Finite Group Databasebeta

← #29689740

Subgroups — #29689740

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic
3#382×D61C211yesmaximal, characteristic
4#21446790C212yesmaximal, complemented
5#21446637C212yesmaximal, complemented
6#21446125C212yesmaximal
7#21446123C212yesmaximal
8#21445835C211yesmaximal, characteristic, complemented
9#21445834C211yesmaximal, characteristic, complemented
10#21445831C211yesmaximal, characteristic, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12#16840316—24no—
13#16840316C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519753C2211yescharacteristic
20C2×C4×Dic122—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519734—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839818—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519689C2212yes—
31C2×#14519687C2212yes—
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×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776C2×C414yescomplemented
55#14519775—22no—
56#14519775C2×C414yescomplemented
57#14519767—28no—
58#14519757—22no—
59#14519757C2×C414yes—
60#14519755—28no—
61#14519753—22no—
62#14519753C2×C414yes—
63C4×Dic122—22no—
64C4×Dic122—28no—
65C4×Dic122C2×C414yescomplemented
66#14519734—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—22no—
76C4⋊C4×D61—42no—
77C4⋊C4×D61—42no—
78C2×#11758198—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#14519701—42no—
89#14519700—42no—
90#14519689—24no—
91#14519687—24no—
92#14519684—21no—
93#14519684—21no—
94#14519684—22no—
95#14519684—22no—
96C22×Q8×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61C2×C414yescomplemented
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106#103×C61—42no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C2×C4×D61—24no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—42no—
134C2×C4×D61—44no—
135C2×C4×D61—44no—
136C2×C4×D61—44no—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—24no—
147C2×Dic122—44no—
148C2×Dic122C22×C412yes—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61C22×C412yes—
157C42×C61—24no—
158C42×C61C22×C412yes—
159C42×C61C2×D412yes—
160#11758198—24no—
161#11758198—24no—
162C4⋊Dic61—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—44no—
173C4×Dic61—24no—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22⋊C414yes—
176C4×Dic61C22×C412yes—
177C4×Dic61C2×D412yes—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—22no—
185C22×D61C2×C22⋊C412yes—
186C22×D61#13714yes—
187Q8×C61—22no—
188Q8×C61—22no—
189Q8×C61—41no—
190Q8×C61—41no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61H(5,𝔽2)11yescharacteristic
206C2×C4×C61H(5,𝔽2)11yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61C2×C22⋊C412yes—
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C2×C22⋊C412yes—
224C4×D61—28no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—44no—
230C4×D61—88no—
231Dic122—22no—
232Dic122—22no—
233Dic122—48no—
234Dic122—41no—
235Dic122—41no—
236C8×C61—42no—
237#92418—42no—
238#21866—611nomaximal, Sylow
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61#53511yescharacteristic
241C22×C61#53611yescharacteristic
242C22×C61C2×#13712yes—
243C22×C61#57512yes—
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—24no—
247C2×D61—24no—
248C2×D61#41914yes—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—42no—
252C4×C61—42no—
253C4×C61—82no—
254C4×C61#52011yescharacteristic
255C4×C61#52011yescharacteristic
256C4×C61#53711yescharacteristic
257C4×C61#53711yescharacteristic
258Dic61—22no—
259Dic61—22no—
260Dic61—22no—
261Dic61—42no—
262Dic61—42no—
263Dic61—82no—
264C2×#533—611no—
265C2×#432—611no—
266C2×#382—611no—
267#1993—612no—
268#1960—612no—
269#1872—612no—
270#1872—612no—
271#1805—611no—
272#1805—611no—
273#1805—611no—
274#1805D6111yescharacteristic, complemented
275C2×C61#231111yescharacteristic
276C2×C61C2×#41911yescharacteristic
277C2×C61#289611yescharacteristic
278C2×C61#321512yes—
279C2×C61#323612yes—
280D61—24no—
281#533—618no—
282#533—1224no—
283C2×C4⋊Q8—611no—
284C2×C4⋊Q8—611no—
285C2×#125—1221no—
286C2×C22⋊Q8—611no—
287C2×C22⋊Q8—611no—
288C2×C22⋊Q8—1221no—
289C2×C4×Q8—611no—
290C2×C4×Q8—1221no—
291C2×C4×Q8C2×D6111yescharacteristic
292C2×#117—1221no—
293C22×C4⋊C4—612no—
294C22×C42—611no—
295#432—1224no—
296C2×C4⋊C8—611no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#103—612no—
299C2×#103—612no—
300C2×#103—612no—
301C2×#103C2×D6112yes—
302C2×C22⋊C8—612no—
303C2×#97—612no—
304#382—612no—
305#382—611no—
306#382—1221no—
307#382—1221no—
308#382C2×D6111yescharacteristic
309C2×#95—1221no—
310C2×#95—1221no—
311#340—612no—
312#340—612no—
313#340—612no—
314#340C2×D6112yes—
315C61#2186611yescharacteristic, complemented, Sylow
316C22×Q8—611no—
317C22×Q8—1221no—
318C22×Q8C22×D6111yescharacteristic
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—1222no—
323C22×C8—612no—
324C22×C8C22×D6112yes—
325C4⋊Q8—614no—
326C4⋊Q8—614no—
327C4⋊Q8—1222no—
328C4⋊Q8—1222no—
329#125—1228no—
330C22⋊Q8—614no—
331C22⋊Q8—614no—
332C22⋊Q8—1222no—
333C22⋊Q8—1228no—
334C22⋊Q8—1222no—
335C4×Q8—22no—
336C4×Q8—614no—
337C4×Q8—1228no—
338C4×Q8—1222no—
339C4×Q8C4×D6114yescomplemented
340#117—1228no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—2442no—
352C2×C4⋊C4—2442no—
353C2×C4⋊C4C22×D6112yes—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C42—21no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—614no—
362C2×C42—614no—
363C2×C42—1221no—
364C2×C42C22×D6111yescharacteristic
365C4⋊C8—22no—
366C4⋊C8—1222no—
367#103—42no—
368#103—2442no—
369#103—2442no—
370#103—2442no—
371C22⋊C8—1224no—
372#97—1224no—
373#95—21no—
374#95—21no—
375#95—1222no—
376#95—1222no—
377#95—1221no—
378#95—1221no—
379C24—611no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—612no—
386C2×Q8—1224no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8—1221no—
392C2×Q8—2444no—
393C2×Q8C2×C4×D6112yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—614no—
403C22×C4—614no—
404C22×C4—1224no—
405C22×C4—1221no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4—2442no—
413C22×C4—2444no—
414C22×C4—2444no—
415C22×C4—2444no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—21no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—612no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C4⋊C4—1221no—
436C4⋊C4—1224no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—2444no—
440C4⋊C4C2×C4×D6112yes—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C42—24no—
444C42—612no—
445C42—612no—
446C42—614no—
447C42—614no—
448C42—1224no—
449C42C2×C4×D6112yes—
450C42D4×D6112yes—
451C23—612no—
452C23—614no—
453C23—1222no—
454C23—1221no—
455C23—1221no—
456C23C2×D4×D6111yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—41no—
460Q8—41no—
461Q8—1222no—
462Q8—1222no—
463Q8—2448no—
464Q8—2441no—
465Q8—2441no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—1228no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—4888no—
491C2×C4—2444no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4C22⋊C4×D6112yes—
499C2×C4C22⋊C4×D6112yes—
500C2×C4C22×C4×D6111yescharacteristic
501C2×C4C2×D4×D6111yescharacteristic
502C2×C4#1451982211yescharacteristic
503C2×C4#1451982311yescharacteristic
504C2×C4#1451982411yescharacteristic
505C2×C4#1451982511yescharacteristic
506C8—42no—
507C8—2442no—
508C22—614no—
509C22—1224no—
510C22—1224no—
511C22—1222no—
512C22—1222no—
513C22C2×C22⋊C4×D6111yescharacteristic
514C22#1684032811yescharacteristic
515C22#1684033511yescharacteristic
516C22#137×D6112yes—
517C22#1684055912yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—1222no—
524C4—1222no—
525C4—1222no—
526C4—4882no—
527C4—2442no—
528C4—2442no—
529C4#1684024211yescharacteristic
530C4#1684024311yescharacteristic
531C4#1684034111yescharacteristic
532C4#1684034211yescharacteristic
533C2—1224no—
534C2#2144840811yescharacteristic
535C2#419×D6111yescharacteristic
536C2#2145433411yescharacteristic
537C2#2145682012yes—
538C2#2145702312yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#21866
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #21446790C2semidirect
5. #21446637C2semidirect
8. #21445835C2semidirect
9. #21445834C2semidirect
10. #21445831C2semidirect
11. C61×#1805C2semidirect
13. #16840316C4semidirect
54. #14519776C2×C4semidirect
56. #14519775C2×C4semidirect
65. C4×Dic122C2×C4semidirect
99. C4×Q8×C61C2×C4semidirect
274. #1805D61semidirect
315. C61#21866semidirect
339. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168398182
#382×D612
#214467902
#214466372
#214461252
#214461232
#214458352
#214458342
#214458312
C61×#18052
#2186661