Finite Group Databasebeta

← #29575578

Subgroups — #29575578

Order 15616.

1062 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
1#21454101C211yesmaximal, characteristic
2C2×#16839497C212yesmaximal, complemented
3#21444042C211yesmaximal, characteristic, complemented
4#21443699C212yesmaximal, complemented
5C22×C22⋊C8×C61C211yesmaximal, characteristic, complemented
6C2×#14519751C411yescharacteristic, complemented
7C2×#14519751C411yescharacteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519746—21no—
10C22×C22⋊Dic61—21no—
11#16839641—21no—
12#16839641—21no—
13#16839641—21no—
14#16839622C411yescharacteristic, complemented
15#16839622C411yescharacteristic, complemented
16#16839622C2211yescharacteristic
17C2×#14519684—21no—
18C2×#14519684C2212yes—
19#16839497—24no—
20C24×C4×C61C2211yescharacteristic
21C23×C8×C61—21no—
22C2×C22⋊C8×C61—21no—
23C2×C22⋊C8×C61—24no—
24C2×C22⋊C8×C61C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C22×C22⋊D61—21no—
29C23×Dic61—22no—
30#14519751—22no—
31#14519751—22no—
32#14519751—42no—
33#14519751—42no—
34#14519746—42no—
35#14519746—42no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—42no—
44C2×C22⋊Dic61—42no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—22no—
52#14519684—22no—
53#14519684—42no—
54C25×C61D411yescharacteristic
55C23×C4×C61—22no—
56C23×C4×C61—22no—
57C23×C4×C61—42no—
58C23×C4×C61—42no—
59C23×C4×C61C2×C411yescharacteristic
60C23×C4×C61C2×C411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61—24no—
66C22×C8×C61—22no—
67C22×C8×C61—42no—
68C22×C8×C61—42no—
69C22⋊C8×C61—24no—
70C22⋊C8×C61—22no—
71C22⋊C8×C61D414yescomplemented
72C24×C61—22no—
73C24×C61—22no—
74C24×C61—22no—
75C24×C61—22no—
76C24×C61—41no—
77C24×C61—41no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C22⋊C411yescharacteristic
80C24×C61D812yes—
81C24×C61SD1612yes—
82C24×C61C2×D411yescharacteristic
83C23×D61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—48no—
104C22×C4×C61—48no—
105C22×C4×C61C22⋊C411yescharacteristic
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×C4×C61#4112yes—
111C2×C22⋊D61—42no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—42no—
121C22×Dic61—24no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—42no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61—42no—
131C2×D244—22no—
132C2×D244—41no—
133C2×D244—41no—
134C2×C8×C61—24no—
135C2×C8×C61—48no—
136C2×C8×C61—42no—
137C2×C8×C61—42no—
138C22⋊Dic61—48no—
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141C22⋊Dic61—41no—
142C22⋊Dic61—41no—
143#11758198—42no—
144#11758198—42no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—22no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—24no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—44no—
171C23×C61—44no—
172C23×C61—44no—
173C23×C61—44no—
174C23×C61#9912yes—
175C23×C61#10212yes—
176C23×C61#10212yes—
177C23×C61#10412yes—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61#12112yes—
180C23×C61C2×D811yescharacteristic
181C23×C61C2×SD1611yescharacteristic
182C23×C61#13512yes—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—44no—
205C2×C4×C61—44no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#9912yes—
209C2×C4×C61C4×D412yes—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12311yescharacteristic
212C22⋊D61—84no—
213C22⋊D61—84no—
214C22⋊D61—82no—
215C22⋊D61—82no—
216C2×Dic61—44no—
217C2×Dic61—44no—
218C2×Dic61—44no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—82no—
229C2×Dic61—82no—
230C2×Dic61—82no—
231C2×Dic61—82no—
232D244—82no—
233C8×C61—44no—
234#9113—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—48no—
253C22×C61—48no—
254C22×C61—41no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61#32714yescomplemented
259C22×C61#38611yescharacteristic
260C22×C61C2×#9911yescharacteristic
261C22×C61C2×#9911yescharacteristic
262C22×C61C2×#10211yescharacteristic
263C22×C61#41611yescharacteristic
264C22×C61C2×#10411yescharacteristic
265C22×C61#42111yescharacteristic
266C22×C61#46512yes—
267C22×C61#46612yes—
268C2×D61—81no—
269C2×D61—81no—
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C4×C61—21no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—41no—
280Dic61—84no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284C22×C22⋊C8D6111yescharacteristic, complemented
285#2847—611no—
286C2×#327—612no—
287C22⋊#95—611no—
288#1294—612no—
289C2×C61—22no—
290C2×C61—24no—
291C2×C61—22no—
292C2×C61—42no—
293C2×C61C2×#32711yescharacteristic
294C2×C61#150111yescharacteristic
295C2×C61#151611yescharacteristic
296C2×C61#151911yescharacteristic
297C2×C61#187111yescharacteristic
298C2×C61#189311yescharacteristic
299C2×C61#189811yescharacteristic
300D61—161no—
301C24×C4C2×D6111yescharacteristic
302C23×C8—21no—
303C2×#121—611no—
304C2×#121—611no—
305C2×#121—611no—
306C2×#120—1221no—
307C22×C22⋊C4—1221no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C2×D6112yes—
311#386—1221no—
312#386—1221no—
313#386—1221no—
314#380—611no—
315#380—611no—
316#380—611no—
317C2×#95—612no—
318C2×#95—1221no—
319#327—1224no—
320C61#911311yescharacteristic, complemented, Sylow
321C25D24411yescharacteristic
322C22×D4—611no—
323C22×D4—1221no—
324C22×D4—1221no—
325C22×D4—1221no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—1222no—
331C23×C4C4×D6111yescharacteristic
332C23×C4C4×D6111yescharacteristic
333C23×C4D24411yescharacteristic
334C23×C4C22⋊D6111yescharacteristic
335C23×C4C22⋊D6111yescharacteristic
336C23×C4C22×D6111yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—1222no—
342#121—1222no—
343#121—2442no—
344#121—2442no—
345#120—2442no—
346#120—2442no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C22⋊C4—1224no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—2442no—
358C2×C22⋊C4—2442no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8C22⋊D6114yescomplemented
362#95—1221no—
363#95—1221no—
364#95—1221no—
365#95—1222no—
366#95—1222no—
367#95—2442no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1221no—
375C24#1175819012yes—
376C24D48812yes—
377C24#1175819811yescharacteristic
378C24#1175819811yescharacteristic
379C24C2×D24411yescharacteristic
380C2×D4—1222no—
381C2×D4—2441no—
382C2×D4—2441no—
383C2×D4—2442no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2442no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—1224no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4#1175819811yescharacteristic
425C22×C4#1175819811yescharacteristic
426C22×C4C2×C4×D6111yescharacteristic
427C22×C4#1175822212yes—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6112yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—1222no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C22⋊C4—2442no—
438C22⋊C4—2448no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2442no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23#1451969112yes—
478C23#1451969912yes—
479C23#1451969912yes—
480C23#1451970312yes—
481C23C2×#1175819811yescharacteristic
482C23#1451975112yes—
483C23C2×D48811yescharacteristic
484C23C2×#1175819011yescharacteristic
485C23#1451979912yes—
486D4—4882no—
487D4—4884no—
488D4—4882no—
489D4—4882no—
490D4—4884no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—4882no—
510C2×C4—4882no—
511C2×C4—4882no—
512C2×C4—4882no—
513C2×C4—2444no—
514C2×C4—2444no—
515C2×C4—2444no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4#1451969112yes—
526C2×C4C4×C22⋊D6112yes—
527C2×C4#1451974611yescharacteristic
528C2×C4#1451976011yescharacteristic
529C8—44no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4881no—
554C22—4881no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22—4881no—
559C22—4881no—
560C22#1683949714yescomplemented
561C22#1683964111yescharacteristic
562C22C2×#1451969111yescharacteristic
563C22C2×#1451969111yescharacteristic
564C22C2×#1451969911yescharacteristic
565C22#1683975911yescharacteristic
566C22C2×#1451970311yescharacteristic
567C22#1683977811yescharacteristic
568C22#1683999112yes—
569C22#1683999812yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—4884no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—9761no—
584C2C2×#1683949711yescharacteristic
585C2#2144431111yescharacteristic
586C2#2144437611yescharacteristic
587C2#2144438811yescharacteristic
588C2#2144612111yescharacteristic
589C2#2144624711yescharacteristic
590C2#2144627211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC22×C22⋊C8×C61C2
socleC23×C61C2×C22⋊C4
prime radical#21454101C2
cosocleC23×C4×C61C23
hypercenterC22×C22⋊C8D61
hypocenterC61#9113
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16839497C2semidirect
3. #21444042C2semidirect
4. #21443699C2semidirect
5. C22×C22⋊C8×C61C2semidirect
6. C2×#14519751C4semidirect
7. C2×#14519751C4semidirect
14. #16839622C4semidirect
15. #16839622C4semidirect
71. C22⋊C8×C61D4semidirect
258. C22×C61#327semidirect
284. C22×C22⋊C8D61semidirect
320. C61#9113semidirect
361. C22⋊C8C22⋊D61semidirect
560. C22#16839497semidirect

Maximal subgroups

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

subgroupindex
#214541012
C2×#168394972
#214440422
#214436992
C22×C22⋊C8×C612
#911361