Finite Group Databasebeta

← #29736663

Subgroups — #29736663

Order 15616.

1320 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840123C212yesmaximal, complemented
3#21454123C211yesmaximal, characteristic
4#21446960C211yesmaximal, characteristic, complemented
5#21446956C211yesmaximal, characteristic, complemented
6#21446952C211yesmaximal, characteristic, complemented
7#21446790C212yesmaximal, complemented
8#21446135C212yesmaximal
9#21446123C212yesmaximal, complemented
10#21445580C211yesmaximal, characteristic, complemented
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×Q8×D61—21no—
13C2×#14519777C2211yescharacteristic
14C2×#14519776—21no—
15C2×#14519776C2212yes—
16C2×#14519775—21no—
17C4⋊Q8×D61—28no—
18C2×#14519766C2211yescharacteristic
19C2×#14519753C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840123—28no—
25C2×#14519701C2212yes—
26C2×#14519699C2212yes—
27C2×#14519697C2212yes—
28C2×#14519689C2212yes—
29C2×#14519687C2212yes—
30#16839664C2211yescharacteristic
31#16839651—21no—
32#16839651—21no—
33#16839640—21no—
34#16839640—21no—
35#16839627C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839517C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C61×#391C2211yescharacteristic
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×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C22×#92418C2312yes—
57#14519777—24no—
58#14519776—28no—
59#14519776—24no—
60#14519776D418yescomplemented
61#14519775—28no—
62#14519766—24no—
63#14519753—24no—
64#14519752—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—21no—
71C4⋊C4×D61—28no—
72C4⋊C4×D61—24no—
73C4⋊C4×D61—24no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—24no—
84#14519701—24no—
85#14519699—24no—
86#14519697—24no—
87#14519689—24no—
88#14519687—24no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—24no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99#103×C61—24no—
100#95×C61—21no—
101#95×C61—21no—
102C23×D61—21no—
103C23×D61C2×D411yescharacteristic
104C2×Q8×C61—24no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—21no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122Q8×D61—432no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—42no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—28no—
140C2×C4×D61—44no—
141C2×C4×D61—44no—
142C2×C4×D61#4114yes—
143C2×Dic122—24no—
144C2×Dic122—28no—
145C2×Dic122—44no—
146C2×Dic122—44no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61C2×D414yes—
156C42×C61—22no—
157#11758198—42no—
158#11758198—42no—
159#11758198—42no—
160#11758198—42no—
161#11758198—42no—
162#11758198—42no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D414yes—
167#11758196—28no—
168#11758196—28no—
169#11758196—42no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61C2×D414yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61C22×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61#12412yes—
189C22×D61AGL(1,ℤ/8)12yes—
190C22×D61#13712yes—
191Q8×C61—48no—
192Q8×C61—42no—
193Q8×C61—42no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—24no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61#13712yes—
209C2×C4×C61C22×D412yes—
210C2×C4×C61C2×#4112yes—
211C2×C4×C612−1+411yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61#12012yes—
226C2×Dic61#12012yes—
227C2×Dic61#12412yes—
228D244—42no—
229D244—42no—
230D244—84no—
231C4×D61—28no—
232C4×D61—28no—
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—416no—
236Dic122—416no—
237Dic122—48no—
238C8×C61—42no—
239#92418—42no—
240#25908—611nomaximal, Sylow
241C22×C61C2×#12011yescharacteristic
242C22×C61C2×#12411yescharacteristic
243C22×C61#53611yescharacteristic
244C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61C2×#13711yescharacteristic
247C22×C61C2×#13711yescharacteristic
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—22no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61#48914yescomplemented
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—44no—
265C4×C61—81no—
266C4×C61—81no—
267C4×C61#54812yes—
268C4×C61#54812yes—
269Dic61—22no—
270Dic61—22no—
271Dic61—22no—
272Dic61—44no—
273Dic61—81no—
274C22×C4⋊Q8—611no—
275C2×#489—612no—
276#2850—611no—
277#2024—611no—
278#2024D6111yescharacteristic, complemented
279#2023—611no—
280#2023—611no—
281#1993—612no—
282#1874—612no—
283#1872—612no—
284#1749—611no—
285C2×C61#243611yescharacteristic
286C2×C61#300311yescharacteristic
287C2×C61#300411yescharacteristic
288C2×C61C2×#48911yescharacteristic
289C2×C61#314111yescharacteristic
290C2×C61#323811yescharacteristic
291C2×C61#323911yescharacteristic
292D61—24no—
293D61—161no—
294C23×Q8—1221no—
295C2×C4⋊Q8—612no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—1221no—
298C2×C4⋊Q8—1221no—
299C2×C4⋊Q8—1228no—
300C2×C4⋊Q8C2×D6111yescharacteristic
301C2×#124—611no—
302C2×C22⋊Q8—611no—
303C2×#121—611no—
304C22×C4⋊C4—612no—
305C22×C4⋊C4—1221no—
306C22×C42—611no—
307#489—1228no—
308C2×#103—612no—
309C2×#103C2×D6112yes—
310C2×#102—612no—
311C2×#102—612no—
312C2×C22⋊C8—612no—
313C2×#97—612no—
314#391C2×D6111yescharacteristic
315#390—611no—
316#388—1221no—
317#388—1221no—
318#386—1221no—
319#386—1221no—
320#382—611no—
321#340—612no—
322#340—612no—
323#340—612no—
324#340C2×D6112yes—
325#336—611no—
326#336C2×D6111yescharacteristic
327C61#2590811yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×Q8—1222no—
330C22×Q8—1228no—
331C22×Q8—1221no—
332C22×Q8—2444no—
333C22×Q8—2444no—
334C22×Q8C22×D6111yescharacteristic
335C22×D4—611no—
336C22×D4—1221no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1222no—
341C22×C8—612no—
342C22×C8C22×D6112yes—
343C4⋊Q8—24no—
344C4⋊Q8—618no—
345C4⋊Q8—1224no—
346C4⋊Q8—1224no—
347C4⋊Q8—1228no—
348C4⋊Q8—1228no—
349#124—1224no—
350C22⋊Q8—1224no—
351#121—1224no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—1224no—
356C2×C4⋊C4—1228no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1221no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—1224no—
374C2×C42C22×D6111yescharacteristic
375#103—24no—
376#103—1224no—
377#102—1224no—
378#102—1224no—
379C22⋊C8—1224no—
380#97—1224no—
381#95—21no—
382#95—21no—
383#95—1221no—
384#95—1221no—
385C24—611no—
386C24—1221no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—1228no—
395C2×Q8—1224no—
396C2×Q8—2442no—
397C2×Q8—2444no—
398C2×Q8—2444no—
399C2×Q8—24432no—
400C2×Q8—2442no—
401C2×D4—1222no—
402C2×D4—1222no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—2442no—
406C2×D4—2442no—
407C2×D4—2442no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
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—1221no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—2444no—
428C22×C4—2444no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—614no—
442C4⋊C4—1228no—
443C4⋊C4—1228no—
444C4⋊C4—1224no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—2442no—
448C4⋊C4D4×D6114yes—
449C22⋊C4—2442no—
450C22⋊C4—2442no—
451C22⋊C4—2442no—
452C22⋊C4—2442no—
453C22⋊C4—2442no—
454C22⋊C4—2442no—
455C42—22no—
456C42—614no—
457C42—1222no—
458C42—1222no—
459C23—612no—
460C23—612no—
461C23—612no—
462C23—1221no—
463C23—1221no—
464C23—1221no—
465C23—1221no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23C2×D4×D6111yescharacteristic
474Q8—48no—
475Q8—42no—
476Q8—42no—
477Q8—2448no—
478Q8—24416no—
479D4—4884no—
480D4—2442no—
481D4—2442no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—1228no—
499C2×C4—1228no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—24416no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4#1451980412yes—
516C2×C4#1451980912yes—
517C2×C4C2×D4×D6112yes—
518C2×C4#41×D6112yes—
519C2×C4#1451982411yescharacteristic
520C2×C4#1451982511yescharacteristic
521C8—42no—
522C8—2442no—
523C22—614no—
524C22—1222no—
525C22—1222no—
526C22—1222no—
527C22—1222no—
528C22—1222no—
529C22—1222no—
530C22—4881no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22#120×D6111yescharacteristic
538C22#124×D6111yescharacteristic
539C22#1684033511yescharacteristic
540C22AGL(1,ℤ/8)×D6111yescharacteristic
541C22C2×#1451980411yescharacteristic
542C22#137×D6111yescharacteristic
543C22C2×#1451980911yescharacteristic
544C4—22no—
545C4—22no—
546C4—44no—
547C4—81no—
548C4—81no—
549C4—1222no—
550C4—1222no—
551C4—1222no—
552C4—4881no—
553C4—2444no—
554C4#1684042012yes—
555C4#1684042212yes—
556C2—1224no—
557C2—9761no—
558C2#2144947211yescharacteristic
559C2#2145504711yescharacteristic
560C2#2145505411yescharacteristic
561C2#489×D6111yescharacteristic
562C2#2145621411yescharacteristic
563C2#2145704411yescharacteristic
564C2#2145705311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#2024D61
hypocenterC61#25908
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840123C2semidirect
4. #21446960C2semidirect
5. #21446956C2semidirect
6. #21446952C2semidirect
7. #21446790C2semidirect
9. #21446123C2semidirect
10. #21445580C2semidirect
11. C61×#2024C2semidirect
60. #14519776D4semidirect
261. C2×D61#489semidirect
278. #2024D61semidirect
327. C61#25908semidirect

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×#168401232
#214541232
#214469602
#214469562
#214469522
#214467902
#214461352
#214461232
#214455802
C61×#20242
#2590861