Finite Group Databasebeta

← #29572086

Subgroups — #29572086

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840432C211yesmaximal, characteristic
2#21445821C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic, complemented
4C2×#16839506C211yesmaximal, characteristic, complemented
5C2×#16839494C211yesmaximal, characteristic, complemented
6C2×#16839491C211yesmaximal, characteristic, complemented
7C61×#1957C211yesmaximal, characteristic, complemented
8C22×#11758224—21no—
9#16840432—22no—
10#16840432—22no—
11#16840432—22no—
12#16840432C414yescomplemented
13C2×#14519773—21no—
14C2×#14519749—22no—
15C2×#14519748—21no—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×D4×Dic61—21no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519700—21no—
22C2×#14519697—21no—
23#16839524—21no—
24#16839524—21no—
25#16839506—22no—
26#16839494—22no—
27#16839491—22no—
28C2×#127×C61C2211yescharacteristic
29C2×#102×C61—22no—
30C2×C4×C8×C61C2211yescharacteristic
31C61×#384C2211yescharacteristic
32C2×#11758224—24no—
33C2×#11758224—24no—
34C2×#11758224—42no—
35C2×#11758224—42no—
36C2×#11758224—42no—
37C2×#11758224—42no—
38C22×Dic122D411yescharacteristic
39C22×D244D411yescharacteristic
40C22×C22⋊D61—22no—
41C22×C22⋊D61—22no—
42C23×Dic61—22no—
43C22×C4×D61—21no—
44C22×#92418—21no—
45C22×#92418—21no—
46#14519773—24no—
47#14519773—24no—
48#14519749—28no—
49#14519749—28no—
50#14519748—24no—
51#14519748—24no—
52C4×Dic122—21no—
53C4×Dic122—21no—
54C4×Dic122C2×C412yes—
55C4×Dic122D412yes—
56C4×D244—21no—
57C4×D244—21no—
58C4×D244C2×C412yes—
59C4×D244D412yes—
60D4×Dic61—24no—
61D4×Dic61—24no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×#11758196—21no—
65C2×C22⋊Dic61—22no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72#14519700—42no—
73#14519697—42no—
74C22×D4×C61—21no—
75C22×D4×C61—21no—
76C22×D4×C61D412yes—
77C22×C8×C61—21no—
78#127×C61—21no—
79#127×C61—21no—
80#127×C61C2×C412yes—
81#127×C61D412yes—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C42×C61C2311yescharacteristic
86#102×C61—44no—
87C4×C8×C61—21no—
88C4×C8×C61—21no—
89C24×C61—22no—
90C23×D61—21no—
91C2×D4×C61—22no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—42no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61C22⋊C414yes—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—28no—
112C2×C22⋊D61—28no—
113C2×C22⋊D61—44no—
114C2×C22⋊D61—44no—
115C2×C22⋊D61—82no—
116C2×C22⋊D61—82no—
117C22×Dic61—21no—
118C22×Dic61—28no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—44no—
122C22×Dic61—44no—
123#11758224—48no—
124#11758224—48no—
125#11758224—48no—
126#11758224—48no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244C22⋊C412yes—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122C22⋊C412yes—
143C2×Dic122D812yes—
144C2×Dic122SD1612yes—
145C2×C8×C61—22no—
146C2×C8×C61—41no—
147C2×C8×C61—41no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61—41no—
153C42×C61C22×C411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C22⋊Dic61—28no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C22⋊C412yes—
163C4⋊Dic61C22⋊C412yes—
164#11758196—22no—
165#11758196—22no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C2×#92418—41no—
169C2×#92418—41no—
170C2×#92418—41no—
171C2×#92418—41no—
172C2×#92418—41no—
173C2×#92418—41no—
174C23×C61—24no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61#12011yescharacteristic
182C22×D61—22no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189D4×C61—44no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—82no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—81no—
202C2×C4×C61—81no—
203C2×C4×C61—81no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61C2×D811yescharacteristic
211C2×C4×C61C2×SD1611yescharacteristic
212C2×C4×C61#13511yescharacteristic
213C2×C4×C61#13511yescharacteristic
214C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
215C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
216C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
217C2×C4×C61#13711yescharacteristic
218C22⋊D61—416no—
219C22⋊D61—88no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—416no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—42no—
238Dic122—22no—
239Dic122—22no—
240Dic122—22no—
241Dic122#10214yescomplemented
242C8×C61—42no—
243#92418—81no—
244#92418—81no—
245#8444—611nomaximal, Sylow
246C22×C61—44no—
247C22×C61—44no—
248C22×C61—44no—
249C22×C61—82no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61#37911yescharacteristic
254C22×C61#44711yescharacteristic
255C22×C61#44911yescharacteristic
256C22×C61#45011yescharacteristic
257C22×C61#45211yescharacteristic
258C22×C61#45311yescharacteristic
259C22×C61#45511yescharacteristic
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—161no—
270C4×C61C2×#10211yescharacteristic
271C4×C61#41611yescharacteristic
272C4×C61#41711yescharacteristic
273C4×C61#41811yescharacteristic
274C4×C61#44711yescharacteristic
275C4×C61#44911yescharacteristic
276C4×C61#45011yescharacteristic
277C4×C61#45211yescharacteristic
278Dic61—24no—
279Dic61—42no—
280Dic61—42no—
281C2×#550—611no—
282#1957D6111yescharacteristic, complemented
283#1803—611no—
284#1802—611no—
285C2×#331—611no—
286C2×Q8⋊C8—611no—
287C2×D4⋊C8—611no—
288C2×C61—84no—
289C2×C61#162111yescharacteristic
290C2×C61#163111yescharacteristic
291C2×C61#164111yescharacteristic
292C2×C61#164211yescharacteristic
293C2×C61#178611yescharacteristic
294C2×C61#178811yescharacteristic
295C2×C61#179211yescharacteristic
296D61—82no—
297C22×#41—1221no—
298#550—614no—
299#550—1222no—
300#550—1222no—
301#550—1222no—
302C2×#127—1221no—
303C2×#127C2×D6111yescharacteristic
304C2×#121—1221no—
305C2×#121—1222no—
306C2×C4×Q8—611no—
307C2×C4×D4—611no—
308C2×C4×D4—1221no—
309C2×C4⋊C8—611no—
310C2×C4⋊C8—611no—
311C2×#103—1221no—
312C2×#102—22no—
313C2×#102—1221no—
314C2×C4×C8C2×D6111yescharacteristic
315#384C2×D6111yescharacteristic
316#340—1221no—
317#340—1221no—
318#331—1222no—
319Q8⋊C8—1222no—
320D4⋊C8—1222no—
321C61#844411yescharacteristic, complemented, Sylow
322C2×#41—1224no—
323C2×#41—1224no—
324C2×#41—2442no—
325C2×#41—2442no—
326C2×#41—2442no—
327C2×#41—2442no—
328C22×Q8—611no—
329C22×D4—21no—
330C22×D4—21no—
331C22×D4—611no—
332C22×D4—1222no—
333C22×D4—1222no—
334C22×D4C22⋊D6112yes—
335C23×C4—1222no—
336C23×C4—1221no—
337C22×C8—21no—
338C22×C8—1221no—
339C22×C8—1221no—
340#127—21no—
341#127—21no—
342#127—1224no—
343#127—1224no—
344#127C4×D6112yes—
345#127D24412yes—
346#121—1224no—
347#121—1224no—
348#121—1228no—
349#121—1228no—
350C4×Q8—612no—
351C4×Q8—612no—
352C4×Q8—1221no—
353C4×Q8—1221no—
354C4×D4—612no—
355C4×D4—612no—
356C4×D4—1224no—
357C4×D4—1224no—
358C4×D4—1221no—
359C4×D4—1221no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1221no—
368C2×C42—1221no—
369C2×C42C22×D6111yescharacteristic
370C4⋊C8—1221no—
371C4⋊C8—1221no—
372C4⋊C8—1221no—
373C4⋊C8—1221no—
374#103—2442no—
375#102—44no—
376#102—2442no—
377C4×C8—21no—
378C4×C8—21no—
379C24—22no—
380C24—1221no—
381#41—2448no—
382#41—2448no—
383#41—2448no—
384#41—2448no—
385C2×Q8—612no—
386C2×Q8—612no—
387C2×Q8—612no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—612no—
407C2×D4—1228no—
408C2×D4—1228no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—4882no—
415C2×D4—4882no—
416C2×D4—2444no—
417C2×D4—2444no—
418C2×D4#1175819814yes—
419C22×C4—41no—
420C22×C4—1221no—
421C22×C4—1228no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—2444no—
427C22×C4—2444no—
428C22×C4—2441no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4—2441no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—2441no—
439C2×C8—2441no—
440C2×C8—2441no—
441C2×C8—2441no—
442C2×C8—2441no—
443C2×C8—2441no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C22⋊C4—1228no—
456C22⋊C4—1222no—
457C22⋊C4—1222no—
458C42—1222no—
459C42—1222no—
460C42C2×C4×D6111yescharacteristic
461C42C2×D24411yescharacteristic
462C42D4×D6111yescharacteristic
463C42D4×D6111yescharacteristic
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—1222no—
472C23—2441no—
473C23—2441no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23#1451974511yescharacteristic
479Q8—614no—
480Q8—1222no—
481Q8—1222no—
482Q8—1222no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—1222no—
494D4—1222no—
495D4—4888no—
496D4—24416no—
497D4—2441no—
498D4—2441no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—24416no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4C22⋊C4×D6111yescharacteristic
519C2×C4C22⋊C4×D6111yescharacteristic
520C2×C4C2×#1175819811yescharacteristic
521C2×C4#1451974511yescharacteristic
522C2×C4#1451974711yescharacteristic
523C2×C4#1451974711yescharacteristic
524C2×C4D8×D6111yescharacteristic
525C2×C4SD16×D6111yescharacteristic
526C2×C4#1451979611yescharacteristic
527C2×C4#1451979811yescharacteristic
528C2×C4#1451980111yescharacteristic
529C2×C4#1451980111yescharacteristic
530C2×C4#1451980411yescharacteristic
531C2×C4#1451980611yescharacteristic
532C8—42no—
533C8—4881no—
534C8—4881no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22#1683961711yescharacteristic
550C22#1683989411yescharacteristic
551C22#1683990411yescharacteristic
552C22#1683991111yescharacteristic
553C22#1683992111yescharacteristic
554C22#1683992911yescharacteristic
555C22#1683993911yescharacteristic
556C4—21no—
557C4—21no—
558C4—161no—
559C4—1224no—
560C4—2442no—
561C4—2442no—
562C4#102×D6111yescharacteristic
563C4#1683975511yescharacteristic
564C4#1683976011yescharacteristic
565C4#1683976511yescharacteristic
566C4#1683989711yescharacteristic
567C4#1683990811yescharacteristic
568C4#1683991411yescharacteristic
569C4#1683992511yescharacteristic
570C2—84no—
571C2—4882no—
572C2#2144488111yescharacteristic
573C2#2144494311yescharacteristic
574C2#2144500911yescharacteristic
575C2#2144501611yescharacteristic
576C2#2144574811yescharacteristic
577C2#2144575611yescharacteristic
578C2#2144577211yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1957C2
socleC23×C61#120
prime radicalC2×#16840432C2
cosocleC2×C42×C61C23
hypercenter#1957D61
hypocenterC61#8444
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. #21445821C2semidirect
3. #21445816C2semidirect
4. C2×#16839506C2semidirect
5. C2×#16839494C2semidirect
6. C2×#16839491C2semidirect
7. C61×#1957C2semidirect
12. #16840432C4semidirect
241. Dic122#102semidirect
282. #1957D61semidirect
321. C61#8444semidirect

Maximal subgroups

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

subgroupindex
C2×#168404322
#214458212
#214458162
C2×#168395062
C2×#168394942
C2×#168394912
C61×#19572
#844461