Finite Group Databasebeta

← #30602410

Subgroups — #30602410

Order 15616.

1550 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519856C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3C2×#16840634C214yesmaximal, complemented
4#407×D61C211yesmaximal, characteristic
5C2×#14519856—28no—
6C2×#14519856—21no—
7C2×#14519856—21no—
8C2×#14519856C412yescomplemented
9C2×#14519856C2212yes—
10C22×C4⋊#6923C412yescomplemented
11C22×C22⋊#6923—21no—
12C22×C22⋊#6923—21no—
13C22×C4×#6923—21no—
14C22×C4×#6923C2212yes—
15C4×C22⋊#6923—21no—
16C4×C22⋊#6923—21no—
17C4×C22⋊#6923C412yescomplemented
18C4×C22⋊#6923C412yescomplemented
19C2×(C2×#6923)⋊C4—21no—
20#16840634—28no—
21C23×C4×D61C2211yescharacteristic
22C2×M16×D61C2212yes—
23#16839726C411yescharacteristic, complemented
24#16839725C412yescomplemented
25#16839724—21no—
26#16839724—21no—
27C22⋊C8×D61—21no—
28C22⋊C8×D61—21no—
29C61×#407C411yescharacteristic, complemented
30C23×#6923—22no—
31C23×#6923—22no—
32#14519856—28no—
33#14519856—28no—
34#14519856—24no—
35#14519856—28no—
36#14519856D418yes—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—28no—
39C2×C4⋊#6923C2×C414yes—
40C2×C22⋊#6923—28no—
41C2×C22⋊#6923—28no—
42C2×C22⋊#6923—22no—
43C2×C22⋊#6923—22no—
44C2×C22⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923—28no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—24no—
50C2×C4×#6923—24no—
51C2×C4×#6923—22no—
52C2×C4×#6923C2×C414yescomplemented
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55C24×D61D411yescharacteristic
56C23×Dic61Q811yescharacteristic
57C22×C4×D61—24no—
58C22×C4×D61—24no—
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61C2×C411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61D411yescharacteristic
69C22×C4×D61Q811yescharacteristic
70C22×C4×D61C2311yescharacteristic
71C2×#11758194C2×C412yescomplemented
72C2×#11758189—22no—
73M16×D61—24no—
74M16×D61—24no—
75C2×C8×D61—22no—
76C22⋊#92418—21no—
77C22⋊#92418—21no—
78#14519689—22no—
79#14519689—22no—
80C23×C4×C61C2×C411yescharacteristic
81C2×M16×C61C2×C412yescomplemented
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C24×C61C4⋊C411yescharacteristic
85C23×D61—24no—
86C23×D61—22no—
87C23×D61—22no—
88C23×D61—42no—
89C23×D61C22⋊C411yescharacteristic
90C23×D61C22⋊C411yescharacteristic
91C23×D61C22⋊C411yescharacteristic
92C23×D61C22⋊C411yescharacteristic
93C23×D61C4⋊C411yescharacteristic
94C23×D61C2×D411yescharacteristic
95C22×#6923—28no—
96C22×#6923—28no—
97C22×#6923—24no—
98C22×#6923—24no—
99C22×#6923—44no—
100C22×#6923—44no—
101C22×#6923—44no—
102C22×#6923—42no—
103C22×#6923—42no—
104C22×C4×C61—24no—
105C22×C4×C61C4211yescharacteristic
106C22×C4×C61C4211yescharacteristic
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C22⋊C411yescharacteristic
109C22×C4×C61C4⋊C411yescharacteristic
110C22×C4×C61C22×C411yescharacteristic
111C22×Dic61—24no—
112C22×Dic61C4⋊C411yescharacteristic
113C22×Dic61C4⋊C411yescharacteristic
114C22×Dic61C4⋊C411yescharacteristic
115C22×Dic61C4⋊C411yescharacteristic
116C22×Dic61C4⋊C411yescharacteristic
117C22×Dic61C2×Q811yescharacteristic
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—24no—
121C2×C4×D61—24no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—24no—
127C2×C4×D61—24no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61C4211yescharacteristic
135C2×C4×D61C4211yescharacteristic
136C2×C4×D61C22⋊C411yescharacteristic
137C2×C4×D61C22⋊C411yescharacteristic
138C2×C4×D61C22⋊C411yescharacteristic
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C4⋊C411yescharacteristic
143C2×C4×D61C4⋊C411yescharacteristic
144C2×C4×D61C4⋊C411yescharacteristic
145C2×C4×D61C4⋊C411yescharacteristic
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D414yes—
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61#4114yes—
152C22⋊#6923—28no—
153C22⋊#6923—28no—
154C22⋊#6923—28no—
155C22⋊#6923—28no—
156C4⋊#6923—28no—
157C4⋊#6923—24no—
158C4⋊#6923C22⋊C418yescomplemented
159C4×#6923—28no—
160C4×#6923—28no—
161C4×#6923—28no—
162C4×#6923—44no—
163M16×C61—24no—
164C2×C8×C61—22no—
165#11758194—24no—
166C2×#92418—22no—
167#11758189—44no—
168C8×D61—42no—
169C8×D61—42no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—41no—
173C23×C61#9511yescharacteristic
174C23×C61#9511yescharacteristic
175C23×C61C2×C4⋊C411yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—48no—
195C22×D61—48no—
196C22×D61#9511yescharacteristic
197C22×D61#9511yescharacteristic
198C22×D61#10414yes—
199C22×D61C2×C22⋊C411yescharacteristic
200C22×D61C2×C22⋊C411yescharacteristic
201C22×D61#12012yes—
202C22×D61#12312yes—
203C2×#6923—28no—
204C2×#6923—28no—
205C2×#6923—44no—
206C2×#6923—48no—
207C2×#6923—48no—
208C2×#6923—416no—
209C2×#6923—44no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61#9511yescharacteristic
218C2×C4×C61#9511yescharacteristic
219C2×C4×C61C2×C4211yescharacteristic
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C4×D414yes—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61#9511yescharacteristic
228C2×Dic61#9511yescharacteristic
229C2×Dic61C2×C4⋊C411yescharacteristic
230C2×Dic61C2×C4⋊C411yescharacteristic
231C2×Dic61C22⋊Q814yes—
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61C2×C22⋊C412yes—
247C4×D61#11712yes—
248C4×D61C4×D414yes—
249C4×D61#12012yes—
250C4×D61#12114yes—
251C4×D61#12312yes—
252C8×C61—42no—
253#92418—42no—
254#10733—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—44no—
263C22×C61#33912yescomplemented
264C22×C61C2×#9511yescharacteristic
265C22×C61#38512yes—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—24no—
271C2×D61—24no—
272C2×D61—24no—
273C2×D61—24no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—44no—
278C2×D61—44no—
279C2×D61—44no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61#33912yescomplemented
283C2×D61#37911yescharacteristic
284C2×D61#38111yescharacteristic
285C2×D61#38612yes—
286C2×D61C2×#10412yes—
287C2×D61#42112yes—
288#6923—48no—
289#6923—48no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61C4×C22⋊C412yes—
293C4×C61#38612yes—
294Dic61—21no—
295Dic61—21no—
296Dic61#38012yes—
297Dic61#38512yes—
298C22×#117—611no—
299C2×#407—611no—
300C2×C4×C22⋊C4—611no—
301C2×#339—614no—
302C2×C61—22no—
303C2×C61—22no—
304C2×C61—41no—
305C2×C61C22⋊#9511yescharacteristic
306C2×C61C2×#33911yescharacteristic
307C2×C61#173311yescharacteristic
308D61—22no—
309D61—22no—
310D61—41no—
311D61#178912yes—
312D61#182612yes—
313C24×C4—611no—
314C22×M16—612no—
315C2×#117—612no—
316C2×#117—612no—
317C2×#117—1228no—
318C2×#117—1221no—
319C2×#117—1221no—
320C22×C4⋊C4—612no—
321C22×C22⋊C4—1221no—
322C22×C22⋊C4—1221no—
323C22×C42—612no—
324C22×C42—1221no—
325#407—611no—
326#407—612no—
327#407—1221no—
328#407—1221no—
329#407#692311yescharacteristic, complemented
330C2×C22⋊C8—1221no—
331C2×C22⋊C8—1221no—
332C4×C22⋊C4—612no—
333C4×C22⋊C4—612no—
334C4×C22⋊C4—1221no—
335C4×C22⋊C4—1221no—
336C2×#95—1221no—
337#339—1228no—
338C61#1073311yescharacteristic, complemented, Sylow
339C25—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—611no—
352C23×C4—611no—
353C23×C4—1224no—
354C23×C4—1224no—
355C23×C4—1222no—
356C23×C4—1222no—
357C23×C4C2×#692311yescharacteristic
358C2×M16—612no—
359C2×M16—1222no—
360C2×M16—1224no—
361C2×M16—1224no—
362C2×M16C2×#692312yescomplemented
363C22×C8—1222no—
364#117—618no—
365#117—1228no—
366#117—1224no—
367#117—1228no—
368#117—1228no—
369C2×C4⋊C4—614no—
370C2×C4⋊C4—1228no—
371C2×C4⋊C4—1222no—
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1228no—
377C2×C22⋊C4—1228no—
378C2×C42—614no—
379C2×C42—1222no—
380C2×C42—1222no—
381C2×C42—1228no—
382C2×C42—1224no—
383C2×C42—1224no—
384C2×C42—1222no—
385C22⋊C8—21no—
386C22⋊C8—21no—
387C22⋊C8—1222no—
388C22⋊C8—1222no—
389C22⋊C8—1221no—
390C22⋊C8—1221no—
391#95—1222no—
392#95—1222no—
393C24—611no—
394C24—611no—
395C24—611no—
396C24—611no—
397C24—611no—
398C24—611no—
399C24—1222no—
400C24—1224no—
401C24—1222no—
402C24—2442no—
403C24C4⋊#692311yescharacteristic
404C22×C4—24no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—614no—
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—1224no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1228no—
449C22×C4—1228no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4C4×#692311yescharacteristic
456C22×C4C4×#692311yescharacteristic
457C22×C4C4⋊#692311yescharacteristic
458C22×C4C22⋊#692311yescharacteristic
459C22×C4C22⋊#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461M16—24no—
462M16—1224no—
463M16—2444no—
464C2×C8—22no—
465C2×C8—1222no—
466C2×C8—2442no—
467C2×C8—2442no—
468C4⋊C4—618no—
469C4⋊C4—1224no—
470C4⋊C4—1228no—
471C22⋊C4—1228no—
472C22⋊C4—1228no—
473C22⋊C4—1228no—
474C22⋊C4—1228no—
475C42—1228no—
476C42—1228no—
477C42—1228no—
478C42—2444no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—611no—
483C23—612no—
484C23—612no—
485C23—614no—
486C23—611no—
487C23—611no—
488C23—611no—
489C23—1221no—
490C23—1221no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—1221no—
495C23—1222no—
496C23—1222no—
497C23—1221no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2448no—
508C23—2448no—
509C23(C2×#6923)⋊C411yescharacteristic
510C23(C2×#6923)⋊C411yescharacteristic
511C23C2×C4⋊#692311yescharacteristic
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—614no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—612no—
521C2×C4—614no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—614no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1228no—
548C2×C4—1228no—
549C2×C4—2444no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—24416no—
553C2×C4—2448no—
554C2×C4—2448no—
555C2×C4—2444no—
556C2×C4(C2×#6923)⋊C411yescharacteristic
557C2×C4(C2×#6923)⋊C411yescharacteristic
558C2×C4C2×C4×#692311yescharacteristic
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4D4×#692314yes—
561C8—42no—
562C8—2442no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—44no—
571C22—611no—
572C22—611no—
573C22—612no—
574C22—612no—
575C22—612no—
576C22—612no—
577C22—1224no—
578C22—1224no—
579C22—1224no—
580C22—1224no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22—2444no—
589C22—2444no—
590C22—2442no—
591C22—2442no—
592C22—2444no—
593C22#1684063412yescomplemented
594C22C2×(C2×#6923)⋊C411yescharacteristic
595C22#1684070912yes—
596C4—21no—
597C4—21no—
598C4—612no—
599C4—612no—
600C4—1221no—
601C4—1221no—
602C4—2448no—
603C4—2448no—
604C4C22⋊C4×#692312yes—
605C4#1684071012yes—
606C2—22no—
607C2—22no—
608C2—41no—
609C2—612no—
610C2—612no—
611C2—1222no—
612C2—1222no—
613C2—2441no—
614C2#2146099211yescharacteristic
615C2C2×#1684063411yescharacteristic
616C2#2146150111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#407C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#407#6923
hypocenterC61#10733
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. C2×#16840634C2semidirect
8. C2×#14519856C4semidirect
10. C22×C4⋊#6923C4semidirect
17. C4×C22⋊#6923C4semidirect
18. C4×C22⋊#6923C4semidirect
23. #16839726C4semidirect
24. #16839725C4semidirect
29. C61×#407C4semidirect
52. C2×C4×#6923C2×C4semidirect
71. C2×#11758194C2×C4semidirect
81. C2×M16×C61C2×C4semidirect
158. C4⋊#6923C22⋊C4semidirect
263. C22×C61#339semidirect
282. C2×D61#339semidirect
329. #407#6923semidirect
338. C61#10733semidirect
362. C2×M16C2×#6923semidirect
593. C22#16840634semidirect

Maximal subgroups

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

subgroupindex
C22×#145198562
C2×C4×C22⋊#69232
C2×#168406342
#407×D612
#1073361