Finite Group Databasebeta

← #29663794

Subgroups — #29663794

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#21456379C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21445240C212yesmaximal, complemented
5#21445230C212yesmaximal, complemented
6#21445196C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444574C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11C61×#3077C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C23×#92418—21no—
14#16840312C2211yescharacteristic
15#16840158C2211yescharacteristic, complemented
16#16840136C2211yescharacteristic, complemented
17#16840112C2211yescharacteristic, complemented
18#16840110C2211yescharacteristic, complemented
19#16839993—21no—
20#16839993—21no—
21#16839977—22no—
22#16839967C2212yes—
23#16839957C2212yes—
24C4×#11758201—21no—
25C4×#11758201—21no—
26D4×#92418—21no—
27D4×#92418—21no—
28D4×#92418C2212yes—
29D4×#92418C2212yes—
30#16839786C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839761—21no—
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839512C2212yescomplemented
36#16839512C2212yescomplemented
37#16839511C2212yescomplemented
38#16839509C2212yescomplemented
39#16839508C2212yescomplemented
40#16839505C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#478—22no—
44C61×#476C2212yes—
45C61×#474C2212yes—
46C61×#428—21no—
47C61×#423C2211yescharacteristic
48C61×#418—21no—
49C2×#11758222—21no—
50#14519803—44no—
51C2×#11758200—22no—
52C2×#11758201—21no—
53C2×#11758201—21no—
54C2×#11758199—22no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418—44no—
60C22×#92418D411yescharacteristic
61C22×#92418D411yescharacteristic
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519708—21no—
70#14519708—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519701—21no—
76#14519701—21no—
77#14519700—22no—
78#14519697—22no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—22no—
82C22⋊#92418—22no—
83C4×#92418D411yescharacteristic
84C4×#92418D411yescharacteristic
85C4×#92418C2311yescharacteristic
86C4×#92418C2311yescharacteristic
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C2×M16×C61—21no—
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D412yes—
100C4×D4×C61C2312yes—
101#117×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#107×C61—22no—
106#106×C61—22no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#102×C61—22no—
110#102×C61—22no—
111C24×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61SD1614yescomplemented
117C2×D4×C61C2×D412yes—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C2×C22⋊D61—41no—
127#11758222—42no—
128#11758222—42no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×C4×D61—41no—
133C2×Dic122—21no—
134M16×C61—42no—
135C2×C8×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—41no—
147C42×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61#4111yescharacteristic
151C42×C61C2411yescharacteristic
152#11758201—42no—
153#11758201—42no—
154#11758200—44no—
155#11758199—44no—
156#11758198—41no—
157#11758198—41no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758194—42no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418—41no—
165C2×#92418—44no—
166C2×#92418—42no—
167C2×#92418—44no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61#12111yescharacteristic
177C22×D61—41no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—81no—
181D4×C61—22no—
182D4×C61—24no—
183D4×C61C2×SD1614yes—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—44no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61C2×SD1612yes—
198C2×C4×C61#13512yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C22⋊D61—82no—
203C2×Dic61—41no—
204D244—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—81no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212C8×C61—42no—
213#92418—21no—
214#92418—21no—
215#92418—21no—
216#92418—42no—
217#92418—42no—
218#92418AGL(1,ℤ/8)12yescomplemented
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—81no—
236C4×C61—81no—
237C4×C61#46512yes—
238C4×C61C22×SD1611yescharacteristic
239C4×C61C2×#13511yescharacteristic
240C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C61#57511yescharacteristic
242Dic61—81no—
243#3167—611no—
244#3077D6111yescharacteristic, complemented
245#3036—611no—
246C2×C8×D4—611no—
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—1221no—
262#533C2×D6111yescharacteristic
263#532—611no—
264C2×C4×D4C2×D6111yescharacteristic
265#498—611no—
266#492—611no—
267#487—611no—
268#486—611no—
269#478—22no—
270#476C2×D6112yes—
271#474C2×D6112yes—
272#465—1221no—
273#465—1221no—
274#463—1222no—
275#461—612no—
276#459—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279C8×D4—612no—
280C8×D4—612no—
281C8×D4—1221no—
282C8×D4—1221no—
283#428—21no—
284#423—611no—
285#423C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287#418—21no—
288#417—1221no—
289C2×C22⋊C8—1221no—
290C2×C4×C8—611no—
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×D4C22⋊D6111yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—2444no—
302C2×SD16—1221no—
303C2×SD16—1221no—
304C2×SD16—1222no—
305C2×D8—1222no—
306C2×M16—21no—
307C2×M16—1221no—
308C22×C8—611no—
309C22×C8—611no—
310C22×C8—1222no—
311C22×C8—1222no—
312C22×C8—1221no—
313C22×C8—2444no—
314C4⋊Q8—611no—
315C4⋊Q8C22×D6111yescharacteristic
316C4⋊Q8C22×D6111yescharacteristic
317#127—611no—
318#125—21no—
319#124—1221no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—1221no—
323#121—1221no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4C22⋊D6112yes—
331C4×D4C22×D6112yes—
332#117—21no—
333C2×C4⋊C4C22⋊D6111yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C22×D6111yescharacteristic
336#107—22no—
337#106—22no—
338#106—1221no—
339#106—1221no—
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—22no—
350#102—22no—
351#102—1221no—
352#102—1221no—
353#102—1222no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—611no—
360C24—21no—
361#41—2442no—
362#41—2442no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—2441no—
372C2×D4#1175820114yescomplemented
373C2×D4C2×C22⋊D6112yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—2441no—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383SD16—2442no—
384SD16—2442no—
385SD16—2444no—
386D8—2444no—
387M16—42no—
388M16—2442no—
389C2×C8—22no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—2444no—
398C2×C8—2441no—
399C2×C8—2444no—
400C2×C8—2442no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4C2×C22⋊D6112yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—41no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—2441no—
426C23#1451974911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—2441no—
431Q8—2441no—
432D4—24no—
433D4—22no—
434D4—4881no—
435D4—4882no—
436D4—2441no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4C2×#1175820114yes—
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—41no—
450C2×C4—41no—
451C2×C4—4882no—
452C2×C4—2441no—
453C2×C4#1451974911yescharacteristic
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4C2×#1175820112yes—
457C2×C4#1451979712yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C8—42no—
462C8—612no—
463C8—1221no—
464C8—1221no—
465C8—1221no—
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683999312yes—
478C22C2×#1451974911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—4881no—
486C4#1683999312yes—
487C4C22×#1175820111yescharacteristic
488C4C2×#1451979711yescharacteristic
489C4AGL(1,ℤ/8)×D6111yescharacteristic
490C4#1684055811yescharacteristic
491C2—22no—
492C2—44no—
493C2—9761no—
494C2#2145528011yescharacteristic
495C2C2×#1683999311yescharacteristic
496C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenter#3077D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
3. C2×D4×#92418C2semidirect
4. #21445240C2semidirect
5. #21445230C2semidirect
6. #21445196C2semidirect
7. #21445162C2semidirect
8. #21444574C2semidirect
9. #21444509C2semidirect
10. #21444497C2semidirect
11. C61×#3077C2semidirect
15. #16840158C22semidirect
16. #16840136C22semidirect
17. #16840112C22semidirect
18. #16840110C22semidirect
35. #16839512C22semidirect
36. #16839512C22semidirect
37. #16839511C22semidirect
38. #16839509C22semidirect
39. #16839508C22semidirect
40. #16839505C22semidirect
116. C2×D4×C61SD16semidirect
218. #92418AGL(1,ℤ/8)semidirect
244. #3077D61semidirect
297. C61#19254semidirect
372. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214552832
C2×D4×#924182
#214452402
#214452302
#214451962
#214451622
#214445742
#214445092
#214444972
C61×#30772
#1925461