Finite Group Databasebeta

← #29655211

Subgroups — #29655211

Order 15616.

908 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456960C211yesmaximal, characteristic, complemented
3#21451759C211yesmaximal, characteristic, complemented
4#21444833C214yesmaximal, complemented
5#21444810C214yesmaximal, complemented
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444171C211yesmaximal, characteristic, complemented
9C61×#3561C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—21no—
12#16840158—21no—
13#16840121—21no—
14#16840112—21no—
15#16840090C2214yes—
16#16840090C2212yes—
17#16840073C2212yes—
18#16840057—21no—
19#16840052—21no—
20#16840047—22no—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839687C2214yes—
24#16839674—22no—
25#16839669—21no—
26#16839665—21no—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839499C2214yes—
30#16839496C2212yes—
31#16839486C2214yes—
32C61×#563—21no—
33C61×#562—21no—
34C61×#556—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37C61×#548—21no—
38C61×#546—21no—
39C61×#543C2212yes—
40C61×#542—21no—
41C61×#542—22no—
42C61×#538C2212yes—
43C61×#536—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#124×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C22⋊Dic61—24no—
55C2×C4×Dic61—21no—
56#14519710—22no—
57#14519708—24no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—24no—
62#14519701—22no—
63#14519698—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—24no—
69#14519684—21no—
70#14519684C2312yes—
71C2×#41×C61—21no—
72C22×Q8×C61—21no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61D411yescharacteristic
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79#126×C61—22no—
80#126×C61—22no—
81#126×C61—22no—
82#126×C61—21no—
83#125×C61—21no—
84#125×C61D411yescharacteristic
85#124×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#124×C61—21no—
90#124×C61—21no—
91#124×C61—21no—
92#124×C61D411yescharacteristic
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—22no—
96#123×C61—21no—
97#123×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2312yes—
108C22⋊Q8×C61C2311yescharacteristic
109#121×C61—21no—
110#121×C61—22no—
111#121×C61—22no—
112#121×C61C2311yescharacteristic
113#120×C61—22no—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61D412yes—
117C4×Q8×C61D411yescharacteristic
118C4×D4×C61—21no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61D411yescharacteristic
123#117×C61—22no—
124#117×C61—21no—
125#117×C61—21no—
126C2×C22⋊C4×C61—22no—
127C2×C22⋊C4×C61D412yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C24×C61C2×D412yes—
131#41×C61—82no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—42no—
137C2×Q8×C61—82no—
138C2×Q8×C61C2×D411yescharacteristic
139C2×Q8×C61C2×D412yes—
140C2×D4×C61—21no—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61C2×D411yescharacteristic
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C22×Dic61—22no—
157C2×Dic122—41no—
158C2×Dic122—41no—
159C2×Dic122—82no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—21no—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—21no—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C22⋊Dic61—84no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4×Dic61—42no—
204C2×#92418—22no—
205C2×#92418—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—24no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61C22×D411yescharacteristic
212Q8×C61—41no—
213Q8×C61—42no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—81no—
217Q8×C61—82no—
218D4×C61—41no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—81no—
222D4×C61—82no—
223D4×C61—82no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—82no—
236C2×C4×C61—82no—
237C2×C4×C61—82no—
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249Dic122—82no—
250Dic122—82no—
251#92418—42no—
252#92418—42no—
253#18438—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—41no—
257C22×C61—42no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61—84no—
263C22×C61C2×#12011yescharacteristic
264C22×C61#57712yes—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61#45512yes—
277Dic61—82no—
278Dic61—82no—
279#3561D6111yescharacteristic
280#3238—611no—
281#3229—611no—
282#2640—611no—
283#1612—614no—
284#1608—614no—
285#1506—611no—
286#1500—611no—
287#1469—611no—
288C2×C61—22no—
289C2×C61—81no—
290C2×C61—82no—
291C2×C61#301011yescharacteristic
292C2×C61C2×#45511yescharacteristic
293C2×C61#302811yescharacteristic
294#563—21no—
295#562—21no—
296#556—21no—
297#551—22no—
298D4×Q8—21no—
299#548—21no—
300#546—21no—
301#543C2×D6112yes—
302#542—21no—
303#542—22no—
304#538C2×D6112yes—
305#536—21no—
306#533C2×D6111yescharacteristic
307C2×C4⋊Q8—611no—
308#529C2×D6111yescharacteristic
309C2×#124C2×D6111yescharacteristic
310#501—1221no—
311#498—1221no—
312#489—1221no—
313#487—1221no—
314#483—612no—
315#483—614no—
316#480—612no—
317#477—1221no—
318#476—1221no—
319#475—1222no—
320#432—611no—
321#431—611no—
322#397—614no—
323#393—1222no—
324#391—1221no—
325#390—1221no—
326#376—611no—
327#328—612no—
328#328—614no—
329#327—612no—
330#323—614no—
331C61#1843811yescharacteristic, complemented, Sylow
332C2×#41—21no—
333C22×Q8—21no—
334C22×Q8—1221no—
335C22×Q8C22⋊D6111yescharacteristic
336C22×D4C22⋊D6111yescharacteristic
337C4⋊Q8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—1221no—
340C4⋊Q8—1221no—
341C4⋊Q8—1221no—
342C4⋊Q8—1221no—
343C4⋊Q8C22⋊D6111yescharacteristic
344C4⋊Q8C22⋊D6111yescharacteristic
345#126—22no—
346#126—22no—
347#126—22no—
348#126—21no—
349#125—21no—
350#125C22⋊D6111yescharacteristic
351#124—21no—
352#124—22no—
353#124—22no—
354#124—21no—
355#124—21no—
356#124—21no—
357#124—21no—
358#124C22⋊D6111yescharacteristic
359#123—22no—
360#123—22no—
361#123—22no—
362#123—21no—
363#123—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8C22×D6111yescharacteristic
374C22⋊Q8C22×D6112yes—
375#121—21no—
376#121—22no—
377#121—22no—
378#121C22×D6111yescharacteristic
379#120—22no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C22⋊D6111yescharacteristic
383C4×Q8C22⋊D6112yes—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4—22no—
388C4×D4C22⋊D6111yescharacteristic
389#117—22no—
390#117—21no—
391#117—21no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—1221no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4C22⋊D6112yes—
397C2×C22⋊C4C22×D6112yes—
398C2×C42—1221no—
399C2×C42C22×D6111yescharacteristic
400#107—1222no—
401#106—1224no—
402#106—1222no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405#103—1224no—
406#103—1222no—
407#102—1222no—
408C22⋊C8—612no—
409C22⋊C8—612no—
410#97—1221no—
411C4×C8—1221no—
412#95—612no—
413#95—1221no—
414#95—1224no—
415C24C2×C22⋊D6112yes—
416#41—82no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—42no—
422C2×Q8—82no—
423C2×Q8—4882no—
424C2×Q8—2441no—
425C2×Q8—2441no—
426C2×Q8C2×C22⋊D6111yescharacteristic
427C2×Q8C2×C22⋊D6112yes—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4C2×C22⋊D6111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6112yes—
444C22×C4C23×D6111yescharacteristic
445C2×C8—1222no—
446C2×C8—1222no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C4⋊C4C2×C22⋊D6112yes—
466C4⋊C4C2×C22⋊D6111yescharacteristic
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—4884no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—2442no—
487C42C2×C22⋊D6111yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C2×C22⋊D6111yescharacteristic
490C42C2×C22⋊D6111yescharacteristic
491C23—22no—
492C23—21no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23C22×C22⋊D6111yescharacteristic
497Q8—41no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—81no—
502Q8—82no—
503Q8—4882no—
504Q8—4882no—
505D4—41no—
506D4—42no—
507D4—42no—
508D4—81no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2444no—
530C2×C4#1451974611yescharacteristic
531C2×C4#1451974611yescharacteristic
532C2×C4#1451974611yescharacteristic
533C2×C4#1451974611yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C8—2442no—
537C8—2442no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—42no—
545C22—41no—
546C22—84no—
547C22C2×#1451974611yescharacteristic
548C22#1684057012yes—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—42no—
560C4—4882no—
561C4—4882no—
562C4#1683994012yes—
563C2—22no—
564C2—82no—
565C2—81no—
566C2#2145510511yescharacteristic
567C2C2×#1683994011yescharacteristic
568C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#18438
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456960C2semidirect
3. #21451759C2semidirect
4. #21444833C2semidirect
5. #21444810C2semidirect
6. #21444331C2semidirect
7. #21444305C2semidirect
8. #21444171C2semidirect
331. C61#18438semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214569602
#214517592
#214448332
#214448102
#214443312
#214443052
#214441712
C61×#35612
#1843861