Finite Group Databasebeta

← #29689702

Subgroups — #29689702

Order 15616.

1166 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840294C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#382×D61C211yesmaximal, characteristic, complemented
4#21446782C212yesmaximal, complemented
5#21446629C212yesmaximal, complemented
6#21446133C212yesmaximal
7#21446125C212yesmaximal, complemented
8#21445835C211yesmaximal, characteristic, complemented
9#21445831C211yesmaximal, characteristic, complemented
10#21445829C211yesmaximal, characteristic, complemented
11C61×#1804C211yesmaximal, characteristic, complemented
12#16840294—24no—
13#16840294C418yescomplemented
14C2×#14519770—21no—
15C2×#14519768C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519759—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×C22⋊D61—21no—
22C2×#14519736—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839628C2211yescharacteristic
33#16839627C2212yes—
34#16839626—21no—
35#16839626—21no—
36#95×D61—21no—
37#95×D61—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519770—28no—
54#14519768—22no—
55#14519768C2×C414yescomplemented
56#14519764—22no—
57#14519764C2×C414yescomplemented
58#14519759—28no—
59#14519757—22no—
60#14519757C2×C414yes—
61#14519748—22no—
62#14519748C2×C414yes—
63C4×Dic122—22no—
64C4×Dic122C2×C414yescomplemented
65C4×C22⋊D61—28no—
66#14519736—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—22no—
74C4⋊C4×D61—42no—
75C4⋊C4×D61—42no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61C2×C414yes—
85C42×D61D414yes—
86C4⋊#92418—22no—
87#14519702—42no—
88#14519700—42no—
89#14519698—42no—
90#14519689—24no—
91C8×Dic61—24no—
92#14519684—21no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C22×C8×C61C2312yes—
99C4×D4×C61—22no—
100C4×D4×C61C2×C414yescomplemented
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#102×C61—42no—
107#95×C61—21no—
108#95×C61—21no—
109C24×C61—21no—
110C23×D61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—24no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C2×C4×D61—24no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—42no—
138C2×C4×D61—44no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61C22⋊C414yes—
142C2×C4×D61C22⋊C414yes—
143C2×C4×D61C22×C412yes—
144C2×C4×D61C2×D412yes—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122C22×C412yes—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61C22×C412yes—
158C22⋊C4×C61—24no—
159C42×C61C22×C412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—24no—
162C22⋊Dic61—24no—
163C22⋊Dic61—24no—
164#11758198—24no—
165C4⋊Dic61—21no—
166C4⋊Dic61—24no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61C22×C412yes—
171#11758196—24no—
172#11758196—24no—
173#11758196—44no—
174C4×Dic61—24no—
175C4×Dic61C22⋊C414yes—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22×C412yes—
178C4×Dic61C2×D412yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—22no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61C22×D411yescharacteristic
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—22no—
191C22×D61C2×C22⋊C412yes—
192C22×D61#13514yes—
193D4×C61—22no—
194D4×C61—22no—
195D4×C61—41no—
196D4×C61—41no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—48no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61C2×C22⋊C412yes—
206C2×C4×C61C2×C22⋊C412yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61H(5,𝔽2)11yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C22⋊D61—48no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61C2×C22⋊C412yes—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C4×D61—28no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—44no—
235C4×D61—88no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240C8×C61—42no—
241#92418—42no—
242#21861—611nomaximal, Sylow
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61#53611yescharacteristic
251C22×C61C2×#13512yes—
252C22×C61#57512yes—
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—24no—
256C2×D61—24no—
257C2×D61#41614yes—
258C4×C61—22no—
259C4×C61—22no—
260C4×C61—42no—
261C4×C61—82no—
262C4×C61#52011yescharacteristic
263C4×C61#52011yescharacteristic
264C4×C61#53711yescharacteristic
265C4×C61#53711yescharacteristic
266Dic61—22no—
267Dic61—22no—
268Dic61—22no—
269Dic61—42no—
270Dic61—42no—
271Dic61—82no—
272C2×#529—611no—
273C2×#431—611no—
274C2×#382—611no—
275#1991—612no—
276#1958—612no—
277#1874—612no—
278#1872—612no—
279#1805—611no—
280#1805—611no—
281#1804—611no—
282#1804D6111yescharacteristic, complemented
283C2×C61—82no—
284C2×C61#231111yescharacteristic
285C2×C61C2×#41611yescharacteristic
286C2×C61#289611yescharacteristic
287C2×C61#321112yes—
288C2×C61#323212yes—
289D61—24no—
290#529—618no—
291#529—1224no—
292C2×#126—1221no—
293C2×#125—611no—
294C2×#124—611no—
295C2×#123—1221no—
296C2×C22⋊Q8—611no—
297C2×#121—611no—
298C2×C4×Q8—611no—
299C2×C4×D4—1221no—
300C2×C4×D4C2×D6111yescharacteristic
301C2×#117—1221no—
302C22×C4⋊C4—612no—
303C22×C42—611no—
304#431—1224no—
305C2×C4⋊C8—611no—
306C2×C4⋊C8C2×D6111yescharacteristic
307C2×#103—612no—
308C2×#103—612no—
309C2×#102—612no—
310C2×#102C2×D6112yes—
311C2×C22⋊C8—612no—
312C2×C4×C8—612no—
313#382—612no—
314#382—611no—
315#382—1221no—
316#382—1221no—
317#382C2×D6111yescharacteristic
318C2×#95—1221no—
319C2×#95—1221no—
320#340—612no—
321#340—612no—
322#340—612no—
323#340C2×D6112yes—
324C61#2186111yescharacteristic, complemented, Sylow
325C22×Q8—611no—
326C22×D4—1221no—
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—1222no—
333C22×C8—612no—
334C22×C8C22×D6112yes—
335#126—1228no—
336#125—614no—
337#125—1222no—
338#124—614no—
339#124—1222no—
340#123—1228no—
341C22⋊Q8—614no—
342C22⋊Q8—1222no—
343#121—614no—
344#121—1222no—
345C4×Q8—614no—
346C4×Q8—1222no—
347C4×D4—22no—
348C4×D4—1228no—
349C4×D4C4×D6114yescomplemented
350#117—1228no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—2442no—
359C2×C4⋊C4—2442no—
360C2×C4⋊C4C22×D6112yes—
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—614no—
371C2×C42—614no—
372C2×C42—1221no—
373C2×C42C22×D6111yescharacteristic
374C4⋊C8—22no—
375C4⋊C8—1222no—
376#103—2442no—
377#103—2442no—
378#102—42no—
379#102—2442no—
380C22⋊C8—1224no—
381C4×C8—1224no—
382#95—21no—
383#95—21no—
384#95—1222no—
385#95—1222no—
386#95—1221no—
387#95—1221no—
388C24—21no—
389C24—611no—
390C2×Q8—612no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—1224no—
402C2×D4—2444no—
403C2×D4C2×C4×D6112yes—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—44no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—614no—
414C22×C4—614no—
415C22×C4—1224no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—2442no—
424C22×C4—2444no—
425C22×C4—2444no—
426C22×C4—2444no—
427C22×C4D4×D6111yescharacteristic
428C22×C4D4×D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—1222no—
434C2×C8—1222no—
435C2×C8—1222no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—612no—
440C4⋊C4—1224no—
441C4⋊C4—1224no—
442C4⋊C4—1224no—
443C4⋊C4—1221no—
444C4⋊C4—1224no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—2444no—
448C4⋊C4C2×C4×D6112yes—
449C22⋊C4—24no—
450C22⋊C4—1224no—
451C22⋊C4—1224no—
452C22⋊C4—1224no—
453C22⋊C4—1224no—
454C42—612no—
455C42—612no—
456C42—614no—
457C42—614no—
458C42—1224no—
459C42C2×C4×D6112yes—
460C42D4×D6112yes—
461C23—22no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—612no—
467C23—614no—
468C23—1222no—
469C23—1221no—
470C23—1221no—
471C23C2×D4×D6111yescharacteristic
472Q8—1222no—
473Q8—1222no—
474Q8—2441no—
475Q8—2441no—
476D4—22no—
477D4—22no—
478D4—41no—
479D4—41no—
480D4—2448no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—612no—
492C2×C4—1228no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—4888no—
504C2×C4—2444no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4C22⋊C4×D6112yes—
512C2×C4C22⋊C4×D6112yes—
513C2×C4C22×C4×D6111yescharacteristic
514C2×C4C2×D4×D6111yescharacteristic
515C2×C4#1451982211yescharacteristic
516C2×C4#1451982311yescharacteristic
517C2×C4#1451982411yescharacteristic
518C2×C4#1451982511yescharacteristic
519C8—42no—
520C8—2442no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—614no—
527C22—1224no—
528C22—1224no—
529C22—1222no—
530C22—1222no—
531C22C2×C22⋊C4×D6111yescharacteristic
532C22#1684032811yescharacteristic
533C22#1684033511yescharacteristic
534C22#135×D6112yes—
535C22#1684055812yes—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—82no—
540C4—1222no—
541C4—1222no—
542C4—1222no—
543C4—4882no—
544C4—2442no—
545C4—2442no—
546C4#1684024211yescharacteristic
547C4#1684024311yescharacteristic
548C4#1684034111yescharacteristic
549C4#1684034211yescharacteristic
550C2—82no—
551C2—1224no—
552C2#2144840811yescharacteristic
553C2#416×D6111yescharacteristic
554C2#2145433311yescharacteristic
555C2#2145678212yes—
556C2#2145698512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1804D61
hypocenterC61#21861
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. #382×D61C2semidirect
4. #21446782C2semidirect
5. #21446629C2semidirect
7. #21446125C2semidirect
8. #21445835C2semidirect
9. #21445831C2semidirect
10. #21445829C2semidirect
11. C61×#1804C2semidirect
13. #16840294C4semidirect
55. #14519768C2×C4semidirect
57. #14519764C2×C4semidirect
64. C4×Dic122C2×C4semidirect
100. C4×D4×C61C2×C4semidirect
282. #1804D61semidirect
324. C61#21861semidirect
349. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402942
C2×#168398142
#382×D612
#214467822
#214466292
#214461332
#214461252
#214458352
#214458312
#214458292
C61×#18042
#2186161