Finite Group Databasebeta

← #29597525

Subgroups — #29597525

Order 15616.

820 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839736C211yesmaximal, characteristic, complemented
2#21449134C211yesmaximal, characteristic
3C2×#16839550C212yesmaximal, complemented
4#21443938C212yesmaximal, complemented
5C61×#1785C211yesmaximal, characteristic, complemented
6C2×#14519745C411yescharacteristic, complemented
7C2×#14519745C2211yescharacteristic
8C2×#14519736—21no—
9C22×#11758198C411yescharacteristic, complemented
10#16839736—24no—
11C2×#14519692—21no—
12C2×#14519692C2212yes—
13#16839673—21no—
14#16839670—21no—
15#16839642C411yescharacteristic, complemented
16#16839642C2211yescharacteristic
17#16839641—21no—
18#16839641—21no—
19#16839625C411yescharacteristic, complemented
20#16839550—24no—
21C22×#41×C61C2211yescharacteristic
22C2×#101×C61—21no—
23C2×#100×C61C2212yes—
24C61×#407—21no—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C23×Dic61C2×C411yescharacteristic
29#14519745—22no—
30#14519745—22no—
31#14519745—42no—
32#14519736—24no—
33#14519736—42no—
34C2×C4⋊Dic61—41no—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61C2×C411yescharacteristic
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—41no—
45C2×#11758198—41no—
46C2×C4×Dic61—41no—
47#14519692—24no—
48#14519692—22no—
49#14519692D414yescomplemented
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54C2×#41×C61—22no—
55C2×#41×C61—22no—
56C2×#41×C61—41no—
57C2×#41×C61—41no—
58C2×#41×C61—42no—
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61C2×C411yescharacteristic
61C22×D4×C61D411yescharacteristic
62C22×D4×C61C2311yescharacteristic
63C23×C4×C61C2×C411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C2×M16×C61—22no—
67#101×C61—42no—
68#100×C61—22no—
69#100×C61—22no—
70C22⋊C8×C61—41no—
71C22⋊C8×C61—41no—
72C24×C61C22⋊C411yescharacteristic
73C24×C61C22×C411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C23×D61—21no—
76C23×D61—22no—
77C23×D61—42no—
78C23×D61—41no—
79C23×D61—41no—
80C23×D61—81no—
81#41×C61—44no—
82#41×C61—44no—
83#41×C61—44no—
84#41×C61—44no—
85C2×Q8×C61—22no—
86C2×Q8×C61—22no—
87C2×Q8×C61—41no—
88C2×D4×C61—22no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—22no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61C2×D412yes—
98C2×D4×C61#4112yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—81no—
122C2×C22⋊D61—81no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—42no—
127C22×Dic61—41no—
128C22×Dic61—81no—
129M16×C61—42no—
130M16×C61—42no—
131C2×C8×C61—42no—
132C22⋊Dic61—22no—
133C22⋊Dic61—24no—
134C22⋊Dic61—42no—
135C22⋊Dic61—41no—
136C22⋊Dic61—81no—
137C22⋊Dic61—81no—
138#11758198—42no—
139#11758198—41no—
140#11758198—41no—
141#11758198—42no—
142#11758198—42no—
143C4⋊Dic61—44no—
144C4×Dic61—44no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61C2×C22⋊C411yescharacteristic
159C23×C61C4×D412yes—
160C23×C61#12011yescharacteristic
161C23×C61#12311yescharacteristic
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—42no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—42no—
171C22×D61—84no—
172C22×D61—84no—
173C22×D61—82no—
174C22×D61—82no—
175Q8×C61—44no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—44no—
179D4×C61—42no—
180D4×C61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61#9911yescharacteristic
199C2×C4×C61#9911yescharacteristic
200C2×C4×C61#9912yes—
201C2×C4×C61#12112yes—
202C22⋊D61—82no—
203C22⋊D61—82no—
204C22⋊D61—84no—
205C2×Dic61—44no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—82no—
216C2×Dic61—82no—
217C2×Dic61—82no—
218C8×C61—82no—
219#13087—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—22no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61#38611yescharacteristic
240C22×C61C2×#9911yescharacteristic
241C22×C61C2×#9911yescharacteristic
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—84no—
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—84no—
251C4×C61—22no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—22no—
255C4×C61—41no—
256Dic61—81no—
257Dic61—81no—
258Dic61—84no—
259Dic61—81no—
260C2×#410—611no—
261#2404—611no—
262C2×AΣL(1,𝔽16)—612no—
263#1785D6111yescharacteristic, complemented
264#1390—612no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—41no—
268C2×C61—42no—
269C2×C61—41no—
270C2×C61—81no—
271C2×C61#189811yescharacteristic
272C2×C61#212011yescharacteristic
273C2×C61#212211yescharacteristic
274D61—82no—
275C22×#41C2×D6111yescharacteristic
276C2×#120—611no—
277C2×#120—611no—
278C2×#117—1221no—
279C22×C22⋊C4—611no—
280C2×#101—21no—
281C2×#100C2×D6112yes—
282#410—1224no—
283C2×#99—612no—
284C2×#99—1221no—
285#407—21no—
286#393—1221no—
287#392—1221no—
288#386—611no—
289#386—611no—
290#386—1221no—
291#386—1221no—
292#381—611no—
293AΣL(1,𝔽16)—1224no—
294C61#1308711yescharacteristic, complemented, Sylow
295C25—611no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—41no—
299C2×#41—41no—
300C2×#41—42no—
301C22×Q8C22⋊D6111yescharacteristic
302C22×D4—1221no—
303C22×D4—1221no—
304C22×D4C4×D6111yescharacteristic
305C22×D4D24411yescharacteristic
306C22×D4C22×D6111yescharacteristic
307C23×C4—611no—
308C23×C4C4×D6111yescharacteristic
309C23×C4D24411yescharacteristic
310C23×C4C22⋊D6111yescharacteristic
311C2×M16—22no—
312#120—1222no—
313#120—1222no—
314#120—2442no—
315#117—1224no—
316#117—2442no—
317C2×C4⋊C4—2441no—
318C2×C22⋊C4—611no—
319C2×C22⋊C4—611no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—2441no—
328C2×C22⋊C4—2441no—
329C2×C42—2441no—
330#101—42no—
331#100—22no—
332#100—22no—
333#99—614no—
334#99—1224no—
335#99—1222no—
336C22⋊C8—41no—
337C22⋊C8—41no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342C24—1221no—
343C24—1222no—
344C24—4881no—
345C24—2441no—
346C24—2441no—
347C24—2442no—
348C24#1175819811yescharacteristic
349C24C2×C4×D6111yescharacteristic
350C24C2×D24411yescharacteristic
351#41—44no—
352#41—44no—
353#41—44no—
354#41—44no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—4881no—
368C2×D4—4881no—
369C2×D4—2442no—
370C2×D4—2442no—
371C2×D4—2442no—
372C2×D4—2441no—
373C2×D4—2441no—
374C2×D4D4×D6112yes—
375C2×D4#1175822412yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—4881no—
393C22×C4—2442no—
394C22×C4—2441no—
395C22×C4#1175819811yescharacteristic
396C22×C4#1175819811yescharacteristic
397C22×C4#1175819811yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—2444no—
403C22⋊C4—1222no—
404C22⋊C4—1224no—
405C22⋊C4—4881no—
406C22⋊C4—4881no—
407C22⋊C4—2442no—
408C22⋊C4—2442no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2442no—
412C22⋊C4—2442no—
413C22⋊C4—2441no—
414C42—2444no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—1222no—
429C23—4882no—
430C23—4882no—
431C23—4884no—
432C23—4884no—
433C23—2441no—
434C23—2441no—
435C23—2441no—
436C23—2442no—
437C23—2442no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C2×#1175819811yescharacteristic
442C23#1451973712yes—
443C23#1451974711yescharacteristic
444C23#1451976111yescharacteristic
445Q8—44no—
446D4—42no—
447D4—42no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—4882no—
452D4—4882no—
453D4—4884no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—4882no—
472C2×C4—4882no—
473C2×C4—4882no—
474C2×C4—2444no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4#1451969112yes—
485C2×C4#1451969111yescharacteristic
486C2×C4#1451969111yescharacteristic
487C2×C4#1451974912yes—
488C8—82no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—81no—
507C22—81no—
508C22—4881no—
509C22—4881no—
510C22—4884no—
511C22—4884no—
512C22—4881no—
513C22—4881no—
514C22—2442no—
515C22—2442no—
516C22—2442no—
517C22#1683963911yescharacteristic
518C22C2×#1451969111yescharacteristic
519C22C2×#1451969111yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—41no—
525C4—4881no—
526C4—4881no—
527C4—4884no—
528C4—4881no—
529C2—21no—
530C2—21no—
531C2—41no—
532C2—42no—
533C2—41no—
534C2—81no—
535C2—4882no—
536C2#2144627011yescharacteristic
537C2#2144746111yescharacteristic
538C2#2144747011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1785C2
socleC22×C61C2×#99
prime radical#21449134C2
cosocleC22×D4×C61C23
hypercenter#1785D61
hypocenterC61#13087
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
1. C2×#16839736C2semidirect
3. C2×#16839550C2semidirect
4. #21443938C2semidirect
5. C61×#1785C2semidirect
6. C2×#14519745C4semidirect
9. C22×#11758198C4semidirect
15. #16839642C4semidirect
19. #16839625C4semidirect
49. #14519692D4semidirect
263. #1785D61semidirect
294. C61#13087semidirect

Maximal subgroups

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

subgroupindex
C2×#168397362
#214491342
C2×#168395502
#214439382
C61×#17852
#1308761