Finite Group Databasebeta

← #30602426

Subgroups — #30602426

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840733C211yesmaximal, characteristic
2#21462263C211yesmaximal, characteristic
3C2×#16840645C214yesmaximal, complemented
4C22×M16×D61C211yesmaximal, characteristic
5C22×#11758213C2212yes—
6C2×#14519856C2212yes—
7#16840733—21no—
8#16840733—21no—
9#16840733—21no—
10#16840733—21no—
11C2×#14519834C412yescomplemented
12C2×C22⋊#92420C412yescomplemented
13#16840705C412yes—
14#16840704—21no—
15#16840704—21no—
16#16840703C412yes—
17C4×C22⋊#6923—21no—
18C4×C22⋊#6923—21no—
19#16840645—28no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758194C411yescharacteristic, complemented
22C22×#11758189—21no—
23C2×M16×D61—28no—
24C2×M16×D61—21no—
25C2×M16×D61—21no—
26C2×M16×D61C412yescomplemented
27C2×M16×D61C2212yes—
28C22×C8×D61—21no—
29C22×M16×C61C411yescharacteristic, complemented
30C2×#11758213—22no—
31C2×#11758213—22no—
32C2×#11758213—24no—
33C2×#11758213—24no—
34C22×#92420C2×C414yes—
35#14519856—24no—
36#14519856—24no—
37C2×C4⋊#6923—22no—
38C2×C22⋊#6923—22no—
39C2×C22⋊#6923—22no—
40C2×C22⋊#6923C2×C414yes—
41C2×C4×#6923—22no—
42#14519834—22no—
43#14519834—22no—
44C22⋊#92420—22no—
45C22⋊#92420—22no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48C24×D61C2×C411yescharacteristic
49C23×Dic61C2×C411yescharacteristic
50C22×C4×D61—24no—
51C22×C4×D61—24no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61Q811yescharacteristic
62C22×C4×D61Q811yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758194—21no—
65C2×#11758194—24no—
66C2×#11758194C2×C412yes—
67C2×#11758189—28no—
68C2×#11758189—22no—
69C2×#11758189—22no—
70M16×D61—28no—
71M16×D61—28no—
72M16×D61—24no—
73M16×D61—24no—
74M16×D61—24no—
75M16×D61D418yes—
76C22×#92418—21no—
77C2×C8×D61—28no—
78C2×C8×D61—22no—
79C2×C8×D61—22no—
80C23×C4×C61C2×C411yescharacteristic
81C2×M16×C61—21no—
82C2×M16×C61—24no—
83C2×M16×C61C2×C412yes—
84C22×C8×C61—21no—
85C24×C61C4211yescharacteristic
86C23×D61—24no—
87C23×D61—42no—
88C23×D61—42no—
89C23×D61C4211yescharacteristic
90C23×D61C22⋊C411yescharacteristic
91C23×D61C22⋊C411yescharacteristic
92C23×D61C4⋊C411yescharacteristic
93C23×D61C4⋊C411yescharacteristic
94C23×D61C22×C411yescharacteristic
95C22×#6923—24no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61C4211yescharacteristic
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C4⋊C411yescharacteristic
102C22×C4×C61C4⋊C411yescharacteristic
103C22×C4×C61C22×C411yescharacteristic
104C22×Dic61—24no—
105C22×Dic61C4211yescharacteristic
106C22×Dic61C22⋊C411yescharacteristic
107C22×Dic61C22⋊C411yescharacteristic
108C22×Dic61C4⋊C411yescharacteristic
109C22×Dic61C4⋊C411yescharacteristic
110C22×Dic61C22×C411yescharacteristic
111C2×C4×D61—24no—
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61—24no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—24no—
120C2×C4×D61—24no—
121C2×C4×D61—24no—
122C2×C4×D61—24no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C22⋊C411yescharacteristic
133C2×C4×D61C4⋊C411yescharacteristic
134C2×C4×D61C4⋊C411yescharacteristic
135C2×C4×D61C4⋊C411yescharacteristic
136C2×C4×D61C4⋊C411yescharacteristic
137C2×C4×D61C4⋊C411yescharacteristic
138C2×C4×D61C4⋊C411yescharacteristic
139C2×C4×D61C2×D414yes—
140C2×C4×D61C2×D411yescharacteristic
141C2×C4×D61C2×D411yescharacteristic
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2×Q811yescharacteristic
144C2×C4×D61#4114yes—
145C22⋊#6923—28no—
146C22⋊#6923—44no—
147C22⋊#6923—44no—
148C2×#92420—24no—
149C2×#92420—28no—
150C4⋊#6923—42no—
151C4⋊#6923—42no—
152C4×#6923—44no—
153#11758213—42no—
154#11758213—44no—
155#11758213—42no—
156M16×C61—24no—
157M16×C61—22no—
158M16×C61C22⋊C414yescomplemented
159C2×C8×C61—24no—
160C2×C8×C61—22no—
161#11758194—24no—
162#11758194—22no—
163#11758194C22⋊C414yescomplemented
164C2×#92418—24no—
165C2×#92418—22no—
166#11758189—28no—
167#11758189—28no—
168C8×D61—28no—
169C8×D61—28no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—41no—
173C23×C61#9511yescharacteristic
174C23×C61#9511yescharacteristic
175C23×C61C2×C4211yescharacteristic
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—416no—
194C22×D61#9511yescharacteristic
195C22×D61#9511yescharacteristic
196C22×D61C2×C22⋊C411yescharacteristic
197C22×D61C2×C4⋊C411yescharacteristic
198C22×D61C4×D414yes—
199C2×#6923—44no—
200C2×#6923—44no—
201C2×#6923—44no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61#9511yescharacteristic
210C2×C4×C61#9511yescharacteristic
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C4⋊C411yescharacteristic
214C2×C4×C61#11712yes—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61#9511yescharacteristic
221C2×Dic61#9511yescharacteristic
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C4⋊C411yescharacteristic
224C2×Dic61C4×D414yes—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61#12012yes—
240C4×D61#12012yes—
241C4×D61#12114yes—
242C4×D61C22⋊Q814yes—
243C4×D61#12312yes—
244C4×D61#12312yes—
245#92420—44no—
246C8×C61—24no—
247#92418—24no—
248#10737—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—44no—
257C22×C61#34412yescomplemented
258C22×C61C2×#9511yescharacteristic
259C22×C61C4×C22⋊C412yes—
260C2×D61—28no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—21no—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—44no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61#34412yescomplemented
276C2×D61#38512yes—
277C2×D61#38612yes—
278#6923—84no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61#37911yescharacteristic
282C4×C61#38012yes—
283C4×C61#38111yescharacteristic
284Dic61—21no—
285Dic61—21no—
286Dic61#38512yes—
287Dic61#38612yes—
288C23×M16—611no—
289C2×#407—611no—
290#2280—611no—
291C2×#344—614no—
292C2×C61—22no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61C22⋊#9511yescharacteristic
296C2×C61C2×#34411yescharacteristic
297C2×C61C2×#34411yescharacteristic
298D61—22no—
299D61—22no—
300D61—22no—
301D61—41no—
302C24×C4—611no—
303C22×M16—612no—
304C22×M16—612no—
305C22×M16—611no—
306C22×M16—612no—
307C22×M16—1221no—
308C22×M16—1228no—
309C22×M16—1221no—
310C22×M16—1221no—
311C22×M16#692311yescharacteristic, complemented
312C23×C8—1221no—
313C2×#117—612no—
314#407—1221no—
315#407—1221no—
316#407—1221no—
317#407—1221no—
318C2×C22⋊C8—612no—
319C2×C22⋊C8—612no—
320#381—612no—
321#380—1221no—
322#380—1221no—
323#379—612no—
324C4×C22⋊C4—1221no—
325C4×C22⋊C4—1221no—
326#344—1228no—
327C61#1073711yescharacteristic, complemented, Sylow
328C25—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—1224no—
343C23×C4—1224no—
344C23×C4C2×#692311yescharacteristic
345C2×M16—21no—
346C2×M16—24no—
347C2×M16—612no—
348C2×M16—618no—
349C2×M16—1222no—
350C2×M16—1222no—
351C2×M16—1228no—
352C2×M16—1224no—
353C2×M16—1224no—
354C2×M16—1228no—
355C2×M16—1228no—
356C2×M16—1224no—
357C2×M16—1221no—
358C2×M16—1224no—
359C2×M16—1224no—
360C2×M16—1224no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16C2×#692312yes—
364C22×C8—21no—
365C22×C8—614no—
366C22×C8—1222no—
367C22×C8—1222no—
368C22×C8—1228no—
369C22×C8—1221no—
370#117—1224no—
371#117—1224no—
372C2×C4⋊C4—1222no—
373C2×C22⋊C4—614no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C42—1222no—
377C22⋊C8—1222no—
378C22⋊C8—1222no—
379C22⋊C8—1222no—
380C22⋊C8—1222no—
381#95—1222no—
382#95—1222no—
383C24—611no—
384C24—611no—
385C24—611no—
386C24—611no—
387C24—611no—
388C24—611no—
389C24—1224no—
390C24—2442no—
391C24—2442no—
392C24C4×#692311yescharacteristic
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—614no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—614no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1224no—
424C22×C4—1224no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1224no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4C4×#692311yescharacteristic
438C22×C4C4⋊#692311yescharacteristic
439C22×C4C4⋊#692311yescharacteristic
440C22×C4C22⋊#692311yescharacteristic
441C22×C4C22⋊#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443M16—24no—
444M16—22no—
445M16—614no—
446M16—1228no—
447M16—1228no—
448M16—1224no—
449M16—1222no—
450M16—2444no—
451M16—2442no—
452M16—2442no—
453M16C22⋊#692314yescomplemented
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—1228no—
457C2×C8—1228no—
458C2×C8—1224no—
459C2×C8—1222no—
460C2×C8—1224no—
461C2×C8—1228no—
462C4⋊C4—2442no—
463C4⋊C4—2442no—
464C22⋊C4—1228no—
465C22⋊C4—2444no—
466C22⋊C4—2444no—
467C42—2444no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—611no—
472C23—611no—
473C23—611no—
474C23—614no—
475C23—611no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—24416no—
494C23(C2×#6923)⋊C411yescharacteristic
495C23(C2×#6923)⋊C411yescharacteristic
496C23C2×C4×#692311yescharacteristic
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—614no—
507C2×C4—614no—
508C2×C4—614no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—611no—
514C2×C4—611no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—2444no—
535C2×C4—2444no—
536C2×C4—2444no—
537C2×C4(C2×#6923)⋊C411yescharacteristic
538C2×C4(C2×#6923)⋊C411yescharacteristic
539C2×C4C2×C22⋊#692312yes—
540C2×C4C2×C22⋊#692311yescharacteristic
541C2×C4C2×C4⋊#692311yescharacteristic
542C2×C4#1451985612yes—
543C8—24no—
544C8—1224no—
545C8—2444no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—44no—
554C22—612no—
555C22—612no—
556C22—612no—
557C22—1228no—
558C22—1221no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—2444no—
568C22—2444no—
569C22—2444no—
570C22—2444no—
571C22—2444no—
572C22#1684064512yescomplemented
573C22C2×(C2×#6923)⋊C411yescharacteristic
574C22C4×C22⋊#692312yes—
575C4—21no—
576C4—21no—
577C4—612no—
578C4—612no—
579C4—1221no—
580C4—1221no—
581C4—4884no—
582C4#1684070311yescharacteristic
583C4#1684070412yes—
584C4#1684070511yescharacteristic
585C2—22no—
586C2—22no—
587C2—41no—
588C2—1222no—
589C2—1222no—
590C2—1222no—
591C2—2441no—
592C2#2146099311yescharacteristic
593C2C2×#1684064511yescharacteristic
594C2#2146153111yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61C2×C4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#10737
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16840645C2semidirect
11. C2×#14519834C4semidirect
12. C2×C22⋊#92420C4semidirect
21. C22×#11758194C4semidirect
26. C2×M16×D61C4semidirect
29. C22×M16×C61C4semidirect
158. M16×C61C22⋊C4semidirect
163. #11758194C22⋊C4semidirect
257. C22×C61#344semidirect
275. C2×D61#344semidirect
311. C22×M16#6923semidirect
327. C61#10737semidirect
453. M16C22⋊#6923semidirect
572. C22#16840645semidirect

Maximal subgroups

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

subgroupindex
C2×#168407332
#214622632
C2×#168406452
C22×M16×D612
#1073761