Finite Group Databasebeta

← #30602408

Subgroups — #30602408

Order 15616.

1510 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3C2×#16840634C214yesmaximal, complemented
4C22×M16×D61C211yesmaximal, characteristic
5C2×#14519856C2212yes—
6C22×C22⋊#6923—21no—
7C22×C4×#6923C2212yes—
8#16840705—21no—
9#16840704C412yescomplemented
10#16840704C412yescomplemented
11#16840703—21no—
12C4×C22⋊#6923—21no—
13C4×C22⋊#6923—21no—
14C4×C22⋊#6923—21no—
15C4×C22⋊#6923—21no—
16C4×C22⋊#6923C412yescomplemented
17C4×C22⋊#6923C412yescomplemented
18C2×(C2×#6923)⋊C4—21no—
19#16840634—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
30C23×#6923—22no—
31#14519856—24no—
32#14519856—24no—
33C2×C4⋊#6923C2×C414yes—
34C2×C22⋊#6923—28no—
35C2×C22⋊#6923—22no—
36C2×C22⋊#6923—22no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923—22no—
39C2×C22⋊#6923—22no—
40C2×C4×#6923—24no—
41C2×C4×#6923—24no—
42C2×C4×#6923—22no—
43C2×C4×#6923—22no—
44C2×C4×#6923C2×C414yes—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—22no—
49C24×D61D411yescharacteristic
50C23×Dic61Q811yescharacteristic
51C22×C4×D61—24no—
52C22×C4×D61—24no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61Q811yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758194—21no—
66C2×#11758194—24no—
67C2×#11758194C2×C412yes—
68C2×#11758189—28no—
69C2×#11758189—22no—
70C2×#11758189—22no—
71M16×D61—28no—
72M16×D61—28no—
73M16×D61—24no—
74M16×D61—24no—
75M16×D61—24no—
76M16×D61D418yes—
77C22×#92418—21no—
78C2×C8×D61—28no—
79C2×C8×D61—22no—
80C2×C8×D61—22no—
81C23×C4×C61C2×C411yescharacteristic
82C2×M16×C61—21no—
83C2×M16×C61—24no—
84C2×M16×C61C2×C412yes—
85C22×C8×C61—21no—
86C24×C61C4⋊C411yescharacteristic
87C23×D61—24no—
88C23×D61—24no—
89C23×D61—42no—
90C23×D61C22⋊C411yescharacteristic
91C23×D61C22⋊C411yescharacteristic
92C23×D61C22⋊C411yescharacteristic
93C23×D61C22⋊C411yescharacteristic
94C23×D61C4⋊C411yescharacteristic
95C23×D61C2×D411yescharacteristic
96C22×#6923—28no—
97C22×#6923—24no—
98C22×#6923—24no—
99C22×#6923—44no—
100C22×#6923—44no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61C4211yescharacteristic
104C22×C4×C61C4211yescharacteristic
105C22×C4×C61C22⋊C411yescharacteristic
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C4⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×Dic61—24no—
110C22×Dic61C4⋊C411yescharacteristic
111C22×Dic61C4⋊C411yescharacteristic
112C22×Dic61C4⋊C411yescharacteristic
113C22×Dic61C4⋊C411yescharacteristic
114C22×Dic61C4⋊C411yescharacteristic
115C22×Dic61C2×Q811yescharacteristic
116C2×C4×D61—24no—
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—24no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61C4211yescharacteristic
133C2×C4×D61C4211yescharacteristic
134C2×C4×D61C22⋊C411yescharacteristic
135C2×C4×D61C22⋊C411yescharacteristic
136C2×C4×D61C22⋊C411yescharacteristic
137C2×C4×D61C22⋊C411yescharacteristic
138C2×C4×D61C22⋊C411yescharacteristic
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C4⋊C411yescharacteristic
141C2×C4×D61C4⋊C411yescharacteristic
142C2×C4×D61C4⋊C411yescharacteristic
143C2×C4×D61C4⋊C411yescharacteristic
144C2×C4×D61C22×C411yescharacteristic
145C2×C4×D61C22×C411yescharacteristic
146C2×C4×D61C2×D414yes—
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61#4114yes—
150C22⋊#6923—28no—
151C22⋊#6923—28no—
152C22⋊#6923—44no—
153C22⋊#6923—44no—
154C22⋊#6923—44no—
155C4⋊#6923—28no—
156C4×#6923—28no—
157C4×#6923—42no—
158C4×#6923—42no—
159C4×#6923—42no—
160C4×#6923—42no—
161M16×C61—24no—
162M16×C61—22no—
163M16×C61C22⋊C414yescomplemented
164C2×C8×C61—24no—
165C2×C8×C61—22no—
166#11758194—24no—
167#11758194—22no—
168#11758194C22⋊C414yescomplemented
169C2×#92418—24no—
170C2×#92418—22no—
171#11758189—28no—
172#11758189—28no—
173C8×D61—28no—
174C8×D61—28no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61#9511yescharacteristic
179C23×C61#9511yescharacteristic
180C23×C61C2×C4⋊C411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—416no—
198C22×D61#9511yescharacteristic
199C22×D61#9511yescharacteristic
200C22×D61#10414yes—
201C22×D61C2×C22⋊C411yescharacteristic
202C22×D61C2×C22⋊C411yescharacteristic
203C22×D61#12114yes—
204C2×#6923—28no—
205C2×#6923—44no—
206C2×#6923—416no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—44no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61#9511yescharacteristic
219C2×C4×C61#9511yescharacteristic
220C2×C4×C61C2×C4211yescharacteristic
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61#11712yes—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61#9511yescharacteristic
230C2×Dic61#9511yescharacteristic
231C2×Dic61C2×C4⋊C411yescharacteristic
232C2×Dic61C2×C4⋊C411yescharacteristic
233C2×Dic61C22⋊Q814yes—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61C4×D414yes—
249C4×D61C4×D414yes—
250C4×D61#12012yes—
251C4×D61#12012yes—
252C4×D61#12312yes—
253C4×D61#12312yes—
254C8×C61—24no—
255#92418—24no—
256#10732—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—44no—
265C22×C61#33912yescomplemented
266C22×C61C2×#9511yescharacteristic
267C22×C61#38012yes—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—28no—
273C2×D61—28no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—44no—
278C2×D61—44no—
279C2×D61—44no—
280C2×D61—44no—
281C2×D61#33912yescomplemented
282C2×D61#38612yes—
283C2×D61#38612yes—
284C2×D61C2×#10412yes—
285C2×D61#42112yes—
286#6923—48no—
287#6923—84no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61C4×C22⋊C412yes—
291C4×C61#37911yescharacteristic
292C4×C61#38111yescharacteristic
293Dic61—21no—
294Dic61—21no—
295Dic61#38512yes—
296Dic61#38512yes—
297C23×M16—611no—
298#2280—611no—
299C2×C4×C22⋊C4—611no—
300C2×#339—614no—
301C2×C61—22no—
302C2×C61—22no—
303C2×C61—41no—
304C2×C61C22⋊#9511yescharacteristic
305C2×C61C2×#33911yescharacteristic
306C2×C61#173311yescharacteristic
307D61—22no—
308D61—22no—
309D61—41no—
310D61#187114yes—
311C24×C4—611no—
312C22×M16—612no—
313C22×M16—611no—
314C22×M16—612no—
315C22×M16—1221no—
316C22×M16—1228no—
317C22×M16—1221no—
318C22×M16—1221no—
319C22×M16#692311yescharacteristic, complemented
320C23×C8—1221no—
321C2×#117—612no—
322C22×C22⋊C4—1221no—
323C22×C42—612no—
324#381—1221no—
325#380—612no—
326#380—612no—
327#379—1221no—
328C4×C22⋊C4—612no—
329C4×C22⋊C4—612no—
330C4×C22⋊C4—1221no—
331C4×C22⋊C4—1221no—
332C4×C22⋊C4—1221no—
333C4×C22⋊C4—1221no—
334C2×#95—1221no—
335#339—1228no—
336C61#1073211yescharacteristic, complemented, Sylow
337C25—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1224no—
352C23×C4—1224no—
353C23×C4—1222no—
354C23×C4C2×#692311yescharacteristic
355C2×M16—21no—
356C2×M16—24no—
357C2×M16—612no—
358C2×M16—618no—
359C2×M16—1222no—
360C2×M16—1222no—
361C2×M16—1228no—
362C2×M16—1224no—
363C2×M16—1224no—
364C2×M16—1228no—
365C2×M16—1228no—
366C2×M16—1224no—
367C2×M16—1221no—
368C2×M16—1224no—
369C2×M16C2×#692312yes—
370C22×C8—21no—
371C22×C8—1222no—
372C22×C8—1222no—
373C22×C8—1228no—
374C22×C8—1221no—
375#117—1224no—
376#117—1224no—
377C2×C4⋊C4—614no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1228no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—614no—
385C2×C42—1224no—
386C2×C42—1224no—
387C2×C42—1222no—
388C2×C42—1222no—
389#95—1222no—
390#95—1222no—
391#95—1222no—
392#95—1222no—
393C24—611no—
394C24—611no—
395C24—611no—
396C24—611no—
397C24—611no—
398C24—611no—
399C24—1224no—
400C24—1224no—
401C24—2442no—
402C24C4⋊#692311yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—614no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—614no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1228no—
448C22×C4—1224no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4C4×#692311yescharacteristic
452C22×C4C4×#692311yescharacteristic
453C22×C4C4⋊#692311yescharacteristic
454C22×C4C22⋊#692311yescharacteristic
455C22×C4C22⋊#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457M16—24no—
458M16—22no—
459M16—614no—
460M16—1228no—
461M16—1228no—
462M16—1224no—
463M16—1222no—
464M16C22⋊#692314yescomplemented
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—1228no—
468C2×C8—1228no—
469C2×C8—1224no—
470C2×C8—1222no—
471C4⋊C4—1228no—
472C22⋊C4—1228no—
473C22⋊C4—1228no—
474C22⋊C4—2444no—
475C22⋊C4—2444no—
476C22⋊C4—2444no—
477C42—1228no—
478C42—2442no—
479C42—2442no—
480C42—2442no—
481C42—2442no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—611no—
486C23—614no—
487C23—614no—
488C23—611no—
489C23—611no—
490C23—611no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—1222no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—24416no—
508C23(C2×#6923)⋊C411yescharacteristic
509C23(C2×#6923)⋊C411yescharacteristic
510C23C2×C4⋊#692311yescharacteristic
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—614no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—614no—
524C2×C4—614no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1228no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—24416no—
554C2×C4—2444no—
555C2×C4(C2×#6923)⋊C411yescharacteristic
556C2×C4(C2×#6923)⋊C411yescharacteristic
557C2×C4C2×C4×#692311yescharacteristic
558C2×C4C2×C22⋊#692312yes—
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4#1451985612yes—
561C8—24no—
562C8—1224no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—44no—
571C22—612no—
572C22—612no—
573C22—612no—
574C22—612no—
575C22—612no—
576C22—1228no—
577C22—1228no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22—2444no—
586C22—2444no—
587C22—2444no—
588C22—2444no—
589C22#1684063412yescomplemented
590C22C2×(C2×#6923)⋊C411yescharacteristic
591C22#1684070412yes—
592C4—21no—
593C4—21no—
594C4—612no—
595C4—612no—
596C4—1221no—
597C4—1221no—
598C4—4884no—
599C4—2448no—
600C4C4×C22⋊#692312yes—
601C4#1684070311yescharacteristic
602C4#1684070511yescharacteristic
603C2—22no—
604C2—22no—
605C2—41no—
606C2—614no—
607C2—1222no—
608C2—1222no—
609C2—2441no—
610C2#2146099311yescharacteristic
611C2C2×#1684063411yescharacteristic
612C2#2146150111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#10732
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16840634C2semidirect
9. #16840704C4semidirect
10. #16840704C4semidirect
16. C4×C22⋊#6923C4semidirect
17. C4×C22⋊#6923C4semidirect
21. C22×#11758194C4semidirect
26. C2×M16×D61C4semidirect
29. C22×M16×C61C4semidirect
163. M16×C61C22⋊C4semidirect
168. #11758194C22⋊C4semidirect
265. C22×C61#339semidirect
281. C2×D61#339semidirect
319. C22×M16#6923semidirect
336. C61#10732semidirect
464. M16C22⋊#6923semidirect
589. C22#16840634semidirect

Maximal subgroups

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

subgroupindex
#214622632
C2×C4×C22⋊#69232
C2×#168406342
C22×M16×D612
#1073261