Finite Group Databasebeta

← #30615455

Subgroups — #30615455

Order 15616.

1646 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D61⋊M16C211yesmaximal, characteristic
2C2×#16840767C211yesmaximal, characteristic, complemented
3C2×#16840766C211yesmaximal, characteristic, complemented
4C2×#16840733C211yesmaximal, characteristic
5#21461766C214yesmaximal, complemented
6#21461764C214yesmaximal, complemented
7#21461483C211yesmaximal, characteristic, complemented
8#21461483C211yesmaximal, characteristic, complemented
9C4×C22⋊C4×D61C211yesmaximal, characteristic
10C22×#11758213C2211yescharacteristic
11C22×#11758213C2211yescharacteristic
12C22×#11758213C2211yescharacteristic
13C22×#11758213C2211yescharacteristic
14C2×D61⋊M16—24no—
15C2×D61⋊M16—24no—
16C2×D61⋊M16—24no—
17C22×#11758217C2212yes—
18C22×D61⋊C8C2212yes—
19#16840767—24no—
20#16840766—24no—
21C2×#14519844C2214yescomplemented
22C2×C4⋊#92420C2214yescomplemented
23#16840733—24no—
24C2×#14519834C2212yes—
25C2×C22⋊#92420C2212yes—
26#16840628C2214yescomplemented
27#16840628C2214yescomplemented
28C23×C4×D61C2211yescharacteristic
29C2×C22⋊C4×D61C412yes—
30C2×C42×D61C2211yescharacteristic
31C2×C42×D61C2211yescharacteristic
32C22⋊C4×Dic61C2212yes—
33C4×#11758198C411yescharacteristic
34C4×#11758198C411yescharacteristic
35C4×C22⋊Dic61C411yescharacteristic
36#16839609C2212yes—
37#95×D61C412yes—
38C4×C22⋊C4×C61C411yescharacteristic
39C2×#11758213—22no—
40C2×#11758213—22no—
41C2×#11758213—22no—
42C2×#11758213—22no—
43C2×#11758213—22no—
44C2×#11758213—22no—
45C2×#11758213—22no—
46C2×#11758213—22no—
47C2×#11758213—22no—
48C2×#11758213—22no—
49C2×#11758213D414yes—
50C2×#11758213D414yes—
51D61⋊M16—232no—
52C2×#11758217—24no—
53C2×#11758217—24no—
54C2×#11758217—24no—
55C2×D61⋊C8—24no—
56C2×D61⋊C8—24no—
57C2×D61⋊C8—24no—
58C22×#92420C2312yes—
59C22×#92420C2312yes—
60C22×#92420C2312yes—
61C22×#92420C2312yes—
62#14519844—28no—
63C4⋊#92420—28no—
64#14519834—24no—
65C22⋊#92420—24no—
66C24×D61C2×C411yescharacteristic
67C23×Dic61C2311yescharacteristic
68C22×C4×D61—22no—
69C22×C4×D61—22no—
70C22×C4×D61—22no—
71C22×C4×D61—22no—
72C22×C4×D61C2×C412yes—
73C22×C4×D61C2×C412yes—
74C22×C4×D61C2×C411yescharacteristic
75C22×C4×D61C2×C411yescharacteristic
76C22×C4×D61D414yes—
77C22×C4×D61Q814yes—
78C22×C4×D61C2311yescharacteristic
79C22×C4×D61C2311yescharacteristic
80C2×C22⋊Dic61C2×C412yes—
81C2×#11758198C2×C412yes—
82C2×#11758198C2×C412yes—
83C22⋊C4×D61—28no—
84C2×C4×Dic61C2×C411yescharacteristic
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61C2312yes—
87C2×C4×Dic61C2312yes—
88C42×D61—22no—
89C42×D61—22no—
90C42×D61—22no—
91C42×D61—22no—
92#14519684C2×C412yes—
93#14519684C2×C412yes—
94#14519684C2×C412yes—
95C23×C4×C61C2×C411yescharacteristic
96C2×C22⋊C4×C61C2×C412yes—
97C2×C42×C61C2×C411yescharacteristic
98C2×C42×C61C2×C411yescharacteristic
99#95×C61C2×C412yes—
100C24×C61C22×C411yescharacteristic
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—24no—
104C23×D61—42no—
105C23×D61C4⋊C414yes—
106C23×D61C22×C411yescharacteristic
107C23×D61C22×C411yescharacteristic
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61C4⋊C414yes—
111C22×C4×C61C22×C412yes—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61C22×C412yes—
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C412yes—
122C22×Dic61C2×D412yes—
123C22×Dic61C2×Q812yes—
124C22×Dic61C2411yescharacteristic
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—28no—
134C2×C4×D61—28no—
135C2×C4×D61—28no—
136C2×C4×D61—28no—
137C2×C4×D61—24no—
138C2×C4×D61—24no—
139C2×C4×D61—24no—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61C4⋊C414yes—
144C2×C4×D61C4⋊C414yes—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×Q812yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61#4112yes—
152C2×C4×D61#4112yes—
153#11758217—216no—
154C2×#92420—22no—
155C2×#92420—22no—
156C2×#92420—22no—
157C2×#92420—22no—
158C2×#92420—22no—
159C2×#92420—22no—
160C2×#92420—22no—
161C2×#92420—22no—
162C2×#92420—22no—
163C2×#92420—22no—
164C2×#92420C2×D414yes—
165C2×#92420C2×D414yes—
166#11758213—28no—
167#11758213—28no—
168#11758213—44no—
169#11758213—44no—
170D61⋊C8—216no—
171C22⋊C4×C61—24no—
172C42×C61—22no—
173C42×C61—22no—
174C22⋊Dic61—24no—
175#11758198—24no—
176#11758198M1618yescomplemented
177C4×Dic61—22no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61C2×C4⋊C412yes—
188C23×C61C23×C411yescharacteristic
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—44no—
197C22×D61—48no—
198C22×D61—48no—
199C22×D61C2×C4⋊C412yes—
200C22×D61C2×C4⋊C412yes—
201C22×D61C4×D412yes—
202C22×D61C4×D412yes—
203C22×D61C2×M1614yes—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—28no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61C2×C4⋊C412yes—
213C2×C4×C61C2×C4⋊C412yes—
214C2×C4×C61C4×D412yes—
215C2×C4×C61C4×D412yes—
216C2×C4×C61C2×M1614yescomplemented
217C2×Dic61—22no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61C2×C4⋊C412yes—
229C2×Dic61C2×M1614yescomplemented
230C2×Dic61C22×D411yescharacteristic
231C2×Dic61C22×D411yescharacteristic
232C2×Dic61C22×D411yescharacteristic
233C2×Dic61C22×Q811yescharacteristic
234C2×Dic61C2×#4111yescharacteristic
235C2×Dic61C2×#4111yescharacteristic
236C4×D61—28no—
237C4×D61—28no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—48no—
243C4×D61—48no—
244C4×D61#12014yes—
245C4×D61C22⋊Q814yes—
246C4×D61C22⋊Q814yes—
247C4×D61#12314yes—
248#92420—24no—
249#92420—24no—
250#92420—42no—
251#92420—42no—
252#20837—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61C22×C4⋊C411yescharacteristic
259C22×C61C2×C4×D411yescharacteristic
260C22×C61C2×C4×D411yescharacteristic
261C22×C61C22×M1612yes—
262C22×C61#56812yes—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—24no—
267C2×D61—24no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—22no—
271C2×D61—416no—
272C2×D61#38514yes—
273C2×D61#42314yescomplemented
274C4×C61—24no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61#38514yes—
278Dic61—24no—
279Dic61—22no—
280Dic61—42no—
281Dic61C2×#12011yescharacteristic
282Dic61C2×C22⋊Q811yescharacteristic
283Dic61C2×C22⋊Q811yescharacteristic
284Dic61C2×#12311yescharacteristic
285C23×M16—611no—
286C2×#423—611no—
287C2×#423—611no—
288C2×#407—611no—
289C2×C4×C22⋊C4—611no—
290#1851—614no—
291#1850—614no—
292#1728—611no—
293#1728—611no—
294C2×C61—42no—
295C2×C61C2×#38511yescharacteristic
296C2×C61C2×#42311yescharacteristic
297C2×C61#290511yescharacteristic
298C2×C61D4×M1612yes—
299C2×C61D4×M1612yes—
300D61—24no—
301D61—24no—
302C24×C4—611no—
303C22×M16—611no—
304C22×M16—611no—
305C22×M16—612no—
306C22×M16—611no—
307C22×M16—611no—
308C22×M16—1224no—
309C22×M16—1224no—
310C22×M16—1224no—
311C23×C8—612no—
312C22×C22⋊C4—612no—
313C22×C42—611no—
314C22×C42—611no—
315#423—1224no—
316#423—1224no—
317C2×C4⋊C8—614no—
318C2×C4⋊C8—614no—
319#407—1224no—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C4×C22⋊C4—611no—
323C4×C22⋊C4—612no—
324C4×C22⋊C4—611no—
325C4×C22⋊C4—611no—
326C4×C22⋊C4—612no—
327C4×C22⋊C4#692311yescharacteristic
328C2×#95—612no—
329#336—614no—
330#336—614no—
331C61#2083711yescharacteristic, complemented, Sylow
332C25—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—614no—
336C23×C4—614no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—612no—
341C23×C4—612no—
342C23×C4—1222no—
343C23×C4—1222no—
344C23×C4—1222no—
345C23×C4—1222no—
346C23×C4C2×#692311yescharacteristic
347C2×M16—614no—
348C2×M16—614no—
349C2×M16—1224no—
350C2×M16—1224no—
351C2×M16—1224no—
352C2×M16—1222no—
353C2×M16—1222no—
354C2×M16—1222no—
355C2×M16—1222no—
356C2×M16—12232no—
357C2×M16—1222no—
358C2×M16—1222no—
359C2×M16—1222no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1222no—
363C22×C8—612no—
364C22×C8—612no—
365C22×C8—612no—
366C22×C8—612no—
367C22×C8—1224no—
368C22×C8—1224no—
369C22×C8—1224no—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—1228no—
374C2×C22⋊C4C2×#692312yes—
375C2×C42—612no—
376C2×C42—612no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—1222no—
380C2×C42—1222no—
381C2×C42—1222no—
382C2×C42—1222no—
383C2×C42C2×#692311yescharacteristic
384C2×C42C2×#692311yescharacteristic
385C4⋊C8—1228no—
386C4⋊C8—1228no—
387C22⋊C8—1224no—
388C22⋊C8—1224no—
389#95—612no—
390#95—612no—
391#95—612no—
392#95C2×#692312yes—
393C24—611no—
394C24—611no—
395C24—614no—
396C24—1224no—
397C24—1222no—
398C24—1222no—
399C24—2442no—
400C24C22×#692311yescharacteristic
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—611no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—611no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—614no—
416C22×C4—614no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1228no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1228no—
425C22×C4—1228no—
426C22×C4—1228no—
427C22×C4—1224no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1224no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1224no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4C4⋊#692314yes—
442C22×C4C22×#692312yes—
443C22×C4C22×#692311yescharacteristic
444C22×C4C22×#692312yes—
445C22×C4C22×#692311yescharacteristic
446M16—1228no—
447M16—1228no—
448M16—12216no—
449M16—2444no—
450M16—2444no—
451C2×C8—614no—
452C2×C8—614no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—12216no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C22⋊C4—24no—
465C22⋊C4—618no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C42—22no—
469C42—22no—
470C42—1222no—
471C42—1222no—
472C42—1222no—
473C42—1222no—
474C42—1222no—
475C42—1222no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—612no—
481C23—612no—
482C23—612no—
483C23—612no—
484C23—614no—
485C23—1228no—
486C23—1228no—
487C23—1221no—
488C23—1221no—
489C23—1222no—
490C23—1222no—
491C23—1222no—
492C23—2444no—
493C23—2448no—
494C23—2448no—
495C23C2×C4⋊#692312yes—
496C23C23×#692311yescharacteristic
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—614no—
512C2×C4—614no—
513C2×C4—614no—
514C2×C4—614no—
515C2×C4—612no—
516C2×C4—614no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1228no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—1228no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—2448no—
535C2×C4—2448no—
536C2×C4C2×C4⋊#692312yes—
537C2×C4C2×C4⋊#692312yes—
538C2×C4D4×#692312yes—
539C2×C4D4×#692312yes—
540C2×C4D61⋊M1614yescomplemented
541C8—1224no—
542C8—1224no—
543C8—2442no—
544C8—2442no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—614no—
551C22—614no—
552C22—1224no—
553C22—1224no—
554C22—1224no—
555C22—1224no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—24416no—
561C22C22×C4⋊#692311yescharacteristic
562C22C2×D4×#692311yescharacteristic
563C22C2×D4×#692311yescharacteristic
564C22C2×D61⋊M1612yes—
565C22#1684084112yes—
566C4—24no—
567C4—22no—
568C4—42no—
569C4—611no—
570C4—611no—
571C4—611no—
572C4—611no—
573C4—1224no—
574C4—1222no—
575C4—2442no—
576C4#1684070914yes—
577C2—42no—
578C2—1224no—
579C2—1224no—
580C2C2×#1684070911yescharacteristic
581C2D61⋊#42311yescharacteristic
582C2#2146246211yescharacteristic
583C2#2146255812yes—
584C2#2146256012yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC4×C22⋊C4×C61C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC4×C22⋊C4#6923
hypocenterC61#20837
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16840767C2semidirect
3. C2×#16840766C2semidirect
5. #21461766C2semidirect
6. #21461764C2semidirect
7. #21461483C2semidirect
8. #21461483C2semidirect
21. C2×#14519844C22semidirect
22. C2×C4⋊#92420C22semidirect
26. #16840628C22semidirect
27. #16840628C22semidirect
176. #11758198M16semidirect
216. C2×C4×C61C2×M16semidirect
229. C2×Dic61C2×M16semidirect
273. C2×D61#423semidirect
331. C61#20837semidirect
540. C2×C4D61⋊M16semidirect

Maximal subgroups

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

subgroupindex
C22×D61⋊M162
C2×#168407672
C2×#168407662
C2×#168407332
#214617662
#214617642
#214614832
C4×C22⋊C4×D612
#2083761