Finite Group Databasebeta

← #29733324

Subgroups — #29733324

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×#16840472C211yesmaximal, characteristic, complemented
2C2×#16840121C211yesmaximal, characteristic, complemented
3#21450759C211yesmaximal, characteristic
4#21446826C212yesmaximal, complemented
5#21446819C212yesmaximal, complemented
6#21446770C211yesmaximal, characteristic, complemented
7#21446616C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446123C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#464C211yesmaximal, characteristic, complemented
12#16840472—24no—
13#16840472—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753C2212yes—
18C2×#14519753C2212yes—
19C2×#14519748C2212yes—
20C2×C4×D244—21no—
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23#16840121—24no—
24C2×#14519701C2212yes—
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29C2×#14519688C2211yescharacteristic
30#16839698—21no—
31#16839671—21no—
32#16839662—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839524C2212yes—
37C22×Q16×C61C2211yescharacteristic
38C22×SD16×C61C2211yescharacteristic
39C2×#124×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#464—24no—
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C22×#92418C2312yes—
51#14519777—22no—
52#14519777D414yescomplemented
53#14519769—28no—
54#14519755—216no—
55#14519753—24no—
56#14519753—24no—
57#14519753—24no—
58#14519753—42no—
59#14519748—24no—
60#14519748—42no—
61C4×D244—28no—
62C4×D244—22no—
63C4×D244—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C4⋊C4×D61—24no—
72C4⋊C4×D61—24no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198D412yes—
77C2×C4×Dic61—21no—
78#14519701—24no—
79#14519701—24no—
80#14519698—24no—
81#14519689—42no—
82#14519689—42no—
83#14519688—22no—
84#14519684—21no—
85C22×Q8×C61C2311yescharacteristic
86C22×Q8×C61C2311yescharacteristic
87C22×D4×C61C2311yescharacteristic
88C2×Q16×C61—22no—
89C2×Q16×C61—42no—
90C2×SD16×C61—22no—
91C2×SD16×C61—42no—
92C22×C8×C61C2311yescharacteristic
93C22×C8×C61C2311yescharacteristic
94#124×C61—22no—
95#124×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61D414yescomplemented
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#103×C61—22no—
106#102×C61—22no—
107C24×C61—21no—
108C23×D61C2×D412yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61C2×D412yes—
133C2×D244—24no—
134C2×D244—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—28no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×Dic122—24no—
145C2×Dic122—81no—
146C2×Dic122—81no—
147Q16×C61—44no—
148SD16×C61—44no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—42no—
157C22⋊C4×C61—42no—
158C42×C61—24no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—44no—
162#11758198—28no—
163#11758198—24no—
164#11758198—44no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C2×D414yes—
170C4⋊Dic61#4114yes—
171#11758196—28no—
172#11758196—28no—
173#11758196—44no—
174C4×Dic61—81no—
175C2×#92418—22no—
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—22no—
192Q8×C61—22no—
193Q8×C61—42no—
194Q8×C61—41no—
195Q8×C61—41no—
196Q8×C61—42no—
197D4×C61—41no—
198D4×C61—41no—
199D4×C61—42no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C61H(5,𝔽2)12yes—
216C22⋊D61—88no—
217C2×Dic61—22no—
218C2×Dic61—24no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226D244—28no—
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—88no—
232Dic122—84no—
233C8×C61—41no—
234C8×C61—41no—
235#92418—42no—
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—42no—
257C4×C61—42no—
258C4×C61—81no—
259C4×C61#53211yescharacteristic
260C4×C61#53511yescharacteristic
261C4×C61#55012yes—
262Dic61—42no—
263Dic61—44no—
264Dic61—44no—
265C2×#554—611no—
266C2×#489—611no—
267C2×#464D6111yescharacteristic, complemented
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×Q16C2×D6111yescharacteristic
284C22×SD16C2×D6111yescharacteristic
285#554—1224no—
286#554—1224no—
287C2×C4⋊Q8—611no—
288C2×#125—1221no—
289C2×#124C2×D6111yescharacteristic
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—612no—
292C2×C22⋊Q8—1222no—
293C2×#121—612no—
294C2×C4×Q8C2×D6111yescharacteristic
295C2×C4×D4—611no—
296C2×C4×D4—1221no—
297C22×C4⋊C4—612no—
298#489—1224no—
299#464—24no—
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#103—612no—
302C2×#103—612no—
303C2×#103C2×D6111yescharacteristic
304C2×#102—612no—
305C2×#102C2×D6111yescharacteristic
306C2×C22⋊C8—612no—
307C2×C22⋊C8—612no—
308C2×#97—611no—
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—1221no—
318C22×Q8C22×D6111yescharacteristic
319C22×Q8C22×D6111yescharacteristic
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4C22×D6111yescharacteristic
323C23×C4—612no—
324C23×C4—1222no—
325C2×Q16—22no—
326C2×Q16—42no—
327C2×SD16—22no—
328C2×SD16—42no—
329C22×C8—612no—
330C22×C8C22×D6111yescharacteristic
331C22×C8C22×D6111yescharacteristic
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#125—1228no—
335#124—22no—
336#124—22no—
337C22⋊Q8—12216no—
338C22⋊Q8—1224no—
339C22⋊Q8—1224no—
340C22⋊Q8—1224no—
341C22⋊Q8—2442no—
342#121—1224no—
343#121—2442no—
344C4×Q8—22no—
345C4×Q8C22⋊D6114yescomplemented
346C4×D4—1228no—
347C4×D4—1222no—
348C4×D4—1222no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—612no—
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×D6111yescharacteristic
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—21no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—22no—
370#103—22no—
371#103—1224no—
372#103—1224no—
373#102—22no—
374#102—1224no—
375C22⋊C8—2442no—
376C22⋊C8—2442no—
377#97—1222no—
378#95—1221no—
379C24—21no—
380C24—612no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—1224no—
389C2×Q8—4881no—
390C2×Q8—4881no—
391C2×Q8C2×C22⋊D6112yes—
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—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—612no—
405C22×C4—1228no—
406C22×C4—1222no—
407C22×C4—1222no—
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—44no—
420SD16—44no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C4⋊C4—21no—
429C4⋊C4—24no—
430C4⋊C4—614no—
431C4⋊C4—614no—
432C4⋊C4—1228no—
433C4⋊C4—1228no—
434C4⋊C4—1222no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—2444no—
439C4⋊C4C2×C22⋊D6112yes—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—1228no—
443C22⋊C4—1224no—
444C22⋊C4—2444no—
445C22⋊C4—2444no—
446C42—24no—
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—22no—
464Q8—22no—
465Q8—42no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—4884no—
470D4—41no—
471D4—41no—
472D4—42no—
473D4—1228no—
474D4—4888no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—1224no—
486C2×C4—1222no—
487C2×C4—1224no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—4888no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
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#1451982212yes—
505C8—41no—
506C8—41no—
507C8—2442no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—1224no—
514C22—1224no—
515C22—1224no—
516C22—2442no—
517C22—2442no—
518C22—2442no—
519C22—2442no—
520C22C2×#1451974511yescharacteristic
521C22#1684040912yes—
522C22#137×D6112yes—
523C22#1684056411yescharacteristic
524C22#1684056911yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—81no—
530C4—2444no—
531C4—2442no—
532C4—2444no—
533C4#1684030811yescharacteristic
534C4#1684032911yescharacteristic
535C4#1684043012yes—
536C2—161no—
537C2—2444no—
538C2#2144943911yescharacteristic
539C2#2145505111yescharacteristic
540C2#2145511911yescharacteristic
541C2#2145790712yes—
542C2#2145793712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#464C2
socleC23×C61C22×D4
prime radical#21450759C2
hypercenterC2×#464D61
hypocenterC61#25579
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840121C2semidirect
4. #21446826C2semidirect
5. #21446819C2semidirect
6. #21446770C2semidirect
7. #21446616C2semidirect
8. #21446133C2semidirect
9. #21446123C2semidirect
10. #21445649C2semidirect
11. C2×C61×#464C2semidirect
52. #14519777D4semidirect
97. C4×Q8×C61D4semidirect
267. C2×#464D61semidirect
316. C61#25579semidirect
345. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168401212
#214507592
#214468262
#214468192
#214467702
#214466162
#214461332
#214461232
#214456492
C2×C61×#4642
#2557961