Finite Group Databasebeta

← #29733319

Subgroups — #29733319

Order 15616.

1206 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×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3C2×#16839981C211yesmaximal, characteristic, complemented
4#21446827C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446771C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446137C212yesmaximal, complemented
9#21446127C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2540C211yesmaximal, characteristic
12C22×#11758202C2211yescharacteristic
13C22×#11758200C2211yescharacteristic
14#16840469—24no—
15#16840469—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519768—21no—
18C2×#14519756C2212yes—
19C2×#14519754—22no—
20C2×C4×Dic122C2211yescharacteristic
21C2×D4×Dic61—21no—
22C22×C4⋊Dic61C2212yes—
23#16840121—24no—
24#16839981—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2212yes—
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2211yescharacteristic
31C2×C22⋊#92418C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37C2×#124×C61C2211yescharacteristic
38C2×C22⋊Q8×C61C2212yes—
39C2×#121×C61C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C61×#399—21no—
42C61×#392—21no—
43C61×#390—22no—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C23×Dic61—22no—
48C2×#11758202—22no—
49C2×#11758202—42no—
50C2×#11758200—22no—
51C2×#11758200—42no—
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yescomplemented
57#14519768—28no—
58#14519756—24no—
59#14519756—24no—
60#14519754—216no—
61C4×Dic122—22no—
62C4×Dic122D414yescomplemented
63D4×Dic61—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—21no—
71C2×#11758196—22no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—21no—
74C4⋊#92418—22no—
75#14519701—22no—
76#14519701—24no—
77#14519700—24no—
78#14519698—22no—
79#14519698—24no—
80C22⋊#92418—42no—
81C22⋊#92418—42no—
82#14519688—22no—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61—22no—
85C22×D4×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88#124×C61—22no—
89#124×C61—22no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—42no—
92#121×C61—24no—
93#121×C61—42no—
94C4×D4×C61—22no—
95C4×D4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#95×C61—21no—
107C24×C61—21no—
108C24×C61C2×D412yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×D4×C61—24no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—44no—
120C2×D4×C61—44no—
121C2×D4×C61—82no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—28no—
138C22×Dic61—82no—
139C22×Dic61—82no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—24no—
144C2×Dic122—81no—
145C2×Dic122—81no—
146C2×Dic122C2×D412yes—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—44no—
150C4⋊C4×C61#4114yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—44no—
153C22⋊C4×C61—44no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61—42no—
156C42×C61—81no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—28no—
160#11758202—44no—
161#11758200—44no—
162C4⋊Dic61—28no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D412yes—
167C4⋊Dic61C2×D414yes—
168#11758196—28no—
169#11758196—24no—
170C4×Dic61—24no—
171C2×#92418—22no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61#13714yes—
190C23×C61C22×D411yescharacteristic
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—42no—
194D4×C61—28no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61—42no—
198D4×C61—88no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
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—41no—
213C2×C4×C61—42no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61H(5,𝔽2)12yes—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—88no—
229Dic122—22no—
230Dic122—22no—
231Dic122—42no—
232Dic122—84no—
233#92418—41no—
234#92418—41no—
235#92418—42no—
236#25579—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—24no—
239C22×C61—24no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#54612yes—
251C22×C61C2×#13712yes—
252C22×C61#57611yescharacteristic
253C22×C61#57711yescharacteristic
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—42no—
257C4×C61—44no—
258C4×C61—81no—
259C4×C61#53211yescharacteristic
260C4×C61#53511yescharacteristic
261C4×C61#55012yes—
262Dic61—42no—
263Dic61—44no—
264Dic61—42no—
265C2×#554—611no—
266C2×#489—611no—
267C2×#464—611no—
268#2540D6111yescharacteristic
269#2001—612no—
270#2000—612no—
271#1988—611no—
272#1955—611no—
273#1874—612no—
274#1872—612no—
275#1763—611no—
276C2×C61—44no—
277C2×C61—161no—
278C2×C61#243111yescharacteristic
279C2×C61#300411yescharacteristic
280C2×C61#301211yescharacteristic
281C2×C61#331412yes—
282C2×C61#331612yes—
283C22×Q16—611no—
284C22×SD16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×#124C2×D6111yescharacteristic
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—1222no—
292C2×C22⋊Q8C2×D6112yes—
293C2×#121C2×D6112yes—
294C2×C4×Q8—611no—
295C2×C4×D4—1221no—
296C2×C4×D4C2×D6111yescharacteristic
297C22×C4⋊C4—612no—
298#489—1224no—
299#464—1224no—
300C2×C4⋊C8—611no—
301C2×#103—612no—
302C2×#103—611no—
303C2×#103—612no—
304C2×#102—612no—
305C2×#102—611no—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
309#399—21no—
310#392—21no—
311#390—22no—
312C4×C4⋊C4C2×D6111yescharacteristic
313#340—612no—
314#340—612no—
315#340—612no—
316C61#2557911yescharacteristic, complemented, Sylow
317C22×Q8—611no—
318C22×Q8—1221no—
319C22×Q8C22×D6111yescharacteristic
320C22×D4—22no—
321C22×D4C22⋊D6111yescharacteristic
322C22×D4C22×D6111yescharacteristic
323C23×C4—1222no—
324C23×C4C22×D6112yes—
325C2×Q16—1222no—
326C2×Q16—2442no—
327C2×SD16—1222no—
328C2×SD16—2442no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8—612no—
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#125—1228no—
335#124—22no—
336#124—22no—
337C22⋊Q8—24no—
338C22⋊Q8—42no—
339C22⋊Q8—12216no—
340C22⋊Q8—1224no—
341C22⋊Q8—1224no—
342#121—24no—
343#121—42no—
344C4×Q8—614no—
345C4×Q8—1222no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—1228no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1224no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C4⋊C4C22×D6112yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4C22⋊D6112yes—
366C2×C42—21no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—1222no—
370#103—1222no—
371#103—1224no—
372#103—1224no—
373#102—1222no—
374#102—1224no—
375C22⋊C8—2442no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—21no—
379C24—21no—
380C24C2×C22⋊D6112yes—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—612no—
386C2×Q8—1221no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—4881no—
391C2×Q8—4881no—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—1228no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—4882no—
414C22×C4—4882no—
415C22×C4C2×C22⋊D6112yes—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419Q16—2444no—
420SD16—2444no—
421C2×C8—1222no—
422C2×C8—1221no—
423C2×C8—1221no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—612no—
432C4⋊C4—614no—
433C4⋊C4—1228no—
434C4⋊C4—1228no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4#1175822414yes—
440C22⋊C4—24no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—1228no—
446C42—81no—
447C42—1224no—
448C42D4×D6112yes—
449C42D4×D6112yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23#1451980714yes—
462C23C22×C22⋊D6111yescharacteristic
463Q8—41no—
464Q8—41no—
465Q8—42no—
466Q8—1222no—
467Q8—1222no—
468Q8—4884no—
469Q8—2442no—
470D4—28no—
471D4—41no—
472D4—41no—
473D4—42no—
474D4—88no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1224no—
493C2×C4—4888no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4#1451974512yes—
500C2×C4#1451974512yes—
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C2×D4×D6111yescharacteristic
503C2×C4C2×#1175822412yes—
504C2×C4#1451982212yes—
505C8—2441no—
506C8—2441no—
507C8—2442no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—82no—
520C22C2×#1451974511yescharacteristic
521C22#1684040612yes—
522C22C2×#1451980712yes—
523C22#1684056511yescharacteristic
524C22#1684057011yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—44no—
529C4—81no—
530C4—2442no—
531C4—2444no—
532C4—2442no—
533C4#1684031011yescharacteristic
534C4#1684032511yescharacteristic
535C4#1684043212yes—
536C2—44no—
537C2—161no—
538C2#2144943511yescharacteristic
539C2#2145505211yescharacteristic
540C2#2145512011yescharacteristic
541C2#2145790012yes—
542C2#2145793012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2540C2
socleC23×C61C22×D4
hypercenter#2540D61
hypocenterC61#25579
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840121C2semidirect
3. C2×#16839981C2semidirect
4. #21446827C2semidirect
5. #21446822C2semidirect
6. #21446771C2semidirect
7. #21446618C2semidirect
8. #21446137C2semidirect
9. #21446127C2semidirect
10. #21445652C2semidirect
56. #14519777D4semidirect
62. C4×Dic122D4semidirect
316. C61#25579semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168401212
C2×#168399812
#214468272
#214468222
#214467712
#214466182
#214461372
#214461272
#214456522
C61×#25402
#2557961