Finite Group Databasebeta

← #29721557

Subgroups — #29721557

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839976C212yesmaximal
2C2×#16839968C211yesmaximal, characteristic, complemented
3C2×C4×#11758199C211yesmaximal, characteristic
4#21450811C211yesmaximal, characteristic, complemented
5#21448719C211yesmaximal, characteristic, complemented
6#21447215C212yesmaximal, complemented
7#21447058C212yesmaximal, complemented
8#21446620C211yesmaximal, characteristic, complemented
9#21446535C211yesmaximal, characteristic, complemented
10#21446154C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C22×#11758199C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15C2×#14519764C2212yes—
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C2212yes—
18C2×C4⋊C4×D61—21no—
19#16839976—28no—
20#16839968—24no—
21C4×#11758199—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2212yes—
24C2×#14519702C2212yes—
25C2×#14519700C2212yes—
26C2×#14519700C2212yes—
27C2×#14519698C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839687—22no—
31#16839667C2212yes—
32#16839658C2212yes—
33#16839633—22no—
34#16839622—21no—
35#16839622—21no—
36#16839619—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839516C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×#102×C61C2212yes—
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×C4×D61—22no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C2×#11758190—24no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C2×#11758199D414yes—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56#14519777—22no—
57#14519777—22no—
58#14519764—24no—
59#14519764—42no—
60C4×D244—22no—
61C4×D244D414yes—
62#14519739—24no—
63#14519739—42no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—22no—
70C4⋊C4×D61—44no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—21no—
74C2×#11758198—22no—
75C2×C4×Dic61C2312yes—
76#14519710—22no—
77#14519706—24no—
78#14519702—24no—
79#14519700—24no—
80#14519700—42no—
81#14519698—22no—
82#14519697—22no—
83C4×#92418—22no—
84#14519684—22no—
85#14519684—22no—
86#14519684—22no—
87#14519684D412yes—
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C22×C8×C61C2312yes—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—22no—
100#102×C61—24no—
101C24×C61—21no—
102C23×D61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61C2×D412yes—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D412yes—
120C22×Dic61C2×D412yes—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×D244C2×D412yes—
126C2×D244#4112yes—
127C2×C4×D61—24no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—44no—
131C2×C4×D61—44no—
132C2×C4×D61—82no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—81no—
148C22⋊C4×C61—42no—
149C42×C61C2×D412yes—
150C42×C61#4112yes—
151C22⋊Dic61—44no—
152C22⋊Dic61—44no—
153#11758199—28no—
154#11758198—24no—
155#11758198—44no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—42no—
160C4⋊Dic61—42no—
161C4⋊Dic61C2×D412yes—
162#11758196—44no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—21no—
170C2×#92418C2×D412yes—
171#11758190—48no—
172C23×C61—22no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—41no—
190D4×C61—41no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—84no—
200C2×C4×C61#12112yes—
201C2×C4×C61#12112yes—
202C2×C4×C61#13512yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222D244—22no—
223D244—22no—
224D244—22no—
225D244#12114yes—
226C4×D61—48no—
227C4×D61—88no—
228Dic122—42no—
229Dic122—42no—
230Dic122—42no—
231Dic122—42no—
232C8×C61—42no—
233#92418—22no—
234#92418—41no—
235#24727—611nomaximal, Sylow
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61C2×#12111yescharacteristic
244C22×C61D4211yescharacteristic
245C22×C61#54811yescharacteristic
246C22×C61C2×#13511yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C61#57711yescharacteristic
249C22×C61#57811yescharacteristic
250C2×D61—24no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C4×C61—22no—
258C4×C61—82no—
259C4×C61—81no—
260C4×C61#46314yescomplemented
261C4×C61C2×#12111yescharacteristic
262C4×C61#53711yescharacteristic
263C4×C61#54711yescharacteristic
264C4×C61#55311yescharacteristic
265Dic61—42no—
266Dic61—42no—
267Dic61—42no—
268Dic61—82no—
269C2×#463—612no—
270C2×#461—611no—
271C2×C4×D8—611no—
272#2547—611no—
273D4×C4⋊C4—611no—
274#2068—612no—
275#2042—612no—
276#1956—611no—
277#1939—611no—
278#1877—612no—
279#1802D6111yescharacteristic, complemented
280C2×C61—82no—
281C2×C61#241611yescharacteristic
282C2×C61C2×#46311yescharacteristic
283C2×C61#304611yescharacteristic
284C2×C61#327211yescharacteristic
285C2×C61#328311yescharacteristic
286C2×C61#336011yescharacteristic
287C2×C61#336711yescharacteristic
288D61—44no—
289C22×SD16—612no—
290C22×D8—611no—
291C2×C4⋊Q8—611no—
292C2×#124—612no—
293C2×C4×D4—612no—
294C2×C4×D4—611no—
295C2×C4×D4C2×D6111yescharacteristic
296C22×C4⋊C4—1221no—
297#463—1228no—
298#461—1224no—
299C4×D8—1224no—
300C2×#107—611no—
301C2×C4⋊C8—612no—
302C2×C4⋊C8C2×D6111yescharacteristic
303C2×#103—612no—
304C2×#103—612no—
305C2×#103—612no—
306C2×#102—611no—
307C2×#102—611no—
308C2×#102C2×D6112yes—
309C2×C4×C8—611no—
310#397—1222no—
311#391—612no—
312#389—612no—
313#385—1222no—
314#384C2×D6111yescharacteristic
315#380—1221no—
316#380—1221no—
317#380—1221no—
318C4×C4⋊C4—611no—
319#340C2×D6112yes—
320#336—612no—
321C61#2472711yescharacteristic, complemented, Sylow
322C22×Q8—612no—
323C22×D4—611no—
324C22×D4C22×D6111yescharacteristic
325C23×C4—21no—
326C23×C4—1222no—
327C23×C4—1221no—
328C2×SD16—1224no—
329C2×SD16—1224no—
330C2×SD16—1224no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6112yes—
337C4⋊Q8—1222no—
338C4⋊Q8—1222no—
339#124—1224no—
340#124—2442no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—614no—
344C4×D4—1224no—
345C4×D4—1222no—
346C4×D4—2442no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—2444no—
356C2×C4⋊C4C22×D6112yes—
357C2×C4⋊C4C22×D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1222no—
363C2×C42—612no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366C4⋊C8—22no—
367C4⋊C8—1224no—
368#103—1224no—
369#103—1224no—
370#103—2442no—
371#102—24no—
372#102—1222no—
373#102—1222no—
374C4×C8—1222no—
375#95—612no—
376#95—1222no—
377#95—1222no—
378#95—1222no—
379C24—21no—
380C24—1221no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—612no—
393C2×D4—612no—
394C2×D4—612no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4D4×D6112yes—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—1221no—
407C22×C4—1224no—
408C22×C4—1222no—
409C22×C4—4882no—
410C22×C4—2444no—
411C22×C4—2444no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—2448no—
418D8—1228no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—612no—
423C2×C8—1221no—
424C2×C8—1221no—
425C2×C8—1221no—
426C2×C8—1221no—
427C2×C8—1221no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—612no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—2444no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C22⋊C4—42no—
442C22⋊C4—1224no—
443C22⋊C4—2444no—
444C22⋊C4—2444no—
445C22⋊C4—2444no—
446C42—1222no—
447C42—1222no—
448C42#1175822212yes—
449C42D4×D6112yes—
450C23—22no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23C2×D4×D6111yescharacteristic
465Q8—2442no—
466Q8—2442no—
467Q8—2442no—
468Q8—2442no—
469D4—22no—
470D4—22no—
471D4—41no—
472D4—41no—
473D4—614no—
474D4—1222no—
475D4—1222no—
476D4—1222no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—84no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—4888no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2448no—
500C2×C4#1451975012yes—
501C2×C4#1451975012yes—
502C2×C4#1451979812yes—
503C2×C4#1451980112yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4#41×D6111yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4#1451982311yescharacteristic
509C2×C4#1451982511yescharacteristic
510C8—42no—
511C8—1222no—
512C8—2441no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—1224no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22C2×#1451975011yescharacteristic
528C22#1684039611yescharacteristic
529C22#1684042211yescharacteristic
530C22C2×#1451979811yescharacteristic
531C22C2×#1451980111yescharacteristic
532C22#1684057211yescharacteristic
533C22#1684057811yescharacteristic
534C4—22no—
535C4—82no—
536C4—81no—
537C4—4882no—
538C4—2442no—
539C4—2442no—
540C4—2442no—
541C4#1683997614yescomplemented
542C4#121×D6111yescharacteristic
543C4#1684034211yescharacteristic
544C4#1684041311yescharacteristic
545C4#1684045611yescharacteristic
546C2—82no—
547C2—2444no—
548C2#2144926111yescharacteristic
549C2C2×#1683997611yescharacteristic
550C2#2145536811yescharacteristic
551C2#2145741411yescharacteristic
552C2#2145753611yescharacteristic
553C2#2145846711yescharacteristic
554C2#2145853611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#1802D61
hypocenterC61#24727
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839968C2semidirect
4. #21450811C2semidirect
5. #21448719C2semidirect
6. #21447215C2semidirect
7. #21447058C2semidirect
8. #21446620C2semidirect
9. #21446535C2semidirect
10. #21446154C2semidirect
11. C61×#1802C2semidirect
260. C4×C61#463semidirect
279. #1802D61semidirect
321. C61#24727semidirect
541. C4#16839976semidirect

Maximal subgroups

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

subgroupindex
C2×#168399762
C2×#168399682
C2×C4×#117581992
#214508112
#214487192
#214472152
#214470582
#214466202
#214465352
#214461542
C61×#18022
#2472761