Finite Group Databasebeta

← #29656602

Subgroups — #29656602

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457210C211yesmaximal, characteristic, complemented
2#21456199C211yesmaximal, characteristic, complemented
3#21455890C211yesmaximal, characteristic
4#21445040C212yesmaximal, complemented
5#21445040C212yesmaximal, complemented
6#21445033C212yesmaximal, complemented
7#21445026C212yesmaximal, complemented
8#21444500C211yesmaximal, characteristic, complemented
9#21444258C211yesmaximal, characteristic
10#21444240C211yesmaximal, characteristic, complemented
11C61×#3503C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840130—22no—
14#16840112C2212yes—
15#16840112C2212yes—
16#16840103—21no—
17#16840047C2211yescharacteristic
18#16840047C2212yes—
19#16840044—21no—
20#16840042C2211yescharacteristic
21#16840018—22no—
22#16839972C2211yescharacteristic
23#16839972C2212yes—
24#16839969—21no—
25#16839967C2211yescharacteristic
26#16839924—21no—
27C4×#11758202—21no—
28C4×#11758200—21no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839771—22no—
32C4×#11758194C2211yescharacteristic
33#16839511C2212yes—
34#16839511C2212yes—
35Q8⋊#92418C2212yes—
36Q8⋊#92418C2212yes—
37Q8⋊#92418C2212yes—
38D4⋊#92418C2212yes—
39Q82×C61C2212yes—
40C61×#557—21no—
41C61×#554—22no—
42C61×#554C2212yes—
43D4×Q8×C61—22no—
44D4×Q8×C61—21no—
45C61×#544—22no—
46C61×#544—21no—
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C2×C4⋊Q8×C61—21no—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520—21no—
52C4×#41×C61C2211yescharacteristic
53C2×#11758202—21no—
54C2×#11758202—22no—
55C2×#11758200—21no—
56C2×#11758194—21no—
57C2×#11758194—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—21no—
67#14519708—22no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73#14519701—22no—
74#14519701—24no—
75#14519701—24no—
76#14519701—21no—
77#14519700—21no—
78#14519700—22no—
79#14519700—21no—
80#14519698—21no—
81C22⋊#92418—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61D411yescharacteristic
86C22×Q8×C61—22no—
87C22×Q8×C61D412yes—
88C4⋊Q8×C61—22no—
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61—24no—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61D412yes—
96C4⋊Q8×C61D411yescharacteristic
97C4⋊Q8×C61D411yescharacteristic
98C4⋊Q8×C61C2312yes—
99#125×C61—22no—
100#125×C61—24no—
101#125×C61—22no—
102#125×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—24no—
105C22⋊Q8×C61—24no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—24no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61D412yes—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C4×Q8×C61—21no—
114C4×Q8×C61D411yescharacteristic
115C4×Q8×C61C2312yes—
116C4×Q8×C61C2311yescharacteristic
117C4×D4×C61—24no—
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C4×D4×C61C2311yescharacteristic
123#117×C61—22no—
124#117×C61—21no—
125#117×C61—21no—
126#117×C61D411yescharacteristic
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—22no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#41×C61—22no—
134#41×C61—22no—
135C2×Q8×C61—24no—
136C2×Q8×C61—22no—
137C2×Q8×C61—21no—
138C2×Q8×C61—44no—
139C2×Q8×C61—84no—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×D411yescharacteristic
142C2×Q8×C61C2×D414yes—
143C2×D4×C61—21no—
144C2×D4×C61C2×D411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—24no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—24no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—22no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—42no—
189#11758202—82no—
190#11758200—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—41no—
198#11758194—42no—
199#11758194—81no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61#12011yescharacteristic
206Q8×C61—21no—
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—44no—
210Q8×C61—44no—
211Q8×C61—42no—
212Q8×C61—84no—
213Q8×C61#13712yes—
214D4×C61—24no—
215D4×C61—21no—
216D4×C61#13712yes—
217C2×C4×C61—24no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—82no—
233C2×C4×C61—84no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×Dic61—41no—
241C2×Dic61—41no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246#92418—42no—
247#92418—41no—
248#92418—41no—
249#92418—42no—
250#18575—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—22no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61C2×#12011yescharacteristic
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—41no—
263C4×C61—42no—
264C4×C61—41no—
265C4×C61—44no—
266C4×C61—42no—
267C4×C61#45512yes—
268C4×C61C2×#13711yescharacteristic
269C4×C61C2×#13711yescharacteristic
270C4×C61#57711yescharacteristic
271C4×C61#57811yescharacteristic
272Dic61—81no—
273Dic61—81no—
274#3503D6111yescharacteristic
275#3255—611no—
276#3139—611no—
277#3107—611no—
278#1645—612no—
279#1645—612no—
280#1644—612no—
281#1643—612no—
282#1548—611no—
283#1489—611no—
284#1485—611no—
285C2×C61—41no—
286C2×C61—42no—
287C2×C61#300411yescharacteristic
288C2×C61#301311yescharacteristic
289C2×C61C2×#45511yescharacteristic
290Q82C2×D6112yes—
291#557—21no—
292#554—22no—
293#554C2×D6112yes—
294D4×Q8—22no—
295D4×Q8—21no—
296#544—22no—
297#544—21no—
298#533—21no—
299#533—611no—
300#533C2×D6111yescharacteristic
301C2×C4⋊Q8—21no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303#520—21no—
304C4×#41C2×D6111yescharacteristic
305#491—1222no—
306#487—612no—
307#487—612no—
308#485—1221no—
309#475—611no—
310#475—612no—
311#475—1221no—
312#474—611no—
313#470—1222no—
314#462—611no—
315#462—612no—
316#462—1221no—
317#461—611no—
318#452—1221no—
319C4×Q16—1221no—
320C4×SD16—1221no—
321#431—611no—
322#423—611no—
323#419—1222no—
324C4×M16—611no—
325#333—612no—
326#333—612no—
327Q8⋊C8—612no—
328Q8⋊C8—612no—
329Q8⋊C8—612no—
330D4⋊C8—612no—
331C61#1857511yescharacteristic, complemented, Sylow
332C2×#41C22⋊D6111yescharacteristic
333C22×Q8—22no—
334C22×Q8C22⋊D6112yes—
335C2×Q16—1222no—
336C2×Q16—1221no—
337C2×SD16—1221no—
338C2×M16—1221no—
339C2×M16—1221no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—24no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—611no—
348C4⋊Q8—611no—
349C4⋊Q8C22⋊D6112yes—
350C4⋊Q8C22⋊D6111yescharacteristic
351C4⋊Q8C22⋊D6111yescharacteristic
352C4⋊Q8C22×D6112yes—
353#125—22no—
354#125—24no—
355#125—22no—
356#125—21no—
357#125—1221no—
358C22⋊Q8—22no—
359C22⋊Q8—24no—
360C22⋊Q8—24no—
361C22⋊Q8—22no—
362C22⋊Q8—24no—
363C22⋊Q8—21no—
364C22⋊Q8—1221no—
365C22⋊Q8—1221no—
366C22⋊Q8C22⋊D6112yes—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—21no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8C22⋊D6111yescharacteristic
374C4×Q8C22×D6111yescharacteristic
375C4×Q8C22×D6112yes—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22⋊D6111yescharacteristic
381C4×D4C22×D6111yescharacteristic
382#117—22no—
383#117—21no—
384#117—21no—
385#117—1221no—
386#117C22⋊D6111yescharacteristic
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#106—1221no—
395#106—1222no—
396C4⋊C8—612no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—1221no—
400#103—1224no—
401#103—1224no—
402#103—1221no—
403#103—1222no—
404#103—1221no—
405#103—1221no—
406#103—1222no—
407#102—1221no—
408C22⋊C8—1221no—
409#97—1221no—
410C4×C8—612no—
411C4×C8—1221no—
412#41—22no—
413#41—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—44no—
418C2×Q8—84no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8C2×C22⋊D6112yes—
422C2×Q8C2×C22⋊D6111yescharacteristic
423C2×Q8C2×C22⋊D6114yes—
424C2×D4—21no—
425C2×D4C2×C22⋊D6111yescharacteristic
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434Q16—4882no—
435Q16—2442no—
436SD16—2442no—
437M16—4881no—
438M16—2442no—
439C2×C8—1222no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1222no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4C2×C22⋊D6112yes—
465C4⋊C4C2×C22⋊D6111yescharacteristic
466C4⋊C4C2×C22⋊D6111yescharacteristic
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—2441no—
473C42—24no—
474C42—21no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—2441no—
479C42C2×C22⋊D6111yescharacteristic
480C42C2×C22⋊D6111yescharacteristic
481C42C2×C22⋊D6111yescharacteristic
482C42C23×D6111yescharacteristic
483C23—21no—
484C23#1451974611yescharacteristic
485Q8—21no—
486Q8—22no—
487Q8—22no—
488Q8—44no—
489Q8—44no—
490Q8—42no—
491Q8—84no—
492Q8—2441no—
493Q8—2441no—
494Q8—2441no—
495Q8—2441no—
496Q8#1451980812yes—
497D4—24no—
498D4—21no—
499D4#1451980812yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4#1451974611yescharacteristic
520C2×C4#1451974611yescharacteristic
521C2×C4#1451974611yescharacteristic
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C8—2442no—
526C8—2441no—
527C8—2441no—
528C8—2442no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22C2×#1451974611yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—44no—
544C4—42no—
545C4—4881no—
546C4—4881no—
547C4#1683994012yes—
548C4C2×#1451980811yescharacteristic
549C4C2×#1451980811yescharacteristic
550C4#1684057011yescharacteristic
551C4#1684057711yescharacteristic
552C2—41no—
553C2—42no—
554C2#2145505311yescharacteristic
555C2#2145513411yescharacteristic
556C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radicalC2×#41×C61D4
hypercenter#3503D61
hypocenterC61#18575
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21457210C2semidirect
2. #21456199C2semidirect
4. #21445040C2semidirect
5. #21445040C2semidirect
6. #21445033C2semidirect
7. #21445026C2semidirect
8. #21444500C2semidirect
10. #21444240C2semidirect
331. C61#18575semidirect

Maximal subgroups

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

subgroupindex
#214572102
#214561992
#214558902
#214450402
#214450332
#214450262
#214445002
#214442582
#214442402
C61×#35032
#1857561