Finite Group Databasebeta

← #29582353

Subgroups — #29582353

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic, complemented
2#21450732C211yesmaximal, characteristic
3#21446622C211yesmaximal, characteristic, complemented
4C2×D4⋊#92418C211yesmaximal, characteristic, complemented
5C2×#16839489C211yesmaximal, characteristic, complemented
6#21443703C211yesmaximal, characteristic, complemented
7C2×C61×#476C211yesmaximal, characteristic, complemented
8C2×#14519751—21no—
9C2×#14519748—21no—
10C2×C4×D244C2211yescharacteristic
11#16840049—22no—
12#16840049—22no—
13C2×#14519708—21no—
14C2×C4⋊#92418C2211yescharacteristic
15C2×#14519701—22no—
16C2×#14519697—21no—
17C2×C4×#92418C2211yescharacteristic
18#16839678—22no—
19#16839632C2211yescharacteristic
20#16839622—21no—
21#16839619—21no—
22D4⋊#92418—22no—
23#16839489—22no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×C4×D4×C61C2211yescharacteristic
26C61×#476—22no—
27C61×#476—22no—
28C2×#106×C61—21no—
29C2×C4⋊C8×C61C2211yescharacteristic
30C2×#102×C61—21no—
31C22×D244D411yescharacteristic
32C22×C22⋊D61—22no—
33C22×C4×D61—21no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519751—42no—
37#14519751—42no—
38#14519748—42no—
39#14519748—42no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×C4⋊Dic61D412yes—
48C2×C22⋊Dic61—22no—
49C2×#11758198—22no—
50C2×#11758198—21no—
51#14519708—24no—
52C4⋊#92418—21no—
53C4⋊#92418D412yescomplemented
54#14519701—44no—
55#14519697—24no—
56C4×#92418—21no—
57C4×#92418—21no—
58#14519684—22no—
59C22×Q8×C61—21no—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C22×C8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61D412yescomplemented
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#106×C61—24no—
77C4⋊C8×C61—21no—
78C4⋊C8×C61D412yescomplemented
79#102×C61—24no—
80C24×C61—21no—
81C23×D61—21no—
82C2×Q8×C61—22no—
83C2×Q8×C61—42no—
84C2×Q8×C61—42no—
85C2×Q8×C61—41no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61SD1612yes—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—44no—
104C2×C22⋊D61—44no—
105C2×C22⋊D61—82no—
106C22×Dic61—21no—
107C22×Dic61—22no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244SD1612yes—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C8×C61—22no—
121C2×C8×C61—41no—
122C2×C8×C61—41no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C42×C61#4111yescharacteristic
135C22⋊Dic61—44no—
136#11758198—44no—
137#11758198—41no—
138#11758198—41no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—41no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144C4⋊Dic61—41no—
145C4⋊Dic61—42no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—41no—
149C2×#92418—41no—
150C2×#92418—41no—
151C2×#92418—41no—
152C23×C61—22no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61#12011yescharacteristic
160C22×D61—22no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167Q8×C61—44no—
168Q8×C61—82no—
169D4×C61—22no—
170D4×C61—22no—
171D4×C61—41no—
172D4×C61—41no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—81no—
180C2×C4×C61—81no—
181C2×C4×C61—81no—
182C2×C4×C61—81no—
183C2×C4×C61—81no—
184C2×C4×C61—81no—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12711yescharacteristic
191C2×C4×C61C2×SD1611yescharacteristic
192C2×C4×C61C2×SD1611yescharacteristic
193C2×C4×C61#13511yescharacteristic
194C2×C4×C61#13511yescharacteristic
195C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C61#13711yescharacteristic
198C2×C4×C61#13711yescharacteristic
199C22⋊D61—88no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206D244—22no—
207D244—22no—
208D244—41no—
209D244—41no—
210C4×D61—81no—
211C4×D61—81no—
212C4×D61—81no—
213C4×D61—81no—
214C8×C61—42no—
215#92418—42no—
216#92418—42no—
217#10235—611nomaximal, Sylow
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61#39211yescharacteristic
226C22×C61#45011yescharacteristic
227C22×C61#45011yescharacteristic
228C22×C61#45211yescharacteristic
229C22×C61#45211yescharacteristic
230C22×C61#45311yescharacteristic
231C22×C61#45411yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—81no—
242C4×C61—82no—
243C4×C61#44811yescharacteristic
244C4×C61#44811yescharacteristic
245C4×C61#44911yescharacteristic
246C4×C61#44911yescharacteristic
247C4×C61#46511yescharacteristic
248C4×C61#46511yescharacteristic
249C4×C61#46811yescharacteristic
250C4×C61#46811yescharacteristic
251Dic61—82no—
252Dic61—81no—
253C2×#476—611no—
254C2×#476D6111yescharacteristic, complemented
255#2537—611no—
256#1956—611no—
257C2×D4⋊C8—611no—
258C2×D4⋊C8—611no—
259#1296—611no—
260C2×C61—82no—
261C2×C61#162911yescharacteristic
262C2×C61#163411yescharacteristic
263C2×C61#163511yescharacteristic
264C2×C61#163511yescharacteristic
265C2×C61#200011yescharacteristic
266C2×C61#200011yescharacteristic
267C2×C61#200611yescharacteristic
268D61—82no—
269C2×C4⋊Q8C2×D6111yescharacteristic
270C2×#121—1221no—
271C2×#121—1221no—
272C2×C4×D4—611no—
273C2×C4×D4C2×D6111yescharacteristic
274#476—22no—
275#476—22no—
276#476—1222no—
277#476—1222no—
278C2×#106—21no—
279C2×#106—1221no—
280C2×C4⋊C8—611no—
281C2×C4⋊C8C2×D6111yescharacteristic
282C2×#103—1222no—
283C2×#102—21no—
284C2×#102—1221no—
285C2×C4×C8—611no—
286#394—1222no—
287#384—611no—
288#380—1221no—
289#380—1221no—
290D4⋊C8—1222no—
291D4⋊C8—1222no—
292C61#1023511yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×D4—611no—
295C22×D4—1222no—
296C22×D4C22⋊D6111yescharacteristic
297C23×C4—21no—
298C23×C4—1221no—
299C22×C8—21no—
300C22×C8—1221no—
301C22×C8—1221no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C22⋊D6112yescomplemented
306#121—2442no—
307#121—2442no—
308#121—2442no—
309#121—2442no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—1221no—
315C4×D4—1221no—
316C4×D4—1221no—
317C4×D4—1221no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4C22⋊D6111yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1221no—
329C2×C42C22×D6111yescharacteristic
330#106—24no—
331#106—1224no—
332C4⋊C8—21no—
333C4⋊C8—612no—
334C4⋊C8—1221no—
335C4⋊C8C22⋊D6112yescomplemented
336#103—2444no—
337#102—24no—
338#102—1224no—
339C4×C8—1221no—
340C4×C8—1221no—
341#95—1222no—
342C24—21no—
343C24—1221no—
344C2×Q8—22no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—41no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—612no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—1221no—
358C2×D4—1221no—
359C2×D4—4882no—
360C2×D4—2444no—
361C2×D4—2444no—
362C2×D4#1175820112yes—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—1221no—
372C22×C4—1222no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—2441no—
388C2×C8—2441no—
389C2×C8—2441no—
390C2×C8—2441no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—2442no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—2444no—
407C22⋊C4—2444no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C42D4×D6111yescharacteristic
411C42#1175822411yescharacteristic
412C42C2×C22⋊D6111yescharacteristic
413C42C2×C22⋊D6111yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—1222no—
422C23—2441no—
423C23—2441no—
424C23—2441no—
425C23—2441no—
426C23—2441no—
427C23—2441no—
428C23#1451974511yescharacteristic
429Q8—44no—
430Q8—82no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—1222no—
436D4—1222no—
437D4—4888no—
438D4—2441no—
439D4—2441no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—4881no—
453C2×C4—4881no—
454C2×C4—4881no—
455C2×C4—4881no—
456C2×C4—2442no—
457C2×C4—2441no—
458C2×C4—2442no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4#1451974511yescharacteristic
463C2×C4#1451974611yescharacteristic
464C2×C4#1451974811yescharacteristic
465C2×C4#1451974911yescharacteristic
466C2×C4#1451974911yescharacteristic
467C2×C4#1451977311yescharacteristic
468C2×C4SD16×D6111yescharacteristic
469C2×C4C2×#1175820111yescharacteristic
470C2×C4#1451979711yescharacteristic
471C2×C4#1451979811yescharacteristic
472C2×C4#1451980211yescharacteristic
473C2×C4#1451980411yescharacteristic
474C2×C4#1451980611yescharacteristic
475C2×C4#1451980811yescharacteristic
476C8—42no—
477C8—2442no—
478C8—2442no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22—4881no—
490C22—2442no—
491C22—2442no—
492C22—2442no—
493C22#1683967011yescharacteristic
494C22#1683991111yescharacteristic
495C22#1683991211yescharacteristic
496C22#1683992111yescharacteristic
497C22#1683992211yescharacteristic
498C22#1683992811yescharacteristic
499C22#1683993311yescharacteristic
500C4—21no—
501C4—21no—
502C4—81no—
503C4—82no—
504C4—4882no—
505C4—4881no—
506C4#1683989911yescharacteristic
507C4#1683990111yescharacteristic
508C4#1683990611yescharacteristic
509C4#1683990711yescharacteristic
510C4#1683998811yescharacteristic
511C4#1683999311yescharacteristic
512C4#1684000511yescharacteristic
513C4#1684001011yescharacteristic
514C2—82no—
515C2—4882no—
516C2#2144493411yescharacteristic
517C2#2144496311yescharacteristic
518C2#2144496811yescharacteristic
519C2#2144496911yescharacteristic
520C2#2144681911yescharacteristic
521C2#2144682011yescharacteristic
522C2#2144685911yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×#476C2
socleC23×C61#120
prime radical#21450732C2
hypercenterC2×#476D61
hypocenterC61#10235
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
3. #21446622C2semidirect
4. C2×D4⋊#92418C2semidirect
5. C2×#16839489C2semidirect
6. #21443703C2semidirect
7. C2×C61×#476C2semidirect
53. C4⋊#92418D4semidirect
66. C4⋊Q8×C61D4semidirect
78. C4⋊C8×C61D4semidirect
254. C2×#476D61semidirect
292. C61#10235semidirect
305. C4⋊Q8C22⋊D61semidirect
335. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
#214507322
#214466222
C2×D4⋊#924182
C2×#168394892
#214437032
C2×C61×#4762
#1023561