Finite Group Databasebeta

← #29595144

Subgroups — #29595144

Order 15616.

844 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#21448324C211yesmaximal, characteristic
2C2×#16839550C211yesmaximal, characteristic
3C2×#16839548C211yesmaximal, characteristic
4#100×Dic61C211yesmaximal, characteristic, complemented
5#99×Dic61C211yesmaximal, characteristic, complemented
6#21443937C211yesmaximal, characteristic, complemented
7C61×#1390C211yesmaximal, characteristic, complemented
8C23×C4×D61C411yescharacteristic, complemented
9C2×#14519745C2211yescharacteristic
10C2×D4×Dic61C2211yescharacteristic
11C2×#14519737—21no—
12C2×#14519694C2211yescharacteristic
13C2×#14519692C2211yescharacteristic
14M16×Dic61—21no—
15#16839638—21no—
16#16839636—21no—
17#16839635C411yescharacteristic, complemented
18#16839617C411yescharacteristic, complemented
19C22⋊C4×Dic61—21no—
20C22⋊C4×Dic61C411yescharacteristic, complemented
21#16839609—21no—
22#16839550C414yescomplemented
23#16839548C414yescomplemented
24#16839530—21no—
25#16839528—21no—
26C2×#100×C61C2211yescharacteristic
27C2×#99×C61C2211yescharacteristic
28C61×#386C2211yescharacteristic
29C24×D61C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C23×Dic61C2×C411yescharacteristic
32C23×Dic61D411yescharacteristic
33C22×C4×D61—21no—
34C22×C4×D61—21no—
35C22×C4×D61—44no—
36C22×C4×D61—41no—
37C22×C4×D61—41no—
38C2×#11758194—21no—
39#14519745—21no—
40#14519745—21no—
41#14519745C2×C412yes—
42#14519745C2×C412yes—
43D4×Dic61—41no—
44D4×Dic61—41no—
45#14519737—44no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61C2×C411yescharacteristic
50C2×C22⋊Dic61C2×C411yescharacteristic
51C2×C22⋊Dic61D411yescharacteristic
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61—41no—
57C2×C4×Dic61D411yescharacteristic
58#14519694—21no—
59#14519694C2×C412yescomplemented
60#14519692—21no—
61#14519692C2×C412yescomplemented
62#14519687—41no—
63C8×Dic61—41no—
64#14519684—21no—
65#14519684—21no—
66#14519684—41no—
67C22×D4×C61C2311yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69C2×M16×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61C2×C411yescharacteristic
73#100×C61—21no—
74#100×C61C2×C412yescomplemented
75#99×C61—21no—
76#99×C61C2×C412yescomplemented
77#95×C61—21no—
78C24×C61C22×C411yescharacteristic
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C23×D61—21no—
82C23×D61—42no—
83C23×D61—42no—
84C23×D61—41no—
85C23×D61—41no—
86C23×D61C4212yescomplemented
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61C22×C411yescharacteristic
90C2×D4×C61C22×C411yescharacteristic
91C2×D4×C61#4111yescharacteristic
92C2×D4×C61#4111yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—42no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C2×C22⋊D61—22no—
100C2×C22⋊D61—22no—
101C2×C22⋊D61—42no—
102C2×C22⋊D61—42no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—42no—
109C22×Dic61—42no—
110C22×Dic61—41no—
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113C22×Dic61C22⋊C411yescharacteristic
114C22×Dic61C22⋊C411yescharacteristic
115C2×C4×D61—24no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×C4×D61—44no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—44no—
126M16×C61—22no—
127M16×C61—41no—
128C2×C8×C61—41no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—42no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊Dic61—21no—
134C22⋊Dic61—22no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—41no—
138C22⋊Dic61C4212yescomplemented
139#11758198—22no—
140#11758198—22no—
141#11758198—42no—
142#11758198—42no—
143C4⋊Dic61—41no—
144#11758196—42no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61—42no—
148C4×Dic61—82no—
149#11758194—22no—
150#11758194—41no—
151C2×#92418—41no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61C2×C4211yescharacteristic
161C23×C61C2×C22⋊C411yescharacteristic
162C23×C61#11711yescharacteristic
163C23×C61C4×D411yescharacteristic
164C23×C61C4×D411yescharacteristic
165C22×D61—22no—
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—44no—
169C22×D61—44no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—48no—
175C22×D61—42no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—41no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—81no—
197C2×C4×C61—81no—
198C2×C4×C61C4×D411yescharacteristic
199C2×C4×C61C4×D411yescharacteristic
200C22⋊D61—44no—
201C22⋊D61—44no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—81no—
217C2×Dic61—81no—
218C2×Dic61—81no—
219C2×Dic61#9911yescharacteristic
220C2×Dic61#9911yescharacteristic
221C4×D61—44no—
222C4×D61—48no—
223C4×D61—44no—
224C8×C61—42no—
225#92418—42no—
226C4×AΣL(1,𝔽16)—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—81no—
243C22×C61C4×C22⋊C411yescharacteristic
244C22×C61C2×#9911yescharacteristic
245C22×C61#41011yescharacteristic
246C2×D61—24no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—44no—
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—44no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—42no—
260C4×C61—81no—
261C4×C61—82no—
262Dic61—21no—
263Dic61—42no—
264Dic61—41no—
265Dic61—41no—
266Dic61—41no—
267Dic61—82no—
268Dic61AΣL(1,𝔽16)12yescomplemented
269C4×#120—611no—
270C2×AΣL(1,𝔽16)—611no—
271C2×AΣL(1,𝔽16)—611no—
272C4×#100—611no—
273C4×#99—611no—
274#1390—611no—
275#1390D6111yescharacteristic, complemented
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—41no—
279C2×C61—41no—
280C2×C61—41no—
281C2×C61—41no—
282C2×C61C4×#9911yescharacteristic
283C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
284C2×C61#212011yescharacteristic
285D61—44no—
286C24×C4—611no—
287C2×#120—611no—
288C2×C4×D4—611no—
289C2×C4×D4—1221no—
290C2×#100—611no—
291C2×#100C2×D6111yescharacteristic
292C2×#99—611no—
293C2×#99C2×D6111yescharacteristic
294C4×M16—1221no—
295#386—1221no—
296#386C2×D6111yescharacteristic
297#385—611no—
298#385—1221no—
299#379—611no—
300C4×C22⋊C4—611no—
301C4×C22⋊C4—1221no—
302C4×C22⋊C4—1221no—
303AΣL(1,𝔽16)—614no—
304AΣL(1,𝔽16)—614no—
305#343—1221no—
306#342—1221no—
307C61C4×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
308C25—611no—
309C22×D4—1221no—
310C22×D4C22×D6111yescharacteristic
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—1221no—
314C23×C4—1221no—
315C23×C4—2444no—
316C23×C4—2441no—
317C23×C4—2441no—
318C23×C4C4×D6111yescharacteristic
319C2×M16—21no—
320C2×M16—1221no—
321#120—612no—
322#120—612no—
323#120—1221no—
324#120—1221no—
325C4×D4—2441no—
326C4×D4—2441no—
327C4×D4—2444no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—1221no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—611no—
333C2×C22⋊C4—611no—
334C2×C22⋊C4—611no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4C4×D6111yescharacteristic
340C2×C42—611no—
341C2×C42—1221no—
342C2×C42—2441no—
343#100—21no—
344#100—612no—
345#100—1221no—
346#100C4×D6112yescomplemented
347#99—21no—
348#99—612no—
349#99—1221no—
350#99C4×D6112yescomplemented
351#97—2441no—
352C4×C8—2441no—
353#95—21no—
354#95—1221no—
355#95—1221no—
356#95—2441no—
357C24—612no—
358C24—1221no—
359C24—1221no—
360C24—2441no—
361C24—2441no—
362C24—2442no—
363C24—2442no—
364C24C2×C4×D6111yescharacteristic
365C24D4×D6111yescharacteristic
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—2442no—
371C2×D4—2442no—
372C2×D4C2×C4×D6111yescharacteristic
373C2×D4C2×C4×D6111yescharacteristic
374C2×D4#1175822411yescharacteristic
375C2×D4#1175822611yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—1224no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—2442no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2442no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4—2444no—
398C22×C4—2444no—
399C22×C4—2444no—
400C22×C4—2444no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2441no—
404C22×C4D4×D6111yescharacteristic
405M16—22no—
406M16—41no—
407M16—1222no—
408M16—2441no—
409C2×C8—41no—
410C2×C8—2441no—
411C4⋊C4—2441no—
412C4⋊C4—2442no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—612no—
418C22⋊C4—1221no—
419C22⋊C4—1222no—
420C22⋊C4—1222no—
421C22⋊C4—1222no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C42—1221no—
428C42—1221no—
429C42—4882no—
430C42—2442no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—1222no—
440C23—1222no—
441C23—1222no—
442C23—2442no—
443C23—2441no—
444C23—2441no—
445C23—2442no—
446C23—2444no—
447C23—2444no—
448C23—2442no—
449C23—2442no—
450C23—2442no—
451C23—2448no—
452C23—2442no—
453C23—2441no—
454C23—2441no—
455C23C42×D6111yescharacteristic
456C23C22⋊C4×D6111yescharacteristic
457C23#1451973511yescharacteristic
458C23#1451973711yescharacteristic
459C23#1451973911yescharacteristic
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—2444no—
465D4—2444no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—611no—
480C2×C4—611no—
481C2×C4—1222no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1222no—
486C2×C4—4881no—
487C2×C4—4881no—
488C2×C4—4881no—
489C2×C4—2444no—
490C2×C4—2444no—
491C2×C4—2448no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4#1451973711yescharacteristic
502C2×C4#1451973911yescharacteristic
503C8—42no—
504C8—2442no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—81no—
521C22—1224no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2444no—
526C22—2444no—
527C22—2444no—
528C22—2444no—
529C22—2442no—
530C22—2442no—
531C22—2442no—
532C22#1683960911yescharacteristic
533C22#99×D6111yescharacteristic
534C22#1683973411yescharacteristic
535C4—21no—
536C4—21no—
537C4—42no—
538C4—81no—
539C4—82no—
540C4—612no—
541C4—1221no—
542C4—4882no—
543C4—2442no—
544C4—2441no—
545C4—2441no—
546C4—2441no—
547C2—21no—
548C2—21no—
549C2—41no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—2444no—
554C2#2144560611yescharacteristic
555C2AΣL(1,𝔽16)×D6111yescharacteristic
556C2#2144745911yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC23×C61C2×C42
FrattiniC22×D4C22×D61
FittingC61×#1390C2
socleC22×C61C2×#99
prime radicalC2×#14519745C22
cosocleC22×D4×C61C23
hypercenter#1390D61
hypocenterC61C4×AΣL(1,𝔽16)
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
4. #100×Dic61C2semidirect
5. #99×Dic61C2semidirect
6. #21443937C2semidirect
7. C61×#1390C2semidirect
8. C23×C4×D61C4semidirect
17. #16839635C4semidirect
18. #16839617C4semidirect
20. C22⋊C4×Dic61C4semidirect
22. #16839550C4semidirect
23. #16839548C4semidirect
59. #14519694C2×C4semidirect
61. #14519692C2×C4semidirect
74. #100×C61C2×C4semidirect
76. #99×C61C2×C4semidirect
86. C23×D61C42semidirect
138. C22⋊Dic61C42semidirect
268. Dic61AΣL(1,𝔽16)semidirect
275. #1390D61semidirect
307. C61C4×AΣL(1,𝔽16)semidirect
346. #100C4×D61semidirect
350. #99C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214483242
C2×#168395502
C2×#168395482
#100×Dic612
#99×Dic612
#214439372
C61×#13902
C4×AΣL(1,𝔽16)61