Finite Group Databasebeta

← #29667694

Subgroups — #29667694

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456401C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic, complemented
4#21456144C211yesmaximal, characteristic, complemented
5#21456141C211yesmaximal, characteristic, complemented
6#21445432C211yesmaximal, characteristic, complemented
7#21445430C211yesmaximal, characteristic, complemented
8#21445427C211yesmaximal, characteristic, complemented
9#21445423C211yesmaximal, characteristic, complemented
10#21445421C211yesmaximal, characteristic, complemented
11#21445418C211yesmaximal, characteristic, complemented
12#21445414C211yesmaximal, characteristic, complemented
13#21445409C211yesmaximal, characteristic, complemented
14C2×C8×#92418C211yesmaximal, characteristic, complemented
15C61×#3167C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840160C2211yescharacteristic
21#16840141C2211yescharacteristic
22#16840138C2211yescharacteristic
23#16840135C2211yescharacteristic
24#16840119—22no—
25#16840116—22no—
26#16840114C2211yescharacteristic
27#16840114C2211yescharacteristic
28#16840114C2211yescharacteristic
29#16840113C2211yescharacteristic
30#16840112C2211yescharacteristic
31#16840112C2211yescharacteristic
32#16840112C2211yescharacteristic
33#16840110C2211yescharacteristic
34#16840110C2211yescharacteristic
35#16840110C2211yescharacteristic
36#16840109C2211yescharacteristic
37#16840108C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840001—22no—
41#16839996—22no—
42#16839991—22no—
43C4×#11758202—22no—
44C4×#11758190—22no—
45C4×#11758199—22no—
46#16839758—22no—
47#16839756—22no—
48C2×C4×#92418C2211yescharacteristic
49C2×C4×#92418C2211yescharacteristic
50C8×#92418C2211yescharacteristic
51C8×#92418C2211yescharacteristic
52C8×#92418C2211yescharacteristic
53C8×#92418C2211yescharacteristic
54C61×#533C2211yescharacteristic
55C61×#532C2211yescharacteristic
56C61×#498C2211yescharacteristic
57C61×#492C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#486C2211yescharacteristic
60C61×#465—22no—
61C4×SD16×C61—22no—
62C2×C4×C8×C61C2211yescharacteristic
63C2×#11758222—21no—
64C2×#11758202—22no—
65C2×#11758190—22no—
66C2×#11758199—22no—
67C22×#92418D412yes—
68C22×#92418D412yes—
69#14519777C2311yescharacteristic
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519774C2311yescharacteristic
73#14519769—21no—
74#14519765—21no—
75#14519756—21no—
76#14519756—21no—
77#14519751—21no—
78#14519751—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×D244—21no—
82C4×D244—21no—
83#14519736—21no—
84#14519710—22no—
85#14519710—22no—
86#14519709—22no—
87#14519702—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—22no—
91#14519700—22no—
92#14519700—22no—
93#14519700—22no—
94#14519699—22no—
95#14519698—22no—
96#14519698—22no—
97#14519698—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101C4×#92418D412yes—
102C4×#92418D412yes—
103C4×#92418C2311yescharacteristic
104C4×#92418C2311yescharacteristic
105C4×#92418C2311yescharacteristic
106C4×#92418C2311yescharacteristic
107C2×#41×C61—21no—
108C2×SD16×C61—22no—
109C22×C8×C61D412yes—
110C4⋊Q8×C61C2311yescharacteristic
111C4⋊Q8×C61C2311yescharacteristic
112C4⋊Q8×C61C2311yescharacteristic
113#127×C61C2311yescharacteristic
114#125×C61—21no—
115#124×C61—21no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118#121×C61—21no—
119#121×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124#117×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—22no—
127#103×C61—22no—
128#102×C61—22no—
129C4×C8×C61D412yes—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132#41×C61—44no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145#11758222—44no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×C4×D61—41no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153SD16×C61—44no—
154C2×C8×C61—24no—
155C2×C8×C61C2×D412yes—
156C2×C8×C61C2×D412yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—41no—
164C4⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C22⋊C4×C61—41no—
167C22⋊C4×C61—41no—
168C42×C61—41no—
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2411yescharacteristic
173C22⋊Dic61—41no—
174#11758202—44no—
175#11758199—44no—
176#11758198—41no—
177#11758198—41no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187C2×#92418—24no—
188C2×#92418—24no—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193#11758190—44no—
194C23×C61—41no—
195C23×C61#12711yescharacteristic
196C22×D61—41no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—82no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—82no—
210C2×C4×C61#12711yescharacteristic
211C2×C4×C61#12711yescharacteristic
212C2×C4×C61#12711yescharacteristic
213C2×C4×C61#13514yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C22⋊D61—82no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—42no—
222D244—42no—
223D244—42no—
224C4×D61—82no—
225Dic122—42no—
226Dic122—42no—
227Dic122—42no—
228C8×C61—22no—
229C8×C61#13514yes—
230#92418—22no—
231#92418—22no—
232#92418—22no—
233#92418#13514yes—
234#19600—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61#49512yes—
241C22×C61C2×#12711yescharacteristic
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—81no—
248C4×C61—81no—
249C4×C61—81no—
250C4×C61#49512yes—
251C4×C61C2×#13512yes—
252C4×C61C2×#13512yes—
253C4×C61C2×#13512yes—
254Dic61—81no—
255Dic61—81no—
256Dic61—81no—
257#3170—611no—
258#3169—611no—
259#3167—611no—
260#3167D6111yescharacteristic, complemented
261#3131—611no—
262#3131—611no—
263#1711—611no—
264#1711—611no—
265#1710—611no—
266#1709—611no—
267#1708—611no—
268#1708—611no—
269#1707—611no—
270#1706—611no—
271C2×C82—611no—
272C2×C61—22no—
273C2×C61—161no—
274C2×C61C2×#49511yescharacteristic
275C2×C61#314711yescharacteristic
276C2×C61#314711yescharacteristic
277D61—161no—
278#533—611no—
279#533C2×D6111yescharacteristic
280#532—611no—
281#532C2×D6111yescharacteristic
282#500—611no—
283#500—611no—
284#498—611no—
285#498C2×D6111yescharacteristic
286#494—611no—
287#493—611no—
288#492—611no—
289#492C2×D6111yescharacteristic
290#488—1222no—
291#488—1222no—
292#487—611no—
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487—611no—
299#487C2×D6111yescharacteristic
300#486—611no—
301#486—611no—
302#486—611no—
303#486—611no—
304#486—611no—
305#486—611no—
306#486—611no—
307#486C2×D6111yescharacteristic
308#467—1222no—
309#466—1222no—
310#465—22no—
311#465—1222no—
312C4×Q16—1222no—
313C4×SD16—22no—
314C4×SD16—1222no—
315C4×D8—1222no—
316#416—1222no—
317#416—1222no—
318C2×C4×C8—611no—
319C2×C4×C8—611no—
320C2×C4×C8C2×D6111yescharacteristic
321C82—611no—
322C82—611no—
323C82—611no—
324C82—611no—
325C61#1960011yescharacteristic, complemented, Sylow
326C2×#41—21no—
327C2×#41—1221no—
328C2×Q16—1222no—
329C2×SD16—22no—
330C2×SD16—1222no—
331C2×D8—1222no—
332C22×C8—612no—
333C22×C8—612no—
334C22×C8C22⋊D6112yes—
335C4⋊Q8—611no—
336C4⋊Q8—611no—
337C4⋊Q8—611no—
338C4⋊Q8C22×D6111yescharacteristic
339C4⋊Q8C22×D6111yescharacteristic
340C4⋊Q8C22×D6111yescharacteristic
341#127—611no—
342#127C22×D6111yescharacteristic
343#125—21no—
344#125—1221no—
345#124—21no—
346#124—1221no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—1221no—
350C22⋊Q8—1221no—
351#121—21no—
352#121—21no—
353#121—1221no—
354#121—1221no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—1221no—
358C4×Q8—1221no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—1221no—
362C4×D4—1221no—
363#117—21no—
364#117—1221no—
365C2×C42C22×D6111yescharacteristic
366#107—1222no—
367#107—1222no—
368#106—22no—
369#106—1222no—
370#103—22no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#103—1222no—
377#103—1222no—
378#102—22no—
379#102—1222no—
380#102—1222no—
381#102—1222no—
382#102—1222no—
383#102—1222no—
384#102—1222no—
385#102—1222no—
386C4×C8—612no—
387C4×C8—612no—
388C4×C8—611no—
389C4×C8—611no—
390C4×C8—611no—
391C4×C8—611no—
392C4×C8C22⋊D6112yes—
393C4×C8C22×D6111yescharacteristic
394C4×C8C22×D6111yescharacteristic
395#41—44no—
396#41—2444no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—41no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—2441no—
411C22×C4—41no—
412C22×C4—2441no—
413C22×C4D4×D6111yescharacteristic
414C22×C4D4×D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416Q16—2444no—
417SD16—44no—
418SD16—2444no—
419D8—2444no—
420C2×C8—24no—
421C2×C8—612no—
422C2×C8—612no—
423C2×C8—612no—
424C2×C8—612no—
425C2×C8—1224no—
426C2×C8—1224no—
427C2×C8C2×C22⋊D6112yes—
428C2×C8C2×C22⋊D6112yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1221no—
440C4⋊C4—1221no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2441no—
444C4⋊C4—2441no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C42—41no—
452C42—2441no—
453C42D4×D6111yescharacteristic
454C42D4×D6111yescharacteristic
455C42C2×C22⋊D6111yescharacteristic
456C42C23×D6111yescharacteristic
457C23—41no—
458C23—2441no—
459C23#1451977311yescharacteristic
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—2442no—
464Q8—2442no—
465Q8—2442no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—82no—
470D4—4882no—
471D4—2442no—
472D4—2442no—
473D4—2442no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—4882no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4#1451977311yescharacteristic
485C2×C4#1451977311yescharacteristic
486C2×C4#1451977311yescharacteristic
487C2×C4#1451979814yes—
488C2×C4C2×D4×D6111yescharacteristic
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C8—22no—
492C8—614no—
493C8—1222no—
494C8—1222no—
495C8—1222no—
496C8#1451979714yes—
497C22—21no—
498C22—21no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22#1684014312yes—
506C22C2×#1451977311yescharacteristic
507C4—21no—
508C4—21no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—4881no—
513C4—4881no—
514C4—4881no—
515C4#1684014312yes—
516C4#135×D6112yes—
517C4C2×#1451979712yes—
518C4C2×#1451979812yes—
519C2—22no—
520C2—161no—
521C2—9761no—
522C2C2×#1684014311yescharacteristic
523C2#2145624311yescharacteristic
524C2#2145624511yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#127
prime radical#21456394C2
hypercenter#3167D61
hypocenterC61#19600
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
3. #21456378C2semidirect
4. #21456144C2semidirect
5. #21456141C2semidirect
6. #21445432C2semidirect
7. #21445430C2semidirect
8. #21445427C2semidirect
9. #21445423C2semidirect
10. #21445421C2semidirect
11. #21445418C2semidirect
12. #21445414C2semidirect
13. #21445409C2semidirect
14. C2×C8×#92418C2semidirect
15. C61×#3167C2semidirect
260. #3167D61semidirect
325. C61#19600semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214563942
#214563782
#214561442
#214561412
#214454322
#214454302
#214454272
#214454232
#214454212
#214454182
#214454142
#214454092
C2×C8×#924182
C61×#31672
#1960061