Finite Group Databasebeta

← #29648652

Subgroups — #29648652

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic
3#423×D61C211yesmaximal, characteristic
4#21444548C214yesmaximal, complemented
5#21444508C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444499C211yesmaximal, characteristic, complemented
8#21444491C211yesmaximal, characteristic, complemented
9C61×#1546C211yesmaximal, characteristic, complemented
10C2×M16×D61—21no—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317—21no—
15#16840317C412yes—
16#16840317C412yes—
17#16840317C2212yes—
18#16840317C2212yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519768—21no—
21C2×#14519757C412yes—
22C2×#14519753—21no—
23C2×Q8×Dic61—21no—
24C2×#14519742C412yes—
25C2×#14519735—21no—
26C2×C42×D61C2211yescharacteristic
27#16839818C2212yes—
28#16839816—21no—
29#16839815C2211yescharacteristic
30#16839813—21no—
31#16839786C2211yescharacteristic
32#16839785—21no—
33#16839784—21no—
34#16839784C2212yes—
35#16839784C2212yes—
36C4⋊C8×D61—21no—
37C4⋊C8×D61C2212yes—
38#16839724—21no—
39C4×#11758189—21no—
40#16839510C2214yescomplemented
41#16839510C2214yescomplemented
42#16839508C2214yescomplemented
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#332C2214yescomplemented
47C22×Dic122C2×C412yes—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C2×#11758194—21no—
52C2×#11758189—22no—
53C2×#11758189—22no—
54M16×D61—44no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2×C412yes—
59#14519777C2312yes—
60#14519768—22no—
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519757—22no—
65#14519757—22no—
66#14519757C2×C414yes—
67#14519757C2×C414yes—
68#14519753—22no—
69#14519753—22no—
70#14519753—22no—
71#14519753—22no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76#14519742—22no—
77#14519742—22no—
78#14519742C2×C414yes—
79#14519742C2×C414yes—
80#14519735—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—22no—
87C2×#11758196—22no—
88C2×#11758198—22no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C42×D61C2×C411yescharacteristic
94C42×D61C2×C411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61C2311yescharacteristic
100C42×D61C2311yescharacteristic
101C4⋊#92418—21no—
102C4⋊#92418C2312yes—
103C4⋊#92418C2312yes—
104#14519706—22no—
105#14519706—22no—
106#14519706—22no—
107C22⋊#92418—21no—
108#14519689—22no—
109#14519687—22no—
110C4×#92418—21no—
111C22×Q8×C61—21no—
112C2×M16×C61—21no—
113C4⋊Q8×C61—21no—
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61C2×C412yes—
116C4⋊Q8×C61C2312yes—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2×C412yes—
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—21no—
121C4⋊C8×C61C2312yes—
122C4⋊C8×C61C2312yes—
123C22⋊C8×C61—21no—
124C4×C8×C61—21no—
125C23×D61C22⋊C411yescharacteristic
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—42no—
129C2×Q8×C61—42no—
130C22×C4×C61—22no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C22⋊C411yescharacteristic
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61C22⋊C411yescharacteristic
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C22×C411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×Dic122—22no—
155C2×Dic122—44no—
156C2×Dic122C22×C414yes—
157M16×C61—42no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C22×C414yes—
164C42×C61C22×C411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168#11758198—24no—
169#11758198—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—24no—
175C4⋊Dic61—24no—
176#11758196—24no—
177#11758196—24no—
178C4×Dic61—24no—
179C4×Dic61—24no—
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22×C411yescharacteristic
187C4×Dic61C22×C411yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C4×Dic61C2×D411yescharacteristic
192#11758194—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195#11758189—44no—
196#11758189—44no—
197C8×D61—44no—
198C23×C61C2×C22⋊C411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—41no—
202C22×D61C2×C22⋊C411yescharacteristic
203C22×D61C2×C22⋊C411yescharacteristic
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—44no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—44no—
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61#13712yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—41no—
223C2×Dic61—44no—
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C2×C22⋊C411yescharacteristic
227C2×Dic61C2×C22⋊C411yescharacteristic
228C2×Dic61C2×C22⋊C411yescharacteristic
229C2×Dic61C2×C22⋊C411yescharacteristic
230C4×D61—22no—
231C4×D61—21no—
232C4×D61—21no—
233C4×D61—22no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61#13714yes—
244Dic122—44no—
245Dic122—44no—
246C8×C61—42no—
247C8×C61—42no—
248#92418—42no—
249#92418—42no—
250#17749—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61#41712yes—
254C22×C61C22×C22⋊C411yescharacteristic
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—44no—
261C2×D61#41912yes—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—44no—
265C4×C61#41312yes—
266C4×C61#41712yes—
267C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C4×C61C2×#13712yes—
269C4×C61C2×#13711yescharacteristic
270Dic61—21no—
271Dic61—21no—
272Dic61—21no—
273Dic61—21no—
274Dic61—21no—
275Dic61—44no—
276Dic61#41912yes—
277C2×#533—611no—
278#2920—611no—
279C2×#423—611no—
280#1559—614no—
281#1550—614no—
282#1548—611no—
283#1548—611no—
284#1546—611no—
285#1546D6111yescharacteristic, complemented
286C2×C61—22no—
287C2×C61C2×#41311yescharacteristic
288C2×C61C2×#41911yescharacteristic
289C2×C61C2×#41711yescharacteristic
290D61—22no—
291D61—41no—
292C22×M16—1221no—
293#533—612no—
294#533—612no—
295#533—612no—
296#533—612no—
297#533—1221no—
298#533—1221no—
299#533—1221no—
300#533—1221no—
301C2×C4⋊Q8—611no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303C2×#125—1221no—
304C2×C22⋊Q8—612no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8—612no—
307C2×C4×Q8—1221no—
308C2×#117—1221no—
309C22×C42—611no—
310#432—612no—
311#432—1221no—
312#431—611no—
313#431—1221no—
314#431C2×D6111yescharacteristic
315#423—611no—
316#423—612no—
317#423—612no—
318#423—1221no—
319#423—1221no—
320#423C2×D6111yescharacteristic
321C2×C4⋊C8—612no—
322C2×C4⋊C8—1221no—
323#407—1221no—
324C4×M16—1221no—
325#333—614no—
326#333—614no—
327#332—614no—
328#332C2×D6114yescomplemented
329C61#1774911yescharacteristic, complemented, Sylow
330C22×Q8—21no—
331C22×Q8—612no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C2×M16—21no—
336C2×M16—1222no—
337C2×M16—1222no—
338C2×M16—1221no—
339C2×M16—2444no—
340C22×C8—1222no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—612no—
344C4⋊Q8—612no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8C4×D6112yes—
348C4⋊Q8C22×D6112yes—
349#125—1222no—
350#125—1222no—
351#125—1222no—
352#125—1222no—
353C22⋊Q8—614no—
354C22⋊Q8—614no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C22⋊Q8—1222no—
360C22⋊Q8—1222no—
361C4×Q8—614no—
362C4×Q8—614no—
363C4×Q8—1222no—
364C4×Q8—1222no—
365C4×Q8—1222no—
366C4×Q8—1222no—
367C4×Q8—1222no—
368C4×Q8—1222no—
369#117—1222no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C4×D6112yes—
379C2×C22⋊C4—1222no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—1222no—
392C2×C42C22×D6111yescharacteristic
393C4⋊C8—21no—
394C4⋊C8—612no—
395C4⋊C8—612no—
396C4⋊C8—1222no—
397C4⋊C8—1222no—
398C4⋊C8—1222no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6112yes—
401C4⋊C8C22×D6112yes—
402C22⋊C8—21no—
403C22⋊C8—1222no—
404C22⋊C8—1221no—
405#97—1222no—
406C4×C8—21no—
407C4×C8—1221no—
408C24—611no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—614no—
414C2×Q8—1222no—
415C2×Q8—2444no—
416C22×C4—22no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1224no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—2442no—
437C22×C4C2×C4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440M16—42no—
441M16—2444no—
442M16—2444no—
443M16—2442no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—2444no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—1224no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4C2×C4×D6114yes—
461C22⋊C4—1224no—
462C22⋊C4—1224no—
463C42—611no—
464C42—611no—
465C42—611no—
466C42—611no—
467C42—611no—
468C42—611no—
469C42—611no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—611no—
474C42—611no—
475C42—1224no—
476C42—1224no—
477C42C2×C4×D6111yescharacteristic
478C42D4×D6111yescharacteristic
479C42D4×D6111yescharacteristic
480C42C23×D6111yescharacteristic
481C23—611no—
482C23—611no—
483C23—1222no—
484C23—1222no—
485C23—2441no—
486C23C22⋊C4×D6111yescharacteristic
487Q8—44no—
488Q8—42no—
489Q8—42no—
490Q8—2444no—
491Q8—2444no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—44no—
496C2×C4—611no—
497C2×C4—614no—
498C2×C4—611no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—611no—
502C2×C4—611no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1222no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1224no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2444no—
521C2×C4C22⋊C4×D6111yescharacteristic
522C2×C4C22⋊C4×D6111yescharacteristic
523C2×C4C22⋊C4×D6111yescharacteristic
524C2×C4#1451980112yes—
525C2×C4#1451980612yes—
526C2×C4C22×C4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C8—42no—
530C8—42no—
531C8—2442no—
532C8—2442no—
533C22—21no—
534C22—21no—
535C22—612no—
536C22—1221no—
537C22—1221no—
538C22—1221no—
539C22—1221no—
540C22—1221no—
541C22—2444no—
542C22#1683976012yes—
543C22C2×C22⋊C4×D6111yescharacteristic
544C4—21no—
545C4—21no—
546C4—44no—
547C4—612no—
548C4—1221no—
549C4—1221no—
550C4—1221no—
551C4—1221no—
552C4—1221no—
553C4—2444no—
554C4#1683974512yes—
555C4#1683976012yes—
556C4C2×#1451980111yescharacteristic
557C4#137×D6112yes—
558C4C2×#1451980611yescharacteristic
559C2—22no—
560C2—1222no—
561C2—2441no—
562C2C2×#1683974511yescharacteristic
563C2#419×D6111yescharacteristic
564C2C2×#1683976011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1546D61
hypocenterC61#17749
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #21444548C2semidirect
5. #21444508C2semidirect
6. #21444501C2semidirect
7. #21444499C2semidirect
8. #21444491C2semidirect
9. C61×#1546C2semidirect
40. #16839510C22semidirect
41. #16839510C22semidirect
42. #16839508C22semidirect
46. C61×#332C22semidirect
285. #1546D61semidirect
328. #332C2×D61semidirect
329. C61#17749semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544472
#423×D612
#214445482
#214445082
#214445012
#214444992
#214444912
C61×#15462
#1774961