Finite Group Databasebeta

← #29713864

Subgroups — #29713864

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×#16840452C211yesmaximal, characteristic
2C2×#16840092C211yesmaximal, characteristic, complemented
3C2×#16839802C211yesmaximal, characteristic, complemented
4#21447157C212yesmaximal, complemented
5#21447022C212yesmaximal, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446117C211yesmaximal, characteristic
8#21446086C212yesmaximal, complemented
9#21446011C212yesmaximal, complemented
10#21445911C211yesmaximal, characteristic, complemented
11C61×#2487C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840452—24no—
14#16840452—24no—
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519758—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25#16840092—24no—
26#16839802—24no—
27C2×#14519710C2212yes—
28C2×#14519708C2212yes—
29C2×#14519708C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#400—21no—
41C61×#399—21no—
42C61×#389C2212yes—
43C61×#386—21no—
44C61×#385—21no—
45C61×#380C2211yescharacteristic
46C61×#380C2212yes—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C2×#11758194—22no—
50C2×#11758194—24no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519770—28no—
54#14519769—22no—
55#14519769—22no—
56#14519767—28no—
57#14519758—28no—
58#14519756—22no—
59#14519756—22no—
60#14519756—22no—
61#14519756D414yes—
62C4×Dic122—22no—
63C4×Dic122—22no—
64#14519736—22no—
65#14519736D414yes—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83#14519710—24no—
84#14519708—24no—
85#14519708—24no—
86#14519700—24no—
87#14519700—24no—
88C22⋊#92418—22no—
89C22⋊#92418—24no—
90C22×D4×C61D411yescharacteristic
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#95×C61—22no—
109#95×C61—21no—
110#95×C61—22no—
111C24×C61C2×D411yescharacteristic
112C24×C61C2×D411yescharacteristic
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—41no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—82no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C2×D412yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61—82no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—82no—
155C42×C61—21no—
156C42×C61—21no—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166C4⋊Dic61C2×D412yes—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61—24no—
173C4×Dic61—24no—
174#11758194—28no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418#4114yes—
181C23×C61—21no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12012yes—
190C23×C61#13712yes—
191C23×C61C22×D411yescharacteristic
192D4×C61—44no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—84no—
214C2×C4×C61#12012yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C612−1+412yes—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—48no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230Dic122—24no—
231Dic122—41no—
232Dic122—41no—
233#92418—24no—
234#92418—42no—
235#24111—611nomaximal, Sylow
236C22×C61—24no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61C2×#12011yescharacteristic
247C22×C61#54712yes—
248C22×C61C2×#13711yescharacteristic
249C22×C61#57511yescharacteristic
250C22×C61#57712yes—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—82no—
256C4×C61—82no—
257C4×C61#52511yescharacteristic
258C4×C61#53611yescharacteristic
259C4×C61#54712yes—
260Dic61—42no—
261Dic61—42no—
262Dic61—42no—
263Dic61—42no—
264C2×#552—611no—
265C2×#483—611no—
266C2×#428—611no—
267#2487D6111yescharacteristic
268#2059—612no—
269#2036—612no—
270#1894—611no—
271#1870—611no—
272#1865—612no—
273#1853—612no—
274#1825—611no—
275C2×C61—42no—
276C2×C61—42no—
277C2×C61#240611yescharacteristic
278C2×C61#300711yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#334412yes—
281C2×C61#335112yes—
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×D4C2×D6111yescharacteristic
293C2×#117—611no—
294C2×#117—1221no—
295C22×C4⋊C4—611no—
296C22×C4⋊C4C2×D6111yescharacteristic
297#483—1224no—
298#428—1224no—
299C2×#107—612no—
300C2×#106—612no—
301C2×#106—612no—
302C2×#103—612no—
303C2×#103—612no—
304C2×C22⋊C8—611no—
305C2×C22⋊C8—612no—
306#400—21no—
307#399—21no—
308#389C2×D6112yes—
309#386—21no—
310#385—21no—
311#380C2×D6111yescharacteristic
312#380C2×D6112yes—
313#340—612no—
314#340—612no—
315#340—612no—
316#336—612no—
317C61#2411111yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×D4C22⋊D6111yescharacteristic
320C23×C4—21no—
321C23×C4—1221no—
322C23×C4C22×D6111yescharacteristic
323C23×C4C22×D6111yescharacteristic
324C2×M16—1224no—
325C2×M16—1222no—
326C22×C8—612no—
327C22×C8—612no—
328#126—1228no—
329#125—1222no—
330#125—1222no—
331#125—1228no—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335C22⋊Q8—1222no—
336C22⋊Q8—1222no—
337C4×Q8—1222no—
338C4×Q8—1222no—
339C4×D4—22no—
340C4×D4—22no—
341#117—614no—
342#117—1228no—
343#117—1222no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4C22⋊D6111yescharacteristic
360C2×C4⋊C4C22×D6112yes—
361C2×C4⋊C4C22×D6112yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4C22⋊D6111yescharacteristic
368C2×C22⋊C4C22⋊D6111yescharacteristic
369C2×C42—1221no—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#107—1224no—
373#106—1224no—
374#106—1224no—
375#103—1224no—
376#103—1224no—
377C22⋊C8—1224no—
378C22⋊C8—1222no—
379#95—22no—
380#95—21no—
381#95—22no—
382C24C2×C22⋊D6111yescharacteristic
383C24C2×C22⋊D6111yescharacteristic
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—82no—
401C22×C4—1224no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4D4×D6112yes—
409C22×C4D4×D6112yes—
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415M16—1228no—
416C2×C8—614no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—1222no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—82no—
429C4⋊C4—612no—
430C4⋊C4—612no—
431C4⋊C4—612no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C4⋊C4—1224no—
436C4⋊C4—1224no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1224no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—42no—
444C22⋊C4—82no—
445C22⋊C4—1224no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C42—21no—
449C42—21no—
450C42—1224no—
451C42—1224no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23#1451974512yes—
461C23#1451980812yes—
462C23C22×C22⋊D6111yescharacteristic
463Q8—1224no—
464Q8—2441no—
465Q8—2441no—
466D4—44no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2448no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4#41×D6112yes—
503C2×C4#1451982412yes—
504C8—1224no—
505C8—2442no—
506C22—24no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22C2×#1451974511yescharacteristic
517C22#1684041112yes—
518C22C2×#1451980811yescharacteristic
519C22#1684056011yescharacteristic
520C22#1684057312yes—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—82no—
526C4—82no—
527C4—2442no—
528C4—2442no—
529C4—2442no—
530C4—2442no—
531C4#1684027011yescharacteristic
532C4#1684033411yescharacteristic
533C4#1684041012yes—
534C2—42no—
535C2—42no—
536C2#2144915111yescharacteristic
537C2#2145507711yescharacteristic
538C2#2145510311yescharacteristic
539C2#2145826812yes—
540C2#2145837312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2487C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2487D61
hypocenterC61#24111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. C2×#16840092C2semidirect
3. C2×#16839802C2semidirect
4. #21447157C2semidirect
5. #21447022C2semidirect
6. #21446250C2semidirect
8. #21446086C2semidirect
9. #21446011C2semidirect
10. #21445911C2semidirect
317. C61#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168400922
C2×#168398022
#214471572
#214470222
#214462502
#214461172
#214460862
#214460112
#214459112
C61×#24872
#2411161