Finite Group Databasebeta

← #29689976

Subgroups — #29689976

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840299C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21446773C212yesmaximal, complemented
5#21446748C212yesmaximal, complemented
6#21446256C212yesmaximal
7#21446249C212yesmaximal, complemented
8#21445834C211yesmaximal, characteristic, complemented
9#21445830C211yesmaximal, characteristic, complemented
10#21445826C211yesmaximal, characteristic, complemented
11C61×#1805C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840299—24no—
14#16840299C418yescomplemented
15C2×#14519772—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519762—21no—
19C2×#14519753C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25#16839784—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519706C2212yes—
28C2×#14519701C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31#16839629—21no—
32#16839628C2211yescharacteristic
33#16839624—21no—
34#16839619C2212yes—
35C4⋊C4×Dic61C2212yes—
36#16839609—21no—
37#16839607—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#382C2211yescharacteristic
45C61×#340C2212yes—
46C22×D244—21no—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C2×#11758189—24no—
53C2×#11758189—42no—
54C2×#11758189—42no—
55C22×#92418C2312yes—
56#14519772—28no—
57#14519768—22no—
58#14519768C2×C414yes—
59#14519766—22no—
60#14519766C2×C414yes—
61#14519762—28no—
62#14519753—22no—
63#14519753C2×C414yes—
64#14519752—22no—
65#14519752C2×C414yes—
66C4×D244—28no—
67C4×D244—22no—
68C4×D244C2×C414yes—
69#14519734—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61C2×C414yes—
86C42×D61D414yes—
87C4⋊#92418—22no—
88#14519706—24no—
89#14519701—42no—
90#14519699—42no—
91#14519697—42no—
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×C414yes—
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×D61—21no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61C2×D411yescharacteristic
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—24no—
133C2×D244—21no—
134C2×D244—44no—
135C2×D244C22×C412yes—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—44no—
144C2×C4×D61C22⋊C414yes—
145C2×C4×D61C22⋊C414yes—
146C2×C4×D61C22×C412yes—
147C2×C4×D61C2×D412yes—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61C22×C412yes—
156C42×C61—24no—
157C42×C61C22×C412yes—
158C42×C61C2×D412yes—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164#11758198—44no—
165C4⋊Dic61—21no—
166C4⋊Dic61—24no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C22×C412yes—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61—44no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180#11758189—48no—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61C2×C22⋊C412yes—
192Q8×C61—22no—
193Q8×C61—22no—
194Q8×C61—41no—
195Q8×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61C2×C22⋊C412yes—
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226D244—22no—
227D244—22no—
228D244—48no—
229D244—41no—
230D244—41no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—44no—
236C4×D61—44no—
237C4×D61—48no—
238C8×C61—42no—
239#92418—42no—
240#21895—611nomaximal, Sylow
241C22×C61C22×C22⋊C411yescharacteristic
242C22×C61#53711yescharacteristic
243C22×C61#53711yescharacteristic
244C22×C61#57512yes—
245C22×C61#57512yes—
246C2×D61—22no—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—82no—
255C2×D61—82no—
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—42no—
259C4×C61—42no—
260C4×C61—82no—
261C4×C61#52011yescharacteristic
262C4×C61#52011yescharacteristic
263C4×C61#53511yescharacteristic
264C4×C61#53611yescharacteristic
265Dic61—22no—
266Dic61—22no—
267Dic61—22no—
268Dic61—42no—
269Dic61—82no—
270C2×#529—611no—
271C2×#423—611no—
272#2283—611no—
273#1989—612no—
274#1983—612no—
275#1895—612no—
276#1894—612no—
277#1805—611no—
278#1805D6111yescharacteristic, complemented
279#1804—611no—
280#1804—611no—
281C2×C61#231111yescharacteristic
282C2×C61#289611yescharacteristic
283C2×C61#289611yescharacteristic
284C2×C61#325612yes—
285C2×C61#325612yes—
286D61—42no—
287D61—82no—
288C22×M16—612no—
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×Q8C2×D6111yescharacteristic
298C2×C4×D4—611no—
299C2×C4×D4—1221no—
300C2×#117—1221no—
301C22×C42—611no—
302#423—1224no—
303C2×C4⋊C8—612no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8C2×D6111yescharacteristic
306C2×#103—612no—
307C2×#103C2×D6112yes—
308C2×#102—612no—
309C2×#102—612no—
310#383—1221no—
311#382—611no—
312#382C2×D6111yescharacteristic
313#381—1221no—
314#380—612no—
315C4×C4⋊C4—612no—
316C4×C22⋊C4—1221no—
317#376—1221no—
318#340—612no—
319#340—612no—
320#340—612no—
321#340C2×D6112yes—
322C61#2189511yescharacteristic, complemented, Sylow
323C22×Q8C22×D6111yescharacteristic
324C22×D4—611no—
325C22×D4—1221no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—1221no—
330C2×M16—1224no—
331C2×M16—2442no—
332C2×M16—2442no—
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—22no—
346C4×Q8C4×D6114yes—
347C4×D4—614no—
348C4×D4—1228no—
349C4×D4—1222no—
350#117—1228no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—612no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4C22×D6112yes—
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C22⋊C4—1221no—
364C2×C22⋊C4—1222no—
365C2×C42—21no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—614no—
370C2×C42—614no—
371C2×C42—1222no—
372C2×C42C22×D6111yescharacteristic
373C4⋊C8—22no—
374C4⋊C8—1224no—
375C4⋊C8—1222no—
376#103—42no—
377#103—2442no—
378#102—2442no—
379#102—2442no—
380#95—21no—
381#95—21no—
382#95—1222no—
383#95—1222no—
384#95—1221no—
385#95—1221no—
386C24—611no—
387C24—1221no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8C2×C4×D6112yes—
394C2×D4—612no—
395C2×D4—1224no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2444no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—614no—
411C22×C4—614no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1224no—
418C22×C4—1222no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2444no—
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425M16—2448no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—612no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4—1224no—
440C4⋊C4—1221no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4C2×C4×D6112yes—
444C22⋊C4—1224no—
445C22⋊C4—1224no—
446C22⋊C4—1224no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—2444no—
450C42—24no—
451C42—612no—
452C42—612no—
453C42—614no—
454C42—614no—
455C42—2444no—
456C42C2×C4×D6112yes—
457C42D4×D6112yes—
458C23—612no—
459C23—1222no—
460C23—1222no—
461C23—1221no—
462C23—1221no—
463C23—1222no—
464C23—2442no—
465C23—2442no—
466C23—2441no—
467C23—2441no—
468C23C2×D4×D6111yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—1222no—
474D4—1222no—
475D4—2448no—
476D4—2441no—
477D4—2441no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
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—2444no—
501C2×C4—2444no—
502C2×C4—2448no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4C22⋊C4×D6112yes—
508C2×C4C22⋊C4×D6112yes—
509C2×C4C22×C4×D6111yescharacteristic
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4#1451982211yescharacteristic
512C2×C4#1451982311yescharacteristic
513C2×C4#1451982411yescharacteristic
514C2×C4#1451982511yescharacteristic
515C8—42no—
516C8—2442no—
517C22—1222no—
518C22—1222no—
519C22—1222no—
520C22—4882no—
521C22—4882no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22C2×C22⋊C4×D6111yescharacteristic
528C22#1684034111yescharacteristic
529C22#1684034211yescharacteristic
530C22#1684055912yes—
531C22#1684056112yes—
532C4—22no—
533C4—22no—
534C4—42no—
535C4—42no—
536C4—82no—
537C4—1222no—
538C4—1222no—
539C4—1222no—
540C4—4882no—
541C4—2442no—
542C4#1684024211yescharacteristic
543C4#1684024311yescharacteristic
544C4#1684032811yescharacteristic
545C4#1684033511yescharacteristic
546C2—4882no—
547C2—2442no—
548C2#2144840811yescharacteristic
549C2#2145433411yescharacteristic
550C2#2145433711yescharacteristic
551C2#2145721812yes—
552C2#2145722212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#21895
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
2. C2×#16839784C2semidirect
3. #21448212C2semidirect
4. #21446773C2semidirect
5. #21446748C2semidirect
7. #21446249C2semidirect
8. #21445834C2semidirect
9. #21445830C2semidirect
10. #21445826C2semidirect
11. C61×#1805C2semidirect
14. #16840299C4semidirect
278. #1805D61semidirect
322. C61#21895semidirect

Maximal subgroups

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

subgroupindex
C2×#168402992
C2×#168397842
#214482122
#214467732
#214467482
#214462562
#214462492
#214458342
#214458302
#214458262
C61×#18052
#2189561