Finite Group Databasebeta

← #29713858

Subgroups — #29713858

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×#16840446C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic, complemented
3#21450130C211yesmaximal, characteristic
4#21447159C212yesmaximal, complemented
5#21447020C212yesmaximal, complemented
6#21446253C211yesmaximal, characteristic, complemented
7#21446114C211yesmaximal, characteristic
8#21446085C212yesmaximal, complemented
9#21446010C212yesmaximal, complemented
10#21445915C211yesmaximal, characteristic, complemented
11C2×C61×#483C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840446—24no—
14#16840446—24no—
15C2×#14519770—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519759—21no—
19C2×#14519753C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4×C22⋊D61C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519735—21no—
24C2×C4⋊C4×D61C2211yescharacteristic
25#16839802—24no—
26C2×#14519710C2212yes—
27C2×#14519708C2212yes—
28C2×#14519701C2212yes—
29C2×#14519689C2212yes—
30#16839700—21no—
31#16839697—21no—
32#16839658C2212yes—
33#16839641—21no—
34#16839635—21no—
35#16839622C2211yescharacteristic
36#16839619C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#483—24no—
44C2×#106×C61C2212yes—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×C22⋊D61D411yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61C2311yescharacteristic
50C2×#11758194—22no—
51C2×#11758194—24no—
52C22×#92418C2312yes—
53#14519770—28no—
54#14519768—22no—
55#14519768—22no—
56#14519767—28no—
57#14519759—28no—
58#14519753—22no—
59#14519753—22no—
60Q8×Dic61—22no—
61Q8×Dic61—22no—
62C4×C22⋊D61—22no—
63C4×C22⋊D61—22no—
64#14519736—22no—
65#14519736D414yes—
66#14519735—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C2×#11758196D411yescharacteristic
77C4⋊C4×D61—22no—
78C4⋊C4×D61—22no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×#11758198D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88#14519710—24no—
89#14519708—24no—
90#14519701—24no—
91#14519689—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108#106×C61—24no—
109#103×C61—24no—
110C22⋊C8×C61—22no—
111C24×C61C2×D411yescharacteristic
112C23×D61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—42no—
123C22×C4×C61—82no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—22no—
129C2×C22⋊D61—22no—
130C2×C22⋊D61—41no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×C4×D61—24no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—24no—
154C22⋊Dic61—24no—
155C22⋊Dic61—22no—
156C22⋊Dic61—42no—
157#11758198—24no—
158#11758198—22no—
159#11758198—82no—
160C4⋊Dic61—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D412yes—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—22no—
172#11758196—82no—
173C4×Dic61—24no—
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—
192Q8×C61—24no—
193Q8×C61—41no—
194Q8×C61—41no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—84no—
208C2×C4×C61#12012yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4112yes—
212C2×C4×C612−1+412yes—
213C22⋊D61—44no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—21no—
217C2×Dic61—22no—
218C2×Dic61—21no—
219C2×Dic61—22no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—22no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C4×D61—48no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—42no—
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—41no—
259Dic61—42no—
260Dic61—41no—
261Dic61—41no—
262Dic61—42no—
263Dic61—82no—
264C2×#552—611no—
265C2×#483D6111yescharacteristic, complemented
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#1825—611no—
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⋊Q8C2×D6111yescharacteristic
291C2×C4×Q8—611no—
292C2×C4×D4—611no—
293C2×#117—611no—
294C2×#117—1221no—
295C22×C4⋊C4—611no—
296C22×C4⋊C4C2×D6111yescharacteristic
297#483—24no—
298#428—1224no—
299C2×#107—612no—
300C2×#106—612no—
301C2×#106C2×D6112yes—
302C2×#103—612no—
303C2×#103C2×D6112yes—
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—611no—
312#380—612no—
313#340—612no—
314#340—612no—
315#340—612no—
316#336—612no—
317C61#2411111yescharacteristic, complemented, Sylow
318C22×Q8C22×D6111yescharacteristic
319C22×D4—611no—
320C23×C4—21no—
321C23×C4—611no—
322C23×C4—1221no—
323C23×C4C22×D6111yescharacteristic
324C2×M16—1224no—
325C2×M16—1222no—
326C22×C8—612no—
327C22×C8C22×D6112yes—
328#126—1228no—
329#125—1222no—
330#125—1228no—
331#125—1222no—
332#123—1228no—
333C22⋊Q8—22no—
334C22⋊Q8—1222no—
335C22⋊Q8—1222no—
336C22⋊Q8C22⋊D6114yes—
337C4×Q8—1222no—
338C4×Q8—1222no—
339C4×D4—1222no—
340C4×D4—1222no—
341#117—614no—
342#117—1228no—
343#117—1222no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—612no—
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—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4C22×D6112yes—
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—611no—
364C2×C22⋊C4—611no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C42—611no—
370C2×C42—1221no—
371C2×C42—1221no—
372#107—1224no—
373#106—24no—
374#106—1224no—
375#103—24no—
376#103—1224no—
377C22⋊C8—22no—
378C22⋊C8—1224no—
379#95—1222no—
380#95—1222no—
381#95—1221no—
382C24—611no—
383C24D4×D6111yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D6112yes—
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—42no—
397C22×C4—82no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—1224no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4D4×D6112yes—
412C22×C4D4×D6111yescharacteristic
413C22×C4C2×C22⋊D6112yes—
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—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—612no—
428C4⋊C4—612no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—4882no—
440C4⋊C4C2×C22⋊D6112yes—
441C22⋊C4—24no—
442C22⋊C4—1224no—
443C22⋊C4—1222no—
444C22⋊C4—1224no—
445C22⋊C4—1222no—
446C22⋊C4—4882no—
447C22⋊C4—2442no—
448C42—1224no—
449C42—1221no—
450C42—1221no—
451C42—1224no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—612no—
457C23—1221no—
458C23—1221no—
459C23—1222no—
460C23—1222no—
461C23#1451974512yes—
462C23C2×D4×D6111yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
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—84no—
480C2×C4—1221no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1222no—
488C2×C4—1222no—
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#1451974512yes—
500C2×C4C2×D4×D6111yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
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—2442no—
515C22—2441no—
516C22C2×#1451974511yescharacteristic
517C22#1684041012yes—
518C22#137×D6111yescharacteristic
519C22#1684056111yescharacteristic
520C22#1684056812yes—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—4882no—
526C4—2441no—
527C4—2441no—
528C4—2442no—
529C4—2441no—
530C4—2442no—
531C4#1684027011yescharacteristic
532C4#1684033411yescharacteristic
533C4#1684041112yes—
534C2—42no—
535C2—2442no—
536C2#2144915111yescharacteristic
537C2#2145507511yescharacteristic
538C2#2145510111yescharacteristic
539C2#2145827712yes—
540C2#2145838212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#483C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC2×#483D61
hypocenterC61#24111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
4. #21447159C2semidirect
5. #21447020C2semidirect
6. #21446253C2semidirect
8. #21446085C2semidirect
9. #21446010C2semidirect
10. #21445915C2semidirect
11. C2×C61×#483C2semidirect
265. C2×#483D61semidirect
317. C61#24111semidirect

Maximal subgroups

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

subgroupindex
C2×#168404462
C2×#168398022
#214501302
#214471592
#214470202
#214462532
#214461142
#214460852
#214460102
#214459152
C2×C61×#4832
#2411161