Finite Group Databasebeta

← #29648561

Subgroups — #29648561

Order 15616.

952 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×#16840317C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic
3#432×D61C211yesmaximal, characteristic
4#21444548C214yesmaximal, complemented
5#21444544C211yesmaximal, characteristic, complemented
6#21444544C211yesmaximal, characteristic, complemented
7#21444536C211yesmaximal, characteristic, complemented
8#21444475C214yesmaximal, complemented
9C61×#1556C211yesmaximal, characteristic, complemented
10#16840317—21no—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317C412yes—
15#16840317C412yes—
16#16840317C2212yes—
17#16840317C2212yes—
18C2×#14519777C2211yescharacteristic
19C2×#14519768—21no—
20C2×#14519757C412yes—
21C2×#14519753—21no—
22C2×Q8×Dic61—21no—
23C2×#14519742C412yes—
24C2×#14519735—21no—
25C2×C42×D61C2211yescharacteristic
26#16839819C2211yescharacteristic
27#16839819C2211yescharacteristic
28#16839818C2212yes—
29#16839817—21no—
30#16839817—21no—
31#16839816—21no—
32#16839814C2212yes—
33#16839812—21no—
34#16839784C2212yes—
35C4⋊C8×D61C2212yes—
36#16839724—21no—
37C22⋊C8×D61—21no—
38#16839714—21no—
39#97×D61—21no—
40#16839510C2214yescomplemented
41#16839510C2214yescomplemented
42#16839508C2214yescomplemented
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#432C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C61×#332C2214yescomplemented
47C22×Dic122C2×C412yes—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C2×#11758189—22no—
52C2×C8×D61—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2×C412yes—
56#14519777C2312yes—
57#14519768—22no—
58#14519768—22no—
59#14519768—22no—
60#14519768—22no—
61#14519757—22no—
62#14519757—22no—
63#14519757C2×C414yes—
64#14519757C2×C414yes—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68#14519753—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73#14519742—22no—
74#14519742—22no—
75#14519742C2×C414yes—
76#14519742C2×C414yes—
77#14519735—22no—
78#14519735—22no—
79#14519735—22no—
80#14519735—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—22no—
84C2×#11758196—22no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C42×D61C2×C411yescharacteristic
91C42×D61C2×C411yescharacteristic
92C42×D61D411yescharacteristic
93C42×D61D411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61C2311yescharacteristic
97C42×D61C2311yescharacteristic
98C4⋊#92418C2312yes—
99C4⋊#92418C2312yes—
100#14519706—22no—
101#14519706—22no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104#14519689—22no—
105#14519689—22no—
106#14519688—21no—
107#14519688—21no—
108#14519687—22no—
109C8×Dic61—22no—
110C22×Q8×C61—21no—
111C4⋊Q8×C61—21no—
112C4⋊Q8×C61—21no—
113C4⋊Q8×C61C2×C412yes—
114C4⋊Q8×C61C2312yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2×C412yes—
117C2×C42×C61C2311yescharacteristic
118C4⋊C8×C61C2312yes—
119C4⋊C8×C61C2312yes—
120C22⋊C8×C61—21no—
121C22⋊C8×C61—21no—
122#97×C61—21no—
123#97×C61—21no—
124C23×D61C22⋊C411yescharacteristic
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—44no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61C22⋊C411yescharacteristic
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C22⋊C411yescharacteristic
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22×C411yescharacteristic
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×Dic122—22no—
152C2×Dic122—44no—
153C2×Dic122C22×C414yes—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C22×C414yes—
160C42×C61C22×C411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2411yescharacteristic
164#11758198—24no—
165#11758198—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—24no—
171C4⋊Dic61—24no—
172#11758196—24no—
173#11758196—24no—
174C4×Dic61—24no—
175C4×Dic61—24no—
176C4×Dic61C22⋊C411yescharacteristic
177C4×Dic61C22⋊C411yescharacteristic
178C4×Dic61C22⋊C411yescharacteristic
179C4×Dic61C22⋊C411yescharacteristic
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22×C411yescharacteristic
183C4×Dic61C22×C411yescharacteristic
184C4×Dic61C2×D411yescharacteristic
185C4×Dic61C2×D411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C2×#92418—22no—
189C2×#92418—22no—
190#11758189—44no—
191C8×D61—44no—
192C23×C61C2×C22⋊C411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—41no—
196C22×D61C2×C22⋊C411yescharacteristic
197C22×D61C2×C22⋊C411yescharacteristic
198Q8×C61—44no—
199Q8×C61—44no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—44no—
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61C2×C22⋊C411yescharacteristic
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—41no—
215C2×Dic61—44no—
216C2×Dic61C2×C22⋊C411yescharacteristic
217C2×Dic61C2×C22⋊C411yescharacteristic
218C2×Dic61C2×C22⋊C411yescharacteristic
219C2×Dic61C2×C22⋊C411yescharacteristic
220C2×Dic61C2×C22⋊C411yescharacteristic
221C2×Dic61C2×C22⋊C411yescharacteristic
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—22no—
225C4×D61—22no—
226C4×D61—22no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—41no—
233C4×D61#13714yes—
234Dic122—44no—
235Dic122—44no—
236C8×C61—42no—
237C8×C61—42no—
238#92418—42no—
239#92418—42no—
240#17734—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61#41312yes—
244C22×C61C22×C22⋊C411yescharacteristic
245C2×D61—21no—
246C2×D61—21no—
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—44no—
251C2×D61#41912yes—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—44no—
257C4×C61C2×#13712yes—
258C4×C61#57512yes—
259Dic61—21no—
260Dic61—21no—
261Dic61—21no—
262Dic61—21no—
263Dic61—21no—
264Dic61—44no—
265Dic61#41912yes—
266C2×#533—611no—
267#2920—611no—
268C2×#432—611no—
269#1559—614no—
270#1558—611no—
271#1558—611no—
272#1556—611no—
273#1556D6111yescharacteristic, complemented
274#1542—614no—
275C2×C61—22no—
276C2×C61C2×#41311yescharacteristic
277C2×C61C2×#41911yescharacteristic
278C2×C61#289611yescharacteristic
279D61—22no—
280D61—41no—
281#533—612no—
282#533—612no—
283#533—612no—
284#533—612no—
285#533—1221no—
286#533—1221no—
287#533—1221no—
288#533—1221no—
289C2×C4⋊Q8—611no—
290C2×C4⋊Q8C2×D6111yescharacteristic
291C2×#125—1221no—
292C2×C22⋊Q8—612no—
293C2×C22⋊Q8—1221no—
294C2×C4×Q8—612no—
295C2×C4×Q8—1221no—
296C2×#117—1221no—
297C22×C42—611no—
298#432—611no—
299#432—611no—
300#432—612no—
301#432—1221no—
302#432—1221no—
303#432—1221no—
304#432C2×D6111yescharacteristic
305#432C2×D6111yescharacteristic
306#431—612no—
307#431—1221no—
308#423—612no—
309C2×C4⋊C8—612no—
310#407—1221no—
311C2×C22⋊C8—1221no—
312C4×M16—1221no—
313C2×#97—1221no—
314#333—614no—
315#333—614no—
316#332—614no—
317#332C2×D6114yescomplemented
318C61#1773411yescharacteristic, complemented, Sylow
319C22×Q8—21no—
320C22×Q8—612no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C2×M16—1222no—
325C22×C8—1222no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—612no—
329C4⋊Q8—612no—
330C4⋊Q8—1221no—
331C4⋊Q8—1221no—
332C4⋊Q8C4×D6112yes—
333C4⋊Q8C22×D6112yes—
334#125—1222no—
335#125—1222no—
336#125—1222no—
337#125—1222no—
338C22⋊Q8—614no—
339C22⋊Q8—614no—
340C22⋊Q8—1222no—
341C22⋊Q8—1222no—
342C22⋊Q8—1222no—
343C22⋊Q8—1222no—
344C22⋊Q8—1222no—
345C22⋊Q8—1222no—
346C4×Q8—614no—
347C4×Q8—614no—
348C4×Q8—1222no—
349C4×Q8—1222no—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×Q8—1222no—
353C4×Q8—1222no—
354#117—1222no—
355#117—1222no—
356#117—1222no—
357#117—1222no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C4×D6112yes—
364C2×C22⋊C4—1222no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—611no—
371C2×C42—611no—
372C2×C42—611no—
373C2×C42—611no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—1222no—
377C2×C42C22×D6111yescharacteristic
378C4⋊C8—612no—
379C4⋊C8—612no—
380C4⋊C8—1222no—
381C4⋊C8—1222no—
382C4⋊C8C22×D6112yes—
383C4⋊C8C22×D6112yes—
384C22⋊C8—21no—
385C22⋊C8—21no—
386C22⋊C8—1222no—
387C22⋊C8—1222no—
388C22⋊C8—1221no—
389C22⋊C8—1221no—
390#97—21no—
391#97—21no—
392#97—1222no—
393#97—1221no—
394#97—1221no—
395C4×C8—1222no—
396C24—611no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—614no—
401C2×Q8—1222no—
402C2×Q8—2444no—
403C22×C4—22no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4C2×C4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4D4×D6111yescharacteristic
426M16—2444no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—2444no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—1224no—
436C4⋊C4—1224no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4C2×C4×D6114yes—
444C22⋊C4—1224no—
445C22⋊C4—1224no—
446C42—611no—
447C42—611no—
448C42—611no—
449C42—611no—
450C42—611no—
451C42—611no—
452C42—611no—
453C42—611no—
454C42—611no—
455C42—611no—
456C42—611no—
457C42—611no—
458C42—1224no—
459C42—1224no—
460C42C2×C4×D6111yescharacteristic
461C42D4×D6111yescharacteristic
462C42D4×D6111yescharacteristic
463C42C23×D6111yescharacteristic
464C23—611no—
465C23—611no—
466C23—1222no—
467C23—1222no—
468C23—2441no—
469C23C22⋊C4×D6111yescharacteristic
470Q8—44no—
471Q8—44no—
472Q8—2444no—
473Q8—2444no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—44no—
479C2×C4—611no—
480C2×C4—614no—
481C2×C4—611no—
482C2×C4—611no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1224no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2444no—
502C2×C4C22⋊C4×D6111yescharacteristic
503C2×C4C22⋊C4×D6111yescharacteristic
504C2×C4C22⋊C4×D6111yescharacteristic
505C2×C4C22×C4×D6111yescharacteristic
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C8—42no—
509C8—42no—
510C8—2442no—
511C8—2442no—
512C22—21no—
513C22—21no—
514C22—612no—
515C22—1221no—
516C22—1221no—
517C22—1221no—
518C22—1221no—
519C22—1221no—
520C22—2444no—
521C22#1683974512yes—
522C22C2×C22⋊C4×D6111yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—612no—
529C4—1221no—
530C4—1221no—
531C4—1221no—
532C4—1221no—
533C4—1221no—
534C4—2444no—
535C4#137×D6112yes—
536C4#1684055812yes—
537C2—22no—
538C2—1222no—
539C2—2441no—
540C2C2×#1683974511yescharacteristic
541C2#419×D6111yescharacteristic
542C2#2145433311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1556C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1556D61
hypocenterC61#17734
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. #21444548C2semidirect
5. #21444544C2semidirect
6. #21444544C2semidirect
7. #21444536C2semidirect
8. #21444475C2semidirect
9. C61×#1556C2semidirect
40. #16839510C22semidirect
41. #16839510C22semidirect
42. #16839508C22semidirect
46. C61×#332C22semidirect
273. #1556D61semidirect
317. #332C2×D61semidirect
318. C61#17734semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544492
#432×D612
#214445482
#214445442
#214445362
#214444752
C61×#15562
#1773461