Finite Group Databasebeta

← #29661308

Subgroups — #29661308

Order 15616.

884 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457009C211yesmaximal, characteristic, complemented
2#21456806C211yesmaximal, characteristic
3#21456222C211yesmaximal, characteristic, complemented
4#21445153C214yesmaximal, complemented
5#21445117C214yesmaximal, complemented
6#21444446C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444417C211yesmaximal, characteristic, complemented
9C2×C61×#563C211yesmaximal, characteristic, complemented
10#16840312C2211yescharacteristic
11#16840172C2212yescomplemented
12#16840152C2212yes—
13#16840132C2212yescomplemented
14#16840128C2212yescomplemented
15#16840128C2212yescomplemented
16#16840121C2212yes—
17#16840119—22no—
18#16840080—21no—
19#16840075—21no—
20#16840059—22no—
21#16839974—22no—
22#16839901—21no—
23#16839896—21no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839761—21no—
27#16839761—21no—
28C4×#11758194C2211yescharacteristic
29#16839504C2214yescomplemented
30#16839504C2214yescomplemented
31#16839502C2214yes—
32C61×#563—22no—
33C61×#563—22no—
34C61×#563—22no—
35C61×#563C2214yes—
36C2×C4⋊Q8×C61—21no—
37C2×#126×C61—21no—
38C2×#126×C61—21no—
39C2×#125×C61C2211yescharacteristic
40C2×#125×C61C2211yescharacteristic
41C2×#124×C61C2211yescharacteristic
42C2×#123×C61—21no—
43C2×C22⋊Q8×C61—21no—
44C2×C22⋊Q8×C61—21no—
45C2×#11758222—21no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60#14519701—22no—
61#14519700—22no—
62#14519700—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66#14519697—22no—
67#14519697—22no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70#14519688C2312yes—
71#14519688C2312yes—
72#14519688C2312yes—
73C4×#92418—21no—
74C22×Q8×C61—21no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C4⋊Q8×C61—24no—
79C4⋊Q8×C61—24no—
80#126×C61—24no—
81#126×C61—24no—
82#126×C61—24no—
83#126×C61—24no—
84#125×C61—21no—
85#125×C61—21no—
86#125×C61—21no—
87#125×C61—21no—
88#125×C61D412yes—
89#125×C61D412yes—
90#125×C61C2312yes—
91#125×C61C2312yes—
92#124×C61—21no—
93#124×C61—21no—
94#124×C61D412yes—
95#124×C61C2312yes—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—24no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C24×C61—21no—
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—81no—
123C2×Q8×C61—82no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—21no—
126C2×D4×C61—81no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—82no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C2×C22⋊D61—41no—
139#11758222—44no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×C4×D61—41no—
144C2×Dic122—21no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C4⋊C4×C61C2×D412yes—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—24no—
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2411yescharacteristic
175#11758201—82no—
176#11758199—82no—
177#11758198—41no—
178#11758198—41no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758194—42no—
182C2×#92418—22no—
183C2×#92418—22no—
184C2×#92418—22no—
185C23×C61—22no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61#12011yescharacteristic
190C22×D61—41no—
191Q8×C61—42no—
192Q8×C61—81no—
193Q8×C61—84no—
194D4×C61—42no—
195D4×C61—81no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—84no—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C22⋊D61—82no—
217C2×Dic61—41no—
218D244—42no—
219D244—42no—
220D244—42no—
221C4×D61—82no—
222Dic122—42no—
223#92418—42no—
224#92418—42no—
225#92418—42no—
226#19002—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—84no—
233C22×C61#45412yes—
234C22×C61C2×#12011yescharacteristic
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249Dic61—81no—
250C2×#563D6111yescharacteristic, complemented
251#3234—611no—
252#3213—611no—
253#3142—611no—
254#1664—614no—
255#1657—614no—
256#1534—611no—
257#1528—611no—
258#1527—611no—
259C2×C61—22no—
260C2×C61—82no—
261C2×C61C2×#45411yescharacteristic
262C2×C61#301911yescharacteristic
263C2×C61#302011yescharacteristic
264D61—161no—
265#563—22no—
266#563—22no—
267#563—22no—
268#563C2×D6114yes—
269#532—611no—
270C2×C4⋊Q8—21no—
271C2×#126—21no—
272C2×#126—21no—
273C2×#125C2×D6111yescharacteristic
274C2×#125C2×D6111yescharacteristic
275C2×#124C2×D6111yescharacteristic
276C2×#123—21no—
277C2×C22⋊Q8—21no—
278C2×C22⋊Q8—21no—
279#501—612no—
280#496—612no—
281#491—612no—
282#490—612no—
283#490—612no—
284#489—612no—
285#488—1222no—
286#481—1221no—
287#480—1221no—
288#478—1222no—
289#463—1222no—
290#448—1221no—
291#447—1221no—
292#432—611no—
293#432—611no—
294#417—1221no—
295#417—1221no—
296C4×M16—611no—
297#330—614no—
298#330—614no—
299#329—614no—
300C61#1900211yescharacteristic, complemented, Sylow
301C2×#41—1221no—
302C22×Q8—21no—
303C22×Q8C22⋊D6111yescharacteristic
304C22×D4C22⋊D6111yescharacteristic
305C23×C4—21no—
306C2×SD16—1222no—
307C2×D8—1222no—
308C2×M16—1221no—
309C4⋊Q8—24no—
310C4⋊Q8—24no—
311C4⋊Q8—611no—
312#127—611no—
313#126—24no—
314#126—24no—
315#126—24no—
316#126—24no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125—21no—
321#125C22⋊D6112yes—
322#125C22⋊D6112yes—
323#125C22×D6112yes—
324#125C22×D6112yes—
325#124—21no—
326#124—21no—
327#124—1221no—
328#124C22⋊D6112yes—
329#124C22×D6112yes—
330#123—24no—
331#123—24no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336#121—1221no—
337#121—1221no—
338C4×D4—1221no—
339C4×D4—1221no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C22⋊D6111yescharacteristic
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42—21no—
356C2×C42C22×D6111yescharacteristic
357#107—1222no—
358#106—1222no—
359C4⋊C8—1221no—
360C4⋊C8—1221no—
361#103—1222no—
362#103—1222no—
363#103—1222no—
364#102—1222no—
365#102—1222no—
366#102—1222no—
367#102—1222no—
368#102—1222no—
369C22⋊C8—1221no—
370C22⋊C8—1221no—
371#97—612no—
372#97—612no—
373#97—612no—
374C4×C8—1221no—
375C24—21no—
376#41—2444no—
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—81no—
380C2×Q8—82no—
381C2×Q8—1221no—
382C2×Q8C2×C22⋊D6112yes—
383C2×D4—21no—
384C2×D4—81no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2441no—
389C2×D4C2×C22⋊D6112yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—82no—
397C22×C4—2441no—
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401SD16—4882no—
402D8—4882no—
403M16—2442no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8—1222no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4C2×C22⋊D6112yes—
424C4⋊C4C2×C22⋊D6112yes—
425C4⋊C4C2×C22⋊D6112yes—
426C4⋊C4C2×C22⋊D6112yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C42—24no—
437C42C2×C22⋊D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—22no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—2441no—
446C23#1451974611yescharacteristic
447Q8—42no—
448Q8—81no—
449Q8—84no—
450Q8—2442no—
451D4—42no—
452D4—81no—
453D4—4882no—
454D4—2442no—
455D4—2442no—
456D4—2442no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—4882no—
472C2×C4—2441no—
473C2×C4#1451974611yescharacteristic
474C2×C4#1451974611yescharacteristic
475C2×C4#1451974611yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4C22×C22⋊D6111yescharacteristic
479C8—2442no—
480C8—2442no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—84no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22#1683993512yes—
492C22C2×#1451974611yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—4881no—
505C2—22no—
506C2—82no—
507C2—9761no—
508C2C2×#1683993511yescharacteristic
509C2#2145517611yescharacteristic
510C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#563C2
socleC22×C61C2×#120
prime radical#21456806C2
hypercenterC2×#563D61
hypocenterC61#19002
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #21457009C2semidirect
3. #21456222C2semidirect
4. #21445153C2semidirect
5. #21445117C2semidirect
6. #21444446C2semidirect
7. #21444421C2semidirect
8. #21444417C2semidirect
9. C2×C61×#563C2semidirect
11. #16840172C22semidirect
13. #16840132C22semidirect
14. #16840128C22semidirect
15. #16840128C22semidirect
29. #16839504C22semidirect
30. #16839504C22semidirect
250. C2×#563D61semidirect
300. C61#19002semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214568062
#214562222
#214451532
#214451172
#214444462
#214444212
#214444172
C2×C61×#5632
#1900261