Finite Group Databasebeta

← #29654157

Subgroups — #29654157

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456640C211yesmaximal, characteristic, complemented
2#21456625C211yesmaximal, characteristic
3#21451199C211yesmaximal, characteristic, complemented
4#21444816C214yesmaximal, complemented
5#21444782C214yesmaximal, complemented
6#21444339C211yesmaximal, characteristic, complemented
7#21444335C211yesmaximal, characteristic
8#21444149C211yesmaximal, characteristic, complemented
9C2×C61×#539C211yesmaximal, characteristic
10C2×#14519756C2211yescharacteristic
11#16840097—21no—
12#16840092—21no—
13#16840090C2212yescomplemented
14#16840073C2212yes—
15#16840018—22no—
16#16840006—22no—
17#16839918—21no—
18#16839917C2212yescomplemented
19#16839913—21no—
20#16839900C2212yes—
21#16839726C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839692C2212yescomplemented
24#16839682C2212yescomplemented
25#16839657—22no—
26#16839635—21no—
27#16839635—21no—
28C4×C22⋊Dic61C2211yescharacteristic
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
45C22×Dic122C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758200—22no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50#14519756—21no—
51#14519756—21no—
52#14519756—21no—
53#14519756—21no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—22no—
56C2×#11758196—21no—
57C2×#11758196—22no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C4×Dic61—21no—
61#14519710—22no—
62#14519708—22no—
63#14519701—22no—
64#14519700—22no—
65#14519700—22no—
66#14519698—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418C2312yes—
70C22⋊#92418C2312yes—
71#14519684—22no—
72#14519684—22no—
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—81no—
123C2×Q8×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—44no—
128C2×D4×C61—81no—
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—81no—
140C22×C4×C61—81no—
141C22×C4×C61—82no—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—81no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C42×C61—24no—
177C22⋊Dic61—24no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180#11758202—82no—
181#11758200—82no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184#11758196—44no—
185#11758196—41no—
186#11758196—41no—
187#11758196—82no—
188C4×Dic61—81no—
189#11758194—81no—
190#11758194—81no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61#12011yescharacteristic
201C23×C61#12011yescharacteristic
202C23×C61C22×D411yescharacteristic
203Q8×C61—42no—
204Q8×C61—81no—
205D4×C61—42no—
206D4×C61—44no—
207D4×C61—44no—
208D4×C61—81no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—81no—
230C2×C4×C61—84no—
231C2×C4×C61#12011yescharacteristic
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×Dic61—24no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242Dic122—42no—
243Dic122—42no—
244#92418—42no—
245#92418—42no—
246#18321—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—44no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—84no—
259C22×C61#45612yes—
260C22×C61C2×#12011yescharacteristic
261C22×C61#57512yes—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—42no—
271Dic61—44no—
272Dic61—81no—
273Dic61—81no—
274C2×#539D6111yescharacteristic
275#3195—611no—
276#3193—611no—
277#2583—611no—
278#1609—614no—
279#1603—614no—
280#1508—611no—
281#1507—611no—
282#1463—611no—
283C2×C61—22no—
284C2×C61—22no—
285C2×C61—41no—
286C2×C61—82no—
287C2×C61#300511yescharacteristic
288C2×C61C2×#45611yescharacteristic
289C2×C61#302611yescharacteristic
290#539—22no—
291#539—22no—
292#539—22no—
293#539C2×D6114yes—
294C2×#126—21no—
295C2×#125—21no—
296C2×#123—21no—
297C2×#123—21no—
298C2×C22⋊Q8—21no—
299C2×C22⋊Q8—611no—
300C2×C22⋊Q8C2×D6111yescharacteristic
301C2×#121C2×D6111yescharacteristic
302C2×C4×D4—21no—
303C22×C4⋊C4C2×D6111yescharacteristic
304#484—1221no—
305#483—612no—
306#483—1221no—
307#480—612no—
308#470—1222no—
309#468—1222no—
310#451—612no—
311#451—1221no—
312#450—1221no—
313#448—612no—
314#407—611no—
315#407—611no—
316#398—612no—
317#395—612no—
318#389—1222no—
319#385—1221no—
320#385—1221no—
321C4×C22⋊C4—611no—
322#328—614no—
323#327—614no—
324#324—614no—
325C61#1832111yescharacteristic, complemented, Sylow
326C22×Q8—611no—
327C22×Q8C22⋊D6111yescharacteristic
328C22×D4—21no—
329C22×D4C22⋊D6111yescharacteristic
330C23×C4—21no—
331C23×C4C22⋊D6111yescharacteristic
332C23×C4C22⋊D6111yescharacteristic
333C23×C4C22×D6111yescharacteristic
334C2×Q16—1222no—
335C2×SD16—1222no—
336C2×M16—1221no—
337C2×M16—1221no—
338#126—24no—
339#126—24no—
340#125—24no—
341#125—24no—
342#123—24no—
343#123—24no—
344#123—24no—
345#123—24no—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—1221no—
351C22⋊Q8—1221no—
352C22⋊Q8—1221no—
353C22⋊Q8—1221no—
354C22⋊Q8C22⋊D6112yes—
355C22⋊Q8C22×D6112yes—
356#121—21no—
357#121—21no—
358#121C22⋊D6112yes—
359#121C22×D6112yes—
360C4×D4—24no—
361C4×D4—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4C22⋊D6112yes—
375C2×C4⋊C4C22⋊D6111yescharacteristic
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22×D6112yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—21no—
386C2×C42—1221no—
387#107—1222no—
388#106—1222no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—1222no—
393C22⋊C8—612no—
394C22⋊C8—612no—
395C22⋊C8—1221no—
396C22⋊C8—1221no—
397#95—612no—
398#95—1222no—
399#95—1222no—
400C24—21no—
401C24C2×C22⋊D6111yescharacteristic
402C2×Q8—21no—
403C2×Q8—81no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—4881no—
407C2×Q8C2×C22⋊D6112yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—81no—
413C2×D4C2×C22⋊D6112yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437Q16—4882no—
438SD16—4882no—
439M16—4881no—
440M16—4881no—
441C2×C8—1222no—
442C2×C8—1222no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—4882no—
457C4⋊C4—2444no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4C2×C22⋊D6112yes—
461C4⋊C4C2×C22⋊D6112yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—1224no—
470C22⋊C4—2441no—
471C22⋊C4—2441no—
472C42—24no—
473C42—4881no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23#1451974611yescharacteristic
482C23#1451974611yescharacteristic
483C23C22×C22⋊D6111yescharacteristic
484Q8—42no—
485Q8—81no—
486Q8—2442no—
487Q8—2442no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—81no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—81no—
513C2×C4—84no—
514C2×C4—1224no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451974611yescharacteristic
522C2×C4#1451974611yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C8—2442no—
526C8—2442no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—44no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22#1683994512yes—
540C22C2×#1451974611yescharacteristic
541C22#1684056012yes—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—4881no—
552C4—4881no—
553C4—2444no—
554C2—22no—
555C2—22no—
556C2—41no—
557C2—82no—
558C2#2145506111yescharacteristic
559C2C2×#1683994511yescharacteristic
560C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#539C2
socleC22×C61C2×#120
prime radicalC2×#121×C61C22
hypercenterC2×#539D61
hypocenterC61#18321
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. #21456640C2semidirect
3. #21451199C2semidirect
4. #21444816C2semidirect
5. #21444782C2semidirect
6. #21444339C2semidirect
8. #21444149C2semidirect
13. #16840090C22semidirect
18. #16839917C22semidirect
23. #16839692C22semidirect
24. #16839682C22semidirect
29. #16839499C22semidirect
31. #16839488C22semidirect
325. C61#18321semidirect

Maximal subgroups

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

subgroupindex
#214566402
#214566252
#214511992
#214448162
#214447822
#214443392
#214443352
#214441492
C2×C61×#5392
#1832161