Finite Group Databasebeta

← #29733326

Subgroups — #29733326

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×#16840473C211yesmaximal, characteristic, complemented
2C2×#16839981C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic
4#21446828C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446617C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446123C212yesmaximal, complemented
10#21445650C211yesmaximal, characteristic, complemented
11C2×C61×#489C211yesmaximal, characteristic, complemented
12C22×#11758202C2211yescharacteristic
13C22×#11758200C2211yescharacteristic
14#16840473—24no—
15#16840473—24no—
16C2×#14519768—21no—
17C2×#14519757C2212yes—
18C2×#14519753—22no—
19C2×#14519753C2212yes—
20C2×#14519748C2212yes—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519739—21no—
24C2×C4⋊C4×D61C2212yes—
25#16839981—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839698—21no—
33#16839671—21no—
34#16839662—22no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#124×C61C2211yescharacteristic
41C61×#489—24no—
42C2×#103×C61C2212yes—
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758202—22no—
51C2×#11758202—42no—
52C2×#11758200—22no—
53C2×#11758200—42no—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56#14519768—28no—
57#14519757—24no—
58#14519757—24no—
59#14519753—216no—
60#14519753—24no—
61#14519753—42no—
62#14519748—24no—
63#14519748—42no—
64C4×Dic122—22no—
65C4×Dic122D414yescomplemented
66C4×D244—22no—
67C4×D244—22no—
68#14519739—28no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C4⋊C4×D61—24no—
77C4⋊C4×D61—24no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519702—24no—
86#14519701—22no—
87#14519698—22no—
88#14519689—42no—
89#14519689—42no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×D4×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61D414yescomplemented
97#124×C61—22no—
98#124×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#102×C61—24no—
106#97×C61—22no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—81no—
115C2×Q8×C61—81no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—82no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61C2×D412yes—
132C2×D244—24no—
133C2×D244—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—28no—
141C2×C4×D61—82no—
142C2×C4×D61—82no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—21no—
146C2×Dic122C2×D412yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D414yes—
153C22⋊C4×C61—42no—
154C22⋊C4×C61—42no—
155C42×C61C2×D412yes—
156C42×C61C2×D412yes—
157C22⋊Dic61—44no—
158#11758202—44no—
159#11758200—44no—
160#11758198—28no—
161#11758198—24no—
162#11758198—44no—
163C4⋊Dic61—28no—
164C4⋊Dic61—22no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D412yes—
168C4⋊Dic61#4114yes—
169#11758196—28no—
170#11758196—24no—
171#11758196—44no—
172C4×Dic61—24no—
173C4×Dic61—81no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61#13714yes—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—42no—
194Q8×C61—84no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61—42no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4112yes—
211C2×C4×C61H(5,𝔽2)12yes—
212C22⋊D61—88no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224D244—28no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—88no—
230Dic122—22no—
231Dic122—22no—
232Dic122—42no—
233C8×C61—42no—
234#92418—41no—
235#92418—41no—
236#25579—611nomaximal, Sylow
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61C2×#12011yescharacteristic
243C22×C61#54612yes—
244C22×C61C2×#13712yes—
245C22×C61#57611yescharacteristic
246C22×C61#57711yescharacteristic
247C2×D61—24no—
248C2×D61—24no—
249C2×D61—24no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—44no—
257C4×C61—81no—
258C4×C61#53211yescharacteristic
259C4×C61#53511yescharacteristic
260C4×C61#55012yes—
261Dic61—42no—
262Dic61—42no—
263Dic61—42no—
264Dic61—44no—
265C2×#554—611no—
266C2×#489D6111yescharacteristic, complemented
267C2×#464—611no—
268#2540—611no—
269#2001—612no—
270#2000—612no—
271#1988—611no—
272#1955—611no—
273#1874—612no—
274#1872—612no—
275#1763—611no—
276C2×C61—161no—
277C2×C61#243111yescharacteristic
278C2×C61#300411yescharacteristic
279C2×C61#301211yescharacteristic
280C2×C61#331412yes—
281C2×C61#331612yes—
282D61—44no—
283C22×Q16—611no—
284C22×SD16—611no—
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8C2×D6111yescharacteristic
288C2×#125—1221no—
289C2×#124C2×D6111yescharacteristic
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—612no—
292C2×C22⋊Q8—1222no—
293C2×#121—612no—
294C2×C4×Q8—611no—
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C22×C4⋊C4—612no—
298#489—24no—
299#464—1224no—
300C2×C4⋊C8—611no—
301C2×#103—611no—
302C2×#103—612no—
303C2×#103C2×D6112yes—
304C2×#102—611no—
305C2×#102C2×D6112yes—
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97C2×D6111yescharacteristic
309#399—1221no—
310#392—1221no—
311#390—1222no—
312C4×C4⋊C4—611no—
313#340—612no—
314#340—612no—
315#340—612no—
316C61#2557911yescharacteristic, complemented, Sylow
317C22×Q8—21no—
318C22×Q8—611no—
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4C22×D6111yescharacteristic
323C23×C4—612no—
324C23×C4—1222no—
325C2×Q16—1222no—
326C2×Q16—2442no—
327C2×SD16—1222no—
328C2×SD16—2442no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8C22×D6112yes—
332C4⋊Q8—22no—
333C4⋊Q8C22⋊D6114yescomplemented
334#125—1228no—
335#124—22no—
336#124—22no—
337C22⋊Q8—1224no—
338C22⋊Q8—12216no—
339C22⋊Q8—1224no—
340C22⋊Q8—1224no—
341C22⋊Q8—2442no—
342#121—1224no—
343#121—2442no—
344C4×Q8—614no—
345C4×Q8—1222no—
346C4×D4—1228no—
347C4×D4—1222no—
348C4×D4—1222no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1224no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1224no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4C22×D6112yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—1221no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—1222no—
370#103—24no—
371#103—1224no—
372#103—1222no—
373#102—24no—
374#102—1222no—
375C22⋊C8—2442no—
376C22⋊C8—2442no—
377#97—22no—
378#95—1221no—
379C24—21no—
380C24—612no—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—612no—
389C2×Q8—1221no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—1224no—
397C2×D4—1222no—
398C2×D4—4882no—
399C2×D4—2444no—
400C2×D4—2444no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—612no—
404C22×C4—1228no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—4882no—
415C22×C4—4882no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419Q16—2444no—
420SD16—2444no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1222no—
427C2×C8—1222no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—612no—
431C4⋊C4—614no—
432C4⋊C4—1228no—
433C4⋊C4—1228no—
434C4⋊C4—1221no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—2444no—
439C4⋊C4C2×C22⋊D6114yes—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—1228no—
443C22⋊C4—1224no—
444C22⋊C4—2444no—
445C22⋊C4—2444no—
446C42—1224no—
447C42—4881no—
448C42D4×D6112yes—
449C42C2×C22⋊D6112yes—
450C23—41no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—614no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23C2×D4×D6111yescharacteristic
463Q8—41no—
464Q8—41no—
465Q8—42no—
466Q8—84no—
467Q8—1222no—
468Q8—1222no—
469Q8—2442no—
470D4—41no—
471D4—41no—
472D4—42no—
473D4—1228no—
474D4—4888no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—4888no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4#1451974512yes—
500C2×C4#1451974512yes—
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4C22×C22⋊D6111yescharacteristic
503C2×C4#41×D6112yes—
504C2×C4#1451982312yes—
505C8—42no—
506C8—2441no—
507C8—2441no—
508C22—82no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—1224no—
514C22—1224no—
515C22—1224no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22C2×#1451974511yescharacteristic
521C22#1684040512yes—
522C22#137×D6112yes—
523C22#1684056611yescharacteristic
524C22#1684057211yescharacteristic
525C4—22no—
526C4—22no—
527C4—44no—
528C4—81no—
529C4—2442no—
530C4—2442no—
531C4—2442no—
532C4—2444no—
533C4#1684031111yescharacteristic
534C4#1684033011yescharacteristic
535C4#1684043112yes—
536C2—161no—
537C2—2444no—
538C2#2144944111yescharacteristic
539C2#2145505111yescharacteristic
540C2#2145512211yescharacteristic
541C2#2145789912yes—
542C2#2145792912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#489C2
socleC23×C61C22×D4
prime radical#21450759C2
hypercenterC2×#489D61
hypocenterC61#25579
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×#16840473C2semidirect
2. C2×#16839981C2semidirect
4. #21446828C2semidirect
5. #21446823C2semidirect
6. #21446772C2semidirect
7. #21446617C2semidirect
8. #21446133C2semidirect
9. #21446123C2semidirect
10. #21445650C2semidirect
11. C2×C61×#489C2semidirect
65. C4×Dic122D4semidirect
96. C4⋊Q8×C61D4semidirect
266. C2×#489D61semidirect
316. C61#25579semidirect
333. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168399812
#214507592
#214468282
#214468232
#214467722
#214466172
#214461332
#214461232
#214456502
C2×C61×#4892
#2557961