Finite Group Databasebeta

← #30603592

Subgroups — #30603592

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2#21462300C211yesmaximal, characteristic
3C2×#16840708C211yesmaximal, characteristic
4C2×#16840641C211yesmaximal, characteristic
5#21461508C211yesmaximal, characteristic
6C2×#16840633C211yesmaximal, characteristic
7#339×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2211yescharacteristic
9C2×#14519856C2211yescharacteristic
10C22×C4⋊#6923—22no—
11C22×C4⋊#6923C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13#16840773—22no—
14#16840773—22no—
15C2×#14519846—21no—
16C2×#14519845—21no—
17#16840713—22no—
18#16840709—22no—
19#16840708—22no—
20#16840708—21no—
21#16840708—22no—
22#16840706—21no—
23#16840641—22no—
24#16840635—22no—
25#16840633—22no—
26C2×M16×D61C2211yescharacteristic
27#117×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839523C411yescharacteristic, complemented
30#16839522C411yescharacteristic, complemented
31#16839521C411yescharacteristic, complemented
32C61×#339C411yescharacteristic, complemented
33C23×#6923—21no—
34C23×#6923D412yes—
35C2×#11758213—21no—
36D61⋊M16—21no—
37D61⋊M16—21no—
38D61⋊M16—21no—
39D61⋊M16Q812yes—
40C2×#11758217—21no—
41C2×D61⋊C8—21no—
42#14519856—21no—
43#14519856—21no—
44#14519856—21no—
45#14519856—21no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—21no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—21no—
50C2×C4⋊#6923—28no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—24no—
59C2×C4⋊#6923—48no—
60C2×C4⋊#6923D411yescharacteristic
61C2×C4⋊#6923D411yescharacteristic
62C2×C4⋊#6923D412yes—
63C2×C22⋊#6923—21no—
64C2×C4×#6923—21no—
65#14519846—24no—
66#14519845—24no—
67(C2×#6923)⋊C4—24no—
68C22×C4×D61—21no—
69C22×C4×D61C2311yescharacteristic
70C2×#11758194C2×C411yescharacteristic
71C2×#11758189—21no—
72M16×D61—21no—
73M16×D61—21no—
74M16×D61D412yes—
75M16×D61Q812yes—
76C2×C8×D61—21no—
77#14519736C2×C411yescharacteristic
78#14519735Q812yes—
79#14519734—21no—
80#14519733—21no—
81C4⋊C4×D61D412yes—
82C22⋊C4×D61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2×C411yescharacteristic
85C42×D61—21no—
86C42×D61—21no—
87C42×D61—21no—
88C42×D61D412yes—
89C42×D61Q812yes—
90C2×M16×C61C2×C411yescharacteristic
91#117×C61C2×C411yescharacteristic
92C2×C42×C61C2×C411yescharacteristic
93C23×D61C2×D411yescharacteristic
94C22×#6923—21no—
95C22×#6923—22no—
96C22×#6923—21no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—44no—
101C22×#6923—44no—
102C22×#6923—41no—
103C22×#6923—41no—
104C22×C4×C61—21no—
105C22×C4×C61C22×C411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61#4111yescharacteristic
108C2×C4×D61—24no—
109C2×C4×D61—24no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61C2×D411yescharacteristic
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61C2×D411yescharacteristic
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×Q811yescharacteristic
119C2×C4×D61C2×Q811yescharacteristic
120C2×C4×D61C2×Q811yescharacteristic
121C2×C4×D61C2×Q811yescharacteristic
122C2×C4×D61#4111yescharacteristic
123C2×C4×D61#4111yescharacteristic
124C2×C4×D61#4111yescharacteristic
125C22⋊#6923—41no—
126C22⋊#6923—41no—
127#11758217—22no—
128#11758217—41no—
129C2×#92420—41no—
130C4⋊#6923—21no—
131C4⋊#6923—22no—
132C4⋊#6923—21no—
133C4⋊#6923—22no—
134C4⋊#6923—21no—
135C4⋊#6923—21no—
136C4⋊#6923—416no—
137C4⋊#6923—41no—
138C4⋊#6923—41no—
139C4⋊#6923—41no—
140C4⋊#6923—41no—
141C4⋊#6923—41no—
142C4⋊#6923—41no—
143C4⋊#6923—84no—
144C4×#6923—41no—
145C4×#6923—41no—
146#11758213—22no—
147#11758213—41no—
148D61⋊C8—22no—
149D61⋊C8—41no—
150M16×C61—21no—
151M16×C61C4⋊C412yescomplemented
152C2×C8×C61—21no—
153C4⋊C4×C61C4⋊C412yescomplemented
154C22⋊C4×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61C4⋊C412yescomplemented
158C22⋊Dic61—21no—
159#11758198—22no—
160C4⋊Dic61C4⋊C412yescomplemented
161#11758196—22no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—22no—
167C4×Dic61C4⋊C412yescomplemented
168#11758194—21no—
169#11758194C4⋊C412yescomplemented
170C2×#92418—21no—
171#11758189—22no—
172#11758189—22no—
173C8×D61—22no—
174C8×D61—22no—
175C23×C61C4×D411yescharacteristic
176C22×D61—22no—
177C22×D61—22no—
178C22×D61#12011yescharacteristic
179C22×D61#12011yescharacteristic
180C22×D61#12111yescharacteristic
181C22×D61C22⋊Q811yescharacteristic
182C22×D61C22⋊Q811yescharacteristic
183C22×D61C4⋊Q811yescharacteristic
184C2×#6923—44no—
185C2×#6923—44no—
186C2×#6923—41no—
187C2×#6923—41no—
188C2×#6923—42no—
189C2×#6923—42no—
190C2×#6923—42no—
191C2×#6923—44no—
192C2×#6923—42no—
193C2×#6923—82no—
194C2×#6923—82no—
195C2×#6923—82no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61C2×C4⋊C411yescharacteristic
200C2×C4×C61C2×C4⋊C411yescharacteristic
201C2×C4×C61C2×C4⋊C411yescharacteristic
202C2×C4×C61C4×D411yescharacteristic
203C2×C4×C61C4×D411yescharacteristic
204C2×C4×C61C4×Q811yescharacteristic
205C2×Dic61—24no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61C22⋊Q811yescharacteristic
209C2×Dic61C22⋊Q811yescharacteristic
210C2×Dic61C22⋊Q811yescharacteristic
211C2×Dic61#12311yescharacteristic
212C2×Dic61#12311yescharacteristic
213C2×Dic61#12511yescharacteristic
214C4×D61—24no—
215C4×D61—28no—
216C4×D61—24no—
217C4×D61—42no—
218C4×D61—42no—
219C4×D61#12011yescharacteristic
220C4×D61#12111yescharacteristic
221C4×D61#12111yescharacteristic
222C4×D61C22⋊Q811yescharacteristic
223C4×D61C22⋊Q811yescharacteristic
224C4×D61C22⋊Q811yescharacteristic
225C4×D61C22⋊Q811yescharacteristic
226C4×D61#12311yescharacteristic
227C4×D61#12511yescharacteristic
228C4×D61#12511yescharacteristic
229C4×D61#12711yescharacteristic
230C4×D61C4⋊Q811yescharacteristic
231C4×D61C4⋊Q811yescharacteristic
232C4×D61C4⋊Q811yescharacteristic
233C4×D61C4⋊Q811yescharacteristic
234C4×D61C4⋊Q811yescharacteristic
235#92420—42no—
236C8×C61—22no—
237#92418—22no—
238#11340—611nomaximal, Sylow
239C22×C61—22no—
240C22×C61#38511yescharacteristic
241C22×C61#38511yescharacteristic
242C22×C61#38911yescharacteristic
243C2×D61—22no—
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—21no—
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61#39211yescharacteristic
251C2×D61#39511yescharacteristic
252C2×D61#39511yescharacteristic
253C2×D61#39811yescharacteristic
254C2×D61#45312yes—
255C2×D61#45612yes—
256#6923—81no—
257#6923—82no—
258#6923—82no—
259#6923—81no—
260C4×C61—24no—
261C4×C61—42no—
262C4×C61#38411yescharacteristic
263C4×C61#38511yescharacteristic
264C4×C61#38911yescharacteristic
265C4×C61#38911yescharacteristic
266Dic61—24no—
267Dic61—42no—
268Dic61#39511yescharacteristic
269Dic61#39811yescharacteristic
270Dic61#39811yescharacteristic
271Dic61#39911yescharacteristic
272C2×#428—611no—
273#2317—611no—
274C2×#384—611no—
275C2×#343—611no—
276#1736—611no—
277C2×#339—611no—
278C2×#339—611no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61#144411yescharacteristic
282C2×C61#185912yes—
283D61—21no—
284D61—21no—
285D61#200211yescharacteristic
286D61#200511yescharacteristic
287D61#206012yes—
288C22×M16—611no—
289C22×M16—611no—
290C2×#117—611no—
291C2×#117—611no—
292C22×C4⋊C4—611no—
293C22×C4⋊C4—611no—
294C22×C4⋊C4—1222no—
295C22×C42—611no—
296#428—1222no—
297#428—1222no—
298C2×#107—1221no—
299C2×#106—1221no—
300#389—1222no—
301#385—1222no—
302#384—1222no—
303#384—1221no—
304#384—1222no—
305#382—1221no—
306#343—1222no—
307#340—1222no—
308#339—611no—
309#339—611no—
310#339—611no—
311#339—1222no—
312#339#692311yescharacteristic, complemented
313C61#1134011yescharacteristic, complemented, Sylow
314C23×C4—611no—
315C23×C4—612no—
316C23×C4—1221no—
317C23×C4—1221no—
318C2×M16—611no—
319C2×M16—612no—
320C2×M16—612no—
321C2×M16—612no—
322C2×M16—1221no—
323C2×M16—1221no—
324C2×M16—1221no—
325C2×M16—1221no—
326C2×M16—1221no—
327C2×M16—1221no—
328C2×M16—1221no—
329C2×M16—1221no—
330C2×M16C2×#692311yescharacteristic
331C22×C8—1221no—
332C22×C8—1221no—
333#117—611no—
334#117—612no—
335#117—1221no—
336#117—1221no—
337#117—1221no—
338#117—1221no—
339#117—1221no—
340#117—1221no—
341#117C2×#692311yescharacteristic
342C2×C4⋊C4—612no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1228no—
359C2×C4⋊C4—2448no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C42—611no—
363C2×C42—612no—
364C2×C42—612no—
365C2×C42—1221no—
366C2×C42—1221no—
367C2×C42—1221no—
368C2×C42—1221no—
369C2×C42—1221no—
370C2×C42C2×#692311yescharacteristic
371#107—1224no—
372#106—1224no—
373#95—1224no—
374C24—611no—
375C22×C4—21no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1224no—
392C22×C4—1224no—
393C22×C4—1222no—
394C22×C4—1221no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4—2444no—
404C22×C4—2444no—
405C22×C4C22×#692311yescharacteristic
406M16—21no—
407M16—612no—
408M16—1222no—
409M16—1222no—
410M16—1221no—
411M16—1222no—
412M16—1222no—
413M16—2441no—
414M16—2441no—
415M16C4⋊#692312yescomplemented
416C2×C8—21no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—2441no—
422C2×C8—2441no—
423C4⋊C4—612no—
424C4⋊C4—1222no—
425C4⋊C4—1221no—
426C4⋊C4—1222no—
427C4⋊C4—1221no—
428C4⋊C4—1222no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—4884no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—24416no—
439C4⋊C4C4⋊#692312yescomplemented
440C22⋊C4—21no—
441C22⋊C4—1222no—
442C22⋊C4—1221no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C42—21no—
446C42—21no—
447C42—612no—
448C42—1222no—
449C42—1222no—
450C42—1222no—
451C42—1221no—
452C42—1221no—
453C42—2441no—
454C42—2441no—
455C42C4⋊#692312yescomplemented
456C23—611no—
457C23—611no—
458C23—611no—
459C23—611no—
460C23—611no—
461C23—611no—
462C23—1222no—
463C23—1222no—
464C23D4×#692311yescharacteristic
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—611no—
478C2×C4—611no—
479C2×C4—611no—
480C2×C4—611no—
481C2×C4—611no—
482C2×C4—611no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—611no—
489C2×C4—611no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1228no—
493C2×C4—1224no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—4882no—
497C2×C4—4882no—
498C2×C4—4882no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2444no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4C2×C4⋊#692311yescharacteristic
511C2×C4C2×C4⋊#692311yescharacteristic
512C2×C4C2×C4⋊#692311yescharacteristic
513C2×C4D4×#692311yescharacteristic
514C2×C4D4×#692311yescharacteristic
515C2×C4Q8×#692311yescharacteristic
516C8—22no—
517C8—1222no—
518C8—2442no—
519C22—22no—
520C22—611no—
521C22—611no—
522C22—612no—
523C22—612no—
524C22—611no—
525C22—611no—
526C22—1221no—
527C22—1221no—
528C22—1222no—
529C22—1221no—
530C22—1221no—
531C22—1222no—
532C22—1222no—
533C22#1684070911yescharacteristic
534C22#1684070911yescharacteristic
535C22#1684071311yescharacteristic
536C4—24no—
537C4—42no—
538C4—611no—
539C4—611no—
540C4—611no—
541C4—611no—
542C4—1224no—
543C4—4882no—
544C4—4882no—
545C4—4881no—
546C4—4881no—
547C4—2442no—
548C4#1684070811yescharacteristic
549C4#1684070911yescharacteristic
550C4#1684071311yescharacteristic
551C4#1684071311yescharacteristic
552C2—21no—
553C2—21no—
554C2—611no—
555C2—611no—
556C2—612no—
557C2—1221no—
558C2—1221no—
559C2#2146100711yescharacteristic
560C2#2146177812yes—
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#339C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#339#6923
hypocenterC61#11340
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
29. #16839523C4semidirect
30. #16839522C4semidirect
31. #16839521C4semidirect
32. C61×#339C4semidirect
151. M16×C61C4⋊C4semidirect
153. C4⋊C4×C61C4⋊C4semidirect
157. C42×C61C4⋊C4semidirect
160. C4⋊Dic61C4⋊C4semidirect
167. C4×Dic61C4⋊C4semidirect
169. #11758194C4⋊C4semidirect
312. #339#6923semidirect
313. C61#11340semidirect
415. M16C4⋊#6923semidirect
439. C4⋊C4C4⋊#6923semidirect
455. C42C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
#214623002
C2×#168407082
C2×#168406412
#214615082
C2×#168406332
#339×D612
#1134061