Finite Group Databasebeta

← #29652437

Subgroups — #29652437

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455763C211yesmaximal, characteristic, complemented
2#21455755C211yesmaximal, characteristic
3#21450106C211yesmaximal, characteristic, complemented
4#21444850C214yesmaximal, complemented
5#21444816C214yesmaximal, complemented
6#21444352C211yesmaximal, characteristic, complemented
7#21444348C211yesmaximal, characteristic
8#21444149C211yesmaximal, characteristic, complemented
9C2×C61×#539C211yesmaximal, characteristic
10C2×#14519736C2211yescharacteristic
11#16840100—22no—
12#16840095C2212yescomplemented
13#16840090C2212yescomplemented
14#16840083—22no—
15#16840078C2212yes—
16#16840073C2212yes—
17#16839802—21no—
18#16839802—21no—
19#16839771—21no—
20#16839767—21no—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839701C2212yescomplemented
24#16839692C2212yescomplemented
25#16839643—22no—
26#16839635—21no—
27#16839625—21no—
28#16839625C2211yescharacteristic
29#16839499C2214yescomplemented
30#16839496C2214yes—
31#16839488C2214yescomplemented
32C61×#539—22no—
33C61×#539—22no—
34C61×#539—22no—
35C61×#539C2214yes—
36C2×#126×C61—21no—
37C2×#125×C61—21no—
38C2×#123×C61—21no—
39C2×#123×C61—21no—
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×C4×D4×C61—21no—
44C22×C4⋊C4×C61C2211yescharacteristic
45C2×#11758194—21no—
46C2×#11758194—21no—
47#14519736—21no—
48#14519736—21no—
49#14519736—21no—
50#14519736—21no—
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2311yescharacteristic
53C2×#11758196—22no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—21no—
56C2×C4×Dic61—21no—
57#14519710—22no—
58#14519710—22no—
59#14519708—22no—
60#14519708—22no—
61#14519701—22no—
62#14519701—22no—
63#14519698—22no—
64#14519698—22no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67C22⋊#92418C2312yes—
68C22⋊#92418C2312yes—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684—21no—
73#14519684C2312yes—
74C22×Q8×C61D411yescharacteristic
75C22×D4×C61—21no—
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C23×C4×C61D411yescharacteristic
79C23×C4×C61D411yescharacteristic
80C23×C4×C61C2311yescharacteristic
81#126×C61—24no—
82#126×C61—24no—
83#125×C61—24no—
84#125×C61—24no—
85#123×C61—24no—
86#123×C61—24no—
87#123×C61—24no—
88#123×C61—24no—
89C22⋊Q8×C61—24no—
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61D412yes—
94C22⋊Q8×C61C2312yes—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61D412yes—
98#121×C61C2312yes—
99C4×D4×C61—24no—
100C4×D4×C61—24no—
101C2×C4⋊C4×C61—21no—
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×C61—21no—
109C2×C4⋊C4×C61D412yes—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2312yes—
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×C61—21no—
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C2×Q8×C61—21no—
122C2×Q8×C61—42no—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—42no—
128C2×D4×C61—44no—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—82no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—24no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C42×C61—24no—
174C22⋊Dic61—42no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—44no—
182C4×Dic61—41no—
183C4×Dic61—41no—
184#11758194—42no—
185#11758194—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61#12011yescharacteristic
196C23×C61#12011yescharacteristic
197C23×C61C22×D411yescharacteristic
198Q8×C61—42no—
199Q8×C61—81no—
200D4×C61—42no—
201D4×C61—44no—
202D4×C61—44no—
203D4×C61—81no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—81no—
225C2×C4×C61—84no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237#92418—42no—
238#92418—42no—
239#18139—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—44no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—84no—
252C22×C61#45812yes—
253C22×C61C2×#12011yescharacteristic
254C22×C61#57512yes—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—42no—
264Dic61—82no—
265Dic61—81no—
266Dic61—81no—
267C2×#539D6111yescharacteristic
268#3095—611no—
269#3094—611no—
270#2484—611no—
271#1615—614no—
272#1609—614no—
273#1511—611no—
274#1510—611no—
275#1463—611no—
276C2×C61—22no—
277C2×C61—22no—
278C2×C61—41no—
279C2×C61—82no—
280C2×C61#300511yescharacteristic
281C2×C61#302011yescharacteristic
282C2×C61C2×#45811yescharacteristic
283#539—22no—
284#539—22no—
285#539—22no—
286#539C2×D6114yes—
287C2×#126—21no—
288C2×#125—21no—
289C2×#123—21no—
290C2×#123—21no—
291C2×C22⋊Q8—21no—
292C2×C22⋊Q8C2×D6111yescharacteristic
293C2×#121C2×D6111yescharacteristic
294C2×C4×D4—21no—
295C2×#117—611no—
296C22×C4⋊C4C2×D6111yescharacteristic
297#485—1222no—
298#484—612no—
299#483—612no—
300#482—1222no—
301#481—612no—
302#480—612no—
303#428—1221no—
304#428—1221no—
305#419—1221no—
306#418—1221no—
307#407—611no—
308#407—611no—
309#400—612no—
310#398—612no—
311#387—1222no—
312#385—1221no—
313#381—611no—
314#381—1221no—
315#328—614no—
316#327—614no—
317#324—614no—
318C61#1813911yescharacteristic, complemented, Sylow
319C22×Q8C22⋊D6111yescharacteristic
320C22×D4—21no—
321C22×D4C22⋊D6111yescharacteristic
322C23×C4—21no—
323C23×C4C22⋊D6111yescharacteristic
324C23×C4C22⋊D6111yescharacteristic
325C23×C4C22×D6111yescharacteristic
326C2×M16—1221no—
327C2×M16—1221no—
328#126—24no—
329#126—24no—
330#125—24no—
331#125—24no—
332#123—24no—
333#123—24no—
334#123—24no—
335#123—24no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8C22⋊D6112yes—
341C22⋊Q8C22×D6112yes—
342#121—21no—
343#121—21no—
344#121C22⋊D6112yes—
345#121C22×D6112yes—
346C4×D4—24no—
347C4×D4—24no—
348#117—1221no—
349#117—1221no—
350#117—1221no—
351#117—1221no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22⋊D6112yes—
364C2×C4⋊C4C22⋊D6111yescharacteristic
365C2×C4⋊C4C22⋊D6111yescharacteristic
366C2×C4⋊C4C22×D6112yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C42—21no—
375C2×C42—1221no—
376#107—1222no—
377#107—1222no—
378#106—1222no—
379#106—1222no—
380#103—1222no—
381#103—1222no—
382#102—1222no—
383#102—1222no—
384C22⋊C8—612no—
385C22⋊C8—612no—
386C22⋊C8—1221no—
387C22⋊C8—1221no—
388#95—612no—
389#95—1221no—
390#95—1222no—
391#95—1222no—
392#95—1222no—
393C24—21no—
394C24C2×C22⋊D6111yescharacteristic
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8C2×C22⋊D6112yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4C2×C22⋊D6112yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4C2×C22⋊D6112yes—
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6112yes—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C23×D6111yescharacteristic
427M16—2442no—
428M16—2442no—
429C2×C8—1222no—
430C2×C8—1222no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—2444no—
447C4⋊C4C2×C22⋊D6112yes—
448C4⋊C4C2×C22⋊D6112yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—2442no—
457C22⋊C4—2441no—
458C22⋊C4—2441no—
459C42—24no—
460C42—2441no—
461C42—2441no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23#1451974611yescharacteristic
470C23#1451974611yescharacteristic
471C23C22×C22⋊D6111yescharacteristic
472Q8—42no—
473Q8—81no—
474D4—42no—
475D4—44no—
476D4—44no—
477D4—81no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—81no—
499C2×C4—84no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451974611yescharacteristic
508C2×C4#1451974611yescharacteristic
509C2×C4C22×C22⋊D6111yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C8—2442no—
512C8—2442no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—84no—
525C22#1683995212yes—
526C22C2×#1451974611yescharacteristic
527C22#1684056012yes—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—4882no—
538C4—4881no—
539C4—4881no—
540C2—22no—
541C2—22no—
542C2—41no—
543C2—82no—
544C2#2145506111yescharacteristic
545C2#2145518311yescharacteristic
546C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#539C2
socleC22×C61C2×#120
prime radicalC2×#121×C61C22
hypercenterC2×#539D61
hypocenterC61#18139
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
3. #21450106C2semidirect
4. #21444850C2semidirect
5. #21444816C2semidirect
6. #21444352C2semidirect
8. #21444149C2semidirect
12. #16840095C22semidirect
13. #16840090C22semidirect
23. #16839701C22semidirect
24. #16839692C22semidirect
29. #16839499C22semidirect
31. #16839488C22semidirect
318. C61#18139semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214557552
#214501062
#214448502
#214448162
#214443522
#214443482
#214441492
C2×C61×#5392
#1813961