Finite Group Databasebeta

← #29661275

Subgroups — #29661275

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457025C211yesmaximal, characteristic, complemented
2#21456822C211yesmaximal, characteristic, complemented
3#21456205C211yesmaximal, characteristic, complemented
4#21445142C214yesmaximal, complemented
5#21445136C214yesmaximal, complemented
6#21444443C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444417C211yesmaximal, characteristic, complemented
9C2×C61×#562C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840172C2212yescomplemented
12#16840157C2212yescomplemented
13#16840132C2212yescomplemented
14#16840121C2212yescomplemented
15#16840121C2212yescomplemented
16#16840121C2212yescomplemented
17#16840116—22no—
18#16840097—21no—
19#16840092—21no—
20#16840066—22no—
21#16839981—22no—
22#16839918—21no—
23#16839913—21no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839771—21no—
27#16839767—21no—
28C4×#11758194C2211yescharacteristic
29#16839504C2214yescomplemented
30#16839502C2214yescomplemented
31#16839502C2214yescomplemented
32C61×#562—22no—
33C61×#562—22no—
34C61×#562—22no—
35C61×#562C2214yes—
36C2×#125×C61C2211yescharacteristic
37C2×#124×C61C2211yescharacteristic
38C2×#124×C61C2211yescharacteristic
39C2×#123×C61—21no—
40C2×#123×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C22⋊Q8×C61—21no—
43C2×#121×C61—21no—
44C2×#121×C61—21no—
45C2×#11758202—22no—
46C2×#11758200—22no—
47C2×#11758194—21no—
48#14519777C2311yescharacteristic
49#14519777C2311yescharacteristic
50#14519769—21no—
51#14519756—21no—
52#14519756—21no—
53C4×Dic122—21no—
54C4×Dic122—21no—
55#14519736—21no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60#14519701—22no—
61#14519701—22no—
62#14519701—22no—
63#14519700—22no—
64#14519700—22no—
65#14519698—22no—
66#14519698—22no—
67#14519698—22no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70#14519688C2312yes—
71#14519688C2312yes—
72#14519688C2312yes—
73C4×#92418—21no—
74C22×Q8×C61D411yescharacteristic
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C22×D4×C61D411yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61—21no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61D412yes—
85#125×C61C2312yes—
86#124×C61—21no—
87#124×C61—21no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61D412yes—
91#124×C61D412yes—
92#124×C61C2312yes—
93#124×C61C2312yes—
94#123×C61—24no—
95#123×C61—24no—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—24no—
99C22⋊Q8×C61—24no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—24no—
102#121×C61—24no—
103#121×C61—24no—
104#121×C61—24no—
105#121×C61—24no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
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×C22⋊C4×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C24×C61—21no—
120C24×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—42no—
124C2×Q8×C61—81no—
125C2×Q8×C61C2×D412yes—
126C2×Q8×C61C2×D412yes—
127C2×D4×C61—24no—
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—42no—
132C2×D4×C61—81no—
133C2×D4×C61—82no—
134C2×D4×C61—82no—
135C2×D4×C61C2×D412yes—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—82no—
144C22×C4×C61—82no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122—21no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2411yescharacteristic
174C22⋊Dic61—41no—
175#11758202—82no—
176#11758200—82no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—41no—
182#11758196—41no—
183C4×Dic61—41no—
184#11758194—42no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61#12011yescharacteristic
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—81no—
200Q8×C61—81no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—81no—
204D4×C61—81no—
205D4×C61—84no—
206D4×C61—84no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—84no—
219C2×C4×C61—84no—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×Dic61—41no—
227C2×Dic61—41no—
228Dic122—42no—
229Dic122—42no—
230#92418—42no—
231#92418—42no—
232#92418—42no—
233#18998—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—84no—
243C22×C61—84no—
244C22×C61#45512yes—
245C22×C61C2×#12011yescharacteristic
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—42no—
255C4×C61—42no—
256Dic61—81no—
257Dic61—81no—
258C2×#562D6111yescharacteristic
259#3236—611no—
260#3215—611no—
261#3140—611no—
262#1662—614no—
263#1661—614no—
264#1533—611no—
265#1528—611no—
266#1527—611no—
267C2×C61—22no—
268C2×C61—82no—
269C2×C61—82no—
270C2×C61C2×#45511yescharacteristic
271C2×C61#301911yescharacteristic
272C2×C61#302011yescharacteristic
273#562—22no—
274#562—22no—
275#562—22no—
276#562C2×D6114yes—
277#533—611no—
278C2×#125C2×D6111yescharacteristic
279C2×#124C2×D6111yescharacteristic
280C2×#124C2×D6111yescharacteristic
281C2×#123—21no—
282C2×#123—21no—
283C2×C22⋊Q8—21no—
284C2×C22⋊Q8—21no—
285C2×#121—21no—
286C2×#121—21no—
287#501—612no—
288#497—612no—
289#491—612no—
290#489—612no—
291#489—612no—
292#489—612no—
293#488—1222no—
294#484—1221no—
295#483—1221no—
296#479—1222no—
297#464—1222no—
298#451—1221no—
299#450—1221no—
300#432—611no—
301#432—611no—
302#419—1221no—
303#418—1221no—
304C4×M16—611no—
305#330—614no—
306#329—614no—
307#329—614no—
308C61#1899811yescharacteristic, complemented, Sylow
309C22×Q8C22⋊D6111yescharacteristic
310C22×Q8C22⋊D6111yescharacteristic
311C22×D4—21no—
312C22×D4—21no—
313C22×D4C22⋊D6111yescharacteristic
314C22×D4C22⋊D6111yescharacteristic
315C23×C4—21no—
316C23×C4—21no—
317C2×Q16—1222no—
318C2×SD16—1222no—
319C2×M16—1221no—
320C4⋊Q8—611no—
321C4⋊Q8—611no—
322#125—21no—
323#125—21no—
324#125—1221no—
325#125C22⋊D6112yes—
326#125C22×D6112yes—
327#124—21no—
328#124—21no—
329#124—21no—
330#124—21no—
331#124C22⋊D6112yes—
332#124C22⋊D6112yes—
333#124C22×D6112yes—
334#124C22×D6112yes—
335#123—24no—
336#123—24no—
337#123—24no—
338#123—24no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8—24no—
343C22⋊Q8—1221no—
344C22⋊Q8—1221no—
345#121—24no—
346#121—24no—
347#121—24no—
348#121—24no—
349C4×Q8—1221no—
350C4×Q8—1221no—
351#117—1221no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4C22⋊D6111yescharacteristic
357C2×C4⋊C4C22⋊D6111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366#106—1222no—
367C4⋊C8—1221no—
368C4⋊C8—1221no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#102—1222no—
375#102—1222no—
376#102—1222no—
377C22⋊C8—1221no—
378C22⋊C8—1221no—
379#97—612no—
380#97—612no—
381#97—612no—
382C4×C8—1221no—
383C24—21no—
384C24—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—81no—
389C2×Q8—1221no—
390C2×Q8—1221no—
391C2×Q8C2×C22⋊D6112yes—
392C2×Q8C2×C22⋊D6112yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—81no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4C2×C22⋊D6112yes—
402C2×D4C2×C22⋊D6112yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414Q16—4882no—
415SD16—4882no—
416M16—2442no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—2441no—
433C4⋊C4—2441no—
434C4⋊C4C2×C22⋊D6112yes—
435C4⋊C4C2×C22⋊D6112yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—2441no—
447C42—2441no—
448C42C2×C22⋊D6111yescharacteristic
449C42C2×C22⋊D6111yescharacteristic
450C42C2×C22⋊D6111yescharacteristic
451C42C23×D6111yescharacteristic
452C23—22no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23#1451974611yescharacteristic
461Q8—42no—
462Q8—42no—
463Q8—81no—
464Q8—81no—
465Q8—2442no—
466Q8—2442no—
467D4—42no—
468D4—42no—
469D4—81no—
470D4—81no—
471D4—84no—
472D4—84no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4#1451974611yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4#1451974611yescharacteristic
491C2×C4C22×C22⋊D6111yescharacteristic
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4C22×C22⋊D6111yescharacteristic
494C8—2442no—
495C8—2442no—
496C8—2442no—
497C22—21no—
498C22—21no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—84no—
506C22—84no—
507C22#1683994012yes—
508C22C2×#1451974611yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—4881no—
520C4—4881no—
521C2—22no—
522C2—82no—
523C2—82no—
524C2C2×#1683994011yescharacteristic
525C2#2145517611yescharacteristic
526C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#562C2
socleC22×C61C2×#120
hypercenterC2×#562D61
hypocenterC61#18998
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #21457025C2semidirect
2. #21456822C2semidirect
3. #21456205C2semidirect
4. #21445142C2semidirect
5. #21445136C2semidirect
6. #21444443C2semidirect
7. #21444421C2semidirect
8. #21444417C2semidirect
11. #16840172C22semidirect
12. #16840157C22semidirect
13. #16840132C22semidirect
14. #16840121C22semidirect
15. #16840121C22semidirect
16. #16840121C22semidirect
29. #16839504C22semidirect
30. #16839502C22semidirect
31. #16839502C22semidirect
308. C61#18998semidirect

Maximal subgroups

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

subgroupindex
#214570252
#214568222
#214562052
#214451422
#214451362
#214444432
#214444212
#214444172
C2×C61×#5622
#1899861