Finite Group Databasebeta

← #29663791

Subgroups — #29663791

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21445242C212yesmaximal, complemented
5#21445228C212yesmaximal, complemented
6#21445194C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444575C211yesmaximal, characteristic, complemented
9#21444507C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15#16840160C2211yescharacteristic, complemented
16#16840135C2211yescharacteristic, complemented
17#16840113C2211yescharacteristic, complemented
18#16840109C2211yescharacteristic, complemented
19#16840062—22no—
20#16840052C2212yes—
21#16840042C2212yes—
22#16839991—21no—
23#16839991—21no—
24#16839977—22no—
25#16839967C2212yes—
26#16839957C2212yes—
27C4×#11758190—21no—
28C4×#11758190—21no—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839786C2211yescharacteristic
32#16839767—21no—
33#16839761—21no—
34#16839513C2212yescomplemented
35#16839513C2212yescomplemented
36#16839510C2212yescomplemented
37#16839509C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yescomplemented
40C23×C8×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C8×D4×C61—21no—
43C8×D4×C61—21no—
44C8×D4×C61C2212yes—
45C8×D4×C61C2212yes—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×C22⋊C8×C61—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50#14519803—44no—
51C2×#11758190—21no—
52C2×#11758190—21no—
53C2×#11758200—22no—
54C2×#11758199—22no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519769—21no—
62#14519765—21no—
63#14519756—21no—
64#14519756—21no—
65#14519751—21no—
66#14519751—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×D244—21no—
70C4×D244—21no—
71#14519736—21no—
72#14519710—22no—
73#14519709—21no—
74#14519709—21no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2312yes—
80#14519702—21no—
81#14519702—21no—
82#14519700—22no—
83#14519699—21no—
84#14519699—21no—
85#14519698—22no—
86#14519698—22no—
87#14519697—22no—
88C22×D4×C61D411yescharacteristic
89C23×C4×C61—21no—
90C22×C8×C61—21no—
91C22×C8×C61—22no—
92C22×C8×C61—22no—
93C22×C8×C61—44no—
94C22×C8×C61D411yescharacteristic
95C22×C8×C61D411yescharacteristic
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C4⋊C8×C61C2312yes—
105C22⋊C8×C61—22no—
106C22⋊C8×C61—22no—
107C4×C8×C61D411yescharacteristic
108C4×C8×C61D411yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C4×C8×C61C2311yescharacteristic
111C24×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61SD1614yescomplemented
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125#11758222—42no—
126#11758222—42no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—41no—
137C2×C8×C61—44no—
138C2×C8×C61—42no—
139C2×C8×C61—44no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61#4111yescharacteristic
151C42×C61C2411yescharacteristic
152C22⋊Dic61—41no—
153#11758200—44no—
154#11758199—44no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758196—41no—
164#11758196—41no—
165C4×Dic61—41no—
166#11758194—42no—
167#11758194—42no—
168C2×#92418—22no—
169C2×#92418—22no—
170#11758190—42no—
171#11758190—42no—
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61#12111yescharacteristic
177C22×D61—41no—
178D4×C61—22no—
179D4×C61—24no—
180D4×C61C2×SD1614yes—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—44no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61C2×SD1612yes—
193C2×C4×C61#13512yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C22⋊D61—82no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201D244—41no—
202D244—41no—
203D244—41no—
204D244—41no—
205D244—81no—
206C4×D61—82no—
207Dic122—41no—
208Dic122—41no—
209Dic122—41no—
210Dic122—41no—
211Dic122—81no—
212C8×C61—21no—
213C8×C61—21no—
214C8×C61—21no—
215C8×C61—42no—
216C8×C61AGL(1,ℤ/8)12yescomplemented
217#92418—42no—
218#92418—42no—
219#19254—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—24no—
223C22×C61—48no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61#46512yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61#46512yes—
236C4×C61C22×SD1611yescharacteristic
237C4×C61C2×#13511yescharacteristic
238C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C61#57511yescharacteristic
240Dic61—81no—
241Dic61—81no—
242Dic61—81no—
243#3167—611no—
244#3077—611no—
245#3036—611no—
246C2×C8×D4D6111yescharacteristic, complemented
247#1679—612no—
248#1677—612no—
249#1671—612no—
250#1665—612no—
251#1566—611no—
252#1550—611no—
253#1547—611no—
254C2×C61—22no—
255C2×C61—44no—
256C2×C61#303611yescharacteristic
257C2×C61C2×#46511yescharacteristic
258C2×C61#306711yescharacteristic
259D61—161no—
260C2×AGL(1,ℤ/8)—1221no—
261C23×C8—21no—
262#533—611no—
263#532—611no—
264C2×C4×D4C2×D6111yescharacteristic
265#498—611no—
266#492—611no—
267#487—611no—
268#486—611no—
269#478—1222no—
270#476—612no—
271#474—612no—
272#465—1221no—
273#465—1221no—
274#463—1222no—
275#461—612no—
276#459—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279C8×D4—21no—
280C8×D4—21no—
281C8×D4C2×D6112yes—
282C8×D4C2×D6112yes—
283#428—1221no—
284#423—611no—
285#423—611no—
286C2×C4⋊C8C2×D6111yescharacteristic
287#418—1221no—
288#417—1221no—
289C2×C22⋊C8—21no—
290C2×C4×C8C2×D6111yescharacteristic
291#334—612no—
292#334—612no—
293#333—612no—
294#332—612no—
295#332—612no—
296#331—612no—
297C61#1925411yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×D4D24411yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—2444no—
302C2×SD16—1222no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—1222no—
306C2×M16—1221no—
307C2×M16—1221no—
308C22×C8—21no—
309C22×C8—22no—
310C22×C8—22no—
311C22×C8—44no—
312C22×C8C22⋊D6111yescharacteristic
313C22×C8C22⋊D6111yescharacteristic
314C4⋊Q8—611no—
315C4⋊Q8—611no—
316C4⋊Q8—611no—
317#127—611no—
318#125—1221no—
319#124—1221no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322#121—1221no—
323#121—1221no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4D24412yes—
331C4×D4C22×D6112yes—
332#117—1221no—
333C2×C4⋊C4D24411yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C22×D6111yescharacteristic
336#107—1222no—
337#106—1221no—
338#106—1221no—
339#106—1222no—
340C4⋊C8—21no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6112yes—
346#103—1221no—
347#103—1221no—
348#103—1222no—
349#102—1222no—
350#102—1222no—
351#102—1221no—
352#102—1221no—
353#102—1222no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C4×C8C22⋊D6111yescharacteristic
357C4×C8C22⋊D6111yescharacteristic
358C4×C8C22×D6111yescharacteristic
359C4×C8C22×D6111yescharacteristic
360C24—21no—
361#41—2442no—
362#41—2442no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C2×D4#1175819014yescomplemented
373C2×D4C2×D24412yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—2441no—
380C22×C4C2×D24411yescharacteristic
381C22×C4#1175822211yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383SD16—2442no—
384SD16—2444no—
385SD16—2442no—
386D8—2444no—
387M16—2442no—
388M16—2442no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—41no—
392C2×C8—44no—
393C2×C8—42no—
394C2×C8—44no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8C2×C22⋊D6111yescharacteristic
398C2×C8C2×C22⋊D6111yescharacteristic
399C2×C8C2×C22⋊D6111yescharacteristic
400C2×C8C2×C22⋊D6111yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—2441no—
409C4⋊C4—2441no—
410C4⋊C4C2×D24412yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—2441no—
417C42C2×D24411yescharacteristic
418C42#1175822211yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—2441no—
426C23#1451975111yescharacteristic
427Q8—4881no—
428Q8—2441no—
429Q8—2441no—
430Q8—2441no—
431Q8—2441no—
432D4—24no—
433D4—22no—
434D4—4881no—
435D4—4882no—
436D4—2441no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4C2×#1175819014yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—4882no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4#1451975111yescharacteristic
454C2×C4#1451975111yescharacteristic
455C2×C4#1451975111yescharacteristic
456C2×C4C2×#1175819012yes—
457C2×C4#1451979912yes—
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C22×D24411yescharacteristic
460C2×C4C2×#1175822211yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—42no—
465C8—2442no—
466C8—2442no—
467C8#1451980312yescomplemented
468C22—21no—
469C22—21no—
470C22—24no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683999112yes—
478C22C2×#1451975111yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—4881no—
484C4—4881no—
485C4—4881no—
486C4#1683999112yes—
487C4C22×#1175819011yescharacteristic
488C4C2×#1451979911yescharacteristic
489C4C2×#1451980311yescharacteristic
490C4#1684056011yescharacteristic
491C2—22no—
492C2—44no—
493C2—9761no—
494C2#2145528311yescharacteristic
495C2C2×#1683999111yescharacteristic
496C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C8×D4×C61C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC2×C8×D4D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. #21456378C2semidirect
2. #21455616C2semidirect
4. #21445242C2semidirect
5. #21445228C2semidirect
6. #21445194C2semidirect
7. #21445160C2semidirect
8. #21444575C2semidirect
9. #21444507C2semidirect
10. #21444494C2semidirect
11. C2×C8×D4×C61C2semidirect
15. #16840160C22semidirect
16. #16840135C22semidirect
17. #16840113C22semidirect
18. #16840109C22semidirect
34. #16839513C22semidirect
35. #16839513C22semidirect
36. #16839510C22semidirect
37. #16839509C22semidirect
38. #16839507C22semidirect
39. #16839505C22semidirect
114. C2×D4×C61SD16semidirect
216. C8×C61AGL(1,ℤ/8)semidirect
246. C2×C8×D4D61semidirect
297. C61#19254semidirect
372. C2×D4#11758190semidirect
467. C8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214556162
#214552832
#214452422
#214452282
#214451942
#214451602
#214445752
#214445072
#214444942
C2×C8×D4×C612
#1925461