Finite Group Databasebeta

← #29685271

Subgroups — #29685271

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840294C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446629C212yesmaximal, complemented
6#21446254C212yesmaximal
7#21446251C212yesmaximal, complemented
8#21445652C211yesmaximal, characteristic, complemented
9#21445650C211yesmaximal, characteristic, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C61×#1762C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840294—24no—
14#16840294C418yescomplemented
15C2×#14519770—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519736—21no—
24C2×C42×D61C2211yescharacteristic
25C2×#14519702C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2212yes—
28#16839714—24no—
29C2×#14519688C2211yescharacteristic
30C2×C8×Dic61C2212yes—
31#16839629—21no—
32#16839627C2212yes—
33#16839624—21no—
34#16839619C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839608—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#97×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—24no—
53C2×#11758189—24no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519768—22no—
57#14519768C2×C414yes—
58#14519764—22no—
59#14519764C2×C414yes—
60#14519759—28no—
61#14519757—22no—
62#14519757C2×C414yes—
63#14519748—22no—
64#14519748C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122C2×C414yes—
67C4×C22⋊D61—28no—
68#14519736—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87#14519702—24no—
88#14519700—24no—
89#14519698—24no—
90#14519688—22no—
91C8×Dic61—24no—
92#14519684—21no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C22×C8×C61C2312yes—
99C4×D4×C61—22no—
100C4×D4×C61C2×C414yes—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#102×C61—24no—
107#97×C61—22no—
108#95×C61—21no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C2×C22⋊D61—24no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—42no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—42no—
136C2×C4×D61C22⋊C414yes—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22×C412yes—
139C2×C4×D61C2×D412yes—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C22×C412yes—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61#4114yes—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C22×C412yes—
151C22⋊C4×C61—24no—
152C42×C61—42no—
153C42×C61C22×C412yes—
154C42×C61C2×D412yes—
155C22⋊Dic61—24no—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158#11758198—24no—
159#11758198—44no—
160C4⋊Dic61—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C22×C412yes—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—24no—
169C4×Dic61—42no—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22⋊C414yes—
172C4×Dic61C22×C412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418#4114yes—
177#11758189—216no—
178C23×C61—22no—
179C23×C61—42no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—42no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61C2×C22⋊C412yes—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—42no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—48no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4112yes—
206C2×C4×C61C2×#4112yes—
207C22⋊D61—48no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61C2×C22⋊C412yes—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—24no—
226C4×D61—44no—
227C4×D61—44no—
228Dic122—22no—
229Dic122—22no—
230Dic122—42no—
231C8×C61—24no—
232#92418—24no—
233#21411—611nomaximal, Sylow
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—82no—
238C22×C61—82no—
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#53212yes—
241C22×C61#57512yes—
242C22×C61#57511yescharacteristic
243C22×C61#57511yescharacteristic
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—22no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—82no—
253C4×C61C4×#4111yescharacteristic
254C4×C61C4×#4111yescharacteristic
255C4×C61#52512yes—
256Dic61—22no—
257Dic61—22no—
258Dic61—22no—
259Dic61—42no—
260Dic61—42no—
261Dic61—82no—
262C2×#529—611no—
263C2×C4×M16—611no—
264#2283—611no—
265#1991—612no—
266#1958—612no—
267#1895—612no—
268#1894—612no—
269#1763—611no—
270#1763—611no—
271#1762—611no—
272#1762D6111yescharacteristic, complemented
273C2×C61—82no—
274C2×C61#229711yescharacteristic
275C2×C61#289611yescharacteristic
276C2×C61#289611yescharacteristic
277C2×C61#318312yes—
278C2×C61#318312yes—
279D61—42no—
280C22×M16—612no—
281#529—618no—
282#529—1224no—
283C2×#126—1221no—
284C2×#125—611no—
285C2×#124—611no—
286C2×#123—1221no—
287C2×C22⋊Q8—611no—
288C2×#121—611no—
289C2×C4×Q8—611no—
290C2×C4×D4—1221no—
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117—1221no—
293C22×C42—611no—
294C2×#103—612no—
295C2×#103—612no—
296C2×#102—612no—
297C2×#102C2×D6112yes—
298C4×M16—1224no—
299C2×#97—611no—
300C2×#97C2×D6111yescharacteristic
301C2×C4×C8—612no—
302#383—1221no—
303#382—612no—
304#381—1221no—
305#380—612no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308C4×C22⋊C4—1221no—
309#376—1221no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340C2×D6112yes—
314C61#2141111yescharacteristic, complemented, Sylow
315C22×Q8—611no—
316C22×D4—1221no—
317C22×D4C22×D6111yescharacteristic
318C23×C4—21no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C2×M16—1224no—
323C2×M16—1224no—
324C2×M16—1224no—
325C22×C8—612no—
326C22×C8C22×D6112yes—
327#126—1228no—
328#125—614no—
329#125—1222no—
330#124—614no—
331#124—1222no—
332#123—1228no—
333C22⋊Q8—614no—
334C22⋊Q8—1222no—
335#121—614no—
336#121—1222no—
337C4×Q8—614no—
338C4×Q8—1222no—
339C4×D4—22no—
340C4×D4—1228no—
341C4×D4C4×D6114yes—
342#117—1228no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22×D6112yes—
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1221no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366#103—1224no—
367#103—1224no—
368#102—24no—
369#102—1224no—
370#97—22no—
371#97—1222no—
372C4×C8—1224no—
373#95—21no—
374#95—1222no—
375#95—1222no—
376#95—1222no—
377#95—1221no—
378C24—21no—
379C24—611no—
380C2×Q8—612no—
381C2×Q8—1221no—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—1224no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4C2×C4×D6112yes—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—44no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—614no—
401C22×C4—614no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—2441no—
408C22×C4—2442no—
409C22×C4—2441no—
410C22×C4D4×D6111yescharacteristic
411C22×C4D4×D6111yescharacteristic
412C22×C4C23×D6111yescharacteristic
413M16—12216no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—614no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8#1175822214yes—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—612no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1224no—
427C4⋊C4—1221no—
428C4⋊C4—1224no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4C2×C4×D6112yes—
432C22⋊C4—24no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—2444no—
438C42—42no—
439C42—612no—
440C42—612no—
441C42—614no—
442C42—614no—
443C42—1224no—
444C42—2442no—
445C42C2×C4×D6112yes—
446C42D4×D6112yes—
447C23—22no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—612no—
453C23—1222no—
454C23—1221no—
455C23—1221no—
456C23—1222no—
457C23C2×D4×D6111yescharacteristic
458Q8—1222no—
459Q8—1222no—
460Q8—2442no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—2448no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—1222no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1224no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—2444no—
487C2×C4—2444no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2442no—
494C2×C4C22⋊C4×D6112yes—
495C2×C4C22⋊C4×D6112yes—
496C2×C4C22×C4×D6111yescharacteristic
497C2×C4C2×D4×D6111yescharacteristic
498C2×C4#41×D6112yes—
499C2×C4C2×#1175822212yes—
500C8—24no—
501C8—1224no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—1222no—
508C22—1222no—
509C22—1222no—
510C22—2442no—
511C22—2442no—
512C22C2×C22⋊C4×D6111yescharacteristic
513C22#1684030912yes—
514C22#1684055812yes—
515C22#1684055811yescharacteristic
516C22#1684056111yescharacteristic
517C4—22no—
518C4—22no—
519C4—42no—
520C4—82no—
521C4—1222no—
522C4—1222no—
523C4—1222no—
524C4—4882no—
525C4—2442no—
526C4—2442no—
527C4#1684023011yescharacteristic
528C4C4×#1175822211yescharacteristic
529C4#1684027112yes—
530C2—82no—
531C2—2442no—
532C2#2144830511yescharacteristic
533C2#2145433311yescharacteristic
534C2#2145433511yescharacteristic
535C2#2145654412yes—
536C2#2145654612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21411
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839714C2semidirect
3. #21448215C2semidirect
4. #21446782C2semidirect
5. #21446629C2semidirect
7. #21446251C2semidirect
8. #21445652C2semidirect
9. #21445650C2semidirect
10. #21445644C2semidirect
11. C61×#1762C2semidirect
14. #16840294C4semidirect
272. #1762D61semidirect
314. C61#21411semidirect

Maximal subgroups

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

subgroupindex
C2×#168402942
C2×#168397142
#214482152
#214467822
#214466292
#214462542
#214462512
#214456522
#214456502
#214456442
C61×#17622
#2141161