Finite Group Databasebeta

← #29645897

Subgroups — #29645897

Order 15616.

1646 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
1C22×#14519737C211yesmaximal, characteristic
2#21454448C212yesmaximal, complemented
3#21444158C214yesmaximal, complemented
4#21444146C211yesmaximal, characteristic, complemented
5#21444145C212yesmaximal, complemented
6#21444138C214yesmaximal, complemented
7C61×#1462C211yesmaximal, characteristic, complemented
8C23×C22⋊D61C411yescharacteristic
9C24×Dic61C411yescharacteristic
10C23×C4×D61C411yescharacteristic
11C2×#14519737—22no—
12C2×#14519737—22no—
13C2×#14519737—22no—
14C2×#14519737—22no—
15C2×#14519737C412yes—
16C2×#14519737C412yes—
17C2×#14519737C2212yes—
18C2×#14519737C2212yes—
19C22×#11758196C411yescharacteristic
20C22×#11758198C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16839816—22no—
23#16839816C2214yescomplemented
24#16839812—22no—
25#16839812C2214yescomplemented
26#16839785—22no—
27#16839726C2212yes—
28M16×Dic61—22no—
29#16839486C2214yescomplemented
30#16839485C2218yescomplemented
31C22×C22⋊C4×C61C2211yescharacteristic
32C61×#407C2212yes—
33C61×#323C2214yescomplemented
34C24×D61C2×C411yescharacteristic
35C22×C22⋊D61—28no—
36C22×C22⋊D61—22no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61C2×C412yes—
39C22×C22⋊D61C2×C412yes—
40C23×Dic61—21no—
41C23×Dic61—21no—
42C23×Dic61—21no—
43C23×Dic61—21no—
44C23×Dic61—44no—
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C412yes—
49C23×Dic61C2×C412yes—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61—44no—
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2×C412yes—
57C2×#11758194—22no—
58#14519737—24no—
59#14519737—24no—
60#14519737—24no—
61#14519737—24no—
62#14519737—24no—
63#14519737—24no—
64#14519737—24no—
65#14519737—24no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198C2×C412yes—
73C2×#11758198C2312yes—
74C2×C4×Dic61C2×C411yescharacteristic
75C2×C4×Dic61C2×C411yescharacteristic
76C2×C4×Dic61D412yes—
77C2×C4×Dic61D412yes—
78C2×C4×Dic61D412yes—
79C2×C4×Dic61D412yes—
80C2×C4×Dic61C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82#14519706—24no—
83#14519706—24no—
84C22⋊#92418—22no—
85C22⋊#92418C2314yes—
86#14519687—24no—
87C8×Dic61—24no—
88C25×C61C2×C411yescharacteristic
89C23×C4×C61C2×C411yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61C2×C412yes—
95C2×C22⋊C4×C61C2312yes—
96C22⋊C8×C61—22no—
97C22⋊C8×C61C2314yes—
98C24×C61—21no—
99C24×C61—21no—
100C24×C61—24no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61C22×C411yescharacteristic
104C24×C61C22×C412yes—
105C23×D61—21no—
106C23×D61—21no—
107C23×D61—24no—
108C23×D61—42no—
109C23×D61—42no—
110C23×D61C22×C412yes—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—216no—
121C2×C22⋊D61—216no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61M1618yescomplemented
128C22×Dic61—24no—
129C22×Dic61—24no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61—44no—
139C22×Dic61—44no—
140C22×Dic61—44no—
141C22×Dic61—44no—
142C22×Dic61—42no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C411yescharacteristic
145C22×Dic61C22⋊C411yescharacteristic
146C22×Dic61C22⋊C412yes—
147C22×Dic61C22⋊C412yes—
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C412yes—
153C22×Dic61C22×C411yescharacteristic
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164M16×C61—44no—
165C2×C8×C61—24no—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168#11758198—24no—
169#11758198—24no—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61C22⋊C412yes—
173C4×Dic61C22⋊C412yes—
174C4×Dic61C22⋊C412yes—
175C4×Dic61C22⋊C412yes—
176C4×Dic61C2×D412yes—
177C4×Dic61C2×D412yes—
178C4×Dic61C2×D412yes—
179C4×Dic61C2×D412yes—
180#11758194—44no—
181C2×#92418—24no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—42no—
190C23×C61—416no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C2×C22⋊C411yescharacteristic
197C23×C61C2×C22⋊C411yescharacteristic
198C23×C61C2×M1614yes—
199C23×C61C23×C411yescharacteristic
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—42no—
205C22×D61—416no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61C2×M1614yes—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—41no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C22×D412yes—
222C22⋊D61—216no—
223C22⋊D61—216no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—24no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—44no—
234C2×Dic61—44no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—44no—
238C2×Dic61—44no—
239C2×Dic61—42no—
240C2×Dic61#10014yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×M1614yes—
248C4×D61—44no—
249C4×D61—44no—
250C4×D61—44no—
251C4×D61—44no—
252C8×C61—44no—
253#92418—44no—
254#17364—611nomaximal, Sylow
255C22×C61—28no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—24no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—48no—
266C22×C61—48no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61#41012yes—
270C22×C61C2×#10012yes—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C22×M1612yes—
273C2×D61—24no—
274C2×D61—44no—
275C2×D61—48no—
276C2×D61—48no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—44no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283Dic61—22no—
284Dic61—21no—
285Dic61—21no—
286Dic61—42no—
287Dic61#40714yes—
288Dic61C2×#10012yes—
289Dic61C2×#10012yes—
290C22×C4×D4—611no—
291#2920—612no—
292#1466—614no—
293#1462—612no—
294#1462—611no—
295#1462D6111yescharacteristic, complemented
296#1460—614no—
297C2×C61—22no—
298C2×C61—22no—
299C2×C61—22no—
300C2×C61—44no—
301C2×C61C2×#40711yescharacteristic
302C2×C61C2×#41011yescharacteristic
303C2×C61C22×#10011yescharacteristic
304D61—44no—
305C23×D4—611no—
306C24×C4—611no—
307C24×C4—611no—
308C2×C4×D4—612no—
309C2×C4×D4—612no—
310C2×C4×D4—612no—
311C2×C4×D4—612no—
312C2×C4×D4—1222no—
313C2×C4×D4—1222no—
314C2×C4×D4—1222no—
315C2×C4×D4—1222no—
316C22×C4⋊C4—611no—
317C22×C22⋊C4—611no—
318C22×C22⋊C4C2×D6111yescharacteristic
319C22×C42—611no—
320#432—614no—
321#432—1222no—
322#431—614no—
323#431—1222no—
324#423—1222no—
325#407—612no—
326#407C2×D6112yes—
327C4×M16—1222no—
328#323—614no—
329#323—618no—
330#323C2×D6114yescomplemented
331C61#1736411yescharacteristic, complemented, Sylow
332C25—611no—
333C25C4×D6111yescharacteristic
334C22×D4—612no—
335C22×D4—612no—
336C22×D4—1222no—
337C22×D4—1222no—
338C22×D4—1228no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—1221no—
347C23×C4—1221no—
348C23×C4—1221no—
349C23×C4—1221no—
350C23×C4—1221no—
351C23×C4—1221no—
352C23×C4—1221no—
353C23×C4—1221no—
354C23×C4—2444no—
355C23×C4—2444no—
356C23×C4C4×D6111yescharacteristic
357C23×C4C22×D6111yescharacteristic
358C2×M16—22no—
359C2×M16—1222no—
360C4×D4—1224no—
361C4×D4—1224no—
362C4×D4—1224no—
363C4×D4—1224no—
364C4×D4—1224no—
365C4×D4—1224no—
366C4×D4—1224no—
367C4×D4—1224no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—612no—
375C2×C22⋊C4—612no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4C4×D6112yes—
379C2×C22⋊C4C22×D6112yes—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—612no—
383C2×C42—612no—
384C2×C42—612no—
385C2×C42—612no—
386C2×C42—611no—
387C2×C42—611no—
388C4⋊C8—1224no—
389C4⋊C8—1224no—
390C22⋊C8—22no—
391C22⋊C8—614no—
392C22⋊C8—1222no—
393C22⋊C8C22×D6114yes—
394#97—1224no—
395C4×C8—1224no—
396C24—21no—
397C24—21no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—612no—
402C24—1224no—
403C24—1221no—
404C24—1221no—
405C24—2442no—
406C24—2442no—
407C24C2×C4×D6111yescharacteristic
408C24C2×C4×D6112yes—
409C2×D4—618no—
410C2×D4—1224no—
411C2×D4—1224no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—12216no—
415C2×D4—12216no—
416C2×D4—1224no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1224no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—2444no—
445C22×C4—2444no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2442no—
457C22×C4C2×C4×D6112yes—
458C22×C4C2×C4×D6111yescharacteristic
459C22×C4D4×D6112yes—
460C22×C4D4×D6112yes—
461C22×C4C23×D6111yescharacteristic
462M16—44no—
463M16—2444no—
464C2×C8—24no—
465C2×C8—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C42—612no—
473C42—612no—
474C42—612no—
475C42—612no—
476C42—612no—
477C42—612no—
478C42—612no—
479C42—612no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—416no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—614no—
495C23—1224no—
496C23—1224no—
497C23—1222no—
498C23—1222no—
499C23—2442no—
500C23—2442no—
501C23—2442no—
502C23—24416no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23C22⋊C4×D6111yescharacteristic
507C23C22⋊C4×D6111yescharacteristic
508C23M16×D6114yes—
509C23C22×C4×D6111yescharacteristic
510D4—12216no—
511D4—12216no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—614no—
524C2×C4—614no—
525C2×C4—612no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2444no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4C22⋊C4×D6112yes—
549C2×C4C2×D4×D6112yes—
550C8—44no—
551C8—2444no—
552C22—28no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—24no—
560C22—44no—
561C22—48no—
562C22—48no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—1224no—
567C22—2444no—
568C22—2448no—
569C22—2448no—
570C22—2444no—
571C22—2442no—
572C22—2442no—
573C22#1683973412yes—
574C22#100×D6112yes—
575C22C2×C22⋊C4×D6111yescharacteristic
576C22C2×M16×D6112yes—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—614no—
581C4—612no—
582C4—612no—
583C4—1222no—
584C4—1221no—
585C4—1221no—
586C4—2442no—
587C2—22no—
588C2—22no—
589C2—22no—
590C2—44no—
591C2—2444no—
592C2#407×D6111yescharacteristic
593C2C2×#1683973411yescharacteristic
594C2C2×#100×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1462C2
socleC22×C61C22×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC22×C4×C61C24
hypercenter#1462D61
hypocenterC61#17364
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. #21444158C2semidirect
4. #21444146C2semidirect
5. #21444145C2semidirect
6. #21444138C2semidirect
7. C61×#1462C2semidirect
23. #16839816C22semidirect
25. #16839812C22semidirect
29. #16839486C22semidirect
30. #16839485C22semidirect
33. C61×#323C22semidirect
127. C2×C22⋊D61M16semidirect
295. #1462D61semidirect
330. #323C2×D61semidirect
331. C61#17364semidirect

Maximal subgroups

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

subgroupindex
C22×#145197372
#214544482
#214441582
#214441462
#214441452
#214441382
C61×#14622
#1736461