Finite Group Databasebeta

← #29665421

Subgroups — #29665421

Order 15616.

704 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#21456760C211yesmaximal, characteristic, complemented
2#21455609C211yesmaximal, characteristic, complemented
3#21455295C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic, complemented
5#21445234C212yesmaximal, complemented
6#21445227C212yesmaximal, complemented
7#21445166C212yesmaximal, complemented
8#21445159C212yesmaximal, complemented
9#21444589C211yesmaximal, characteristic, complemented
10#21444507C211yesmaximal, characteristic, complemented
11C2×C61×#331C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13#16840319C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840134C2212yes—
16#16840108C2212yes—
17#16840070—21no—
18#16840063—21no—
19#16840058C2211yescharacteristic, complemented
20#16840053C2211yescharacteristic, complemented
21#16840048C2211yescharacteristic, complemented
22#16840043C2211yescharacteristic, complemented
23#16839959—21no—
24#16839959—21no—
25#16839959—21no—
26#16839959C2212yes—
27#16839954—21no—
28#16839954C2212yes—
29#16839913—22no—
30#16839886C2212yescomplemented
31#16839827C2212yescomplemented
32C8×C22⋊D61—22no—
33#16839815C2211yescharacteristic
34#16839814—21no—
35#16839800—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839767—21no—
39#16839759—21no—
40#16839512C2212yescomplemented
41#16839512C2212yescomplemented
42#16839510C2212yescomplemented
43#16839509C2212yescomplemented
44C2×#127×C61C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#331C2212yes—
48C61×#331C2212yes—
49C2×#11758222D411yescharacteristic
50#14519803—44no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758199—22no—
55C2×#11758194—21no—
56C2×C8×D61—22no—
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519769—21no—
60#14519756—21no—
61#14519756—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122D411yescharacteristic
65C4×Dic122C2311yescharacteristic
66C4×D244D411yescharacteristic
67C4×D244C2311yescharacteristic
68C4×C22⋊D61—21no—
69C4×C22⋊D61—21no—
70#14519736—21no—
71#14519736D411yescharacteristic
72#14519734—21no—
73C42×D61—21no—
74#14519711—21no—
75#14519711—21no—
76#14519709—21no—
77#14519709—21no—
78C4⋊#92418—21no—
79C4⋊#92418D411yescharacteristic
80C4⋊#92418D411yescharacteristic
81C4⋊#92418C2312yes—
82#14519706—22no—
83#14519702—21no—
84#14519702—21no—
85#14519699—21no—
86#14519699—21no—
87#14519698—22no—
88#14519698—22no—
89#14519698—22no—
90#14519698—22no—
91C22⋊#92418D412yes—
92#14519689—22no—
93C4×#92418C2312yes—
94C8×Dic61—22no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×C8×C61—21no—
99C22×C8×C61—21no—
100#127×C61—21no—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108C4⋊C8×C61C2311yescharacteristic
109C4⋊C8×C61C2311yescharacteristic
110C24×C61—41no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—81no—
120C2×D4×C61—81no—
121C2×D4×C61—82no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×C22⋊D61—21no—
126#11758222—22no—
127#11758222SD1614yescomplemented
128C2×D244C2×D411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122C2×D411yescharacteristic
135C2×C8×C61—22no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—42no—
139C2×C8×C61—81no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—21no—
145C22⋊Dic61—41no—
146#11758200—44no—
147#11758200—82no—
148#11758200—82no—
149#11758199—44no—
150#11758198—21no—
151#11758198—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61C2×D411yescharacteristic
157C4⋊Dic61C2×D411yescharacteristic
158#11758196—21no—
159#11758196—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165C4×Dic61—41no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418C2×D412yes—
169C2×#92418C2×D412yes—
170C8×D61—82no—
171C23×C61—42no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61#12111yescharacteristic
176C22×D61—21no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—82no—
182D4×C61—82no—
183D4×C61—82no—
184D4×C61—82no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C2×SD1612yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—24no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202D244—21no—
203D244C2×SD1612yes—
204C4×D61—24no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—81no—
208Dic122—21no—
209Dic122—41no—
210Dic122—41no—
211Dic122—81no—
212Dic122C2×SD1612yes—
213C8×C61—41no—
214C8×C61—41no—
215C8×C61—42no—
216#92418—22no—
217#92418—22no—
218#92418—42no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—82no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61—164no—
226C22×C61#46512yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61#46512yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×#13511yescharacteristic
237C4×C61#57611yescharacteristic
238C4×C61#57711yescharacteristic
239Dic61—22no—
240Dic61—41no—
241Dic61—41no—
242Dic61—81no—
243Dic61—81no—
244#3208—611no—
245#3076—611no—
246#3038—611no—
247#2988—611no—
248#1678—612no—
249#1677—612no—
250#1666—612no—
251#1665—612no—
252#1570—611no—
253#1550—611no—
254C2×#331D6111yescharacteristic, complemented
255C2×C61—22no—
256C2×C61—162no—
257C2×C61#304511yescharacteristic
258C2×C61C2×#46511yescharacteristic
259C2×C61#306811yescharacteristic
260D61—42no—
261C2×AGL(1,ℤ/8)—1221no—
262#533—611no—
263C2×#127C2×D6111yescharacteristic
264C4×#41—611no—
265#492—612no—
266#486—612no—
267#479—1221no—
268#478—1221no—
269#477—611no—
270#476—611no—
271#475—611no—
272#474—611no—
273#460—612no—
274#460—1221no—
275#460—1221no—
276#460—1221no—
277#459—612no—
278#459—1221no—
279#450—1222no—
280C8×Q8—612no—
281C8×D4—612no—
282C8×D4—1222no—
283#431—611no—
284#431—1221no—
285#427—1221no—
286#423—611no—
287C2×C4⋊C8—1221no—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×C4⋊C8C2×D6111yescharacteristic
290#418—1221no—
291#416—1221no—
292#334—612no—
293#334—612no—
294#333—612no—
295#332—612no—
296#331C2×D6112yes—
297#331C2×D6112yes—
298C61#1940911yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—2444no—
304C2×SD16—1222no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—1222no—
308C2×M16—1221no—
309C22×C8—21no—
310C22×C8—21no—
311C22×C8—1222no—
312C4⋊Q8—611no—
313C4⋊Q8—611no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D6112yes—
318#125—1221no—
319C22⋊Q8—1221no—
320C22⋊Q8—1221no—
321C4×Q8—611no—
322C4×Q8—611no—
323C4×Q8—1221no—
324C4×Q8—1221no—
325C4×D4—611no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329#117—611no—
330#117—1221no—
331#117—1221no—
332C2×C42—1221no—
333C2×C42C22×D6111yescharacteristic
334#107—1221no—
335#107—1221no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—612no—
343C4⋊C8—1222no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6112yes—
346C4⋊C8C22×D6111yescharacteristic
347C4⋊C8C22×D6111yescharacteristic
348#103—1221no—
349#103—1221no—
350#102—1222no—
351#102—1222no—
352#102—1221no—
353#102—1221no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C4×C8—612no—
359C4×C8—1222no—
360C24—41no—
361#41—614no—
362#41—1222no—
363C2×Q8—611no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—82no—
377C2×D4—611no—
378C2×D4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4#1175822211yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385SD16—4882no—
386SD16—4882no—
387SD16—2444no—
388D8—2444no—
389M16—2442no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—42no—
394C2×C8—81no—
395C2×C8—612no—
396C2×C8—612no—
397C2×C8—1222no—
398C2×C8—4882no—
399C4⋊C4—611no—
400C4⋊C4—611no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—2441no—
413C42—1221no—
414C42—1221no—
415C42—1221no—
416C42—2441no—
417C42#1175822211yescharacteristic
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—1221no—
426C23#1451975011yescharacteristic
427Q8—612no—
428Q8—1221no—
429Q8—4881no—
430Q8—2441no—
431Q8—2441no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—612no—
441D4—1224no—
442D4—1221no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—41no—
446C2×C4—1221no—
447C2×C4—1224no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—4881no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4#1451975011yescharacteristic
456C2×C4#1451975011yescharacteristic
457C2×C4#1451975011yescharacteristic
458C2×C4SD16×D6112yes—
459C2×C4#1451979612yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×#1175822211yescharacteristic
463C8—41no—
464C8—41no—
465C8—42no—
466C8—1222no—
467C8—1222no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—164no—
475C22—1222no—
476C22—2441no—
477C22—2441no—
478C22#1683999012yes—
479C22C2×#1451975011yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—1222no—
484C4—4881no—
485C4—4881no—
486C4—2441no—
487C4—2441no—
488C4#1683999012yes—
489C4C2×SD16×D6111yescharacteristic
490C4C2×#1451979611yescharacteristic
491C4#1684056311yescharacteristic
492C4#1684056811yescharacteristic
493C2—22no—
494C2—162no—
495C2—2442no—
496C2#2145536311yescharacteristic
497C2C2×#1683999011yescharacteristic
498C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#331C2
socleC22×C61C2×#121
prime radical#21455295C2
hypercenterC2×#331D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455609C2semidirect
4. #21454928C2semidirect
5. #21445234C2semidirect
6. #21445227C2semidirect
7. #21445166C2semidirect
8. #21445159C2semidirect
9. #21444589C2semidirect
10. #21444507C2semidirect
11. C2×C61×#331C2semidirect
19. #16840058C22semidirect
20. #16840053C22semidirect
21. #16840048C22semidirect
22. #16840043C22semidirect
30. #16839886C22semidirect
31. #16839827C22semidirect
40. #16839512C22semidirect
41. #16839512C22semidirect
42. #16839510C22semidirect
43. #16839509C22semidirect
127. #11758222SD16semidirect
254. C2×#331D61semidirect
298. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214556092
#214552952
#214549282
#214452342
#214452272
#214451662
#214451592
#214445892
#214445072
C2×C61×#3312
#1940961