Finite Group Databasebeta

← #30615543

Subgroups — #30615543

Order 15616.

1320 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840840C211yesmaximal, characteristic
2#21462461C212yesmaximal, complemented
3#21462403C211yesmaximal, characteristic
4#21461791C218yesmaximal, complemented
5C2×#16840643C212yesmaximal, complemented
6C2×#16839734C211yesmaximal, characteristic
7#16840840—24no—
8#16840840—22no—
9#16840840C2214yes—
10C2×#14519864—24no—
11C2×D61⋊M16—21no—
12C22×#11758217—21no—
13C22×#11758217C2211yescharacteristic
14C22×#11758217C2212yes—
15C22×#11758217C2211yescharacteristic
16#16840768—24no—
17#16840768C2218yescomplemented
18#16840765—24no—
19#16840735—22no—
20#16840735C2214yes—
21#16840734—21no—
22#16840732—21no—
23#16840643C2218yescomplemented
24C22×#11758224C2211yescharacteristic
25C2×#14519733C2212yes—
26#16839734C414yes—
27#16839734C2214yes—
28C2×#14519692C411yescharacteristic
29C2×#14519691C412yes—
30C2×#99×C61C411yescharacteristic
31#14519864—216no—
32#14519864—216no—
33C2×#11758213—22no—
34D61⋊M16—24no—
35D61⋊M16—24no—
36C2×#11758217—22no—
37C2×#11758217—24no—
38C2×#11758217—22no—
39C2×#11758217—24no—
40C2×#11758217—22no—
41C2×#11758217—22no—
42C2×#11758217—22no—
43C2×#11758217—21no—
44C2×#11758217—21no—
45C2×#11758217—21no—
46C2×#11758217—22no—
47C2×#11758217—22no—
48C2×#11758217—22no—
49C2×#11758217—22no—
50C2×#11758217D414yes—
51C2×#11758217C2312yes—
52C2×#11758217C2314yes—
53C2×D61⋊C8—22no—
54C22×#92420—22no—
55C22×#92420—21no—
56C22×#92420D414yes—
57C22×#92420C2312yes—
58#14519844—28no—
59C4⋊#92420—28no—
60#14519834—24no—
61C22⋊#92420—24no—
62C2×#11758224—22no—
63C2×#11758224—22no—
64C2×#11758224—22no—
65C2×#11758224—22no—
66C2×#11758224C2×C412yes—
67C2×#11758224D412yes—
68C2×#11758224Q812yes—
69C2×#11758224C2312yes—
70C22×Dic122Q811yescharacteristic
71C22×C22⋊D61C2×C411yescharacteristic
72C22×C22⋊D61Q811yescharacteristic
73C23×Dic61D411yescharacteristic
74C23×Dic61C2311yescharacteristic
75C22×C4×D61D411yescharacteristic
76#14519733—22no—
77#14519733—22no—
78#14519733C2×C414yes—
79#14519733C2314yes—
80C2×#11758196—22no—
81C2×C22⋊Dic61C2×C412yes—
82C2×C4×Dic61—22no—
83#14519692C2×C414yes—
84#14519691C2×C418yes—
85C22×D4×C61C2×C411yescharacteristic
86C2×C22⋊C4×C61C2×C412yes—
87#99×C61C2×C414yes—
88C24×C61C4⋊C411yescharacteristic
89C24×C61C22×C411yescharacteristic
90C23×D61C4⋊C411yescharacteristic
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61C4⋊C412yes—
94C2×D4×C61C22×C412yes—
95C22×C4×C61—22no—
96C22×C4×C61C4⋊C411yescharacteristic
97C2×C22⋊D61—22no—
98C2×C22⋊D61—22no—
99C2×C22⋊D61—22no—
100C2×C22⋊D61—22no—
101C2×C22⋊D61C4⋊C412yes—
102C2×C22⋊D61C4⋊C412yes—
103C2×C22⋊D61C22×C412yes—
104C2×C22⋊D61C2×Q812yes—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61C4⋊C411yescharacteristic
111C22×Dic61C4⋊C412yes—
112C22×Dic61C22×C411yescharacteristic
113C22×Dic61C2×D412yes—
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C22×Dic61C2×D412yes—
117C22×Dic61C2×D412yes—
118C22×Dic61C2×Q811yescharacteristic
119C22×Dic61C2×Q811yescharacteristic
120C22×Dic61#4112yes—
121C22×Dic61#4112yes—
122C22×Dic61C2411yescharacteristic
123#11758224—24no—
124#11758224—24no—
125#11758224—24no—
126#11758224—24no—
127#11758224—24no—
128#11758224—24no—
129#11758224—24no—
130#11758224—24no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61C4⋊C412yes—
136C2×C4×D61C2×D412yes—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122C4⋊C412yes—
140C2×Dic122C2×Q812yes—
141#11758217—28no—
142#11758217—24no—
143#11758217—24no—
144#11758217—24no—
145#11758217—24no—
146#11758217—24no—
147#11758217—41no—
148#11758217—42no—
149#11758217—41no—
150#11758217—41no—
151#11758217—41no—
152C2×#92420—28no—
153C2×#92420—24no—
154C2×#92420—24no—
155C2×#92420—22no—
156C2×#92420—22no—
157C2×#92420—24no—
158C2×#92420—42no—
159C2×#92420C2×D418yes—
160#11758213—28no—
161D61⋊C8—28no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61C22×C414yes—
164C22⋊Dic61—22no—
165C22⋊Dic61C22×C414yes—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—24no—
169C4×Dic61—24no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—81no—
173C23×C61—81no—
174C23×C61C2×C4⋊C412yes—
175C23×C61C2×C4⋊C411yescharacteristic
176C23×C61C2×C4⋊C411yescharacteristic
177C23×C61C4×D412yes—
178C23×C61C4×D412yes—
179C23×C61C23×C411yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—81no—
183C22×D61C2×C4⋊C412yes—
184D4×C61—24no—
185D4×C61—24no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—82no—
190C2×C4×C61C2×C4⋊C412yes—
191C22⋊D61—24no—
192C22⋊D61—24no—
193C22⋊D61—24no—
194C22⋊D61—24no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—24no—
202C2×Dic61—82no—
203C2×Dic61C2×C4⋊C412yes—
204C2×Dic61C2×C4⋊C411yescharacteristic
205C2×Dic61C2×C4⋊C411yescharacteristic
206C2×Dic61C4×D412yes—
207C2×Dic61C4×D412yes—
208C2×Dic61#12011yescharacteristic
209C2×Dic61#12012yes—
210C2×Dic61#12011yescharacteristic
211C2×Dic61C22⋊Q812yes—
212C2×Dic61C22⋊Q812yes—
213C2×Dic61C22⋊Q812yes—
214C2×Dic61C22⋊Q812yes—
215C2×Dic61#12312yes—
216C2×Dic61#12311yescharacteristic
217C2×Dic61#12311yescharacteristic
218C2×Dic61C22×D412yes—
219C2×Dic61C22×D411yescharacteristic
220C2×Dic61C22×Q811yescharacteristic
221C2×Dic61C2×#4112yes—
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—22no—
226Dic122—24no—
227Dic122—24no—
228#92420—28no—
229#92420—42no—
230#92420—42no—
231#20892—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—42no—
237C22×C61—41no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—81no—
241C22×C61#38511yescharacteristic
242C22×C61#38511yescharacteristic
243C22×C61#38512yes—
244C22×C61C22×C4⋊C411yescharacteristic
245C22×C61C2×C4×D412yes—
246C2×D61—22no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—81no—
250C4×C61—22no—
251C4×C61—44no—
252Dic61—22no—
253Dic61—44no—
254Dic61#38512yes—
255Dic61#38511yescharacteristic
256Dic61#38511yescharacteristic
257Dic61C2×#12011yescharacteristic
258Dic61C2×C22⋊Q812yes—
259Dic61C2×#12311yescharacteristic
260C2×#568—611no—
261#2905—612no—
262C2×#410—611no—
263#2879—611no—
264#1868—618no—
265C2×#344—612no—
266C2×C61—22no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—22no—
271C2×C61—42no—
272C2×C61C2×#38511yescharacteristic
273D61—42no—
274C22×#41—611no—
275#568—614no—
276#568—1224no—
277#568—1222no—
278C2×#131—1224no—
279C22×M16—612no—
280C22×M16—611no—
281C22×M16—611no—
282C22×M16—1221no—
283C22×M16—1221no—
284C2×#117—612no—
285#424—618no—
286#424—1224no—
287#423—1224no—
288#410—614no—
289#410—614no—
290C2×#99—611no—
291C2×#99—612no—
292C2×#99#692311yescharacteristic
293#408—614no—
294#408—1222no—
295#407—1221no—
296#407—1221no—
297#344—618no—
298C61#2089211yescharacteristic, complemented, Sylow
299C2×#41—612no—
300C2×#41—612no—
301C2×#41—612no—
302C2×#41—612no—
303C2×#41—1222no—
304C2×#41—1222no—
305C2×#41—1222no—
306C2×#41—1222no—
307C22×Q8—611no—
308C22×D4—611no—
309C22×D4—611no—
310C22×D4C2×#692311yescharacteristic
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—611no—
314#131—12216no—
315#131—12216no—
316C2×M16—614no—
317C2×M16—614no—
318C2×M16—612no—
319C2×M16—1222no—
320C2×M16—1222no—
321C2×M16—1222no—
322C2×M16—1222no—
323C2×M16—1224no—
324C2×M16—1224no—
325C2×M16—1222no—
326C2×M16—1224no—
327C2×M16—1224no—
328C2×M16—1222no—
329C2×M16—1221no—
330C2×M16—1221no—
331C2×M16—1221no—
332C2×M16—1222no—
333C2×M16—1222no—
334C2×M16—1222no—
335C2×M16—1222no—
336C22×C8—614no—
337C22×C8—612no—
338C22×C8—1222no—
339C22×C8—1222no—
340C22×C8—1221no—
341#117—614no—
342#117—614no—
343#117—1222no—
344#117—1222no—
345C2×C4⋊C4—1222no—
346C2×C22⋊C4—612no—
347C2×C22⋊C4C2×#692312yes—
348C2×C42—1222no—
349C4⋊C8—1228no—
350C4⋊C8—1228no—
351#99—614no—
352#99—618no—
353#99C2×#692314yes—
354C22⋊C8—1224no—
355C22⋊C8—1224no—
356C24—611no—
357C24C4⋊#692311yescharacteristic
358C24C22×#692311yescharacteristic
359#41—1224no—
360#41—1224no—
361#41—1224no—
362#41—1224no—
363#41—1224no—
364#41—1224no—
365#41—1224no—
366#41—1224no—
367C2×Q8—612no—
368C2×Q8—612no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—612no—
374C2×D4—612no—
375C2×D4—612no—
376C2×D4—612no—
377C2×D4—1222no—
378C2×D4—1222no—
379C2×D4—1222no—
380C2×D4—1222no—
381C2×D4C4⋊#692312yes—
382C2×D4C22×#692312yes—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—612no—
395C22×C4—611no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—1221no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4C4⋊#692311yescharacteristic
409M16—1224no—
410M16—1224no—
411M16—1224no—
412M16—1224no—
413M16—1228no—
414M16—1228no—
415M16—1224no—
416M16—2441no—
417M16—2442no—
418M16—2441no—
419M16—2441no—
420M16—2441no—
421C2×C8—618no—
422C2×C8—1224no—
423C2×C8—1224no—
424C2×C8—1228no—
425C2×C8—1228no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1224no—
429C2×C8—2442no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C22⋊C4—22no—
433C22⋊C4—614no—
434C22⋊C4—1222no—
435C22⋊C4C22×#692314yes—
436C42—1224no—
437C42—1224no—
438C23—21no—
439C23—21no—
440C23—81no—
441C23—81no—
442C23—612no—
443C23—1221no—
444C23—1221no—
445C23—4881no—
446C23C2×C4⋊#692312yes—
447C23C2×C4⋊#692311yescharacteristic
448C23C2×C4⋊#692311yescharacteristic
449C23D4×#692312yes—
450C23D4×#692312yes—
451C23C23×#692311yescharacteristic
452Q8—1224no—
453Q8—1224no—
454D4—24no—
455D4—24no—
456D4—1224no—
457D4—1224no—
458D4—1224no—
459D4—1224no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—82no—
464C2×C4—611no—
465C2×C4—612no—
466C2×C4—611no—
467C2×C4—612no—
468C2×C4—612no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—611no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1224no—
494C2×C4—4882no—
495C2×C4C2×C4⋊#692312yes—
496C8—1228no—
497C8—2442no—
498C8—2442no—
499C22—22no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—42no—
506C22—42no—
507C22—81no—
508C22—1222no—
509C22—4881no—
510C22—2442no—
511C22—2442no—
512C22#1684070911yescharacteristic
513C22#1684070911yescharacteristic
514C22#1684070912yes—
515C22C22×C4⋊#692311yescharacteristic
516C22C2×D4×#692312yes—
517C4—22no—
518C4—44no—
519C4—611no—
520C4—612no—
521C4—611no—
522C4—611no—
523C4—611no—
524C4—612no—
525C4—1222no—
526C4—2444no—
527C2—22no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—22no—
532C2—42no—
533C2—2442no—
534C2C2×#1684070911yescharacteristic
subgroupG/H
centerC2C2×#16840709
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#99×C61C4
socleC2×C61C2×#385
prime radicalC22×#11758224C22
cosocleC22×Dic61C24
hypercenterC2×#99#6923
hypocenterC61#20892
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #21462461C2semidirect
4. #21461791C2semidirect
5. C2×#16840643C2semidirect
17. #16840768C22semidirect
23. #16840643C22semidirect
298. C61#20892semidirect

Maximal subgroups

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

subgroupindex
C2×#168408402
#214624612
#214624032
#214617912
C2×#168406432
C2×#168397342
#2089261