Finite Group Databasebeta

← #29713863

Subgroups — #29713863

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840451C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic, complemented
3C2×#16839805C211yesmaximal, characteristic, complemented
4#21450137C211yesmaximal, characteristic
5#21447156C212yesmaximal, complemented
6#21447025C212yesmaximal, complemented
7#21446250C211yesmaximal, characteristic, complemented
8#21446114C211yesmaximal, characteristic
9#21446088C212yesmaximal, complemented
10#21446013C212yesmaximal, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840451—24no—
14#16840451—24no—
15C2×#14519770—21no—
16C2×#14519769—21no—
17C2×#14519768C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519734—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C2×C4⋊C4×D61C2211yescharacteristic
26#16840093—24no—
27#16839805—24no—
28C2×#14519710C2212yes—
29C2×#14519709C2212yes—
30C2×#14519708C2212yes—
31C2×#14519702C2212yes—
32C2×#14519700C2212yes—
33C2×#14519689C2212yes—
34#16839702—21no—
35#16839699—21no—
36#16839661C2212yes—
37#16839639—21no—
38#16839637—21no—
39#16839619C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×#117×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122C2311yescharacteristic
48C22×C22⋊D61D411yescharacteristic
49C23×Dic61—21no—
50C22×C4×D61—21no—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194—22no—
53C2×#11758194—24no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519769—28no—
57#14519768—22no—
58#14519768—22no—
59#14519759—28no—
60#14519757—22no—
61#14519757—22no—
62#14519756—22no—
63#14519756D414yes—
64#14519742—22no—
65#14519742—22no—
66C4×C22⋊D61—22no—
67C4×C22⋊D61—22no—
68#14519734—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C4⋊C4×D61—22no—
83C4⋊C4×D61—22no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—22no—
88C2×#11758198D411yescharacteristic
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91#14519710—24no—
92#14519709—24no—
93#14519708—24no—
94#14519702—24no—
95#14519700—24no—
96#14519689—24no—
97#14519684—22no—
98#14519684—22no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101#117×C61—22no—
102#117×C61D414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C22⋊C8×C61—22no—
110#95×C61—21no—
111C24×C61C2×D411yescharacteristic
112C23×D61C2×D411yescharacteristic
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—41no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—42no—
129C22×Dic61—82no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C2×C4×D61—24no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—42no—
155C42×C61—24no—
156C22⋊Dic61—24no—
157C22⋊Dic61—22no—
158#11758198—24no—
159#11758198—22no—
160#11758198—82no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61—82no—
168C4⋊Dic61C2×D412yes—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173#11758196—22no—
174C4×Dic61—24no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177#11758194—28no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418#4114yes—
181C23×C61—22no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—22no—
185C23×C61#12012yes—
186C23×C61C22×D411yescharacteristic
187C22×D61—21no—
188C22×D61—22no—
189C22×D61—21no—
190C22×D61—22no—
191C22×D61#13712yes—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61#12012yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4112yes—
208C2×C4×C612−1+412yes—
209C22⋊D61—44no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—84no—
226C4×D61—48no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—42no—
230Dic122—24no—
231Dic122—41no—
232Dic122—41no—
233C8×C61—42no—
234#92418—24no—
235#24111—611nomaximal, Sylow
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61C2×#12011yescharacteristic
242C22×C61#54712yes—
243C22×C61C2×#13711yescharacteristic
244C22×C61#57511yescharacteristic
245C22×C61#57712yes—
246C2×D61—24no—
247C2×D61—22no—
248C2×D61—42no—
249C2×D61—41no—
250C2×D61—41no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—82no—
255C4×C61#52511yescharacteristic
256C4×C61#53611yescharacteristic
257C4×C61#54712yes—
258Dic61—42no—
259Dic61—41no—
260Dic61—42no—
261Dic61—41no—
262Dic61—41no—
263Dic61—82no—
264C2×#552—611no—
265C2×#483—611no—
266C2×#428—611no—
267#2487—611no—
268#2059—612no—
269#2036—612no—
270#1894—611no—
271#1870—611no—
272#1865—612no—
273#1853—612no—
274#1825D6111yescharacteristic, complemented
275C2×C61—42no—
276C2×C61#240611yescharacteristic
277C2×C61#300711yescharacteristic
278C2×C61#301011yescharacteristic
279C2×C61#334412yes—
280C2×C61#335112yes—
281D61—42no—
282C22×M16—611no—
283#552—1224no—
284#552—1224no—
285C2×#126—1221no—
286C2×#125—611no—
287C2×#125—1221no—
288C2×#123—1221no—
289C2×C22⋊Q8—611no—
290C2×C22⋊Q8—611no—
291C2×C4×Q8—611no—
292C2×C4×D4—611no—
293C2×#117—1221no—
294C2×#117C2×D6111yescharacteristic
295C22×C4⋊C4—611no—
296C22×C4⋊C4—611no—
297#483—1224no—
298#428—1224no—
299C2×#107—612no—
300C2×#106—612no—
301C2×#106—612no—
302C2×#103—612no—
303C2×#103—612no—
304C2×C22⋊C8—612no—
305C2×C22⋊C8C2×D6111yescharacteristic
306#400—1221no—
307#399—1221no—
308#389—612no—
309#386—1221no—
310#385—1221no—
311#380—612no—
312#380C2×D6111yescharacteristic
313#340—612no—
314#340C2×D6112yes—
315#340C2×D6112yes—
316#336—612no—
317C61#2411111yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×D4—611no—
320C23×C4—611no—
321C23×C4—1221no—
322C23×C4—1221no—
323C23×C4C22×D6111yescharacteristic
324C2×M16—1224no—
325C2×M16—1222no—
326C22×C8—612no—
327C22×C8C22×D6112yes—
328#126—1228no—
329#125—1228no—
330#125—1222no—
331#125—1222no—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335C22⋊Q8—1222no—
336C22⋊Q8—1222no—
337C4×Q8—1222no—
338C4×Q8—1222no—
339C4×D4—1222no—
340C4×D4—1222no—
341#117—22no—
342#117—1228no—
343#117D24414yes—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—611no—
365C2×C22⋊C4—611no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1222no—
369C2×C42—21no—
370C2×C42—611no—
371C2×C42—1221no—
372#107—1224no—
373#106—1224no—
374#106—1224no—
375#103—1224no—
376#103—1224no—
377C22⋊C8—22no—
378C22⋊C8—1224no—
379#95—21no—
380#95—1222no—
381#95—1222no—
382C24—611no—
383C24D4×D6111yescharacteristic
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—2441no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—1224no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—4882no—
408C22×C4—2442no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4C2×D24412yes—
412C22×C4D4×D6112yes—
413C22×C4D4×D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415M16—1228no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—614no—
420C2×C8—1222no—
421C2×C8—1222no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—612no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1221no—
433C4⋊C4—1222no—
434C4⋊C4—1224no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—4882no—
438C4⋊C4—2442no—
439C4⋊C4C2×D24412yes—
440C4⋊C4C2×D24412yes—
441C22⋊C4—24no—
442C22⋊C4—42no—
443C22⋊C4—1224no—
444C22⋊C4—1222no—
445C22⋊C4—1224no—
446C22⋊C4—1222no—
447C22⋊C4—4882no—
448C42—24no—
449C42—1224no—
450C42—1221no—
451C42—1221no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—612no—
457C23—1221no—
458C23—1221no—
459C23—1222no—
460C23—1222no—
461C23#1451974712yes—
462C23C2×D4×D6111yescharacteristic
463Q8—1224no—
464Q8—2441no—
465Q8—2441no—
466D4—2444no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—4884no—
489C2×C4—2448no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4#1451974712yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×D24411yescharacteristic
502C2×C4#41×D6112yes—
503C2×C4#1451982412yes—
504C8—42no—
505C8—1224no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—1224no—
512C22—1222no—
513C22—2441no—
514C22—2441no—
515C22—2442no—
516C22C2×#1451974711yescharacteristic
517C22#1684041012yes—
518C22#137×D6111yescharacteristic
519C22#1684055811yescharacteristic
520C22#1684056912yes—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—4882no—
526C4—2441no—
527C4—2441no—
528C4—2442no—
529C4—2441no—
530C4—2442no—
531C4#1684027411yescharacteristic
532C4#1684033811yescharacteristic
533C4#1684041012yes—
534C2—42no—
535C2—2442no—
536C2#2144915611yescharacteristic
537C2#2145508011yescharacteristic
538C2#2145510811yescharacteristic
539C2#2145826712yes—
540C2#2145837212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenter#1825D61
hypocenterC61#24111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. C2×#16840093C2semidirect
3. C2×#16839805C2semidirect
5. #21447156C2semidirect
6. #21447025C2semidirect
7. #21446250C2semidirect
9. #21446088C2semidirect
10. #21446013C2semidirect
11. C61×#1825C2semidirect
274. #1825D61semidirect
317. C61#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#168404512
C2×#168400932
C2×#168398052
#214501372
#214471562
#214470252
#214462502
#214461142
#214460882
#214460132
C61×#18252
#2411161