Finite Group Databasebeta

← #29658500

Subgroups — #29658500

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic, complemented
2#21455850C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21445026C214yesmaximal, complemented
5#21444891C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#554C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840136C2212yescomplemented
13#16840112C2212yescomplemented
14#16840080—21no—
15#16840047C2212yescomplemented
16#16839977—22no—
17#16839967C2212yes—
18#16839964—22no—
19#16839962C2212yescomplemented
20#16839957C2212yes—
21#16839901—22no—
22#16839901—21no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839511C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#554—22no—
32C61×#554—22no—
33C61×#554—22no—
34C61×#554C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#125×C61—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758222—21no—
44#14519803—44no—
45C2×#11758200—22no—
46C2×#11758201—22no—
47C2×#11758201—24no—
48C2×#11758199—22no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—24no—
63#14519701—22no—
64#14519700—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D412yes—
73C22×D4×C61D411yescharacteristic
74C23×C4×C61—22no—
75C23×C4×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79C4⋊Q8×C61C2312yes—
80#125×C61—24no—
81#125×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—28no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—28no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D412yes—
89C4×Q8×C61C2312yes—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61D412yes—
95C4×D4×C61C2312yes—
96C2×C4⋊C4×C61—24no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—42no—
113C2×Q8×C61—82no—
114C2×Q8×C61C2×D412yes—
115C2×Q8×C61C2×D414yes—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61SD1614yescomplemented
119C2×D4×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—28no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—44no—
128C22×C4×C61—84no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C2×C22⋊D61—41no—
133#11758222—42no—
134#11758222—42no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×C4×D61—41no—
139C2×Dic122—21no—
140C4⋊C4×C61—28no—
141C4⋊C4×C61—28no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161#11758201—44no—
162#11758201—84no—
163#11758200—44no—
164#11758199—44no—
165#11758198—41no—
166#11758198—41no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61—24no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61#12011yescharacteristic
178C22×D61—41no—
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—42no—
182Q8×C61—44no—
183Q8×C61—82no—
184D4×C61—22no—
185D4×C61—24no—
186D4×C61C2×SD1614yes—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—41no—
193C2×C4×C61—44no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—44no—
197C2×C4×C61—42no—
198C2×C4×C61—44no—
199C2×C4×C61—88no—
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61C2×SD1612yes—
204C2×C4×C61#13712yes—
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C22⋊D61—82no—
209C2×Dic61—41no—
210D244—42no—
211D244—41no—
212D244—41no—
213D244—42no—
214C4×D61—82no—
215Dic122—41no—
216Dic122—41no—
217#92418—42no—
218#92418—42no—
219#92418—42no—
220#18748—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—24no—
224C22×C61—48no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61#44812yes—
228C22×C61C2×#12011yescharacteristic
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—44no—
239C4×C61#44812yes—
240C4×C61C22×SD1611yescharacteristic
241C4×C61C2×#13711yescharacteristic
242C4×C61#57512yes—
243Dic61—81no—
244C2×#554D6111yescharacteristic, complemented
245#3206—611no—
246#3103—611no—
247#3036—611no—
248#1643—614no—
249#1622—614no—
250#1548—611no—
251#1487—611no—
252#1478—611no—
253C2×C61—22no—
254C2×C61—44no—
255C2×C61C2×#44811yescharacteristic
256C2×C61#300511yescharacteristic
257C2×C61#301911yescharacteristic
258D61—161no—
259C2×AGL(1,ℤ/8)—1221no—
260#554—22no—
261#554—22no—
262#554—22no—
263#554C2×D6114yes—
264#532—611no—
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×#125—21no—
267C2×C22⋊Q8—21no—
268C2×C22⋊Q8—22no—
269C2×C4×Q8C2×D6111yescharacteristic
270C2×C4×D4—21no—
271C2×C4×D4C2×D6111yescharacteristic
272C22×C4⋊C4—21no—
273#492—612no—
274#487—612no—
275#481—1221no—
276#475—612no—
277#463—1222no—
278#461—612no—
279#461—1222no—
280#460—612no—
281#459—612no—
282#448—1221no—
283#448—1222no—
284C4×SD16—1222no—
285#431—611no—
286#431—611no—
287#423—611no—
288#417—1221no—
289#333—614no—
290Q8⋊C8—614no—
291D4⋊C8—614no—
292C61#1874811yescharacteristic, complemented, Sylow
293C2×#41—1221no—
294C22×Q8C22⋊D6111yescharacteristic
295C22×Q8C22⋊D6112yes—
296C22×D4C22⋊D6111yescharacteristic
297C23×C4—22no—
298C23×C4—21no—
299AGL(1,ℤ/8)—2444no—
300C2×SD16—1224no—
301C2×SD16—1222no—
302C2×SD16—1222no—
303C2×D8—1222no—
304C2×M16—1221no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—611no—
308C4⋊Q8C22⋊D6112yes—
309C4⋊Q8C22×D6112yes—
310#127—611no—
311#125—24no—
312#125—24no—
313#124—1221no—
314C22⋊Q8—24no—
315C22⋊Q8—28no—
316C22⋊Q8—24no—
317C22⋊Q8—28no—
318#121—1221no—
319#121—1221no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8C22⋊D6112yes—
323C4×Q8C22×D6112yes—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4C22⋊D6112yes—
331C4×D4C22×D6112yes—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C22⋊D6111yescharacteristic
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C42—21no—
344C2×C42C22×D6111yescharacteristic
345#106—1222no—
346C4⋊C8—612no—
347C4⋊C8—612no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350#103—1222no—
351#103—1224no—
352#103—1222no—
353#102—1222no—
354#102—1222no—
355C22⋊C8—1221no—
356C22⋊C8—1221no—
357C4×C8—612no—
358C4×C8—1221no—
359C24—21no—
360#41—2442no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—42no—
365C2×Q8—82no—
366C2×Q8—1221no—
367C2×Q8C2×C22⋊D6112yes—
368C2×Q8C2×C22⋊D6114yes—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—2441no—
375C2×D4#1175820114yescomplemented
376C2×D4C2×C22⋊D6112yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—44no—
385C22×C4—84no—
386C22×C4—2441no—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—4884no—
391SD16—2444no—
392SD16—2444no—
393D8—2444no—
394M16—2442no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4C2×C22⋊D6112yes—
411C4⋊C4C2×C22⋊D6112yes—
412C22⋊C4—28no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C42—22no—
418C42—22no—
419C42C2×C22⋊D6111yescharacteristic
420C42C2×C22⋊D6111yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—2441no—
428C23#1451974611yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—44no—
432Q8—42no—
433Q8—82no—
434Q8—2441no—
435Q8—2441no—
436D4—22no—
437D4—24no—
438D4—4882no—
439D4—2442no—
440D4—2441no—
441D4—2441no—
442D4—2442no—
443D4C2×#1175820114yes—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—88no—
457C2×C4—4882no—
458C2×C4—2441no—
459C2×C4#1451974611yescharacteristic
460C2×C4#1451974611yescharacteristic
461C2×C4#1451974611yescharacteristic
462C2×C4C2×#1175820112yes—
463C2×C4#1451980812yes—
464C2×C4C22×C22⋊D6111yescharacteristic
465C2×C4C22×C22⋊D6111yescharacteristic
466C2×C4C22×C22⋊D6111yescharacteristic
467C8—2442no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—24no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—4881no—
477C22—4881no—
478C22—4881no—
479C22#1683990112yes—
480C22C2×#1451974611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—44no—
486C4—42no—
487C4—42no—
488C4—4881no—
489C4#1683990112yes—
490C4C22×#1175820111yescharacteristic
491C4C2×#1451980811yescharacteristic
492C4#1684056012yes—
493C2—22no—
494C2—44no—
495C2—9761no—
496C2C2×#1683990111yescharacteristic
497C2#2145506111yescharacteristic
498C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radical#21455283C2
hypercenterC2×#554D61
hypocenterC61#18748
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
2. #21455850C2semidirect
4. #21445026C2semidirect
5. #21444891C2semidirect
6. #21444500C2semidirect
7. #21444248C2semidirect
8. #21444210C2semidirect
9. C2×C61×#554C2semidirect
12. #16840136C22semidirect
13. #16840112C22semidirect
15. #16840047C22semidirect
19. #16839962C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
118. C2×D4×C61SD16semidirect
244. C2×#554D61semidirect
292. C61#18748semidirect
375. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214558502
#214552832
#214450262
#214448912
#214445002
#214442482
#214442102
C2×C61×#5542
#1874861