Finite Group Databasebeta

← #29730770

Subgroups — #29730770

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839989C212yesmaximal, complemented
2C2×#16839959C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21450741C211yesmaximal, characteristic, complemented
5#21447005C212yesmaximal, complemented
6#21446822C212yesmaximal, complemented
7#21446626C211yesmaximal, characteristic, complemented
8#21446582C211yesmaximal, characteristic, complemented
9#21446248C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11C61×#2541C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C23×#92418C2212yes—
14C2×#14519756C2212yes—
15C2×C4×Dic122C2211yescharacteristic
16#16839989—28no—
17#16839959—24no—
18D4×#92418—24no—
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839685—22no—
29#16839665—21no—
30#16839658—21no—
31#16839632C2211yescharacteristic
32#16839622C2212yes—
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61C2212yes—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#392—22no—
42C61×#390—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×Dic122C2311yescharacteristic
45C2×#11758200—24no—
46C2×#11758200—24no—
47C2×#11758200—42no—
48C22×#92418—24no—
49C22×#92418—24no—
50C22×#92418—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54#14519756—24no—
55#14519756—42no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—21no—
63C2×#11758196—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C4×Dic61—21no—
67#14519708—24no—
68C4⋊#92418D414yescomplemented
69#14519700—22no—
70#14519700D414yescomplemented
71#14519698—24no—
72#14519698—24no—
73#14519698—42no—
74C22⋊#92418—24no—
75C4×#92418—22no—
76#14519684—22no—
77C22×D4×C61—22no—
78C22×D4×C61—22no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61D411yescharacteristic
82C22×D4×C61C2312yes—
83C23×C4×C61C2312yes—
84#127×C61—22no—
85#127×C61—22no—
86#127×C61—44no—
87#121×C61—24no—
88#121×C61—24no—
89#121×C61—42no—
90#121×C61—42no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#95×C61—21no—
102C24×C61—22no—
103C24×C61C2×D412yes—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—44no—
113C2×D4×C61—44no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C2×D4×C61—82no—
123C2×D4×C61—82no—
124C2×D4×C61SD1614yescomplemented
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61#4112yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—44no—
153C22⋊C4×C61—44no—
154C42×C61—42no—
155C42×C61C2×D412yes—
156C42×C61#4112yes—
157C22⋊Dic61—44no—
158C22⋊Dic61—82no—
159#11758200—48no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61—81no—
166#11758196—44no—
167#11758196—42no—
168C4×Dic61—42no—
169C2×#92418—21no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—48no—
177C2×#92418C2×D414yes—
178C2×#92418C2×D412yes—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61C2×SD1614yes—
193C23×C61C22×D411yescharacteristic
194D4×C61—28no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—84no—
200D4×C61—88no—
201D4×C61—88no—
202D4×C61—84no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61#12112yes—
219C2×C4×C61#12112yes—
220C2×C4×C61C2×SD1612yes—
221C2×C4×C61#13512yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×Dic61—22no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237Dic122—22no—
238Dic122—22no—
239Dic122—41no—
240Dic122—41no—
241#92418—22no—
242#92418—24no—
243#92418—41no—
244#25387—611nomaximal, Sylow
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—24no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61—82no—
259C22×C61C2×#12111yescharacteristic
260C22×C61D4211yescharacteristic
261C22×C61#54611yescharacteristic
262C22×C61C22×SD1611yescharacteristic
263C22×C61C2×#13511yescharacteristic
264C22×C61#57611yescharacteristic
265C22×C61#57711yescharacteristic
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61#46514yescomplemented
271C4×C61#53211yescharacteristic
272C4×C61#54011yescharacteristic
273C4×C61D4211yescharacteristic
274C4×C61#55011yescharacteristic
275Dic61—42no—
276Dic61—82no—
277Dic61—81no—
278C2×#465—612no—
279C2×#460—611no—
280C2×C8×D4—611no—
281#2541D6111yescharacteristic
282#2538—611no—
283#2034—612no—
284#2000—612no—
285#1957—611no—
286#1948—611no—
287#1893—612no—
288C4×#103—611no—
289C2×C61—44no—
290C2×C61—162no—
291C2×C61#242811yescharacteristic
292C2×C61C2×#46511yescharacteristic
293C2×C61#306211yescharacteristic
294C2×C61D4×SD1611yescharacteristic
295C2×C61#328711yescharacteristic
296C2×C61#329511yescharacteristic
297C2×C61#330411yescharacteristic
298C22×SD16—612no—
299C23×C8—612no—
300C2×#127—21no—
301C2×#127C2×D6111yescharacteristic
302C2×C22⋊Q8—612no—
303C2×#121C2×D6112yes—
304C2×#121C2×D6112yes—
305C2×C4×Q8—611no—
306C2×C4×D4C2×D6111yescharacteristic
307#465—1228no—
308#460—1224no—
309C8×D4—1224no—
310C2×#106—612no—
311C2×C4⋊C8—611no—
312C2×#103—611no—
313C2×#103—611no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—612no—
318C2×C4×C8—611no—
319#396—1222no—
320#392—22no—
321#390—21no—
322#390—1221no—
323#389—1221no—
324#384—611no—
325#380—612no—
326C4×C4⋊C4C2×D6111yescharacteristic
327#340—611no—
328#340—612no—
329#340—611no—
330C61#2538711yescharacteristic, complemented, Sylow
331C22×Q8—611no—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C22⋊D6111yescharacteristic
337C22×D4C22×D6112yes—
338C23×C4C22×D6112yes—
339C2×SD16—1224no—
340C2×SD16—1224no—
341C2×SD16—2442no—
342C22×C8—611no—
343C22×C8—612no—
344C22×C8—611no—
345C22×C8—1224no—
346C22×C8—1224no—
347C22×C8—2442no—
348#127—22no—
349#127—22no—
350#127—44no—
351C22⋊Q8—1224no—
352C22⋊Q8—2442no—
353#121—24no—
354#121—24no—
355#121—42no—
356#121—42no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×D4—22no—
360C4×D4—22no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4C22⋊D6111yescharacteristic
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4C22⋊D6112yes—
375C2×C42—21no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378#106—1224no—
379C4⋊C8—614no—
380#103—614no—
381#103—1222no—
382#102—1224no—
383#102—1224no—
384#102—2442no—
385C22⋊C8—1224no—
386C4×C8—1222no—
387#95—21no—
388#95—1222no—
389C24—22no—
390C24C2×C22⋊D6112yes—
391C2×Q8—612no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4#1175820014yescomplemented
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—2448no—
434C2×C8—614no—
435C2×C8—612no—
436C2×C8—1221no—
437C2×C8—1222no—
438C2×C8—1221no—
439C2×C8—1222no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—2448no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—1221no—
448C4⋊C4—1222no—
449C4⋊C4—1221no—
450C4⋊C4—1222no—
451C4⋊C4—4881no—
452C4⋊C4—2444no—
453C4⋊C4—2442no—
454C4⋊C4—2442no—
455C4⋊C4D4×D6112yes—
456C4⋊C4#1175822412yes—
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—4882no—
461C22⋊C4—2444no—
462C42—42no—
463C42—2442no—
464C42D4×D6112yes—
465C42#1175822412yes—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C2×#1175820014yes—
480C23C22×C22⋊D6111yescharacteristic
481Q8—1222no—
482Q8—1222no—
483Q8—2441no—
484Q8—2441no—
485D4—28no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—84no—
491D4—88no—
492D4—88no—
493D4—84no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—1222no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2441no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451974912yes—
519C2×C4#1451974912yes—
520C2×C4C2×#1175820012yes—
521C2×C4#1451979712yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×#1175822411yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982211yescharacteristic
528C8—1222no—
529C8—1224no—
530C8—2441no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×#1451974911yescharacteristic
546C22D4×C22⋊D6111yescharacteristic
547C22#1684040611yescharacteristic
548C22C22×#1175820011yescharacteristic
549C22C2×#1451979711yescharacteristic
550C22#1684056511yescharacteristic
551C22#1684057311yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—4881no—
557C4—4882no—
558C4—2442no—
559C4#1683998914yescomplemented
560C4#1684031011yescharacteristic
561C4#1684036611yescharacteristic
562C4#1684039711yescharacteristic
563C4#1684043211yescharacteristic
564C2—44no—
565C2—162no—
566C2#2144941011yescharacteristic
567C2C2×#1683998911yescharacteristic
568C2#2145549511yescharacteristic
569C2D4×#1175820011yescharacteristic
570C2#2145757511yescharacteristic
571C2#2145764311yescharacteristic
572C2#2145775811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2541C2
socleC23×C61C22×D4
hypercenter#2541D61
hypocenterC61#25387
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16839989C2semidirect
2. C2×#16839959C2semidirect
3. C2×D4×#92418C2semidirect
4. #21450741C2semidirect
5. #21447005C2semidirect
6. #21446822C2semidirect
7. #21446626C2semidirect
8. #21446582C2semidirect
9. #21446248C2semidirect
10. C4×#14519700C2semidirect
68. C4⋊#92418D4semidirect
70. #14519700D4semidirect
124. C2×D4×C61SD16semidirect
270. C4×C61#465semidirect
330. C61#25387semidirect
417. C2×D4#11758200semidirect
559. C4#16839989semidirect

Maximal subgroups

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

subgroupindex
C2×#168399892
C2×#168399592
C2×D4×#924182
#214507412
#214470052
#214468222
#214466262
#214465822
#214462482
C4×#145197002
C61×#25412
#2538761