Finite Group Databasebeta

← #29579141

Subgroups — #29579141

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454122C211yesmaximal, characteristic
2#21445797C211yesmaximal, characteristic, complemented
3C2×#16839499C211yesmaximal, characteristic, complemented
4C2×#16839497C211yesmaximal, characteristic
5C2×#16839495C211yesmaximal, characteristic
6#95×Dic61C211yesmaximal, characteristic, complemented
7C2×C61×#328C211yesmaximal, characteristic, complemented
8C2×#14519766C411yescharacteristic, complemented
9C2×#14519751C2211yescharacteristic
10C22×C4×Dic61—21no—
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519706—21no—
13C2×C22⋊#92418C2211yescharacteristic
14#16839669C411yescharacteristic, complemented
15#16839663C411yescharacteristic, complemented
16#16839652—21no—
17#16839652—21no—
18#16839639—21no—
19#16839639—21no—
20#16839627C411yescharacteristic, complemented
21C2×#14519684—21no—
22C2×#14519684C2211yescharacteristic
23#16839516—21no—
24#16839499—22no—
25#16839497—22no—
26#16839495—22no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×#95×C61C2211yescharacteristic
30C61×#328—22no—
31C22×D244C2×C411yescharacteristic
32C22×C22⋊D61—21no—
33C23×Dic61—21no—
34C23×Dic61D411yescharacteristic
35C23×Dic61D411yescharacteristic
36C22×#92418—21no—
37#14519766—42no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61—24no—
49C2×C4×Dic61—42no—
50C2×C4×Dic61—42no—
51C2×C4×Dic61—42no—
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61C2×C411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56#14519706—42no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59#14519684—24no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684D412yes—
66C22×Q8×C61C2×C411yescharacteristic
67C23×C4×C61—21no—
68C23×C4×C61C2311yescharacteristic
69C22×C8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79#95×C61—21no—
80#95×C61D412yes—
81C24×C61C2×D411yescharacteristic
82C23×D61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—41no—
86C2×Q8×C61—41no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—41no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—41no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61C22⋊C411yescharacteristic
127C22×Dic61C22⋊C411yescharacteristic
128C22×Dic61C22⋊C411yescharacteristic
129C22×Dic61C22⋊C411yescharacteristic
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—41no—
133C2×D244—41no—
134C2×C8×C61—41no—
135C2×C8×C61—41no—
136C2×C8×C61—41no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143#11758198—42no—
144#11758198—41no—
145#11758198—41no—
146#11758198—42no—
147#11758198—42no—
148#11758198—42no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—41no—
154C4×Dic61—41no—
155C4×Dic61—48no—
156C2×#92418—41no—
157C2×#92418—41no—
158C2×#92418—41no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61#12111yescharacteristic
165C23×C61#12111yescharacteristic
166C23×C61#13511yescharacteristic
167C23×C61#13511yescharacteristic
168C23×C61AGL(1,ℤ/8)11yescharacteristic
169C23×C61#13711yescharacteristic
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177Q8×C61—81no—
178Q8×C61—81no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—24no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61C4×D411yescharacteristic
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61#12411yescharacteristic
203C2×C4×C61#12711yescharacteristic
204C22⋊D61—81no—
205C22⋊D61—81no—
206C22⋊D61—81no—
207C22⋊D61—81no—
208C2×Dic61—24no—
209C2×Dic61—24no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—24no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61#9912yes—
232C2×Dic61#10412yes—
233C2×Dic61#10412yes—
234D244—81no—
235D244—81no—
236C8×C61—81no—
237#92418—81no—
238#9713—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61#39011yescharacteristic
252C22×C61C2×#9911yescharacteristic
253C22×C61#41011yescharacteristic
254C22×C61#41611yescharacteristic
255C22×C61#41711yescharacteristic
256C22×C61C2×#10411yescharacteristic
257C22×C61C2×#10411yescharacteristic
258C22×C61#46311yescharacteristic
259C22×C61#46311yescharacteristic
260C22×C61#46411yescharacteristic
261C22×C61#46411yescharacteristic
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—44no—
274C4×C61—81no—
275C4×C61—81no—
276Dic61—22no—
277Dic61—22no—
278Dic61—22no—
279Dic61—44no—
280Dic61—41no—
281Dic61—81no—
282#2850—611no—
283#1797—611no—
284C2×#328—611no—
285C2×#328D6111yescharacteristic, complemented
286C2×#327—611no—
287C2×#327—611no—
288C4×#95—611no—
289C2×C61—21no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61#150011yescharacteristic
294C2×C61#150111yescharacteristic
295C2×C61#150411yescharacteristic
296C2×C61#150911yescharacteristic
297C2×C61#196211yescharacteristic
298C2×C61#196411yescharacteristic
299C2×C61#198311yescharacteristic
300D61—161no—
301C2×#124—611no—
302C2×C22⋊Q8C2×D6111yescharacteristic
303C2×#121—611no—
304C22×C42—611no—
305C22×C42—1221no—
306C2×C4⋊C8—1221no—
307C2×C22⋊C8—611no—
308C2×C22⋊C8C2×D6111yescharacteristic
309#391—611no—
310#390—611no—
311#388—1221no—
312#388—1221no—
313#386—1221no—
314#386—1221no—
315#382—611no—
316C2×#95—611no—
317C2×#95—1221no—
318C2×#95C2×D6111yescharacteristic
319#336—1221no—
320#328—22no—
321#328—1222no—
322#327—1222no—
323#327—1222no—
324C61#971311yescharacteristic, complemented, Sylow
325C22×Q8C4×D6111yescharacteristic
326C22×D4—611no—
327C22×D4—1221no—
328C23×C4—21no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1221no—
332C23×C4C22×D6111yescharacteristic
333C22×C8—21no—
334C22×C8—1221no—
335#124—2442no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340#121—1221no—
341#121—1221no—
342#121—1221no—
343#121—1221no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4C4×D6111yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—1224no—
358C2×C42—2442no—
359C2×C42—2442no—
360C2×C42—2442no—
361C4⋊C8—2442no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C22⋊C8—1221no—
365C22⋊C8—1221no—
366#95—21no—
367#95—612no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371#95—1221no—
372#95—1224no—
373#95—1221no—
374#95C22⋊D6112yes—
375C24—1221no—
376C24D4×D6111yescharacteristic
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—2441no—
384C2×D4—2441no—
385C2×D4—2441no—
386C2×D4—2441no—
387C2×D4—2441no—
388C2×D4—2441no—
389C2×D4—2441no—
390C2×D4—2441no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—2441no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4C2×C4×D6111yescharacteristic
423C22×C4#1175822211yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4D4×D6111yescharacteristic
426C22×C4#1175822411yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—2441no—
432C2×C8—2441no—
433C2×C8—2441no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—2442no—
443C22⋊C4—2442no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C42—1222no—
449C42—1222no—
450C42—2441no—
451C42—2441no—
452C42—2448no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—2441no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23C22⋊C4×D6111yescharacteristic
465C23#1451974911yescharacteristic
466C23#1451975011yescharacteristic
467C23#1451979811yescharacteristic
468C23#1451980011yescharacteristic
469C23#1451980411yescharacteristic
470C23#1451980911yescharacteristic
471Q8—81no—
472Q8—81no—
473D4—4881no—
474D4—4881no—
475D4—4881no—
476D4—4881no—
477D4—4881no—
478D4—4881no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1224no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451973711yescharacteristic
527C2×C4C4×C22⋊D6111yescharacteristic
528C2×C4#1451976311yescharacteristic
529C2×C4#1451977311yescharacteristic
530C8—81no—
531C8—4881no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—4881no—
545C22—4881no—
546C22—4881no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22#1683966211yescharacteristic
552C22#99×D6111yescharacteristic
553C22#1683973411yescharacteristic
554C22#1683975711yescharacteristic
555C22#1683976211yescharacteristic
556C22#104×D6111yescharacteristic
557C22#104×D6111yescharacteristic
558C22#1683997911yescharacteristic
559C22#1683998011yescharacteristic
560C22#1683998611yescharacteristic
561C22#1683998711yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—44no—
567C4—81no—
568C4—81no—
569C4—1222no—
570C4—1222no—
571C4—1222no—
572C4—4881no—
573C4—2444no—
574C4—2441no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—9761no—
580C2#2144430411yescharacteristic
581C2#2144430911yescharacteristic
582C2#2144432211yescharacteristic
583C2#2144434311yescharacteristic
584C2#2144664511yescharacteristic
585C2#2144665311yescharacteristic
586C2#2144674811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#328C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenterC2×#328D61
hypocenterC61#9713
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. #21445797C2semidirect
3. C2×#16839499C2semidirect
6. #95×Dic61C2semidirect
7. C2×C61×#328C2semidirect
8. C2×#14519766C4semidirect
14. #16839669C4semidirect
15. #16839663C4semidirect
20. #16839627C4semidirect
285. C2×#328D61semidirect
324. C61#9713semidirect

Maximal subgroups

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

subgroupindex
#214541222
#214457972
C2×#168394992
C2×#168394972
C2×#168394952
#95×Dic612
C2×C61×#3282
#971361