Finite Group Databasebeta

← #29646061

Subgroups — #29646061

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840363C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21444177C214yesmaximal, complemented
5#21444164C214yesmaximal, complemented
6#21444140C211yesmaximal, characteristic, complemented
7#21444131C211yesmaximal, characteristic, complemented
8#21444131C211yesmaximal, characteristic, complemented
9C61×#1460C211yesmaximal, characteristic, complemented
10#16840363—22no—
11#16840363—22no—
12#16840363—22no—
13#16840363—22no—
14C2×#14519773C411yescharacteristic
15C2×#14519768C411yescharacteristic
16C2×#14519760C412yes—
17C2×#14519759C2212yes—
18C2×#14519752C411yescharacteristic
19C2×#14519751C412yes—
20C2×#14519750C2212yes—
21C2×#14519748C411yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24#16839841—22no—
25#16839839—22no—
26C8×D244—22no—
27#16839827—22no—
28#16839815C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839724C2214yes—
31C22⋊C8×D61C2214yes—
32#16839487C2212yes—
33#16839487C2212yes—
34#16839486C2214yes—
35#16839485C2212yes—
36#16839485C2212yes—
37C2×C4×D4×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#431C2211yescharacteristic
40C61×#323C2214yes—
41C22×D244C2×C411yescharacteristic
42C22×D244C2×C412yes—
43C22×C22⋊D61C2×C412yes—
44C22×C22⋊D61C2×C412yes—
45C22×C22⋊D61C2×C412yes—
46C22×C4×D61C2×C411yescharacteristic
47C22×C4×D61C2312yes—
48C2×#11758189—24no—
49C2×C8×D61—24no—
50#14519773—22no—
51#14519773—22no—
52#14519768—22no—
53#14519768—22no—
54#14519760—24no—
55#14519760—24no—
56#14519759—24no—
57#14519759—24no—
58#14519752—22no—
59#14519752—22no—
60#14519751—24no—
61#14519751—24no—
62#14519750—24no—
63#14519750—24no—
64#14519748—22no—
65#14519748—22no—
66C4×D244D412yes—
67C4×D244D412yes—
68C4×D244D412yes—
69C4×D244D412yes—
70#14519734—21no—
71#14519734—21no—
72#14519734—21no—
73#14519734—21no—
74C2×C4⋊Dic61C2×C412yes—
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C412yes—
78C2×#11758196C2311yescharacteristic
79C2×C22⋊Dic61C2×C412yes—
80C2×#11758198C2×C411yescharacteristic
81C2×#11758198C2×C412yes—
82C2×#11758198C2×C412yes—
83C2×#11758198C2311yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C4⋊#92418—21no—
86C4⋊#92418—21no—
87C22⋊#92418C2312yes—
88C22⋊#92418C2312yes—
89#14519689—24no—
90#14519689—24no—
91C4×#92418—21no—
92C4×#92418—21no—
93C22×D4×C61C2×C411yescharacteristic
94C23×C4×C61C2×C412yes—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C22⋊C4×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61—21no—
104C22⋊C8×C61C2312yes—
105C22⋊C8×C61C2312yes—
106C4×C8×C61—21no—
107C4×C8×C61—21no—
108C24×C61C22×C412yes—
109C23×D61C22×C411yescharacteristic
110C23×D61C22×C412yes—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—82no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—82no—
126C2×C22⋊D61—82no—
127C2×C22⋊D61—82no—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C412yes—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—24no—
135C2×D244—82no—
136C2×D244C22⋊C412yes—
137C2×D244C22⋊C412yes—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—82no—
143C2×C4×D61C2×D414yes—
144C2×C4×D61C2×D414yes—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C22⋊Dic61—24no—
156#11758198—24no—
157#11758198—21no—
158#11758198—21no—
159#11758198—21no—
160#11758198—21no—
161#11758198C22⋊C414yes—
162#11758198C22⋊C414yes—
163C4⋊Dic61—24no—
164C4⋊Dic61C22⋊C412yes—
165C4⋊Dic61C22⋊C412yes—
166#11758196—21no—
167#11758196—21no—
168#11758196—21no—
169#11758196—21no—
170#11758196—24no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175#11758189—84no—
176C8×D61—84no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61C23×C411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61C2×C22⋊C414yes—
193D4×C61—42no—
194D4×C61—42no—
195C2×C4×C61—22no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—82no—
200C2×C4×C61—82no—
201C2×C4×C61C2×C22⋊C411yescharacteristic
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61#13112yes—
206C2×C4×C61#13112yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C22⋊D61—84no—
210C22⋊D61—84no—
211C22⋊D61—84no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61C2×C22⋊C414yes—
221D244—24no—
222D244—24no—
223D244—84no—
224C4×D61—44no—
225C4×D61—44no—
226C4×D61—42no—
227C4×D61—42no—
228C8×C61—42no—
229C8×C61—42no—
230#92418—42no—
231#92418—42no—
232#17389—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61C22×C22⋊C411yescharacteristic
241C22×C61C2×#13111yescharacteristic
242C22×C61C2×#13111yescharacteristic
243C2×D61—24no—
244C2×D61—24no—
245C2×D61—24no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61—41no—
249C2×D61—82no—
250C2×D61—82no—
251C2×D61—81no—
252C2×D61—81no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—42no—
258C4×C61#40812yes—
259C4×C61#40812yes—
260Dic61—24no—
261Dic61—41no—
262Dic61—41no—
263Dic61—41no—
264C2×#540—611no—
265#2975—611no—
266#2972—611no—
267#1471—614no—
268#1468—614no—
269#1460—611no—
270#1460D6111yescharacteristic, complemented
271#1458—611no—
272#1458—611no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61—82no—
276C2×C61C2×#40811yescharacteristic
277C2×C61#288311yescharacteristic
278C2×C61#288811yescharacteristic
279D61—44no—
280D61—81no—
281#540—1222no—
282#540—1222no—
283#540—1222no—
284#540—1222no—
285C2×#127—611no—
286C2×#125—611no—
287C2×#123—612no—
288C2×#123—612no—
289C2×#121—611no—
290C2×#121—612no—
291C2×#121—611no—
292C2×#121—612no—
293C2×C4×D4—611no—
294C2×C4×D4C2×D6111yescharacteristic
295C2×#117—611no—
296#436—1222no—
297#436—1222no—
298C8×D4—1222no—
299C8×D4—1222no—
300#431—611no—
301#431—611no—
302#431C2×D6111yescharacteristic
303#431C2×D6111yescharacteristic
304#407—614no—
305C2×C22⋊C8—614no—
306#324—612no—
307#324—612no—
308#323—614no—
309#323—612no—
310#323—612no—
311#323C2×D6114yes—
312C61#1738911yescharacteristic, complemented, Sylow
313C22×D4—612no—
314C22×D4—612no—
315C22×D4—612no—
316C22×D4—611no—
317C22×D4—612no—
318C22×D4C4×D6111yescharacteristic
319C23×C4—612no—
320C23×C4—611no—
321C23×C4C4×D6112yes—
322C2×M16—1224no—
323C22×C8—1224no—
324#127—1222no—
325#127—1222no—
326#125—1222no—
327#125—1222no—
328#123—1224no—
329#123—1224no—
330#123—1224no—
331#123—1224no—
332#121—1224no—
333#121—1224no—
334#121—1224no—
335#121—1224no—
336#121—1222no—
337#121—1222no—
338#121—1222no—
339#121—1222no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—612no—
345C4×D4—612no—
346C4×D4—612no—
347C4×D4—612no—
348#117—1221no—
349#117—1221no—
350#117—1221no—
351#117—1221no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4C4×D6111yescharacteristic
358C2×C22⋊C4—612no—
359C2×C22⋊C4—612no—
360C2×C22⋊C4—611no—
361C2×C22⋊C4—612no—
362C2×C22⋊C4—611no—
363C2×C22⋊C4C22×D6112yes—
364C2×C42—611no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—21no—
367C4⋊C8—21no—
368C4⋊C8—1221no—
369C4⋊C8—1221no—
370C22⋊C8—612no—
371C22⋊C8—612no—
372C22⋊C8—1224no—
373C22⋊C8—1224no—
374C22⋊C8C22×D6112yes—
375C22⋊C8C22×D6112yes—
376C4×C8—21no—
377C4×C8—21no—
378C4×C8—1221no—
379C4×C8—1221no—
380C24—612no—
381C24—611no—
382C24C2×C4×D6112yes—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—612no—
388C2×D4—612no—
389C2×D4—1224no—
390C2×D4—1224no—
391C2×D4—1224no—
392C2×D4—1224no—
393C2×D4—1222no—
394C2×D4—1222no—
395C2×D4—4882no—
396C2×D4—4882no—
397C2×D4—4882no—
398C2×D4—4882no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—82no—
402C22×C4—612no—
403C22×C4—614no—
404C22×C4—614no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—4882no—
413C22×C4C2×C4×D6112yes—
414C22×C4C2×C4×D6111yescharacteristic
415C22×C4C2×C4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417M16—4884no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—4884no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—612no—
426C4⋊C4—612no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—614no—
436C22⋊C4—614no—
437C22⋊C4—1224no—
438C22⋊C4—1224no—
439C22⋊C4—1221no—
440C22⋊C4—1221no—
441C22⋊C4—1221no—
442C22⋊C4—1221no—
443C42—1221no—
444C42—1221no—
445C42D4×D6111yescharacteristic
446C42D4×D6111yescharacteristic
447C42D4×D6111yescharacteristic
448C42D4×D6111yescharacteristic
449C23—22no—
450C23—22no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—614no—
455C23—1222no—
456C23—1222no—
457C23—1221no—
458C23—1221no—
459C23—1221no—
460C23—2442no—
461C23—2442no—
462C23—2441no—
463C23—2441no—
464C23C22×C4×D6111yescharacteristic
465D4—42no—
466D4—42no—
467D4—1224no—
468D4—1224no—
469D4—4884no—
470D4—4884no—
471D4—4884no—
472D4—4884no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—614no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—2444no—
486C2×C4—2444no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4C22⋊C4×D6111yescharacteristic
493C2×C4C22⋊C4×D6111yescharacteristic
494C2×C4C22⋊C4×D6111yescharacteristic
495C2×C4C22⋊C4×D6111yescharacteristic
496C2×C4#1451978312yes—
497C2×C4#1451978512yes—
498C2×C4C2×D4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C8—42no—
501C8—42no—
502C8—2442no—
503C8—2442no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—82no—
511C22—1224no—
512C22—1224no—
513C22—1224no—
514C22—4881no—
515C22—4881no—
516C22—4882no—
517C22—4882no—
518C22—2441no—
519C22—2441no—
520C22—2441no—
521C22C2×C22⋊C4×D6111yescharacteristic
522C22C2×#1451978311yescharacteristic
523C22C2×#1451978511yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—42no—
529C4—1224no—
530C4—2441no—
531C4—2441no—
532C4—2441no—
533C4#1683972712yes—
534C4#1683972712yes—
535C2—21no—
536C2—21no—
537C2—82no—
538C2—4881no—
539C2—2444no—
540C2C2×#1683972711yescharacteristic
541C2#2145427811yescharacteristic
542C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1460C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840363C2
cosocleC22×C4×C61C24
hypercenter#1460D61
hypocenterC61#17389
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21454842C2semidirect
4. #21444177C2semidirect
5. #21444164C2semidirect
6. #21444140C2semidirect
7. #21444131C2semidirect
8. #21444131C2semidirect
9. C61×#1460C2semidirect
270. #1460D61semidirect
312. C61#17389semidirect

Maximal subgroups

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

subgroupindex
C2×#168403632
#214548592
#214548422
#214441772
#214441642
#214441402
#214441312
C61×#14602
#1738961