612no—
400C2×D4—1224no—
401C2×D4—1222no—
402C2×D4—1221no—
403C2×D4—1222no—
404C2×D4—2442no—
405C2×D4—2442no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—614no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1228no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4—2444no—
429C22×C4—2444no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—614no—
445C4⋊C4—1228no—
446C4⋊C4—1228no—
447C4⋊C4—1224no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4D4×D6114yes—
451C4⋊C4Q8×D6112yescomplemented
452C22⋊C4—1224no—
453C22⋊C4—2442no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C42—22no—
458C42—614no—
459C42—1222no—
460C42—1222no—
461C42—2442no—
462C23—612no—
463C23—612no—
464C23—612no—
465C23—1221no—
466C23—1221no—
467C23—1221no—
468C23—1221no—
469C23—1222no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23C2×D4×D6111yescharacteristic
477Q8—48no—
478Q8—2448no—
479Q8—24416no—
480D4—1224no—
481D4—2448no—
482D4—2442no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—1228no—
502C2×C4—1228no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—4884no—
511C2×C4—24416no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4D8×D6112yes—
518C2×C4#1451980912yes—
519C2×C4C2×D4×D6112yes—
520C2×C4C2×Q8×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982512yes—
523C8—42no—
524C8—2442no—
525C22—614no—
526C22—1222no—
527C22—1222no—
528C22—1222no—
529C22—1222no—
530C22—1222no—
531C22—1222no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22—2442no—
537C22—2442no—
538C22—2442no—
539C22#120×D6111yescharacteristic
540C22C22⋊Q8×D6111yescharacteristic
541C22#1684033711yescharacteristic
542C22C2×D8×D6111yescharacteristic
543C22AGL(1,ℤ/8)×D6111yescharacteristic
544C22#137×D6111yescharacteristic
545C22C2×#1451980911yescharacteristic
546C4—22no—
547C4—22no—
548C4—44no—
549C4—42no—
550C4—81no—
551C4—614no—
552C4—1222no—
553C4—1222no—
554C4—4881no—
555C4—2444no—
556C4#1684042212yes—
557C4#1684042712yes—
558C2—1224no—
559C2—4882no—
560C2#2144917911yescharacteristic
561C2#447×D6111yescharacteristic
562C2#2145505411yescharacteristic
563C2#474×D6111yescharacteristic
564C2#2145563311yescharacteristic
565C2#2145694111yescharacteristic
566C2#2145705011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#2057D61
hypocenterC61#24298
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840041C2semidirect
4. #21447147C2semidirect
5. #21446956C2semidirect
6. #21446952C2semidirect
7. #21446769C2semidirect
9. #21446005C2semidirect
10. #21445798C2semidirect
11. C61×#2057C2semidirect
61. #14519776D4semidirect
64. #14519752Q8semidirect
126. C2×D244C2×Q8semidirect
158. C4⋊C4×C61C2×Q8semidirect
262. C2×D61#474semidirect
279. #2057D61semidirect
328. C61#24298semidirect
451. C4⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168400412
#214483272
#214471472
#214469562
#214469522
#214467692
#214461352
#214460052
#214457982
C61×#20572
#2429861