Finite Group Databasebeta

← #30615202

Subgroups — #30615202

Order 15616.

1382 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
1#21462501C212yesmaximal, complemented
2#21461720C212yesmaximal, complemented
3#21461684C212yesmaximal, complemented
4#21461683C212yesmaximal, complemented
5#21461652C212yesmaximal
6#21461569C212yesmaximal, complemented
7C4×#14519837C212yesmaximal, complemented
8C2×#16840234C211yesmaximal, characteristic
9#16840787—28no—
10#16840765C2214yescomplemented
11C2×#14519837C2214yes—
12C2×#14519837C2214yes—
13#16840734C2214yes—
14C4×#11758217C2214yescomplemented
15#16840642C2214yescomplemented
16#16840642C2214yescomplemented
17C22×D4×D61C2211yescharacteristic
18#16840234—28no—
19C2×C4×D244C411yescharacteristic
20C2×D4×Dic61C411yescharacteristic
21C2×#14519739C411yescharacteristic
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519737C2211yescharacteristic
25C2×#14519737C2211yescharacteristic
26C2×#14519735C2211yescharacteristic
27C2×#14519734C2211yescharacteristic
28C22×#11758198C411yescharacteristic
29C22×#11758198C411yescharacteristic
30C2×C22⋊C4×D61C411yescharacteristic
31C2×C22⋊C4×D61C411yescharacteristic
32C2×C4×D4×C61C411yescharacteristic
33C2×#11758217C2318yes—
34#14519844—24no—
35C4⋊#92420—24no—
36#14519837—28no—
37#14519837—28no—
38#14519834—28no—
39#14519832—24no—
40C4×#92420—24no—
41C24×D61C2×C411yescharacteristic
42C24×D61C2×C411yescharacteristic
43C22×D244C2×C411yescharacteristic
44C22×C22⋊D61C2311yescharacteristic
45C22×C22⋊D61C2311yescharacteristic
46C2×D4×D61—24no—
47C2×D4×D61—24no—
48C2×D4×D61—24no—
49C23×Dic61C2×C411yescharacteristic
50C23×Dic61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C4×D244—24no—
57D4×Dic61—24no—
58#14519739—24no—
59C4×C22⋊D61—24no—
60C4×C22⋊D61—24no—
61#14519737—24no—
62#14519737—24no—
63#14519735D418yes—
64#14519734—24no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196C2311yescharacteristic
67C2×#11758196C2311yescharacteristic
68C2×C22⋊Dic61C2×C411yescharacteristic
69C2×C22⋊Dic61C2×C411yescharacteristic
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×#11758198C2×C411yescharacteristic
73C2×#11758198C2×C411yescharacteristic
74C2×#11758198C2×C411yescharacteristic
75C2×#11758198C2×C411yescharacteristic
76C2×#11758198C2×C411yescharacteristic
77C2×#11758198C2×C411yescharacteristic
78C2×#11758198C2×C411yescharacteristic
79C2×#11758198C2×C411yescharacteristic
80C22⋊C4×D61—24no—
81C22⋊C4×D61—24no—
82C2×C4×Dic61C2×C411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C22×D4×C61C2×C411yescharacteristic
86C23×C4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88C4×D4×C61—24no—
89C2×C4⋊C4×C61C2×C411yescharacteristic
90C2×C22⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61C2×C411yescharacteristic
92C2×C42×C61C2×C411yescharacteristic
93C24×C61C22×C411yescharacteristic
94C24×C61C22×C411yescharacteristic
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—81no—
102C23×D61—81no—
103C23×D61—81no—
104C23×D61—81no—
105C23×D61C22×C411yescharacteristic
106C23×D61C22×C411yescharacteristic
107C23×D61C22×C411yescharacteristic
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—81no—
114C22×C4×C61—81no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C2×C22⋊D61—22no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61#4114yes—
125C2×C22⋊D61#4114yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—81no—
129C22×Dic61—81no—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C2411yescharacteristic
135D4×D61—232no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—81no—
146C2×C4×D61—81no—
147C2×C4×D61—81no—
148C2×C4×D61—81no—
149C2×C4×D61C2×D414yes—
150#11758217—28no—
151#11758217—28no—
152C2×#92420—28no—
153C4⋊C4×C61C22⋊C414yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—22no—
157C22⋊Dic61—22no—
158C22⋊Dic61—22no—
159#11758198—22no—
160#11758198—22no—
161#11758198—22no—
162#11758198—22no—
163#11758198—22no—
164#11758198—22no—
165#11758198C22⋊C414yes—
166#11758198C22⋊C414yes—
167C4⋊Dic61C22⋊C414yes—
168#11758196—22no—
169#11758196—22no—
170C4×Dic61—22no—
171C4×Dic61C2×D414yes—
172C4×Dic61C2×D414yes—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61#11712yes—
182C23×C61#11712yes—
183C23×C61C23×C411yescharacteristic
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—28no—
191C22×D61—28no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—82no—
199C22×D61—82no—
200C22×D61—82no—
201C22×D61—82no—
202C22×D61—84no—
203C22×D61—84no—
204C22×D61—84no—
205C22×D61—84no—
206C22×D61C2×C22⋊C412yes—
207C22×D61#11712yes—
208C22×D61#11712yes—
209D4×C61—28no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—82no—
219C2×C4×C61—82no—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C22⋊D61—28no—
224C22⋊D61—28no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61—82no—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C22×D412yes—
236C2×Dic61C2×#4112yes—
237C2×Dic61C2×#4112yes—
238D244—28no—
239C4×D61—28no—
240C4×D61—82no—
241C4×D61—82no—
242C4×D61—82no—
243C4×D61—82no—
244#92420—48no—
245#20729—611nomaximal, Sylow
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61C22×C22⋊C411yescharacteristic
259C22×C61C2×#11711yescharacteristic
260C22×C61C2×#11711yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—88no—
277C2×D61—88no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283Dic61—22no—
284Dic61—22no—
285Dic61—41no—
286Dic61—41no—
287Dic61#52514yes—
288C2×#518—611no—
289#2921—612no—
290#1822—612no—
291#1800—612no—
292#1800—612no—
293#1783—612no—
294C4×#100—612no—
295C4×#100—612no—
296C2×C61—21no—
297C2×C61—21no—
298C2×C61—42no—
299C2×C61—42no—
300C2×C61#228011yescharacteristic
301C2×C61#288811yescharacteristic
302C2×C61#288811yescharacteristic
303D61—42no—
304D61—42no—
305D61—42no—
306C23×D4—611no—
307#518—1228no—
308C2×C4×D4—611no—
309C2×C4×D4—611no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×C4×D4—611no—
314C2×C4×D4—611no—
315C2×C4×D4#692311yescharacteristic
316C2×#117—611no—
317C2×#117—611no—
318C22×C22⋊C4—611no—
319C22×C22⋊C4—611no—
320C22×C22⋊C4—611no—
321C22×C22⋊C4—611no—
322#433—1228no—
323#423—614no—
324C2×#100—614no—
325C2×#100—614no—
326#407—614no—
327C4×M16—614no—
328#343—614no—
329#343—614no—
330C61#2072911yescharacteristic, complemented, Sylow
331C25—611no—
332C25—611no—
333C22×D4—611no—
334C22×D4—611no—
335C22×D4—611no—
336C22×D4—1224no—
337C22×D4—1224no—
338C22×D4—1224no—
339C22×D4C2×#692311yescharacteristic
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4C2×#692311yescharacteristic
348C23×C4C2×#692311yescharacteristic
349C2×M16—618no—
350C4×D4—24no—
351C4×D4—1224no—
352C4×D4—1224no—
353C4×D4—1224no—
354C4×D4—1224no—
355C4×D4—1224no—
356C4×D4—1224no—
357C4×D4—1224no—
358#117—618no—
359#117—1224no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4C2×#692311yescharacteristic
364C2×C22⋊C4—611no—
365C2×C22⋊C4—611no—
366C2×C22⋊C4—611no—
367C2×C22⋊C4—611no—
368C2×C22⋊C4—611no—
369C2×C22⋊C4—611no—
370C2×C22⋊C4—611no—
371C2×C22⋊C4—611no—
372C2×C22⋊C4—611no—
373C2×C22⋊C4—611no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4C2×#692311yescharacteristic
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42C2×#692311yescharacteristic
384C4⋊C8—1224no—
385C4⋊C8—1224no—
386#100—1228no—
387#100—1228no—
388C22⋊C8—1228no—
389#97—1224no—
390C4×C8—1224no—
391C24—611no—
392C24—611no—
393C24—611no—
394C24—1222no—
395C24—1222no—
396C24—1222no—
397C24—1222no—
398C24—1222no—
399C24—1222no—
400C24—4881no—
401C24—4881no—
402C24—4881no—
403C24—4881no—
404C24C22×#692311yescharacteristic
405C24C22×#692311yescharacteristic
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—614no—
410C2×D4—614no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—12232no—
418C2×D4—1222no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—614no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4—4881no—
441C22×C4—4881no—
442C22×C4—4881no—
443C22×C4C22×#692311yescharacteristic
444C22×C4C22×#692311yescharacteristic
445C22×C4C22×#692311yescharacteristic
446C22×C4C22×#692311yescharacteristic
447C22×C4C22×#692311yescharacteristic
448M16—1228no—
449M16—1228no—
450C2×C8—1228no—
451C4⋊C4—614no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4C22⋊#692314yes—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—614no—
458C22⋊C4—614no—
459C22⋊C4—1222no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—1222no—
464C22⋊C4—1222no—
465C22⋊C4—1222no—
466C22⋊C4—1222no—
467C42—22no—
468C42—614no—
469C42—614no—
470C42—1222no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—612no—
480C23—612no—
481C23—612no—
482C23—1228no—
483C23—1228no—
484C23—1221no—
485C23—1221no—
486C23—1221no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—4882no—
491C23—4882no—
492C23—4882no—
493C23—4882no—
494C23—4884no—
495C23—4884no—
496C23—4884no—
497C23—4884no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2441no—
502C23—2441no—
503C23—2441no—
504C23#1451985612yes—
505C23#1451985612yes—
506C23C23×#692311yescharacteristic
507D4—28no—
508D4—1228no—
509D4—1228no—
510D4—1228no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—1228no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4882no—
530C2×C4—4882no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4C2×C22⋊#692312yes—
540C2×C4C2×C22⋊#692312yes—
541C2×C4C2×C22⋊#692312yes—
542C8—2448no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—4888no—
561C22—4888no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2441no—
569C22—2441no—
570C22—2441no—
571C22—2441no—
572C22C22×C22⋊#692311yescharacteristic
573C22C2×#1451985611yescharacteristic
574C22C2×#1451985611yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—614no—
581C4—1222no—
582C4—1222no—
583C4—2441no—
584C4—2441no—
585C2—21no—
586C2—21no—
587C2—42no—
588C2—42no—
589C2—2442no—
590C2—2442no—
591C2—2442no—
592C2#2146226311yescharacteristic
593C2#2146242911yescharacteristic
594C2#2146243011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC22×Dic61C24
hypercenterC2×C4×D4#6923
hypocenterC61#20729
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21462501C2semidirect
2. #21461720C2semidirect
3. #21461684C2semidirect
4. #21461683C2semidirect
6. #21461569C2semidirect
7. C4×#14519837C2semidirect
10. #16840765C22semidirect
14. C4×#11758217C22semidirect
15. #16840642C22semidirect
16. #16840642C22semidirect
330. C61#20729semidirect

Maximal subgroups

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

subgroupindex
#214625012
#214617202
#214616842
#214616832
#214616522
#214615692
C4×#145198372
C2×#168402342
#2072961