Finite Group Databasebeta

← #29575404

Subgroups — #29575404

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839641C211yesmaximal, characteristic
2#21445729C211yesmaximal, characteristic, complemented
3C2×#16839485C212yesmaximal, complemented
4#21443661C212yesmaximal, complemented
5C2×C61×#407C211yesmaximal, characteristic, complemented
6C23×C22⋊D61C411yescharacteristic
7C22×C22⋊Dic61C411yescharacteristic
8C22×#11758198C411yescharacteristic
9C22×#11758198C2211yescharacteristic
10C2×C22⋊#92418—21no—
11C2×C22⋊#92418—21no—
12C2×C22⋊#92418C2212yes—
13#16839641—21no—
14#16839641—21no—
15#16839641—21no—
16#16839641—21no—
17C2×#14519684C411yescharacteristic
18C2×#14519684C2211yescharacteristic
19#16839485—24no—
20C24×C4×C61C2211yescharacteristic
21C22×M16×C61—21no—
22C61×#407—24no—
23C2×C22⋊C8×C61—21no—
24C2×C22⋊C8×C61C2212yes—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C22×C22⋊D61—22no—
29C22×C22⋊D61—22no—
30C22×C22⋊D61—42no—
31C22×C22⋊D61—42no—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C22×#92418—22no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—42no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—22no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—22no—
45C2×#11758198—42no—
46C22⋊#92418—22no—
47C22⋊#92418—22no—
48C22⋊#92418—42no—
49C22⋊#92418—42no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54C25×C61C2×C411yescharacteristic
55C23×C4×C61—22no—
56C23×C4×C61—22no—
57C23×C4×C61—42no—
58C23×C4×C61—42no—
59C23×C4×C61C2×C411yescharacteristic
60C23×C4×C61D411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C2×M16×C61—24no—
66C2×M16×C61—42no—
67C2×M16×C61—42no—
68C22×C8×C61—22no—
69C22⋊C8×C61—24no—
70C22⋊C8×C61—22no—
71C22⋊C8×C61D414yescomplemented
72C24×C61—22no—
73C24×C61—22no—
74C24×C61—21no—
75C24×C61—21no—
76C24×C61—22no—
77C24×C61—22no—
78C24×C61—41no—
79C24×C61—41no—
80C24×C61C22⋊C411yescharacteristic
81C24×C61C22⋊C411yescharacteristic
82C24×C61C22×C411yescharacteristic
83C23×D61—21no—
84C23×D61—21no—
85C23×D61—22no—
86C23×D61—22no—
87C23×D61—41no—
88C23×D61—41no—
89C23×D61—41no—
90C23×D61—41no—
91C23×D61M1612yescomplemented
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—48no—
112C22×C4×C61—48no—
113C22×C4×C61C22⋊C411yescharacteristic
114C22×C4×C61C22⋊C411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61#4112yes—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—48no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—42no—
124C2×C22⋊D61—48no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—41no—
134C22×Dic61—41no—
135C22×Dic61—41no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138C22×Dic61—41no—
139C22×Dic61M1612yescomplemented
140M16×C61—48no—
141C2×C8×C61—24no—
142C2×C8×C61—42no—
143C2×C8×C61—42no—
144C22⋊Dic61—42no—
145C22⋊Dic61—42no—
146#11758198—42no—
147#11758198—42no—
148#11758198—41no—
149#11758198—42no—
150#11758198—41no—
151C2×#92418—42no—
152C2×#92418—42no—
153C2×#92418—42no—
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—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—44no—
173C23×C61—44no—
174C23×C61—44no—
175C23×C61—44no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61#9912yes—
181C23×C61#10012yes—
182C23×C61C2×C22⋊C411yescharacteristic
183C23×C61C4×D412yes—
184C23×C61C2×M1611yescharacteristic
185C23×C61C2×M1611yescharacteristic
186C23×C61#13112yes—
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—82no—
201C22×D61—82no—
202C22×D61—82no—
203C22×D61—82no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
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—42no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61#9912yes—
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12112yes—
225C2×C4×C61#12311yescharacteristic
226C22⋊D61—48no—
227C22⋊D61—48no—
228C2×Dic61—22no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—82no—
240C2×Dic61—82no—
241C8×C61—44no—
242#92418—82no—
243#9083—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—48no—
259C22×C61—48no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61#38611yescharacteristic
269C22×C61#40711yescharacteristic
270C22×C61#40811yescharacteristic
271C22×C61C2×#9911yescharacteristic
272C22×C61C2×#9911yescharacteristic
273C22×C61C2×#10011yescharacteristic
274C22×C61#41311yescharacteristic
275C22×C61#43512yes—
276C22×C61#43612yes—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—84no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—84no—
286C4×C61—22no—
287C4×C61—21no—
288C4×C61—22no—
289C4×C61—21no—
290C4×C61—41no—
291Dic61—42no—
292Dic61—81no—
293Dic61—81no—
294C2×#407D6111yescharacteristic, complemented
295C2×#386—611no—
296#1779—611no—
297C2×#323—612no—
298#1280—612no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—42no—
302C2×C61—42no—
303C2×C61#146211yescharacteristic
304C2×C61#146511yescharacteristic
305C2×C61#146811yescharacteristic
306C2×C61#147111yescharacteristic
307C2×C61#187011yescharacteristic
308C2×C61#189811yescharacteristic
309C2×C61#190011yescharacteristic
310D61—82no—
311C23×D4—611no—
312C24×C4C2×D6111yescharacteristic
313C22×M16—21no—
314C22×C22⋊C4—611no—
315C22×C22⋊C4—611no—
316C22×C22⋊C4—611no—
317#407—24no—
318C2×C22⋊C8—21no—
319C2×C22⋊C8—612no—
320C2×C22⋊C8—1221no—
321C2×C22⋊C8—1221no—
322C2×C22⋊C8C2×D6112yes—
323#386—1221no—
324#386—1221no—
325#386—1221no—
326#386—1221no—
327C2×#95—611no—
328C2×#95—611no—
329#323—1224no—
330C61#908311yescharacteristic, complemented, Sylow
331C25—611no—
332C25C4×D6111yescharacteristic
333C22×D4—1222no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—2442no—
338C22×D4—2442no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4C4×D6111yescharacteristic
347C23×C4D24411yescharacteristic
348C23×C4D24411yescharacteristic
349C23×C4C22⋊D6111yescharacteristic
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352C2×M16—24no—
353C2×M16—42no—
354C2×M16—42no—
355C22×C8—22no—
356C22×C8—1222no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—2442no—
366C2×C22⋊C4—2442no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369C22⋊C8—1222no—
370C22⋊C8—1222no—
371C22⋊C8—2442no—
372C22⋊C8—2442no—
373C22⋊C8C22⋊D6114yescomplemented
374#95—1221no—
375#95—1221no—
376#95—1221no—
377#95—1221no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—612no—
387C24—1222no—
388C24—1222no—
389C24—1221no—
390C24—1221no—
391C24—2441no—
392C24—2441no—
393C24—2441no—
394C24—2441no—
395C24#1175819811yescharacteristic
396C24#1175819811yescharacteristic
397C24C2×C4×D6111yescharacteristic
398C2×D4—1224no—
399C2×D4—1224no—
400C2×D4—2448no—
401C2×D4—2442no—
402C2×D4—2442no—
403C2×D4—2448no—
404C2×D4—2442no—
405C2×D4—2442no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—612no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—2441no—
435C22×C4—2441no—
436C22×C4—2441no—
437C22×C4—2441no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4#1175819811yescharacteristic
441C22×C4#1175819811yescharacteristic
442C22×C4C2×D24411yescharacteristic
443C22×C4#1175822212yes—
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6112yes—
446M16—48no—
447C2×C8—24no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—2442no—
451C2×C8—2442no—
452C2×C8—2442no—
453C22⋊C4—2442no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C22⋊C4—2442no—
457C22⋊C4—2441no—
458C22⋊C4—2442no—
459C22⋊C4—2441no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—24no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—1224no—
487C23—1222no—
488C23—4882no—
489C23—4882no—
490C23—4882no—
491C23—4882no—
492C23—2442no—
493C23—2442no—
494C23—2442no—
495C23—2442no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23—2441no—
500C23—2441no—
501C23—2442no—
502C23—2441no—
503C23#1451969112yes—
504C23#1451969312yes—
505C23C2×#1175819811yescharacteristic
506C23C4×C22⋊D6112yes—
507C23M16×D6111yescharacteristic
508C23M16×D6111yescharacteristic
509C23#1451978512yes—
510D4—2448no—
511D4—2448no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—1222no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4#1451969112yes—
544C2×C4#1451974611yescharacteristic
545C2×C4#1451975112yes—
546C2×C4#1451976011yescharacteristic
547C8—44no—
548C8—4882no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—48no—
564C22—48no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—4884no—
574C22—4884no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—2442no—
580C22—2442no—
581C22—2442no—
582C22#1683964111yescharacteristic
583C22#1683972511yescharacteristic
584C22#1683973011yescharacteristic
585C22C2×#1451969111yescharacteristic
586C22C2×#1451969111yescharacteristic
587C22C2×#1451969311yescharacteristic
588C22#1683974411yescharacteristic
589C22#1683983112yes—
590C22#1683983812yes—
591C4—21no—
592C4—22no—
593C4—22no—
594C4—21no—
595C4—41no—
596C4—4881no—
597C4—4881no—
598C4—2442no—
599C2—22no—
600C2—22no—
601C2—42no—
602C2—42no—
603C2—4882no—
604C2#2144414511yescharacteristic
605C2#2144415511yescharacteristic
606C2#2144416811yescharacteristic
607C2#2144418011yescharacteristic
608C2#2144611611yescharacteristic
609C2#2144627211yescharacteristic
610C2#2144628211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#407C2
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC23×C4×C61C23
hypercenterC2×#407D61
hypocenterC61#9083
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. #21445729C2semidirect
3. C2×#16839485C2semidirect
4. #21443661C2semidirect
5. C2×C61×#407C2semidirect
71. C22⋊C8×C61D4semidirect
91. C23×D61M16semidirect
139. C22×Dic61M16semidirect
294. C2×#407D61semidirect
330. C61#9083semidirect
373. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396412
#214457292
C2×#168394852
#214436612
C2×C61×#4072
#908361