Finite Group Databasebeta

← #29645923

Subgroups — #29645923

Order 15616.

1210 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840242C211yesmaximal, characteristic
2#21454847C212yesmaximal, complemented
3#21444162C212yesmaximal, complemented
4#21444154C212yesmaximal, complemented
5#21444153C211yesmaximal, characteristic, complemented
6#21444147C212yesmaximal, complemented
7#21444134C212yesmaximal, complemented
8#324×D61C212yesmaximal, complemented
9C61×#1464C211yesmaximal, characteristic, complemented
10C22×#11758222C411yescharacteristic
11#16840242—24no—
12#16840242—24no—
13C2×C4×Dic122C411yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244C411yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519737C411yescharacteristic
18C2×#14519737C411yescharacteristic
19C2×#14519737C2211yescharacteristic
20C2×#14519737C2211yescharacteristic
21C2×#14519735C411yescharacteristic
22C2×#14519735C411yescharacteristic
23C22×C4⋊Dic61C411yescharacteristic
24C2×C4⋊C4×D61C2211yescharacteristic
25C2×C4⋊C4×D61C2211yescharacteristic
26#16839835—24no—
27#16839832—24no—
28#16839819C2212yes—
29#16839724C2214yes—
30C22⋊C8×D61C2214yes—
31#16839488C2212yes—
32#16839487C2214yes—
33#16839487C2214yes—
34#16839486C2212yes—
35C2×#117×C61C2211yescharacteristic
36C61×#432C2212yes—
37C61×#324C2212yes—
38C61×#323C2212yes—
39C2×#11758222—24no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—24no—
43C22×Dic122C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61C2×C411yescharacteristic
46C22×C22⋊D61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—24no—
56C2×C8×D61—24no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×Dic122—22no—
61C4×D244—22no—
62C4×D244—22no—
63C4×D244D414yes—
64C4×D244D414yes—
65#14519737—24no—
66#14519737—24no—
67#14519737—24no—
68#14519737—24no—
69#14519735—24no—
70#14519735—24no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2311yescharacteristic
80C2×#11758196C2311yescharacteristic
81C4⋊C4×D61—24no—
82C4⋊C4×D61—24no—
83C2×#11758198C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2×C411yescharacteristic
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C4⋊#92418—22no—
92C22⋊#92418C2314yes—
93#14519689—24no—
94#14519689—24no—
95#14519688—22no—
96C23×C4×C61C2×C411yescharacteristic
97#117×C61—22no—
98#117×C61—22no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61C2×C411yescharacteristic
102C2×C22⋊C4×C61C2311yescharacteristic
103C2×C42×C61C2×C411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106C22⋊C8×C61C2314yes—
107#97×C61—22no—
108C24×C61C22×C411yescharacteristic
109C23×D61C22×C411yescharacteristic
110C23×D61C22×C411yescharacteristic
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61M1614yescomplemented
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—81no—
130C22×Dic61—81no—
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137#11758222—216no—
138#11758222—216no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244C22⋊C412yes—
142C2×D244C22⋊C412yes—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—81no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61M1614yescomplemented
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D414yes—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×C8×C61—24no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170#11758198—22no—
171#11758198—22no—
172#11758198C22⋊C414yes—
173#11758198C22⋊C414yes—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61C22⋊C412yes—
179C4⋊Dic61C22⋊C412yes—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C2×#92418—24no—
189#11758189—84no—
190C8×D61—84no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C2×M1612yes—
196C23×C61C23×C411yescharacteristic
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61C2×C22⋊C412yes—
204C22×D61C2×C22⋊C412yes—
205C22×D61C2×M1612yes—
206C22×D61C2×M1612yes—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×M1612yes—
223C2×C4×C61C22×D412yes—
224C22⋊D61—28no—
225C22⋊D61—28no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—82no—
237C2×Dic61—82no—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×M1612yes—
241C2×Dic61C2×M1612yes—
242D244—24no—
243D244—24no—
244C4×D61—28no—
245C4×D61—28no—
246C4×D61—44no—
247C4×D61—44no—
248C4×D61—82no—
249C4×D61—82no—
250Dic122—24no—
251Dic122—24no—
252C8×C61—44no—
253#92418—44no—
254#17368—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—22no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C22×C22⋊C411yescharacteristic
263C22×C61C22×M1611yescharacteristic
264C22×C61C22×M1611yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61#40714yes—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282Dic61—22no—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—41no—
287Dic61—41no—
288C2×#520—611no—
289#2973—612no—
290#1467—612no—
291#1465—612no—
292#1464—611no—
293#1464D6111yescharacteristic, complemented
294#1463—612no—
295#1459—612no—
296C2×#324—612no—
297C2×C61—21no—
298C2×C61—21no—
299C2×C61—42no—
300C2×C61C2×#40711yescharacteristic
301C2×C61#288311yescharacteristic
302C2×C61#288911yescharacteristic
303D61—24no—
304D61—42no—
305C22×#41—611no—
306#520—1224no—
307#520—1224no—
308C2×C4×Q8—611no—
309C2×C4×Q8—611no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×C4×D4—611no—
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C2×#117—611no—
317C2×#117—611no—
318C2×#117C2×D6111yescharacteristic
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4—611no—
322#435—1224no—
323#435—1224no—
324#432—612no—
325#432C2×D6112yes—
326#407—614no—
327C2×C22⋊C8—614no—
328#324—612no—
329#324—614no—
330#324—614no—
331#324C2×D6112yes—
332#323—612no—
333#323C2×D6112yes—
334C61#1736811yescharacteristic, complemented, Sylow
335C2×#41—1224no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1224no—
339C22×Q8—611no—
340C22×D4—611no—
341C22×D4—611no—
342C22×D4—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×C4C4×D6111yescharacteristic
352C2×M16—1224no—
353C22×C8—1224no—
354C4×Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×Q8—1222no—
358C4×D4—614no—
359C4×D4—614no—
360C4×D4—1222no—
361C4×D4—1222no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1224no—
365C4×D4—1224no—
366#117—22no—
367#117—22no—
368#117—1224no—
369#117—1224no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1224no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4C4×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4C4×D6111yescharacteristic
389C2×C22⋊C4C22×D6111yescharacteristic
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42C4×D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396C4⋊C8—22no—
397C4⋊C8—1222no—
398C22⋊C8—614no—
399C22⋊C8—1224no—
400C22⋊C8—1224no—
401C22⋊C8C22×D6114yes—
402#97—22no—
403#97—1222no—
404C24—611no—
405C24—611no—
406C24C2×C4×D6111yescharacteristic
407#41—12216no—
408#41—12216no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×D4—614no—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—614no—
432C22×C4—614no—
433C22×C4—614no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—4881no—
446C22×C4—4881no—
447C22×C4—4881no—
448C22×C4—4881no—
449C22×C4—4881no—
450C22×C4—4881no—
451C22×C4C2×C4×D6111yescharacteristic
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4C2×C4×D6111yescharacteristic
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457M16—4884no—
458C2×C8—24no—
459C2×C8—1224no—
460C2×C8—4884no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—614no—
476C22⋊C4—614no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C42—21no—
480C42—21no—
481C42—1222no—
482C42—1222no—
483C42—1222no—
484C42—1222no—
485C42D4×D6112yes—
486C42D4×D6112yes—
487C23—21no—
488C23—21no—
489C23—41no—
490C23—41no—
491C23—612no—
492C23—612no—
493C23—612no—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23M16×D6112yes—
502C23C22×C4×D6111yescharacteristic
503Q8—1224no—
504Q8—1224no—
505D4—1224no—
506D4—1224no—
507D4—1228no—
508D4—1228no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—1228no—
526C2×C4—1228no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6112yes—
546C2×C4M16×D6112yes—
547C2×C4C2×D4×D6112yes—
548C8—44no—
549C8—2444no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—614no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×C22⋊C4×D6111yescharacteristic
568C22C2×M16×D6111yescharacteristic
569C22C2×M16×D6111yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—1222no—
578C4—1222no—
579C4—1222no—
580C4—1222no—
581C4—2441no—
582C4—2441no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—1224no—
587C2—2442no—
588C2#407×D6111yescharacteristic
589C2#2145427811yescharacteristic
590C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1464C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758222C4
cosocleC22×C4×C61C24
hypercenter#1464D61
hypocenterC61#17368
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21454847C2semidirect
3. #21444162C2semidirect
4. #21444154C2semidirect
5. #21444153C2semidirect
6. #21444147C2semidirect
7. #21444134C2semidirect
8. #324×D61C2semidirect
9. C61×#1464C2semidirect
126. C2×C22⋊D61M16semidirect
154. C2×C4×D61M16semidirect
293. #1464D61semidirect
334. C61#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168402422
#214548472
#214441622
#214441542
#214441532
#214441472
#214441342
#324×D612
C61×#14642
#1736861