Finite Group Databasebeta

← #29685211

Subgroups — #29685211

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840294C211yesmaximal, characteristic
2C2×C4×#11758189C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446786C212yesmaximal, complemented
5#21446634C212yesmaximal, complemented
6#21446254C212yesmaximal
7#21446251C212yesmaximal, complemented
8C4×#14519700C211yesmaximal, characteristic, complemented
9C4×#14519702C211yesmaximal, characteristic, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C4×#102×C61C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840294—24no—
14#16840294C418yescomplemented
15C2×#14519770—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519759—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×#14519736—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519698C2212yes—
29C4×#11758189—24no—
30C2×#14519687C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839631C2212yes—
33#16839626—21no—
34#16839621—21no—
35#16839619C2212yes—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—24no—
53C2×#11758189—24no—
54C22×#92418C2312yes—
55#14519770—28no—
56#14519768—22no—
57#14519768C2×C414yes—
58#14519764—22no—
59#14519764C2×C414yes—
60#14519759—28no—
61#14519757—22no—
62#14519757C2×C414yes—
63#14519748—22no—
64#14519748C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122C2×C414yes—
67C4×C22⋊D61—28no—
68#14519736—28no—
69#14519734—44no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90#14519702—24no—
91#14519700—24no—
92#14519698—24no—
93#14519687—24no—
94C4×#92418—22no—
95#14519684—21no—
96#14519684—22no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C22×C8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61C2×C414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#102×C61—24no—
108C4×C8×C61—22no—
109#95×C61—21no—
110C24×C61—21no—
111C23×D61C2×D411yescharacteristic
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
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—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—42no—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—21no—
144C2×Dic122C22×C412yes—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61#4114yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C22×C412yes—
152C22⋊C4×C61—24no—
153C42×C61—42no—
154C42×C61C22×C412yes—
155C42×C61C2×D412yes—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159#11758198—24no—
160#11758198—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C22×C412yes—
167#11758196—24no—
168#11758196—24no—
169#11758196—44no—
170#11758196—44no—
171C4×Dic61—24no—
172C4×Dic61—42no—
173C4×Dic61C22⋊C414yes—
174C4×Dic61C22⋊C414yes—
175C4×Dic61C22×C412yes—
176C4×Dic61C2×D412yes—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418#4114yes—
180#11758189—216no—
181C23×C61—22no—
182C23×C61—42no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—42no—
186C23×C61C22×D411yescharacteristic
187C22×D61—22no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—22no—
191C22×D61C2×C22⋊C412yes—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—42no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—48no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61AGL(1,ℤ/8)12yes—
206C2×C4×C61#13712yes—
207C2×C4×C61C23×C411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4112yes—
210C2×C4×C61C2×#4112yes—
211C22⋊D61—48no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61C2×C22⋊C412yes—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C412yes—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—24no—
229C4×D61—28no—
230C4×D61—44no—
231Dic122—22no—
232Dic122—22no—
233Dic122—42no—
234C8×C61—24no—
235#92418—24no—
236#21403—611nomaximal, Sylow
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—82no—
241C22×C61—82no—
242C22×C61C22×C22⋊C411yescharacteristic
243C22×C61#53212yes—
244C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C61C2×#13711yescharacteristic
246C22×C61#57512yes—
247C2×D61—22no—
248C2×D61—22no—
249C2×D61—22no—
250C2×D61—42no—
251C2×D61—42no—
252C4×C61—22no—
253C4×C61—42no—
254C4×C61—82no—
255C4×C61#41714yes—
256C4×C61C4×#4111yescharacteristic
257C4×C61C4×#4111yescharacteristic
258C4×C61#52512yes—
259Dic61—22no—
260Dic61—22no—
261Dic61—22no—
262Dic61—42no—
263Dic61—42no—
264Dic61—82no—
265C2×#529—611no—
266C2×C4×M16—611no—
267#2285—611no—
268#1992—612no—
269#1959—612no—
270#1895—612no—
271#1894—612no—
272C4×#103—611no—
273C4×#103—611no—
274C4×#102—611no—
275C4×#102D6111yescharacteristic, complemented
276C2×C61—82no—
277C2×C61#229711yescharacteristic
278C2×C61C2×#41711yescharacteristic
279C2×C61#289611yescharacteristic
280C2×C61#317112yes—
281C2×C61#317212yes—
282D61—42no—
283C22×M16—612no—
284#529—618no—
285#529—1224no—
286C2×#126—1221no—
287C2×#125—611no—
288C2×#124—611no—
289C2×#123—1221no—
290C2×C22⋊Q8—611no—
291C2×#121—611no—
292C2×C4×Q8—611no—
293C2×C4×D4—1221no—
294C2×C4×D4C2×D6111yescharacteristic
295C2×#117—1221no—
296C2×#117—1221no—
297C22×C42—611no—
298C2×#103—612no—
299C2×#103—612no—
300C2×#102—612no—
301C2×#102C2×D6112yes—
302C4×M16—1224no—
303C2×#97—612no—
304C2×C4×C8—611no—
305C2×C4×C8C2×D6111yescharacteristic
306#384—612no—
307#382—1221no—
308#380—612no—
309#380—1221no—
310C4×C4⋊C4—611no—
311C4×C4⋊C4—1221no—
312C4×C4⋊C4C2×D6111yescharacteristic
313#340—612no—
314#340—612no—
315#340—612no—
316#340C2×D6112yes—
317C61#2140311yescharacteristic, complemented, Sylow
318C22×Q8—611no—
319C22×D4—1221no—
320C22×D4C22×D6111yescharacteristic
321C23×C4—21no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C2×M16—1224no—
326C2×M16—1224no—
327C2×M16—1224no—
328C22×C8—612no—
329C22×C8C22×D6112yes—
330#126—1228no—
331#125—614no—
332#125—1222no—
333#124—614no—
334#124—1222no—
335#123—1228no—
336C22⋊Q8—614no—
337C22⋊Q8—1222no—
338#121—614no—
339#121—1222no—
340C4×Q8—614no—
341C4×Q8—1222no—
342C4×D4—22no—
343C4×D4—1228no—
344C4×D4C4×D6114yes—
345#117—1228no—
346#117—2444no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4C22×D6112yes—
356C2×C4⋊C4C22×D6111yescharacteristic
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4—1221no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1222no—
363C2×C42—21no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—614no—
368C2×C42—614no—
369C2×C42—1221no—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#103—1224no—
373#103—1224no—
374#102—24no—
375#102—1224no—
376#97—1224no—
377C4×C8—22no—
378C4×C8—1222no—
379#95—21no—
380#95—1222no—
381#95—1221no—
382C24—21no—
383C24—611no—
384C2×Q8—612no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—1224no—
392C2×D4—2444no—
393C2×D4C2×C4×D6112yes—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—612no—
403C22×C4—612no—
404C22×C4—614no—
405C22×C4—614no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—2442no—
414C22×C4D4×D6111yescharacteristic
415C22×C4D4×D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417M16—12216no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—614no—
421C2×C8—1222no—
422C2×C8—1222no—
423C2×C8#1175822214yes—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—612no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1224no—
431C4⋊C4—1221no—
432C4⋊C4—1224no—
433C4⋊C4—1222no—
434C4⋊C4—1222no—
435C4⋊C4—2444no—
436C4⋊C4—2444no—
437C4⋊C4C2×C4×D6112yes—
438C22⋊C4—24no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C22⋊C4—2444no—
444C42—42no—
445C42—612no—
446C42—612no—
447C42—614no—
448C42—614no—
449C42—1224no—
450C42—2442no—
451C42C2×C4×D6112yes—
452C42D4×D6112yes—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—612no—
459C23—1221no—
460C23—1221no—
461C23—1222no—
462C23—1222no—
463C23C2×D4×D6111yescharacteristic
464Q8—1222no—
465Q8—1222no—
466Q8—2442no—
467D4—22no—
468D4—22no—
469D4—42no—
470D4—2448no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1224no—
489C2×C4—1228no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—2444no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4C22⋊C4×D6112yes—
499C2×C4C22⋊C4×D6112yes—
500C2×C4#1451980112yes—
501C2×C4#1451980612yes—
502C2×C4C22×C4×D6111yescharacteristic
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4#41×D6112yes—
505C2×C4C2×#1175822212yes—
506C8—24no—
507C8—1224no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—1222no—
514C22—1222no—
515C22—1222no—
516C22—2442no—
517C22—2442no—
518C22C2×C22⋊C4×D6111yescharacteristic
519C22#1684030912yes—
520C22C2×#1451980111yescharacteristic
521C22C2×#1451980611yescharacteristic
522C22#1684055912yes—
523C4—22no—
524C4—42no—
525C4—82no—
526C4—1222no—
527C4—1222no—
528C4—1222no—
529C4—4882no—
530C4—2442no—
531C4—2442no—
532C4#1683976014yes—
533C4#1684023011yescharacteristic
534C4C4×#1175822211yescharacteristic
535C4#1684027112yes—
536C2—82no—
537C2—2442no—
538C2#2144830511yescharacteristic
539C2C2×#1683976011yescharacteristic
540C2#2145433411yescharacteristic
541C2#2145641212yes—
542C2#2145642112yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenterC4×#102D61
hypocenterC61#21403
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758189C2semidirect
3. #21448223C2semidirect
4. #21446786C2semidirect
5. #21446634C2semidirect
7. #21446251C2semidirect
8. C4×#14519700C2semidirect
9. C4×#14519702C2semidirect
10. C4×#14519698C2semidirect
11. C4×#102×C61C2semidirect
14. #16840294C4semidirect
275. C4×#102D61semidirect
317. C61#21403semidirect

Maximal subgroups

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

subgroupindex
C2×#168402942
C2×C4×#117581892
#214482232
#214467862
#214466342
#214462542
#214462512
C4×#145197002
C4×#145197022
C4×#145196982
C4×#102×C612
#2140361