Finite Group Databasebeta

← #29580740

Subgroups — #29580740

Order 15616.

686 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455279C211yesmaximal, characteristic, complemented
2#21450720C211yesmaximal, characteristic, complemented
3#21444379C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3100C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519751—21no—
10C2×C4×D244C2211yescharacteristic
11#16839974—22no—
12#16839964—21no—
13#16839954—21no—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839763—21no—
17#16839688—21no—
18#16839670—21no—
19#16839638—22no—
20#16839628C2211yescharacteristic
21#16839622—21no—
22#16839496—22no—
23#16839491—22no—
24D4⋊#92418—22no—
25C61×#532C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#480—21no—
28C61×#474—21no—
29C61×#459—21no—
30C61×#447—21no—
31C4×D8×C61—22no—
32C61×#431C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C22⋊D61—22no—
35C22×C4×D61—21no—
36#14519802—42no—
37#14519802—42no—
38C2×#11758201—22no—
39C2×#11758199—22no—
40C2×#11758194—21no—
41#14519751—42no—
42#14519751—42no—
43C4×D244—21no—
44C4×D244—21no—
45C4×D244—21no—
46C4×D244—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61D411yescharacteristic
49C2×C22⋊Dic61—22no—
50C2×#11758198—22no—
51C2×#11758198—21no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418D412yescomplemented
55#14519701—22no—
56#14519698—22no—
57C22⋊#92418—21no—
58C4×#92418—21no—
59#14519684—22no—
60#14519684—21no—
61#14519684D412yes—
62C2×#41×C61—21no—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61—21no—
65C2×D8×C61—22no—
66C4⋊Q8×C61D411yescharacteristic, complemented
67#127×C61D411yescharacteristic, complemented
68#124×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76C4×D4×C61—21no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80#107×C61—22no—
81C4⋊C8×C61—21no—
82#102×C61—22no—
83#102×C61—22no—
84C22⋊C8×C61—21no—
85C4×C8×C61D412yescomplemented
86C24×C61—21no—
87C23×D61—21no—
88#41×C61—41no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92C2×Q8×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—42no—
99C2×D4×C61—41no—
100C22×C4×C61—21no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—44no—
112C2×C22⋊D61—44no—
113C2×C22⋊D61—82no—
114C22×Dic61—21no—
115C22×Dic61—22no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244D812yescomplemented
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61—82no—
125D8×C61—44no—
126C2×C8×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C22⋊Dic61—82no—
140#11758201—44no—
141#11758199—44no—
142#11758198—44no—
143#11758198—41no—
144#11758198—41no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149#11758194—41no—
150#11758194—41no—
151C2×#92418—22no—
152C2×#92418—41no—
153C23×C61—22no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61#12011yescharacteristic
162C22×D61—22no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—41no—
166C22×D61—41no—
167Q8×C61—41no—
168Q8×C61—41no—
169D4×C61—22no—
170D4×C61—22no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—42no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—81no—
177D4×C61—81no—
178D4×C61—81no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—81no—
186C2×C4×C61—81no—
187C2×C4×C61—81no—
188C2×C4×C61—81no—
189C2×C4×C61—81no—
190C2×C4×C61—81no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12711yescharacteristic
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61#13511yescharacteristic
199C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
201C22⋊D61—88no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207D244—22no—
208D244—22no—
209D244—41no—
210D244—41no—
211C4×D61—82no—
212C4×D61—82no—
213C8×C61—42no—
214#92418—42no—
215#92418—81no—
216#9996—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61#39211yescharacteristic
230C22×C61#44711yescharacteristic
231C22×C61#44911yescharacteristic
232C2×D61—44no—
233C2×D61—42no—
234C2×D61—82no—
235C2×D61—81no—
236C2×D61—81no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—81no—
242C4×C61—81no—
243C4×C61#44711yescharacteristic
244C4×C61#44911yescharacteristic
245C4×C61#46611yescharacteristic
246C4×C61#46911yescharacteristic
247Dic61—82no—
248Dic61—81no—
249#3100D6111yescharacteristic, complemented
250#3036—611no—
251#2535—611no—
252#1517—611no—
253#1486—611no—
254#1478—611no—
255#1286—611no—
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—82no—
259C2×C61—161no—
260C2×C61#199911yescharacteristic
261C2×C61#200811yescharacteristic
262C2×C61#201011yescharacteristic
263D61—82no—
264C2×AGL(1,ℤ/8)—1221no—
265#532C2×D6111yescharacteristic
266C2×#121—1221no—
267C2×C4×D4—611no—
268C2×C4×D4C2×D6111yescharacteristic
269#480—21no—
270#474—21no—
271#463—1222no—
272#461—1221no—
273#459—21no—
274#459—1221no—
275#447—21no—
276C4×D8—22no—
277#431—611no—
278#431C2×D6111yescharacteristic
279#423—611no—
280#417—1221no—
281#397—1221no—
282#392—1221no—
283#386—1222no—
284#382—611no—
285#380—1221no—
286#327—1222no—
287D4⋊C8—1222no—
288D4⋊C8—1222no—
289C61#999611yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C22×D4—611no—
292C22×D4—1222no—
293C22×D4C22⋊D6111yescharacteristic
294C23×C4—21no—
295C23×C4—1221no—
296AGL(1,ℤ/8)—2442no—
297AGL(1,ℤ/8)—2442no—
298C2×SD16—1222no—
299C2×D8—22no—
300C2×D8—1222no—
301C2×M16—1221no—
302C4⋊Q8C22⋊D6111yescharacteristic, complemented
303#127C22⋊D6111yescharacteristic, complemented
304#124—21no—
305#121—21no—
306#121—21no—
307#121—2442no—
308#121—2442no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—1221no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4—1221no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
326C2×C42C22×D6111yescharacteristic
327#107—22no—
328C4⋊C8—21no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332#103—1222no—
333#102—22no—
334#102—22no—
335#102—1222no—
336C22⋊C8—21no—
337C22⋊C8—1221no—
338C4×C8—1221no—
339C4×C8C22⋊D6112yescomplemented
340#95—612no—
341#95—1222no—
342#95—1221no—
343C24—21no—
344C24—1221no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349C2×Q8—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—41no—
357C2×D4—612no—
358C2×D4—1221no—
359C2×D4—1221no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—4882no—
364C2×D4—2444no—
365C2×D4—2444no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—1222no—
375C22×C4—1221no—
376C22×C4—4882no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4D4×D6111yescharacteristic
380C22×C4D4×D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382SD16—2444no—
383D8—44no—
384D8—2444no—
385M16—2441no—
386M16—2441no—
387C2×C8—22no—
388C2×C8—1222no—
389C2×C8—2441no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—4882no—
403C22⋊C4—2444no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42D4×D6111yescharacteristic
407C42#1175822411yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—1222no—
419C23—2442no—
420C23—2442no—
421C23—2441no—
422C23—2441no—
423C23#1451974511yescharacteristic
424Q8—41no—
425Q8—41no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430D4—42no—
431D4—41no—
432D4—41no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—1222no—
437D4—1222no—
438D4—4888no—
439D4—2441no—
440D4—2441no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—4882no—
454C2×C4—4882no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2441no—
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974611yescharacteristic
462C2×C4#1451974811yescharacteristic
463C2×C4#1451974911yescharacteristic
464C2×C4#1451974911yescharacteristic
465C2×C4#1451977311yescharacteristic
466C2×C4D8×D6111yescharacteristic
467C2×C4#1451979611yescharacteristic
468C2×C4#1451980111yescharacteristic
469C2×C4#1451980111yescharacteristic
470C8—42no—
471C8—4881no—
472C8—2442no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—4882no—
486C22—4881no—
487C22—4881no—
488C22—2444no—
489C22—2442no—
490C22#1683967011yescharacteristic
491C22#1683989411yescharacteristic
492C22#1683990411yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—81no—
498C4—81no—
499C4—4882no—
500C4—4881no—
501C4#1683989411yescharacteristic
502C4#1683990411yescharacteristic
503C4#1683999511yescharacteristic
504C4#1684001211yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—161no—
509C2—4882no—
510C2#2144681411yescharacteristic
511C2#2144686911yescharacteristic
512C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3100C2
socleC22×C61#392
hypercenter#3100D61
hypocenterC61#9996
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21455279C2semidirect
2. #21450720C2semidirect
3. #21444379C2semidirect
4. #21444244C2semidirect
5. #21444211C2semidirect
6. #21443678C2semidirect
7. C61×#3100C2semidirect
54. C4⋊#92418D4semidirect
66. C4⋊Q8×C61D4semidirect
67. #127×C61D4semidirect
85. C4×C8×C61D4semidirect
121. C2×D244D8semidirect
249. #3100D61semidirect
289. C61#9996semidirect
302. C4⋊Q8C22⋊D61semidirect
303. #127C22⋊D61semidirect
339. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214552792
#214507202
#214443792
#214442442
#214442112
#214436782
C61×#31002
#999661