Finite Group Databasebeta

← #29645924

Subgroups — #29645924

Order 15616.

1210 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840241C211yesmaximal, characteristic
2#21454848C211yesmaximal, characteristic, complemented
3#21444162C212yesmaximal, complemented
4#21444155C212yesmaximal, complemented
5#21444151C212yesmaximal, complemented
6#21444148C212yesmaximal, complemented
7#21444135C212yesmaximal, complemented
8C2×#16839487C212yesmaximal, complemented
9C61×#2973C211yesmaximal, characteristic, complemented
10C22×#11758224C411yescharacteristic
11#16840241—24no—
12#16840241—24no—
13C2×C4×Dic122C2211yescharacteristic
14C2×#14519742C411yescharacteristic
15C2×D4×Dic61C411yescharacteristic
16C2×C4×C22⋊D61C2211yescharacteristic
17C2×C4×C22⋊D61C2211yescharacteristic
18C2×#14519737C411yescharacteristic
19C2×#14519737C411yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×#14519733C411yescharacteristic
22C2×#14519733C411yescharacteristic
23C22×#11758196C2211yescharacteristic
24C22×#11758196C2211yescharacteristic
25C2×C4⋊C4×D61C411yescharacteristic
26#16839834—24no—
27#16839819C2212yes—
28#16839726C2212yes—
29C2×C22⋊#92418C2212yes—
30#16839487C2214yes—
31#16839487C2214yes—
32#16839487C2214yes—
33#16839485C2214yes—
34C2×C4×D4×C61C2211yescharacteristic
35C61×#435—24no—
36C61×#432C2212yes—
37C61×#407C2212yes—
38C2×C22⋊C8×C61C2212yes—
39C2×#11758224—24no—
40C2×#11758224—24no—
41C2×#11758224—24no—
42C22×Dic122C2×C411yescharacteristic
43C22×C22⋊D61C2×C411yescharacteristic
44C22×C22⋊D61C2×C411yescharacteristic
45C23×Dic61C2×C411yescharacteristic
46C23×Dic61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C23×Dic61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C2×#11758194—22no—
53C22×#92418—22no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56#14519742—24no—
57D4×Dic61—24no—
58C4×C22⋊D61—22no—
59C4×C22⋊D61—22no—
60C4×C22⋊D61—22no—
61C4×C22⋊D61—22no—
62#14519737—24no—
63#14519737—24no—
64#14519734—22no—
65#14519734—22no—
66#14519733—24no—
67#14519733—24no—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196C2×C411yescharacteristic
74C2×#11758196C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2311yescharacteristic
79C2×#11758196C2311yescharacteristic
80C2×#11758196C2311yescharacteristic
81C4⋊C4×D61—24no—
82C2×C22⋊Dic61C2×C411yescharacteristic
83C2×C22⋊Dic61C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2311yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C4⋊#92418—22no—
92C22⋊#92418—22no—
93C22⋊#92418—22no—
94C22⋊#92418C2314yes—
95#14519688—22no—
96C22×D4×C61C2×C411yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C22×C8×C61—22no—
101C4×D4×C61D414yes—
102C4×D4×C61D414yes—
103C2×C4⋊C4×C61C2×C411yescharacteristic
104C2×C22⋊C4×C61C2×C411yescharacteristic
105C2×C22⋊C4×C61C2×C411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—22no—
108C22⋊C8×C61—22no—
109C22⋊C8×C61—22no—
110C22⋊C8×C61C2314yes—
111#97×C61—22no—
112C24×C61C22×C411yescharacteristic
113C24×C61C22×C411yescharacteristic
114C23×D61C22×C411yescharacteristic
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61C22⋊C414yes—
118C22×C4×C61—81no—
119C22×C4×C61—81no—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—22no—
130C2×C22⋊D61—22no—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61M1614yescomplemented
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—81no—
143C22×Dic61—81no—
144C22×Dic61M1614yescomplemented
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152#11758224—232no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—81no—
159C2×C4×D61—81no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163M16×C61—82no—
164C2×C8×C61—24no—
165C2×C8×C61—82no—
166C4⋊C4×C61C22⋊C414yes—
167C22⋊C4×C61C22⋊C414yes—
168C22⋊C4×C61C22⋊C414yes—
169C42×C61C2×D412yes—
170C42×C61C2×D412yes—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173#11758198—22no—
174#11758198—21no—
175#11758198—21no—
176C4⋊Dic61—22no—
177#11758196—22no—
178#11758196—22no—
179#11758196—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—21no—
183#11758196—21no—
184#11758196—21no—
185#11758196—21no—
186#11758196—21no—
187#11758196—21no—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193#11758194—82no—
194C2×#92418—24no—
195C2×#92418—82no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61C2×C22⋊C412yes—
203C23×C61C2×C22⋊C412yes—
204C23×C61C2×M1612yes—
205C23×C61C2×M1612yes—
206C23×C61C23×C411yescharacteristic
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61C2×M1612yes—
212D4×C61—28no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C22⋊D61—28no—
230C22⋊D61—28no—
231C2×Dic61—28no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—28no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—82no—
248C2×Dic61—82no—
249C2×Dic61C2×M1612yes—
250C2×Dic61C2×M1612yes—
251C2×Dic61C2×M1612yes—
252C4×D61—28no—
253C4×D61—82no—
254C4×D61—82no—
255Dic122—28no—
256C8×C61—44no—
257#92418—44no—
258#17368—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—22no—
261C22×C61—21no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61#40714yes—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61C22×M1611yescharacteristic
273C22×C61C22×M1611yescharacteristic
274C2×D61—22no—
275C2×D61—42no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—42no—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—22no—
285Dic61—22no—
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292C2×#520—611no—
293#2973—611no—
294#2973D6111yescharacteristic, complemented
295#1467—612no—
296#1465—612no—
297#1464—612no—
298#1463—612no—
299#1459—612no—
300C2×#324—612no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—24no—
304C2×C61—42no—
305C2×C61C2×#40711yescharacteristic
306C2×C61#288311yescharacteristic
307C2×C61#288911yescharacteristic
308D61—42no—
309C22×#41—611no—
310#520—1224no—
311#520—1224no—
312C2×C4×Q8—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C2×C4×D4—611no—
317C2×C4×D4—611no—
318C2×C4×D4—611no—
319C2×C4×D4C2×D6111yescharacteristic
320C2×#117—611no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326#435—24no—
327#435—1224no—
328#432—612no—
329#432C2×D6112yes—
330#407—612no—
331#407C2×D6112yes—
332C2×C22⋊C8—612no—
333C2×C22⋊C8C2×D6112yes—
334#324—614no—
335#324—614no—
336#324—614no—
337#323—614no—
338C61#1736811yescharacteristic, complemented, Sylow
339C2×#41—1224no—
340C2×#41—1224no—
341C2×#41—1224no—
342C22×Q8—611no—
343C22×D4—611no—
344C22×D4—611no—
345C22×D4C4×D6111yescharacteristic
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4C22×D6111yescharacteristic
354C23×C4C22×D6111yescharacteristic
355C2×M16—22no—
356C2×M16—1222no—
357C22×C8—22no—
358C22×C8—1222no—
359C4×Q8—1224no—
360C4×Q8—1222no—
361C4×Q8—1222no—
362C4×D4—1224no—
363C4×D4—1222no—
364C4×D4—1222no—
365C4×D4—1222no—
366C4×D4—1222no—
367C4×D4—1224no—
368C4×D4—1224no—
369C4×D4D24414yes—
370C4×D4C22⋊D6114yes—
371#117—1224no—
372#117—1224no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4C4×D6111yescharacteristic
390C2×C22⋊C4—611no—
391C2×C22⋊C4—611no—
392C2×C22⋊C4—611no—
393C2×C22⋊C4—611no—
394C2×C22⋊C4C4×D6111yescharacteristic
395C2×C22⋊C4C4×D6111yescharacteristic
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42C22×D6111yescharacteristic
402C4⋊C8—22no—
403C4⋊C8—1222no—
404C22⋊C8—22no—
405C22⋊C8—22no—
406C22⋊C8—614no—
407C22⋊C8—1222no—
408C22⋊C8—1222no—
409C22⋊C8C22×D6114yes—
410#97—22no—
411#97—1222no—
412C24—611no—
413C24C2×C4×D6111yescharacteristic
414C24C2×C4×D6111yescharacteristic
415#41—12232no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—614no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4#1175819814yes—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—614no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—4881no—
451C22×C4—4881no—
452C22×C4—4881no—
453C22×C4—4881no—
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6111yescharacteristic
456C22×C4C2×C4×D6111yescharacteristic
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4C2×D24412yes—
459C22×C4C2×D24412yes—
460C22×C4C2×C22⋊D6112yes—
461C22×C4C2×C22⋊D6112yes—
462C22×C4C23×D6111yescharacteristic
463M16—82no—
464M16—4882no—
465C2×C8—24no—
466C2×C8—82no—
467C2×C8—1224no—
468C2×C8—4882no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4#1175819814yes—
482C22⋊C4—1222no—
483C22⋊C4—1222no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1222no—
487C22⋊C4#1175819814yes—
488C22⋊C4#1175819814yes—
489C42—1222no—
490C42—1222no—
491C42—1222no—
492C42—1222no—
493C42—1222no—
494C42C2×D24412yes—
495C42C2×C22⋊D6112yes—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—612no—
503C23—1221no—
504C23—1221no—
505C23—2441no—
506C23—2441no—
507C23C2×#1175819812yes—
508C23C2×#1175819812yes—
509C23M16×D6112yes—
510C23M16×D6112yes—
511C23C22×C4×D6111yescharacteristic
512Q8—1228no—
513D4—28no—
514D4—1228no—
515D4—1228no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—1228no—
530C2×C4—1228no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1228no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—4882no—
543C2×C4—4882no—
544C2×C4—4882no—
545C2×C4—4882no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4C2×#1175819812yes—
551C2×C4C2×#1175819812yes—
552C2×C4C2×#1175819812yes—
553C2×C4C2×#1175819812yes—
554C2×C4C22×C22⋊D6111yescharacteristic
555C2×C4C22×D24411yescharacteristic
556C8—44no—
557C8—2444no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—1222no—
570C22—2442no—
571C22—2442no—
572C22—2441no—
573C22—2441no—
574C22#1683972514yes—
575C22C22×#1175819811yescharacteristic
576C22C2×M16×D6111yescharacteristic
577C22C2×M16×D6111yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—42no—
595C2—2442no—
596C2C2×#1683972511yescharacteristic
597C2#2145427911yescharacteristic
598C2#2145430111yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2973C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758224C4
cosocleC22×C4×C61C24
hypercenter#2973D61
hypocenterC61#17368
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21454848C2semidirect
3. #21444162C2semidirect
4. #21444155C2semidirect
5. #21444151C2semidirect
6. #21444148C2semidirect
7. #21444135C2semidirect
8. C2×#16839487C2semidirect
9. C61×#2973C2semidirect
134. C2×C22⋊D61M16semidirect
144. C22×Dic61M16semidirect
294. #2973D61semidirect
338. C61#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168402412
#214548482
#214441622
#214441552
#214441512
#214441482
#214441352
C2×#168394872
C61×#29732
#1736861