Finite Group Databasebeta

← #29595463

Subgroups — #29595463

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21460530C211yesmaximal, characteristic
2#21447447C211yesmaximal, characteristic, complemented
3#21447446C211yesmaximal, characteristic, complemented
4C2×#16839545C211yesmaximal, characteristic, complemented
5#21447439C211yesmaximal, characteristic, complemented
6#21447432C211yesmaximal, characteristic, complemented
7C2×C61×#350C211yesmaximal, characteristic, complemented
8#16840598C411yescharacteristic
9#16840529C411yescharacteristic
10#16840528—22no—
11#16840528C411yescharacteristic
12#16840528C411yescharacteristic
13#16840527—22no—
14#16840527—22no—
15C2×#14519783—22no—
16C2×#14519783C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#131×D61—22no—
19#131×D61—22no—
20C2×#14519717C2211yescharacteristic, complemented
21C2×#14519717C2211yescharacteristic, complemented
22#16839546—22no—
23#16839545—22no—
24#16839545—22no—
25#16839543—22no—
26#16839541—22no—
27C2×#131×C61C2211yescharacteristic
28C2×M32×C61C2211yescharacteristic, complemented
29C22×C16×C61C2211yescharacteristic, complemented
30C61×#350—22no—
31#14519825—21no—
32#14519824—21no—
33#14519824C812yes—
34#14519823—21no—
35#14519823C812yes—
36#14519822—21no—
37#14519822—22no—
38C2×#11758222—22no—
39C2×#11758222C2×C411yescharacteristic
40C2×#11758222C2×C411yescharacteristic
41#41×D61—22no—
42#41×D61—22no—
43#14519785—24no—
44#14519785—24no—
45#14519785—22no—
46#14519785—22no—
47#14519784—22no—
48#14519784—22no—
49#14519783—24no—
50#14519783—24no—
51#14519783—22no—
52#14519783—21no—
53#14519783—21no—
54#14519783—21no—
55#14519783—21no—
56#14519783—21no—
57#14519783—21no—
58#14519783—22no—
59#14519783D412yes—
60#14519783D412yes—
61C2×#11758194—22no—
62C2×#11758194C2×C411yescharacteristic
63C2×#11758194C2×C411yescharacteristic
64C2×#11758189—24no—
65C2×#11758189—21no—
66C2×#11758189D412yes—
67M16×D61—24no—
68M16×D61—24no—
69M16×D61—22no—
70M16×D61—22no—
71C2×C8×D61—24no—
72C2×C8×D61—21no—
73C2×C8×D61D412yes—
74#14519717—21no—
75#14519717—21no—
76#14519717—21no—
77#14519717—21no—
78C2×#11758185—21no—
79C2×#11758185—21no—
80C2×#41×C61C2×C411yescharacteristic
81#131×C61—21no—
82#131×C61—21no—
83#131×C61D412yes—
84#131×C61D412yes—
85C2×M16×C61—21no—
86C2×M16×C61C2×C411yescharacteristic
87C22×C8×C61—21no—
88C22×C8×C61C2311yescharacteristic
89M32×C61—21no—
90M32×C61—21no—
91C2×C16×C61—21no—
92C2×C16×C61—21no—
93C2×C16×C61—21no—
94C2×C16×C61—21no—
95#41×C61—21no—
96#41×C61—21no—
97#41×C61C22⋊C412yes—
98#41×C61C2×C812yes—
99C2×Q8×C61M1611yescharacteristic
100C2×D4×C61—21no—
101C2×D4×C61M1611yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61C22×C411yescharacteristic
104C2×C22⋊D61—24no—
105C2×C22⋊D61—21no—
106C2×C22⋊D61M1612yes—
107#11758226—22no—
108#11758226—22no—
109Q8×D61—22no—
110Q8×D61—22no—
111#11758224—24no—
112#11758224—22no—
113#11758224—22no—
114#11758224—24no—
115D4×D61—24no—
116D4×D61—22no—
117D4×D61—24no—
118D4×D61—22no—
119#11758222—24no—
120#11758222—24no—
121#11758222—21no—
122#11758222—21no—
123#11758222—21no—
124#11758222—21no—
125#11758222—21no—
126#11758222—21no—
127#11758222—22no—
128#11758222—22no—
129#11758222C22⋊C412yes—
130#11758222C2×C812yes—
131C2×D244—22no—
132C2×D244C2×C811yescharacteristic
133C2×D244M1611yescharacteristic
134C2×C4×D61—24no—
135C2×C4×D61—21no—
136C2×C4×D61C22⋊C412yes—
137C2×Dic122—22no—
138C2×Dic122C2×C811yescharacteristic
139C2×Dic122M1611yescharacteristic
140M16×C61—21no—
141M16×C61—22no—
142M16×C61—22no—
143M16×C61C22⋊C412yes—
144C2×C8×C61—21no—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61C2×D411yescharacteristic
153#11758194—24no—
154#11758194—21no—
155#11758194—21no—
156#11758194—21no—
157#11758194C22⋊C412yes—
158C2×#92418—24no—
159C2×#92418—21no—
160C2×#92418C22⋊C412yes—
161#11758189—24no—
162#11758189—24no—
163#11758189—22no—
164#11758189—22no—
165#11758189—22no—
166#11758189—22no—
167C8×D61—24no—
168C8×D61—24no—
169C8×D61—22no—
170C8×D61—22no—
171C8×D61—22no—
172C8×D61—22no—
173C16×C61—41no—
174C16×C61—41no—
175#11758185—41no—
176#11758185—41no—
177C23×C61—21no—
178C23×C61C2×M1611yescharacteristic
179C22×D61—24no—
180C22×D61—21no—
181C22×D61C22⋊C812yes—
182Q8×C61—21no—
183Q8×C61C22⋊C812yes—
184D4×C61—22no—
185D4×C61—22no—
186D4×C61—21no—
187D4×C61C22⋊C812yes—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C22×C811yescharacteristic
195C2×C4×C61C2×M1611yescharacteristic
196C2×C4×C61C2×M1611yescharacteristic
197C22⋊D61—24no—
198C22⋊D61—24no—
199C22⋊D61—22no—
200C22⋊D61—22no—
201C22⋊D61—22no—
202C22⋊D61—22no—
203C2×Dic61—24no—
204C2×Dic61—21no—
205C2×Dic61C22⋊C812yes—
206D244—24no—
207D244—21no—
208D244—21no—
209D244—21no—
210D244C22⋊C812yes—
211C4×D61—24no—
212C4×D61—24no—
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—22no—
217Dic122—24no—
218Dic122—21no—
219Dic122—21no—
220Dic122—21no—
221Dic122C22⋊C812yes—
222C8×C61—22no—
223C8×C61#12011yescharacteristic
224C8×C61#12011yescharacteristic
225C8×C61#12011yescharacteristic
226C8×C61#12011yescharacteristic
227#92418—22no—
228#92418—22no—
229#12685—611nomaximal, Sylow
230C22×C61—22no—
231C22×C61—41no—
232C22×C61—81no—
233C22×C61C2×C22⋊C811yescharacteristic
234C22×C61#40711yescharacteristic
235C22×C61#40711yescharacteristic
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—42no—
239C2×D61—81no—
240C2×D61—84no—
241C4×C61—22no—
242C4×C61#37911yescharacteristic
243C4×C61C2×C22⋊C811yescharacteristic
244C4×C61C2×C22⋊C811yescharacteristic
245C4×C61#40711yescharacteristic
246Dic61—22no—
247Dic61—22no—
248#3573—611no—
249#2117—611no—
250#2117—611no—
251C2×#350—611no—
252C2×#350D6111yescharacteristic, complemented
253#2115—611no—
254#2113—611no—
255C2×C61—21no—
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—42no—
259C2×C61#177911yescharacteristic
260D61—42no—
261D61—42no—
262#585—611no—
263#568—611no—
264#568—611no—
265#568—611no—
266#568—1222no—
267#568—1222no—
268#568—1222no—
269C2×#131—611no—
270C2×#131—611no—
271C2×#131—1222no—
272C2×#131—1222no—
273C2×#131—1222no—
274C2×#131C2×D6111yescharacteristic
275C2×M32—611no—
276C2×M32—611no—
277C2×M32C2×D6111yescharacteristic, complemented
278C22×C16C2×D6111yescharacteristic, complemented
279#350—22no—
280#350—1222no—
281#350—1222no—
282#350—1222no—
283#349—1222no—
284C22⋊C16—1222no—
285C61#1268511yescharacteristic, complemented, Sylow
2862−1+4—612no—
2872−1+4—1221no—
2882−1+4—1221no—
289H(5,𝔽2)—612no—
290H(5,𝔽2)—1222no—
291H(5,𝔽2)—1221no—
292H(5,𝔽2)—1221no—
293C2×#41—611no—
294C2×#41—611no—
295C2×#41—1222no—
296C2×#41—1222no—
297C2×#41—1222no—
298C2×#41C4×D6111yescharacteristic
299#131—21no—
300#131—21no—
301#131—612no—
302#131—612no—
303#131—1222no—
304#131—1222no—
305#131—1224no—
306#131—1224no—
307#131—1222no—
308#131—1222no—
309#131—1224no—
310#131—1224no—
311#131—1222no—
312#131—1222no—
313#131—1221no—
314#131—1221no—
315#131—1221no—
316#131—1221no—
317#131—1221no—
318#131—1221no—
319#131D24412yes—
320#131C22⋊D6112yes—
321C2×M16—21no—
322C2×M16—611no—
323C2×M16—612no—
324C2×M16—611no—
325C2×M16—1224no—
326C2×M16—1222no—
327C2×M16—1224no—
328C2×M16—1224no—
329C2×M16—1222no—
330C2×M16—1222no—
331C2×M16—1221no—
332C2×M16C4×D6111yescharacteristic
333C22×C8—21no—
334C22×C8—612no—
335C22×C8—1224no—
336C22×C8—1221no—
337C22×C8C22×D6111yescharacteristic
338M32—21no—
339M32—21no—
340M32—1221no—
341M32—1221no—
342M32—1221no—
343M32—1221no—
344C2×C16—21no—
345C2×C16—21no—
346C2×C16—21no—
347C2×C16—21no—
348C2×C16—1221no—
349C2×C16—1221no—
350#41—21no—
351#41—21no—
352#41—612no—
353#41—612no—
354#41—1222no—
355#41—1222no—
356#41—1224no—
357#41—1224no—
358#41—1224no—
359#41—1224no—
360#41—1222no—
361#41—1222no—
362#41—1222no—
363#41—1222no—
364#41—1221no—
365#41—1221no—
366#41—1221no—
367#41—1221no—
368#41—1221no—
369#41—1221no—
370#41C8×D6112yes—
371#41#1175819812yes—
372C2×Q8—611no—
373C2×Q8—611no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8#1175818911yescharacteristic
378C2×D4—21no—
379C2×D4—612no—
380C2×D4—611no—
381C2×D4—611no—
382C2×D4—1222no—
383C2×D4—1224no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—1224no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4#1175818911yescharacteristic
390C22×C4—21no—
391C22×C4—612no—
392C22×C4—1224no—
393C22×C4—1221no—
394C22×C4C2×C4×D6111yescharacteristic
395M16—21no—
396M16—22no—
397M16—22no—
398M16—612no—
399M16—1224no—
400M16—1224no—
401M16—1224no—
402M16—1221no—
403M16—1221no—
404M16—1221no—
405M16—1222no—
406M16—1222no—
407M16—1222no—
408M16—1222no—
409M16#1175819812yes—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—612no—
414C2×C8—1224no—
415C2×C8—1224no—
416C2×C8—1224no—
417C2×C8—1221no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8C2×D24411yescharacteristic
423C2×C8D4×D6111yescharacteristic
424C2×C8D4×D6111yescharacteristic
425C2×C8D4×D6111yescharacteristic
426C2×C8D4×D6111yescharacteristic
427C2×C8C2×C22⋊D6111yescharacteristic
428C16—41no—
429C16—41no—
430C16—2441no—
431C16—2441no—
432C23—21no—
433C23—612no—
434C23—1224no—
435C23—1221no—
436C23M16×D6111yescharacteristic
437Q8—21no—
438Q8—612no—
439Q8—1224no—
440Q8—1221no—
441Q8—1221no—
442Q8—1221no—
443Q8#1451968912yes—
444D4—22no—
445D4—22no—
446D4—21no—
447D4—612no—
448D4—1224no—
449D4—1224no—
450D4—1224no—
451D4—1222no—
452D4—1222no—
453D4—1222no—
454D4—1222no—
455D4—1221no—
456D4—1221no—
457D4—1221no—
458D4#1451968912yes—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—612no—
463C2×C4—1224no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1221no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4C22⋊C4×D6111yescharacteristic
472C2×C4C22⋊C4×D6111yescharacteristic
473C2×C4C2×#1175819811yescharacteristic
474C2×C4C2×C8×D6111yescharacteristic
475C2×C4M16×D6111yescharacteristic
476C2×C4C2×#1175818911yescharacteristic
477C8—22no—
478C8—1222no—
479C8—1222no—
480C8#1451974511yescharacteristic
481C8#1451974511yescharacteristic
482C8#1451974711yescharacteristic
483C8#1451974711yescharacteristic
484C22—22no—
485C22—41no—
486C22—81no—
487C22—1222no—
488C22—1222no—
489C22—4881no—
490C22—4884no—
491C22—2442no—
492C22C22⋊C8×D6111yescharacteristic
493C22#1683972411yescharacteristic
494C22#1683972511yescharacteristic
495C4—22no—
496C4—1222no—
497C4—1222no—
498C4#1683961711yescharacteristic
499C4C22⋊C8×D6111yescharacteristic
500C4C2×#1451968911yescharacteristic
501C4#1683972411yescharacteristic
502C2—21no—
503C2—21no—
504C2—21no—
505C2—42no—
506C2—2442no—
507C2—2442no—
508C2#2144572811yescharacteristic
subgroupG/H
centerC8#14519747
commutatorC2×C4×C61C22×C8
FrattiniC22×C8C22×D61
FittingC2×C61×#350C2
socleC2×C61#1779
prime radical#16840598C4
cosocleC22×C8×C61C23
hypercenterC2×#350D61
hypocenterC61#12685
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. #21447447C2semidirect
3. #21447446C2semidirect
4. C2×#16839545C2semidirect
5. #21447439C2semidirect
6. #21447432C2semidirect
7. C2×C61×#350C2semidirect
20. C2×#14519717C22semidirect
21. C2×#14519717C22semidirect
28. C2×M32×C61C22semidirect
29. C22×C16×C61C22semidirect
252. C2×#350D61semidirect
277. C2×M32C2×D61semidirect
278. C22×C16C2×D61semidirect
285. C61#12685semidirect

Maximal subgroups

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

subgroupindex
#214605302
#214474472
#214474462
C2×#168395452
#214474392
#214474322
C2×C61×#3502
#1268561