Finite Group Databasebeta

← #29662849

Subgroups — #29662849

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456215C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3D4×#11758194C211yesmaximal, characteristic, complemented
4#21445396C212yesmaximal, complemented
5#21445354C212yesmaximal, complemented
6#21445272C212yesmaximal, complemented
7#21445230C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444500C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C2×C61×#476C211yesmaximal, characteristic, complemented
12C22×#11758194—21no—
13#16840312C2211yescharacteristic
14#16840124—22no—
15#16840114C2212yescomplemented
16#16840110C2212yescomplemented
17#16840075—21no—
18#16840042C2211yescharacteristic
19#16840042C2211yescharacteristic
20#16839957C2211yescharacteristic
21#16839957C2211yescharacteristic
22#16839896—21no—
23C4×#11758199—21no—
24C4×#11758199—21no—
25#16839840C2212yes—
26#16839834—22no—
27D4×#92418C2212yes—
28#16839815C2211yescharacteristic
29#16839786C2211yescharacteristic
30#16839761—21no—
31#16839761—21no—
32#16839726—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839511C2212yescomplemented
37#16839511C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yescomplemented
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#476—21no—
43C61×#476—21no—
44C61×#476C2212yes—
45C61×#476C2212yes—
46C2×#106×C61—21no—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61—21no—
49C2×#11758222—21no—
50C2×#11758199—21no—
51C2×#11758199—21no—
52C2×#11758194—21no—
53C2×#11758194—22no—
54C2×#11758194—44no—
55C2×#11758194D411yescharacteristic
56C2×#11758194D411yescharacteristic
57C22×#92418—22no—
58#14519777C2311yescharacteristic
59#14519774C2311yescharacteristic
60#14519765—21no—
61#14519751—21no—
62#14519751—21no—
63C4×D244—21no—
64C4×D244—21no—
65#14519710—21no—
66#14519710—21no—
67C4⋊#92418—21no—
68C4⋊#92418C2311yescharacteristic
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2312yes—
71#14519700—22no—
72#14519700—22no—
73#14519698—21no—
74#14519698—21no—
75#14519697—21no—
76#14519697—21no—
77#14519697—22no—
78#14519697—22no—
79C22⋊#92418—21no—
80C22⋊#92418—22no—
81C22⋊#92418—22no—
82#14519688D412yes—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C22×Q8×C61—21no—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C22×C8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61C2312yes—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61D411yescharacteristic
96C4×D4×C61D411yescharacteristic
97C4×D4×C61C2311yescharacteristic
98C4×D4×C61C2311yescharacteristic
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105#106×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#102×C61—22no—
109#102×C61—22no—
110C24×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—41no—
113C2×Q8×C61—82no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—41no—
117C2×D4×C61C2×D411yescharacteristic
118C2×D4×C61C2×D411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C2×C22⋊D61—41no—
131#11758222—42no—
132#11758222—42no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×C4×D61—41no—
137C2×Dic122—21no—
138C2×C8×C61—22no—
139C2×C8×C61—42no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61C2×D411yescharacteristic
146C4⋊C4×C61C2×D411yescharacteristic
147C22⋊C4×C61—21no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61#4111yescharacteristic
154C42×C61C2411yescharacteristic
155#11758199—42no—
156#11758199—42no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161#11758194—22no—
162#11758194—22no—
163#11758194—44no—
164#11758194—42no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—22no—
168C2×#92418—44no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—81no—
176C23×C61#12111yescharacteristic
177C22×D61—41no—
178Q8×C61—82no—
179Q8×C61—82no—
180D4×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—42no—
184D4×C61#13712yes—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—82no—
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C22⋊D61—82no—
208C2×Dic61—41no—
209D244—41no—
210D244—41no—
211D244—41no—
212D244—41no—
213D244—81no—
214C4×D61—82no—
215Dic122—41no—
216Dic122—41no—
217C8×C61—42no—
218#92418—22no—
219#92418—22no—
220#92418—42no—
221#92418—41no—
222#92418—41no—
223#19168—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—84no—
233C22×C61#46312yes—
234C22×C61C2×#12111yescharacteristic
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—41no—
242C4×C61—41no—
243C4×C61—82no—
244C4×C61#46312yes—
245C4×C61C2×#13511yescharacteristic
246C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C4×C61C2×#13711yescharacteristic
248C4×C61#57511yescharacteristic
249Dic61—81no—
250#3141—611no—
251#3100—611no—
252C2×#476D6111yescharacteristic, complemented
253D4×M16—611no—
254#1703—612no—
255#1697—612no—
256#1683—612no—
257#1677—612no—
258#1565—611no—
259#1548—611no—
260#1547—611no—
261C2×C61—22no—
262C2×C61—42no—
263C2×C61—81no—
264C2×C61C2×#46311yescharacteristic
265C2×C61#305611yescharacteristic
266C2×C61#306811yescharacteristic
267D61—161no—
268C22×M16—1221no—
269#532—611no—
270C2×C4⋊Q8C2×D6111yescharacteristic
271C2×C4×D4C2×D6111yescharacteristic
272#489—1222no—
273#487—612no—
274#486—612no—
275#480—1221no—
276#476—21no—
277#476—21no—
278#476C2×D6112yes—
279#476C2×D6112yes—
280#474—611no—
281#474—611no—
282#459—611no—
283#459—611no—
284#447—1221no—
285C4×D8—1221no—
286C4×D8—1221no—
287#436—612no—
288#435—1222no—
289C8×D4—612no—
290#431—611no—
291C2×#106—21no—
292#423—611no—
293C2×C4⋊C8C2×D6111yescharacteristic
294#417—1221no—
295#417—1221no—
296C2×#102—21no—
297#407—1221no—
298C4×M16—611no—
299#335—612no—
300#334—612no—
301#333—612no—
302#333—612no—
303#332—612no—
304#331—612no—
305C61#1916811yescharacteristic, complemented, Sylow
306C2×#41—1221no—
307C22×Q8—21no—
308C22×D4C22⋊D6111yescharacteristic
309C23×C4—21no—
310C2×D8—1221no—
311C2×D8—1221no—
312C2×M16—611no—
313C2×M16—611no—
314C2×M16—1222no—
315C2×M16—1221no—
316C2×M16—2444no—
317C22×C8—21no—
318C22×C8—1222no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—611no—
323C4⋊Q8C22×D6112yes—
324#127—611no—
325#124—1221no—
326#121—1221no—
327#121—1221no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1221no—
331C4×D4—1221no—
332C4×D4C22⋊D6111yescharacteristic
333C4×D4C22⋊D6111yescharacteristic
334C4×D4C22×D6111yescharacteristic
335C4×D4C22×D6111yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C22⋊D6111yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D6111yescharacteristic
341#107—1221no—
342#107—1221no—
343#106—22no—
344#106—22no—
345C4⋊C8—21no—
346C4⋊C8—612no—
347C4⋊C8—611no—
348C4⋊C8—611no—
349C4⋊C8—1221no—
350C4⋊C8C22×D6112yes—
351#103—1222no—
352#103—1222no—
353#102—22no—
354#102—22no—
355#102—1222no—
356#102—1222no—
357#102—1221no—
358#102—1221no—
359#102—1221no—
360#102—1221no—
361C22⋊C8—1222no—
362C22⋊C8—1222no—
363C22⋊C8—1221no—
364#97—612no—
365C4×C8—612no—
366C4×C8—1221no—
367C24—21no—
368#41—2442no—
369#41—2442no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—82no—
373C2×Q8—1221no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—2441no—
381C2×D4C2×C22⋊D6111yescharacteristic
382C2×D4C2×C22⋊D6111yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—2441no—
392C22×C4D4×D6111yescharacteristic
393C22×C4#1175822411yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395D8—2442no—
396D8—2442no—
397M16—1222no—
398M16—1222no—
399M16—2444no—
400M16—2442no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—612no—
404C2×C8—612no—
405C2×C8—1222no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8—2444no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4C2×C22⋊D6111yescharacteristic
417C4⋊C4C2×C22⋊D6111yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C42D4×D6111yescharacteristic
425C42#1175822411yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C23×D6111yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—2441no—
434C23#1451974911yescharacteristic
435Q8—82no—
436Q8—82no—
437Q8—2441no—
438Q8—2441no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—4881no—
444D4—4882no—
445D4—2441no—
446D4—2441no—
447D4—2441no—
448D4—2441no—
449D4#1451980812yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—4882no—
465C2×C4—2441no—
466C2×C4#1451974911yescharacteristic
467C2×C4#1451974911yescharacteristic
468C2×C4#1451974911yescharacteristic
469C2×C4#1451979812yes—
470C2×C4#1451980412yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C2×C4C2×#1175822411yescharacteristic
474C8—42no—
475C8—1222no—
476C8—1222no—
477C8—2442no—
478C8—2441no—
479C8—2441no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—4881no—
490C22—4881no—
491C22—4881no—
492C22#1683997912yes—
493C22C2×#1451974911yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—82no—
500C4—4881no—
501C4#1683997912yes—
502C4C2×#1451979811yescharacteristic
503C4C2×#1451980411yescharacteristic
504C4C2×#1451980811yescharacteristic
505C4#1684056011yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—9761no—
510C2C2×#1683997911yescharacteristic
511C2#2145544511yescharacteristic
512C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#476C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenterC2×#476D61
hypocenterC61#19168
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #21456215C2semidirect
3. D4×#11758194C2semidirect
4. #21445396C2semidirect
5. #21445354C2semidirect
6. #21445272C2semidirect
7. #21445230C2semidirect
8. #21444569C2semidirect
9. #21444500C2semidirect
10. #21444494C2semidirect
11. C2×C61×#476C2semidirect
15. #16840114C22semidirect
16. #16840110C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839511C22semidirect
37. #16839511C22semidirect
38. #16839507C22semidirect
39. #16839505C22semidirect
252. C2×#476D61semidirect
305. C61#19168semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214558192
D4×#117581942
#214453962
#214453542
#214452722
#214452302
#214445692
#214445002
#214444942
C2×C61×#4762
#1916861