Finite Group Databasebeta

← #29689757

Subgroups — #29689757

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840294C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21448225C211yesmaximal, characteristic, complemented
4#21446786C212yesmaximal, complemented
5#21446634C212yesmaximal, complemented
6#21446138C212yesmaximal
7#21446130C212yesmaximal, 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×#14519735C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839818—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31C2×#14519687C2212yes—
32#16839631C2212yes—
33#16839628C2211yescharacteristic
34#16839621—21no—
35#16839621—21no—
36#16839613—21no—
37#16839613—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×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519770—28no—
53#14519768—22no—
54#14519768C2×C414yes—
55#14519764—22no—
56#14519764C2×C414yes—
57#14519759—28no—
58#14519757—22no—
59#14519757C2×C414yes—
60#14519748—22no—
61#14519748C2×C414yes—
62C4×Dic122—22no—
63C4×Dic122C2×C414yes—
64C4×C22⋊D61—28no—
65#14519736—28no—
66#14519735—42no—
67#14519735—42no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C42×D61C2×C414yes—
87C42×D61D414yes—
88C4⋊#92418—22no—
89#14519702—42no—
90#14519700—42no—
91#14519698—42no—
92#14519689—24no—
93#14519687—24no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C22×C8×C61C2312yes—
99C4×D4×C61—22no—
100C4×D4×C61C2×C414yes—
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—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22×C412yes—
141C2×C4×D61C2×D412yes—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122C22×C412yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C22×C412yes—
155C22⋊C4×C61—24no—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160C22⋊Dic61—24no—
161#11758198—24no—
162#11758198—44no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C22×C412yes—
169#11758196—24no—
170#11758196—24no—
171#11758196—44no—
172#11758196—44no—
173C4×Dic61—24no—
174C4×Dic61—44no—
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×D61—22no—
192C22×D61C2×C22⋊C412yes—
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—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—44no—
234C4×D61—44no—
235Dic122—22no—
236Dic122—22no—
237Dic122—41no—
238Dic122—41no—
239C8×C61—42no—
240#92418—42no—
241#21868—611nomaximal, Sylow
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61C22×C22⋊C411yescharacteristic
248C22×C61#53511yescharacteristic
249C22×C61#53611yescharacteristic
250C22×C61#57512yes—
251C22×C61#57512yes—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—42no—
256C2×D61—42no—
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—82no—
261C4×C61#52011yescharacteristic
262C4×C61#52011yescharacteristic
263C4×C61#53711yescharacteristic
264C4×C61#53711yescharacteristic
265Dic61—22no—
266Dic61—22no—
267Dic61—22no—
268Dic61—42no—
269Dic61—42no—
270Dic61—82no—
271C2×#529—611no—
272C2×#432—611no—
273#2285—611no—
274#1992—612no—
275#1959—612no—
276#1875—612no—
277#1873—612no—
278#1805—611no—
279#1805—611no—
280#1804—611no—
281#1804D6111yescharacteristic, complemented
282C2×C61—82no—
283C2×C61#231111yescharacteristic
284C2×C61#289611yescharacteristic
285C2×C61#289611yescharacteristic
286C2×C61#321912yes—
287C2×C61#324012yes—
288D61—42no—
289#529—618no—
290#529—1224no—
291C2×#126—1221no—
292C2×#125—611no—
293C2×#124—611no—
294C2×#123—1221no—
295C2×C22⋊Q8—611no—
296C2×#121—611no—
297C2×C4×Q8—611no—
298C2×C4×D4—1221no—
299C2×C4×D4C2×D6111yescharacteristic
300C2×#117—612no—
301C2×#117—1221no—
302C22×C42—611no—
303#432—1224no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#103—612no—
307C2×#103—612no—
308C2×#102—612no—
309C2×#102C2×D6112yes—
310C2×C22⋊C8—612no—
311C2×#97—612no—
312#384—612no—
313#382—611no—
314#382C2×D6111yescharacteristic
315#380—1221no—
316#380—1221no—
317C4×C4⋊C4—1221no—
318C4×C4⋊C4—1221no—
319#340—612no—
320#340—612no—
321#340—612no—
322#340C2×D6112yes—
323C61#2186811yescharacteristic, complemented, Sylow
324C22×Q8—611no—
325C22×D4—1221no—
326C22×D4C22×D6111yescharacteristic
327C23×C4—21no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C22×C8—612no—
332C22×C8C22×D6112yes—
333#126—1228no—
334#125—614no—
335#125—1222no—
336#124—614no—
337#124—1222no—
338#123—1228no—
339C22⋊Q8—614no—
340C22⋊Q8—1222no—
341#121—614no—
342#121—1222no—
343C4×Q8—614no—
344C4×Q8—1222no—
345C4×D4—22no—
346C4×D4—1228no—
347C4×D4C4×D6114yes—
348#117—1228no—
349#117—2442no—
350#117—2442no—
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—1222no—
359C2×C4⋊C4C22×D6112yes—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—614no—
371C2×C42—614no—
372C2×C42—1221no—
373C2×C42—1222no—
374C2×C42C22×D6111yescharacteristic
375C4⋊C8—22no—
376C4⋊C8—1222no—
377#103—2442no—
378#103—2442no—
379#102—42no—
380#102—2442no—
381C22⋊C8—1224no—
382#97—1224no—
383#95—21no—
384#95—21no—
385#95—1221no—
386#95—1221no—
387C24—21no—
388C24—611no—
389C2×Q8—612no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—1224no—
401C2×D4—2444no—
402C2×D4C2×C4×D6112yes—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—614no—
413C22×C4—614no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4D4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4C23×D6111yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—612no—
436C4⋊C4—1224no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4—1221no—
440C4⋊C4—1224no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—2444no—
444C4⋊C4—2444no—
445C4⋊C4C2×C4×D6112yes—
446C22⋊C4—24no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—2444no—
452C42—612no—
453C42—612no—
454C42—614no—
455C42—614no—
456C42—1224no—
457C42—2444no—
458C42C2×C4×D6112yes—
459C42D4×D6112yes—
460C23—22no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—612no—
466C23—1222no—
467C23—1222no—
468C23—1221no—
469C23—1221no—
470C23C2×D4×D6111yescharacteristic
471Q8—1222no—
472Q8—1222no—
473Q8—2441no—
474Q8—2441no—
475D4—22no—
476D4—22no—
477D4—41no—
478D4—41no—
479D4—2448no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4C22⋊C4×D6112yes—
510C2×C4C22⋊C4×D6112yes—
511C2×C4C22×C4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4#1451982211yescharacteristic
514C2×C4#1451982311yescharacteristic
515C2×C4#1451982411yescharacteristic
516C2×C4#1451982511yescharacteristic
517C8—42no—
518C8—2442no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—1222no—
525C22—1222no—
526C22—1222no—
527C22—2442no—
528C22—2442no—
529C22C2×C22⋊C4×D6111yescharacteristic
530C22#1684032811yescharacteristic
531C22#1684033511yescharacteristic
532C22#1684055812yes—
533C22#1684055912yes—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—82no—
538C4—1222no—
539C4—1222no—
540C4—1222no—
541C4—4882no—
542C4—2442no—
543C4—2442no—
544C4#1684024211yescharacteristic
545C4#1684024311yescharacteristic
546C4#1684034111yescharacteristic
547C4#1684034211yescharacteristic
548C2—82no—
549C2—2442no—
550C2#2144840811yescharacteristic
551C2#2145433311yescharacteristic
552C2#2145433411yescharacteristic
553C2#2145685512yes—
554C2#2145705812yes—
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#21868
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×#16839818C2semidirect
3. #21448225C2semidirect
4. #21446786C2semidirect
5. #21446634C2semidirect
7. #21446130C2semidirect
8. #21445835C2semidirect
9. #21445831C2semidirect
10. #21445829C2semidirect
11. C61×#1804C2semidirect
13. #16840294C4semidirect
281. #1804D61semidirect
323. C61#21868semidirect

Maximal subgroups

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

subgroupindex
C2×#168402942
C2×#168398182
#214482252
#214467862
#214466342
#214461382
#214461302
#214458352
#214458312
#214458292
C61×#18042
#2186861