Finite Group Databasebeta

← #29713855

Subgroups — #29713855

Order 15616.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic
2C2×#16840098C211yesmaximal, characteristic, complemented
3C2×#16839804C211yesmaximal, characteristic, complemented
4#21450127C211yesmaximal, characteristic
5#21447158C212yesmaximal, complemented
6#21447016C212yesmaximal, complemented
7#21446252C211yesmaximal, characteristic, complemented
8#21446076C212yesmaximal, complemented
9#21446009C212yesmaximal, complemented
10#21445915C211yesmaximal, characteristic, complemented
11C61×#1870C211yesmaximal, characteristic
12#16840452—24no—
13#16840452—24no—
14C2×#14519770—21no—
15C2×#14519769C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519758—21no—
18C2×#14519758—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic
26#16840098—24no—
27#16839804—24no—
28C2×#14519711C2212yes—
29C2×#14519711C2212yes—
30C2×#14519709C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33#16839692—21no—
34#16839687—21no—
35#16839660C2212yes—
36#16839623—21no—
37#16839622C2211yescharacteristic
38#16839622C2212yes—
39#16839525C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C22×M16×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×C22⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C23×Dic61—21no—
50#14519770—28no—
51#14519769—22no—
52#14519769—22no—
53#14519767—28no—
54#14519758—28no—
55#14519758—44no—
56#14519756—22no—
57#14519756—22no—
58#14519756—22no—
59#14519756D414yes—
60C4×Dic122—22no—
61C4×Dic122—22no—
62#14519736—22no—
63#14519736D414yes—
64#14519733—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—22no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89#14519711—24no—
90#14519711—24no—
91#14519709—24no—
92#14519702—24no—
93#14519702—24no—
94#14519684—21no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61D411yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C2×M16×C61—22no—
101C2×M16×C61—24no—
102C22×C8×C61C2312yes—
103C22×C8×C61C2312yes—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61D411yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C22⋊C8×C61—22no—
111C22⋊C8×C61—24no—
112C24×C61C2×D411yescharacteristic
113C24×C61C2×D411yescharacteristic
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—41no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—24no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—21no—
145C2×Dic122C2×D412yes—
146M16×C61—28no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61#4114yes—
153C4⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—21no—
157C42×C61—21no—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160C22⋊Dic61—24no—
161C22⋊Dic61—42no—
162C22⋊Dic61—44no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61C2×D412yes—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×D412yes—
175#11758196—24no—
176#11758196—24no—
177#11758196—24no—
178#11758196—24no—
179#11758196—24no—
180#11758196—44no—
181C4×Dic61—24no—
182C4×Dic61—24no—
183C23×C61—21no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61#12012yes—
192C23×C61AGL(1,ℤ/8)12yes—
193C23×C61C22×D411yescharacteristic
194D4×C61—44no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61#12012yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4112yes—
213C2×C4×C612−1+412yes—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—48no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—84no—
232Dic122—24no—
233Dic122—41no—
234Dic122—41no—
235C8×C61—24no—
236C8×C61—42no—
237#24110—611nomaximal, Sylow
238C22×C61—24no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61C2×#12011yescharacteristic
249C22×C61#54712yes—
250C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C61#57511yescharacteristic
252C22×C61#57812yes—
253C4×C61—22no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—41no—
257C4×C61#52511yescharacteristic
258C4×C61#53611yescharacteristic
259C4×C61#54712yes—
260Dic61—42no—
261Dic61—42no—
262Dic61—42no—
263Dic61—42no—
264Dic61—82no—
265Dic61—82no—
266C2×#552—611no—
267C2×#484—611no—
268C2×#428—611no—
269#2486—611no—
270#2059—612no—
271#2035—612no—
272#1894—611no—
273#1870D6111yescharacteristic
274#1863—612no—
275#1852—612no—
276#1825—611no—
277C2×C61—42no—
278C2×C61—42no—
279C2×C61#240611yescharacteristic
280C2×C61#300611yescharacteristic
281C2×C61#301111yescharacteristic
282C2×C61#334312yes—
283C2×C61#335212yes—
284C22×M16C2×D6111yescharacteristic
285#552—1224no—
286#552—1224no—
287C2×#126—1221no—
288C2×#125—611no—
289C2×#125—1221no—
290C2×#123—1221no—
291C2×#123—1221no—
292C2×C22⋊Q8—611no—
293C2×C22⋊Q8—611no—
294C2×C4×Q8—611no—
295C2×C4×D4C2×D6111yescharacteristic
296C2×#117—611no—
297C2×#117—1221no—
298C22×C4⋊C4—611no—
299C22×C4⋊C4—611no—
300#484—1224no—
301#428—1224no—
302C2×#107—612no—
303C2×#107—612no—
304C2×#106—612no—
305C2×#103—612no—
306C2×#103—612no—
307C2×C22⋊C8C2×D6111yescharacteristic
308C2×C22⋊C8C2×D6112yes—
309#398—1221no—
310#397—1221no—
311#389—612no—
312#381—1221no—
313#380—611no—
314#380—612no—
315#340—612no—
316#340—612no—
317#340—612no—
318#336C2×D6112yes—
319C61#2411011yescharacteristic, complemented, Sylow
320C22×Q8—611no—
321C22×D4D24411yescharacteristic
322C23×C4—1221no—
323C23×C4—1221no—
324C23×C4C22×D6111yescharacteristic
325C23×C4C22×D6111yescharacteristic
326C2×M16—22no—
327C2×M16—24no—
328C22×C8C22×D6112yes—
329C22×C8C22×D6112yes—
330#126—1228no—
331#125—1222no—
332#125—1222no—
333#125—1228no—
334#123—1228no—
335#123—2444no—
336C22⋊Q8—614no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C22⋊Q8—1222no—
340C4×Q8—1222no—
341C4×Q8—1222no—
342C4×D4—22no—
343C4×D4—22no—
344#117—614no—
345#117—1228no—
346#117—1222no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4D24411yescharacteristic
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C22⋊C4—1221no—
369C2×C22⋊C4—1222no—
370C2×C22⋊C4D24411yescharacteristic
371C2×C22⋊C4D24411yescharacteristic
372C2×C42—1221no—
373C2×C42—1221no—
374C2×C42C22×D6111yescharacteristic
375#107—1224no—
376#107—1224no—
377#106—1224no—
378#103—1224no—
379#103—1224no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382#95—1222no—
383#95—1221no—
384#95—1222no—
385C24C2×D24411yescharacteristic
386C24C2×D24411yescharacteristic
387C2×Q8—612no—
388C2×Q8—1221no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—1224no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4—2442no—
412C22×C4C2×D24411yescharacteristic
413C22×C4C2×D24411yescharacteristic
414C22×C4C2×D24411yescharacteristic
415C22×C4C2×D24411yescharacteristic
416C22×C4D4×D6112yes—
417C22×C4D4×D6112yes—
418C22×C4C23×D6111yescharacteristic
419M16—28no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8#1175822414yes—
426C4⋊C4—22no—
427C4⋊C4—612no—
428C4⋊C4—612no—
429C4⋊C4—612no—
430C4⋊C4—1224no—
431C4⋊C4—1224no—
432C4⋊C4—1224no—
433C4⋊C4—1224no—
434C4⋊C4—1224no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—2444no—
444C4⋊C4—2442no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—1224no—
448C22⋊C4—1224no—
449C22⋊C4—1224no—
450C22⋊C4—2442no—
451C22⋊C4—2444no—
452C42—21no—
453C42—21no—
454C42—1224no—
455C42—1224no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23#1451974712yes—
465C23#1451980512yes—
466C23C22×D24411yescharacteristic
467Q8—1224no—
468Q8—2441no—
469Q8—2441no—
470D4—44no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—4884no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4—2442no—
496C2×C4—2448no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4#1451974712yes—
504C2×C4C2×D4×D6111yescharacteristic
505C2×C4C2×D4×D6111yescharacteristic
506C2×C4C2×#1175822412yes—
507C2×C4#1451982412yes—
508C8—24no—
509C8—42no—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—42no—
517C22—41no—
518C22—42no—
519C22—42no—
520C22C2×#1451974711yescharacteristic
521C22#1684041612yes—
522C22C2×#1451980511yescharacteristic
523C22#1684056211yescharacteristic
524C22#1684057912yes—
525C4—22no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—4882no—
530C4—4882no—
531C4—2442no—
532C4—2442no—
533C4—2442no—
534C4—2442no—
535C4#1684026811yescharacteristic
536C4#1684033411yescharacteristic
537C4#1684041212yes—
538C2—42no—
539C2—42no—
540C2#2144915311yescharacteristic
541C2#2145507311yescharacteristic
542C2#2145511611yescharacteristic
543C2#2145826112yes—
544C2#2145839312yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1870C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1870D61
hypocenterC61#24110
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#16840098C2semidirect
3. C2×#16839804C2semidirect
5. #21447158C2semidirect
6. #21447016C2semidirect
7. #21446252C2semidirect
8. #21446076C2semidirect
9. #21446009C2semidirect
10. #21445915C2semidirect
319. C61#24110semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168400982
C2×#168398042
#214501272
#214471582
#214470162
#214462522
#214460762
#214460092
#214459152
C61×#18702
#2411061