Finite Group Databasebeta

← #29663787

Subgroups — #29663787

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#21455612C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3C2×C8×D244C211yesmaximal, characteristic, complemented
4#21445239C212yesmaximal, complemented
5#21445225C212yesmaximal, complemented
6#21445191C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444575C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444496C211yesmaximal, characteristic, complemented
11C61×#3167C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C22×C8×D61—21no—
14C2×C4×D244C2211yescharacteristic
15#16840059—22no—
16#16840049C2212yes—
17#16840039C2212yes—
18#16839974—22no—
19#16839964C2212yes—
20#16839954C2212yes—
21C8×D244—21no—
22C8×D244—21no—
23C8×D244C2212yes—
24C8×D244C2212yes—
25#16839805—21no—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839766—21no—
30#16839763—21no—
31C2×#14519689—21no—
32#16839513C2212yescomplemented
33#16839513C2212yescomplemented
34#16839511C2212yescomplemented
35#16839508C2212yescomplemented
36#16839508C2212yescomplemented
37#16839506C2212yescomplemented
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#498C2211yescharacteristic, complemented
41C61×#492C2211yescharacteristic, complemented
42C61×#487C2211yescharacteristic, complemented
43C61×#486C2211yescharacteristic, complemented
44C61×#465—21no—
45C61×#465—21no—
46C4×SD16×C61—21no—
47C4×SD16×C61—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51#14519802—44no—
52C2×#11758201—22no—
53C2×#11758199—22no—
54C2×#11758194—21no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59C2×C8×D61—44no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244D412yes—
63C4×D244C2312yes—
64C2×C4⋊Dic61D411yescharacteristic
65C2×#11758198—21no—
66#14519710—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519698—22no—
76#14519697—22no—
77#14519697—22no—
78#14519689—22no—
79#14519689—22no—
80C2×#41×C61—21no—
81C2×SD16×C61—21no—
82C2×SD16×C61—21no—
83C22×C8×C61D411yescharacteristic
84C22×C8×C61D411yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61C2311yescharacteristic
89#125×C61—21no—
90#124×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99#117×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#106×C61—21no—
102#106×C61—21no—
103#103×C61—21no—
104#103×C61—21no—
105#102×C61—21no—
106#102×C61—21no—
107C4×C8×C61D411yescharacteristic
108C4×C8×C61D411yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C4×C8×C61C2311yescharacteristic
111C23×D61—21no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C22×Dic61—21no—
126C2×D244—21no—
127C2×D244—22no—
128C2×D244SD1614yescomplemented
129C2×D244C2×D412yes—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134SD16×C61—42no—
135SD16×C61—42no—
136C2×C8×C61—22no—
137C2×C8×C61—41no—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61—41no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61#4111yescharacteristic
157C42×C61C2411yescharacteristic
158#11758201—44no—
159#11758199—44no—
160#11758198—22no—
161#11758198—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61C2×D412yes—
164#11758194—42no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—42no—
170C8×D61—44no—
171C8×D61—44no—
172C23×C61—41no—
173C23×C61#12111yescharacteristic
174C22×D61—22no—
175C22×D61—24no—
176C22×D61—42no—
177C22×D61—42no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—41no—
182Q8×C61—81no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—81no—
188D4×C61—82no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—82no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×SD1612yes—
200C2×C4×C61#13512yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×Dic61—22no—
205C2×Dic61—42no—
206D244—22no—
207D244—24no—
208D244C2×SD1614yes—
209C4×D61—44no—
210C4×D61—42no—
211C4×D61—42no—
212C8×C61—21no—
213C8×C61—21no—
214C8×C61—21no—
215C8×C61AGL(1,ℤ/8)12yescomplemented
216#92418—42no—
217#92418—42no—
218#92418—42no—
219#19254—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61#46512yes—
226C22×C61C2×#12111yescharacteristic
227C2×D61—24no—
228C2×D61—48no—
229C2×D61—42no—
230C2×D61—42no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—81no—
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—42no—
243#3167D6111yescharacteristic, complemented
244#3077—611no—
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—161no—
256C2×C61#303611yescharacteristic
257C2×C61C2×#46511yescharacteristic
258C2×C61#306711yescharacteristic
259D61—44no—
260C2×AGL(1,ℤ/8)—1221no—
261C23×C8—1221no—
262#533C2×D6111yescharacteristic
263#532C2×D6111yescharacteristic
264C2×C4×D4—611no—
265#498C2×D6111yescharacteristic, complemented
266#492C2×D6111yescharacteristic, complemented
267#487C2×D6111yescharacteristic, complemented
268#486C2×D6111yescharacteristic, complemented
269#478—1222no—
270#476—612no—
271#474—612no—
272#465—21no—
273#465—21no—
274#463—1222no—
275#461—612no—
276#459—612no—
277C4×SD16—21no—
278C4×SD16—21no—
279C8×D4—612no—
280C8×D4—612no—
281C8×D4—1221no—
282C8×D4—1221no—
283#428—1221no—
284#423—611no—
285#423—611no—
286C2×C4⋊C8—611no—
287#418—1221no—
288#417—1221no—
289C2×C22⋊C8—1221no—
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—21no—
299C22×D4—611no—
300C23×C4—1221no—
301AGL(1,ℤ/8)—2444no—
302C2×SD16—21no—
303C2×SD16—21no—
304C2×SD16—1222no—
305C2×D8—1222no—
306C2×M16—1221no—
307C2×M16—1221no—
308C22×C8—1221no—
309C22×C8—1222no—
310C22×C8—1222no—
311C22×C8—2444no—
312C22×C8C22⋊D6111yescharacteristic
313C22×C8C22⋊D6111yescharacteristic
314C4⋊Q8C22×D6111yescharacteristic
315C4⋊Q8C22×D6111yescharacteristic
316C4⋊Q8C22×D6111yescharacteristic
317#127C22×D6111yescharacteristic
318#125—21no—
319#124—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—612no—
329C4×D4—612no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—21no—
333C2×C4⋊C4—611no—
334C2×C22⋊C4—1221no—
335C2×C42C22×D6111yescharacteristic
336#107—1222no—
337#106—21no—
338#106—21no—
339#106—1222no—
340C4⋊C8—612no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346#103—21no—
347#103—21no—
348#103—1222no—
349#102—21no—
350#102—21no—
351#102—1222no—
352#102—1222no—
353#102—1222no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356C4×C8C22⋊D6111yescharacteristic
357C4×C8C22⋊D6111yescharacteristic
358C4×C8C22×D6111yescharacteristic
359C4×C8C22×D6111yescharacteristic
360C24—1221no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—612no—
371C2×D4—614no—
372C2×D4—1221no—
373C2×D4—1222no—
374C22×C4—41no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822411yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383SD16—42no—
384SD16—42no—
385SD16—2444no—
386D8—2444no—
387M16—2442no—
388M16—2442no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—2444no—
395C2×C8—2442no—
396C2×C8—2444no—
397C2×C8C2×C22⋊D6111yescharacteristic
398C2×C8C2×C22⋊D6111yescharacteristic
399C2×C8C2×C22⋊D6111yescharacteristic
400C2×C8C2×C22⋊D6111yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—612no—
410C4⋊C4—1221no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42—41no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—41no—
422C23—1224no—
423C23—1222no—
424C23—2442no—
425C23—2442no—
426C23#1451974811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438D4—614no—
439D4—1224no—
440D4—1222no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—1222no—
449C2×C4—2444no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4#1451974811yescharacteristic
454C2×C4#1451974811yescharacteristic
455C2×C4#1451974811yescharacteristic
456C2×C4SD16×D6112yes—
457C2×C4#1451979812yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C2×C4C2×#1175822411yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—2442no—
465C8—2442no—
466C8—2442no—
467C8#1451980212yescomplemented
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—1224no—
474C22—2448no—
475C22—2442no—
476C22—2442no—
477C22#1683998812yes—
478C22C2×#1451974811yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—2442no—
486C4#1683998812yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451979811yescharacteristic
489C4C2×#1451980211yescharacteristic
490C4#1684056011yescharacteristic
491C2—22no—
492C2—161no—
493C2—2444no—
494C2#2145527911yescharacteristic
495C2C2×#1683998811yescharacteristic
496C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3167D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. C2×C8×D244C2semidirect
4. #21445239C2semidirect
5. #21445225C2semidirect
6. #21445191C2semidirect
7. #21445157C2semidirect
8. #21444575C2semidirect
9. #21444509C2semidirect
10. #21444496C2semidirect
11. C61×#3167C2semidirect
32. #16839513C22semidirect
33. #16839513C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
36. #16839508C22semidirect
37. #16839506C22semidirect
40. C61×#498C22semidirect
41. C61×#492C22semidirect
42. C61×#487C22semidirect
43. C61×#486C22semidirect
128. C2×D244SD16semidirect
215. C8×C61AGL(1,ℤ/8)semidirect
243. #3167D61semidirect
265. #498C2×D61semidirect
266. #492C2×D61semidirect
267. #487C2×D61semidirect
268. #486C2×D61semidirect
297. C61#19254semidirect
467. C8#14519802semidirect

Maximal subgroups

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

subgroupindex
#214556122
#214552792
C2×C8×D2442
#214452392
#214452252
#214451912
#214451572
#214445752
#214445092
#214444962
C61×#31672
#1925461