Finite Group Databasebeta

← #29712655

Subgroups — #29712655

Order 15616.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519756C211yesmaximal, characteristic
2C2×#16840093C212yesmaximal, complemented
3C2×#16839622C211yesmaximal, characteristic
4#21447016C214yesmaximal, complemented
5#21446124C212yesmaximal, complemented
6#21446014C212yesmaximal, complemented
7#21445755C212yesmaximal, complemented
8C61×#1779C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519756—28no—
11C2×#14519756—21no—
12C2×#14519756—21no—
13C2×#14519756C2212yes—
14C2×#14519756C2212yes—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758196—21no—
19C22×C22⋊Dic61—21no—
20C22×C22⋊Dic61—21no—
21#16840093—28no—
22C2×#14519709C2214yescomplemented
23C2×#14519702C2214yescomplemented
24C2×#14519702C2214yescomplemented
25#16839622—21no—
26#16839622—21no—
27#16839622C2212yes—
28#16839622C2212yes—
29C2×#14519684—21no—
30#16839525C2214yescomplemented
31C24×C4×C61C2211yescharacteristic
32C2×C22⋊C8×C61C2212yes—
33C2×C22⋊C8×C61C2212yes—
34C2×C22⋊C8×C61C2212yes—
35C22×Dic122—21no—
36C22×Dic122—24no—
37C22×Dic122—24no—
38C22×Dic122—42no—
39C22×Dic122C2312yes—
40C23×Dic61—21no—
41C23×Dic61—21no—
42C23×Dic61—21no—
43C23×Dic61—21no—
44#14519756—28no—
45#14519756—28no—
46#14519756—24no—
47#14519756—24no—
48#14519756—24no—
49#14519756D418yes—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—24no—
52C2×C4⋊Dic61—24no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—42no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2312yes—
59C2×#11758196—28no—
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—28no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—44no—
68#14519709—28no—
69#14519702—28no—
70#14519702—44no—
71#14519684—22no—
72#14519684—22no—
73C25×C61D411yescharacteristic
74C23×C4×C61—24no—
75C23×C4×C61—24no—
76C23×C4×C61—42no—
77C23×C4×C61D411yescharacteristic
78C23×C4×C61D411yescharacteristic
79C23×C4×C61D411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C22×C8×C61C2314yes—
84C22⋊C8×C61—24no—
85C22⋊C8×C61—24no—
86C22⋊C8×C61—42no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—41no—
92C24×C61—41no—
93C24×C61SD1614yes—
94C24×C61C2×D411yescharacteristic
95C24×C61C2×D411yescharacteristic
96C24×C61C2×D411yescharacteristic
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—416no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D414yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4114yes—
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—24no—
121C22×Dic61—24no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C22×Dic61—42no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C2×Dic122—24no—
137C2×Dic122—24no—
138C2×Dic122—24no—
139C2×Dic122—22no—
140C2×Dic122—416no—
141C2×Dic122—42no—
142C2×Dic122—42no—
143C2×Dic122C2×D414yes—
144C2×C8×C61—24no—
145C2×C8×C61—24no—
146C2×C8×C61—24no—
147C22⋊Dic61—28no—
148C22⋊Dic61—28no—
149C22⋊Dic61—44no—
150C22⋊Dic61—44no—
151C4⋊Dic61—24no—
152C4⋊Dic61—22no—
153C4⋊Dic61—24no—
154C4⋊Dic61—24no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157C4⋊Dic61C2×D414yes—
158#11758196—28no—
159#11758196—28no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—28no—
175C23×C61—28no—
176C23×C61—44no—
177C23×C61—44no—
178C23×C61—44no—
179C23×C61—44no—
180C23×C61—44no—
181C23×C61#12012yes—
182C23×C61#12312yes—
183C23×C61C2×SD1612yes—
184C23×C61C2×SD1612yes—
185C23×C61C2×SD1612yes—
186C23×C61#13712yes—
187C23×C61#13712yes—
188C23×C61C22×D411yescharacteristic
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—28no—
194C2×C4×C61—28no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—44no—
199C2×C4×C61—44no—
200C2×C4×C61—44no—
201C2×C4×C61—44no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12312yes—
204C2×C4×C61C22×D412yes—
205C2×C4×C61C2×#4112yes—
206C2×C4×C61H(5,𝔽2)12yes—
207C2×Dic61—24no—
208C2×Dic61—24no—
209C2×Dic61—48no—
210C2×Dic61—48no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—84no—
222C2×Dic61—84no—
223Dic122—28no—
224Dic122—48no—
225Dic122—44no—
226C8×C61—44no—
227#24003—611nomaximal, Sylow
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—416no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61#44814yes—
249C22×C61#48314yescomplemented
250C22×C61C2×#12011yescharacteristic
251C22×C61C2×#12311yescharacteristic
252C22×C61#53411yescharacteristic
253C22×C61C22×SD1611yescharacteristic
254C22×C61C2×#13711yescharacteristic
255C22×C61C2×#13711yescharacteristic
256C22×C61#57511yescharacteristic
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—41no—
260C4×C61#54614yes—
261Dic61—44no—
262Dic61—44no—
263Dic61—82no—
264Dic61—82no—
265C22×C22⋊Q8—611no—
266C2×#483—612no—
267C2×#380—611no—
268#2035—614no—
269#1872—612no—
270#1853—612no—
271#1787—612no—
272#1779D6111yescharacteristic
273C2×C61—24no—
274C2×C61—24no—
275C2×C61—42no—
276C2×C61#240511yescharacteristic
277C2×C61C2×#44811yescharacteristic
278C2×C61#300711yescharacteristic
279C2×C61C2×#48311yescharacteristic
280C2×C61#309511yescharacteristic
281C2×C61#318711yescharacteristic
282C2×C61#319511yescharacteristic
283C23×Q8—611no—
284C24×C4C2×D6111yescharacteristic
285C2×C22⋊Q8—612no—
286C2×C22⋊Q8—612no—
287C2×C22⋊Q8—1228no—
288C2×C22⋊Q8—1221no—
289C2×C22⋊Q8—1221no—
290C22×C4⋊C4—611no—
291C22×C4⋊C4—611no—
292C22×C4⋊C4—611no—
293C22×C4⋊C4—1221no—
294C22×C22⋊C4—1221no—
295C22×C22⋊C4—1221no—
296#483—1228no—
297C2×#106—614no—
298C2×#103—614no—
299C2×#103—614no—
300C2×C22⋊C8C2×D6112yes—
301C2×C22⋊C8C2×D6112yes—
302C2×C22⋊C8C2×D6112yes—
303#380—612no—
304#380—612no—
305#380—1221no—
306#380—1221no—
307C2×#95—1221no—
308#340—614no—
309C61#2400311yescharacteristic, complemented, Sylow
310C25D24411yescharacteristic
311C22×Q8—612no—
312C22×Q8—1224no—
313C22×Q8—1221no—
314C22×Q8—1224no—
315C22×Q8—2442no—
316C23×C4—24no—
317C23×C4—24no—
318C23×C4—42no—
319C23×C4—1221no—
320C23×C4—1221no—
321C23×C4—1221no—
322C23×C4—1221no—
323C23×C4D24411yescharacteristic
324C23×C4D24411yescharacteristic
325C23×C4D24411yescharacteristic
326C23×C4C22×D6111yescharacteristic
327C23×C4C22×D6111yescharacteristic
328C23×C4C22×D6111yescharacteristic
329C22×C8C22×D6114yes—
330C22⋊Q8—618no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333C22⋊Q8—1228no—
334C22⋊Q8—1228no—
335C22⋊Q8—1224no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1224no—
340C2×C4⋊C4—1228no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1224no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—2442no—
348C2×C22⋊C4—1228no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1222no—
353C2×C22⋊C4—2444no—
354#106—1228no—
355#103—1228no—
356#103—2444no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—42no—
360#95—1222no—
361#95—1222no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—41no—
367C24—41no—
368C24#1175819014yes—
369C24C2×D24411yescharacteristic
370C24C2×D24411yescharacteristic
371C24C2×D24411yescharacteristic
372C2×Q8—614no—
373C2×Q8—1224no—
374C2×Q8—1222no—
375C2×Q8—1224no—
376C2×Q8—1224no—
377C2×Q8—24416no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—416no—
397C22×C4—1224no—
398C22×C4—1224no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4C2×D24411yescharacteristic
414C22×C4C2×D24411yescharacteristic
415C22×C4C2×D24411yescharacteristic
416C22×C4D4×D6114yes—
417C22×C4#1175822414yes—
418C22×C4C23×D6111yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C4⋊C4—614no—
423C4⋊C4—1228no—
424C4⋊C4—1224no—
425C4⋊C4—1222no—
426C4⋊C4—1228no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—2442no—
430C4⋊C4—2442no—
431C22⋊C4—1228no—
432C22⋊C4—1228no—
433C22⋊C4—2444no—
434C22⋊C4—2444no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—28no—
450C23—28no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#1451974712yes—
457C23#1451976112yes—
458C23C2×#1175819012yes—
459C23C2×#1175819012yes—
460C23C2×#1175819012yes—
461C23#1451981012yes—
462C23#1451981012yes—
463C23C22×D24411yescharacteristic
464Q8—1228no—
465Q8—2448no—
466Q8—2444no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—1224no—
481C2×C4—1224no—
482C2×C4—4884no—
483C2×C4—4884no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2448no—
491C2×C4—2448no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4#1451974712yes—
497C2×C4#1451976112yes—
498C2×C4C2×D4×D6112yes—
499C2×C4C2×#1175822412yes—
500C2×C4#1451982212yes—
501C8—44no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22#1683990314yes—
523C22#1684009314yescomplemented
524C22C2×#1451974711yescharacteristic
525C22C2×#1451976111yescharacteristic
526C22#1684032411yescharacteristic
527C22C22×#1175819011yescharacteristic
528C22C2×#1451981011yescharacteristic
529C22C2×#1451981011yescharacteristic
530C22#1684056211yescharacteristic
531C4—22no—
532C4—22no—
533C4—41no—
534C4—4882no—
535C4—4882no—
536C4—2444no—
537C4—2444no—
538C4#1684039914yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2#2144914711yescharacteristic
543C2C2×#1683990311yescharacteristic
544C2#2145508111yescharacteristic
545C2C2×#1684009311yescharacteristic
546C2#2145576811yescharacteristic
547C2#2145657711yescharacteristic
548C2#2145664311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1779C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenter#1779D61
hypocenterC61#24003
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
4. #21447016C2semidirect
5. #21446124C2semidirect
6. #21446014C2semidirect
7. #21445755C2semidirect
22. C2×#14519709C22semidirect
23. C2×#14519702C22semidirect
24. C2×#14519702C22semidirect
30. #16839525C22semidirect
249. C22×C61#483semidirect
309. C61#24003semidirect
523. C22#16840093semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
C2×#168400932
C2×#168396222
#214470162
#214461242
#214460142
#214457552
C61×#17792
#2400361