Finite Group Databasebeta

← #29656151

Subgroups — #29656151

Order 15616.

1200 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456215C211yesmaximal, characteristic, complemented
2#21455910C211yesmaximal, characteristic, complemented
3#21455910C211yesmaximal, characteristic, complemented
4#21444970C214yesmaximal, complemented
5#21444908C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444262C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×C61×#547C211yesmaximal, characteristic, complemented
10#16840312C2211yescharacteristic
11#16840124—22no—
12#16840114C2212yescomplemented
13#16840110C2212yescomplemented
14#16840080—21no—
15#16840080—21no—
16#16840052C2212yescomplemented
17#16840052C2212yescomplemented
18#16839957C2212yescomplemented
19#16839957C2212yescomplemented
20#16839896—21no—
21#16839896—21no—
22#16839872—22no—
23#16839872—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839761—22no—
27C4×#11758194C2211yescharacteristic
28#16839508C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#547—22no—
33C61×#547—22no—
34C61×#547—22no—
35C61×#547C2214yes—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#123×C61—22no—
38C2×C22⋊Q8×C61—21no—
39C2×#121×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758222—21no—
44C2×#11758199—22no—
45C2×#11758199—22no—
46C2×#11758194—21no—
47#14519777C2311yescharacteristic
48#14519774C2311yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751—21no—
52C4×D244—21no—
53C4×D244—21no—
54#14519708—22no—
55#14519708—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519700—24no—
59#14519698—22no—
60#14519698—22no—
61#14519697—22no—
62#14519697—22no—
63#14519697—24no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C2×#41×C61—24no—
71C2×#41×C61—24no—
72C2×#41×C61—48no—
73C22×Q8×C61—21no—
74C22×D4×C61—22no—
75C22×D4×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—21no—
79C23×C4×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61D412yes—
83C4⋊Q8×C61C2312yes—
84#123×C61—28no—
85#123×C61—28no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88#121×C61—24no—
89#121×C61—24no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61D412yes—
95C4×D4×C61D412yes—
96C4×D4×C61C2312yes—
97C4×D4×C61C2312yes—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61D412yes—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C24×C61—21no—
111#41×C61—416no—
112#41×C61—48no—
113#41×C61—48no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—44no—
117C2×D4×C61—24no—
118C2×D4×C61—28no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—42no—
122C2×D4×C61—48no—
123C2×D4×C61C2×D412yes—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—28no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—44no—
134C22×C4×C61—44no—
135C22×C4×C61—48no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—41no—
140#11758222—44no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×C4×D61—41no—
145C2×Dic122—21no—
146C4⋊C4×C61—28no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2411yescharacteristic
164#11758199—44no—
165#11758199—44no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61#12011yescharacteristic
183C22×D61—41no—
184Q8×C61—44no—
185Q8×C61—44no—
186D4×C61—24no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—42no—
191D4×C61—416no—
192D4×C61AGL(1,ℤ/8)14yes—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—44no—
200C2×C4×C61—44no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—416no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C22⋊D61—82no—
214C2×Dic61—41no—
215D244—42no—
216D244—42no—
217D244—42no—
218C4×D61—82no—
219Dic122—42no—
220#92418—42no—
221#92418—42no—
222#92418—42no—
223#18533—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—24no—
227C22×C61—48no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—84no—
234C22×C61#45412yes—
235C22×C61C2×#12011yescharacteristic
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—42no—
244C4×C61—44no—
245C4×C61—42no—
246C4×C61C2×AGL(1,ℤ/8)12yes—
247C4×C61#57512yes—
248Dic61—81no—
249C2×#547D6111yescharacteristic, complemented
250#3141—611no—
251#3109—611no—
252#3109—611no—
253#1635—614no—
254#1625—614no—
255#1556—611no—
256#1490—611no—
257#1490—611no—
258C2×C61—22no—
259C2×C61—44no—
260C2×C61—82no—
261C2×C61#300311yescharacteristic
262C2×C61#300511yescharacteristic
263C2×C61C2×#45411yescharacteristic
264D61—161no—
265C22×#41—21no—
266#547—22no—
267#547—22no—
268#547—22no—
269#547C2×D6114yes—
270#532—611no—
271C2×C4⋊Q8C2×D6111yescharacteristic
272C2×#123—22no—
273C2×C22⋊Q8—21no—
274C2×#121—21no—
275C2×C4×D4C2×D6111yescharacteristic
276C2×C4×D4C2×D6111yescharacteristic
277C22×C4⋊C4—21no—
278#489—1222no—
279#487—612no—
280#486—612no—
281#481—1221no—
282#481—1221no—
283#476—612no—
284#476—612no—
285#459—612no—
286#459—612no—
287#447—1221no—
288#447—1221no—
289#442—1222no—
290#442—1222no—
291#432—611no—
292#432—611no—
293#417—1222no—
294C4×M16—611no—
295#332—614no—
296D4⋊C8—614no—
297D4⋊C8—614no—
298C61#1853311yescharacteristic, complemented, Sylow
299C2×#41—24no—
300C2×#41—24no—
301C2×#41—48no—
302C2×#41—1221no—
303C22×Q8—21no—
304C22×D4—22no—
305C22×D4C22⋊D6111yescharacteristic
306C22×D4C22⋊D6111yescharacteristic
307C23×C4—22no—
308C23×C4—21no—
309C23×C4—21no—
310C2×D8—1222no—
311C2×D8—1222no—
312C2×M16—1221no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—611no—
316C4⋊Q8C22⋊D6112yes—
317C4⋊Q8C22×D6112yes—
318#127—611no—
319#124—1221no—
320#123—28no—
321#123—28no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324#121—24no—
325#121—24no—
326#121—1221no—
327#121—1221no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4C22⋊D6112yes—
335C4×D4C22⋊D6112yes—
336C4×D4C22×D6112yes—
337C4×D4C22×D6112yes—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6112yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C42C22×D6111yescharacteristic
349#106—1222no—
350#106—1222no—
351C4⋊C8—612no—
352C4⋊C8—612no—
353#103—1224no—
354#102—1222no—
355#102—1222no—
356#102—1224no—
357#102—1222no—
358#102—1222no—
359C22⋊C8—1221no—
360C22⋊C8—1221no—
361#97—1221no—
362#97—1221no—
363#97—1221no—
364C4×C8—612no—
365C24—21no—
366C24—21no—
367#41—416no—
368#41—48no—
369#41—48no—
370#41—2444no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—1221no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4—2441no—
385C2×D4C2×C22⋊D6112yes—
386C2×D4C2×C22⋊D6112yes—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—2441no—
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402D8—2444no—
403D8—2444no—
404M16—2442no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C4⋊C4—28no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4C2×C22⋊D6114yes—
417C4⋊C4C2×C22⋊D6112yes—
418C4⋊C4C2×C22⋊D6112yes—
419C22⋊C4—28no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C42C2×C22⋊D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
429C42C23×D6111yescharacteristic
430C23—24no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—2441no—
439C23#1451974611yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—2442no—
443D4—24no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—42no—
448D4—416no—
449D4—4882no—
450D4—2442no—
451D4—2442no—
452D4—2442no—
453D4#1451980314yes—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—4882no—
469C2×C4—2441no—
470C2×C4#1451974611yescharacteristic
471C2×C4#1451974611yescharacteristic
472C2×C4#1451974611yescharacteristic
473C2×C4C22×C22⋊D6111yescharacteristic
474C2×C4C22×C22⋊D6111yescharacteristic
475C2×C4C22×C22⋊D6111yescharacteristic
476C8—2442no—
477C8—2442no—
478C8—2442no—
479C22—21no—
480C22—21no—
481C22—24no—
482C22—48no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22#1683993512yes—
493C22C2×#1451974611yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—44no—
500C4—42no—
501C4—4881no—
502C4C2×#1451980312yes—
503C4#1684056012yes—
504C2—22no—
505C2—44no—
506C2—82no—
507C2—9761no—
508C2#2145504611yescharacteristic
509C2#2145506111yescharacteristic
510C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#547C2
socleC22×C61C2×#120
hypercenterC2×#547D61
hypocenterC61#18533
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #21456215C2semidirect
2. #21455910C2semidirect
3. #21455910C2semidirect
4. #21444970C2semidirect
5. #21444908C2semidirect
6. #21444536C2semidirect
7. #21444262C2semidirect
8. #21444262C2semidirect
9. C2×C61×#547C2semidirect
12. #16840114C22semidirect
13. #16840110C22semidirect
16. #16840052C22semidirect
17. #16840052C22semidirect
18. #16839957C22semidirect
19. #16839957C22semidirect
28. #16839508C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
249. C2×#547D61semidirect
298. C61#18533semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214559102
#214449702
#214449082
#214445362
#214442622
C2×C61×#5472
#1853361