Finite Group Databasebeta

← #29655229

Subgroups — #29655229

Order 15616.

904 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21451940C211yesmaximal, characteristic, complemented
4#21444822C214yesmaximal, complemented
5#21444816C214yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—21no—
12#16840165—21no—
13#16840121—21no—
14#16840108—21no—
15#16840090C2212yes—
16#16840073C2214yes—
17#16840073C2212yes—
18#16840057—21no—
19#16840052—21no—
20#16840042—22no—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839692C2214yes—
24#16839682—22no—
25#16839631—21no—
26#16839627—21no—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839496C2214yes—
30#16839496C2212yes—
31#16839488C2214yes—
32C61×#563—21no—
33C61×#562—21no—
34C61×#554—21no—
35C61×#553—21no—
36C61×#553—21no—
37C61×#547—22no—
38C61×#547—21no—
39C61×#544—21no—
40C61×#540—21no—
41C61×#539—22no—
42C61×#539C2212yes—
43C61×#537C2212yes—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×#11758196—24no—
55C2×C4×Dic61—21no—
56#14519710—24no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—22no—
62#14519698—24no—
63#14519698—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—24no—
70#14519684C2312yes—
71C2×#41×C61—21no—
72C2×#41×C61—22no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—21no—
80#125×C61—21no—
81#125×C61—22no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61—21no—
85#125×C61—21no—
86#125×C61D411yescharacteristic
87#124×C61—21no—
88#124×C61D411yescharacteristic
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—21no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2311yescharacteristic
103#121×C61—21no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—21no—
109#121×C61—22no—
110#121×C61C2312yes—
111#121×C61C2311yescharacteristic
112C4×Q8×C61D411yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61D412yes—
120C4×D4×C61D411yescharacteristic
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130#41×C61—82no—
131#41×C61—84no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D411yescharacteristic
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D411yescharacteristic
142C2×D4×C61C2×D412yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—22no—
154C22×Dic61—22no—
155C2×Dic122—41no—
156C2×Dic122—41no—
157C2×Dic122—82no—
158C4⋊C4×C61—22no—
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—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61—21no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200#11758196—84no—
201C4×Dic61—24no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—41no—
208Q8×C61—81no—
209Q8×C61—82no—
210D4×C61—42no—
211D4×C61—41no—
212D4×C61—44no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—81no—
216D4×C61—82no—
217D4×C61—82no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—82no—
231C2×C4×C61—82no—
232C2×C4×C61—82no—
233C2×C4×C61—82no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×Dic61—24no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245Dic122—82no—
246Dic122—82no—
247#92418—42no—
248#92418—42no—
249#18440—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—42no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—41no—
258C22×C61C2×#12011yescharacteristic
259C22×C61#57612yes—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—42no—
270C4×C61—41no—
271C4×C61#45612yes—
272Dic61—44no—
273Dic61—82no—
274#3559D6111yescharacteristic
275#3238—611no—
276#3226—611no—
277#2660—611no—
278#1610—614no—
279#1609—614no—
280#1505—611no—
281#1500—611no—
282#1470—611no—
283C2×C61—22no—
284C2×C61—81no—
285C2×C61—82no—
286C2×C61#300811yescharacteristic
287C2×C61C2×#45611yescharacteristic
288C2×C61#302811yescharacteristic
289#563—21no—
290#562—21no—
291#554—21no—
292#553—21no—
293#553—21no—
294#547—22no—
295#547—21no—
296#544—21no—
297#540—21no—
298#539—22no—
299#539C2×D6112yes—
300#537C2×D6112yes—
301#532C2×D6111yescharacteristic
302C2×C4⋊Q8—611no—
303#529C2×D6111yescharacteristic
304C2×#125C2×D6111yescharacteristic
305#501—1221no—
306#500—1221no—
307#489—1221no—
308#486—1221no—
309#483—612no—
310#480—614no—
311#480—612no—
312#477—1221no—
313#476—1221no—
314#474—1222no—
315#432—611no—
316#431—611no—
317#398—614no—
318#395—1222no—
319#384—1221no—
320#382—1221no—
321#376—611no—
322#328—612no—
323#327—614no—
324#327—612no—
325#324—614no—
326C61#1844011yescharacteristic, complemented, Sylow
327C2×#41—21no—
328C2×#41—22no—
329C22×Q8—1221no—
330C4⋊Q8—21no—
331C4⋊Q8—22no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8—1221no—
336C4⋊Q8C22⋊D6111yescharacteristic
337#127C22⋊D6111yescharacteristic
338#126—22no—
339#126—22no—
340#126—21no—
341#125—21no—
342#125—22no—
343#125—21no—
344#125—21no—
345#125—21no—
346#125—21no—
347#125C22⋊D6111yescharacteristic
348#124—21no—
349#124C22⋊D6111yescharacteristic
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355#123—21no—
356#123—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8C22×D6111yescharacteristic
364#121—21no—
365#121—22no—
366#121—21no—
367#121—22no—
368#121—22no—
369#121—21no—
370#121—22no—
371#121C22×D6111yescharacteristic
372#121C22×D6112yes—
373C4×Q8C22⋊D6111yescharacteristic
374C4×D4—22no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—22no—
379C4×D4—21no—
380C4×D4C22⋊D6111yescharacteristic
381C4×D4C22⋊D6112yes—
382#117—21no—
383#117—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C4⋊C4C22⋊D6112yes—
391C2×C4⋊C4C22⋊D6111yescharacteristic
392C2×C4⋊C4C22×D6112yes—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—1224no—
396#107—1222no—
397#106—1222no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400#103—1222no—
401#102—1224no—
402#102—1222no—
403C22⋊C8—612no—
404C22⋊C8—612no—
405#97—1221no—
406C4×C8—1221no—
407#95—612no—
408#95—1221no—
409#95—1224no—
410#41—82no—
411#41—84no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—4882no—
415C2×Q8—2441no—
416C2×Q8—2441no—
417C2×Q8C2×C22⋊D6111yescharacteristic
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C22⋊D6111yescharacteristic
425C2×D4C2×C22⋊D6112yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4C2×C22⋊D6112yes—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6112yes—
437C22×C4C23×D6111yescharacteristic
438C2×C8—1222no—
439C2×C8—1222no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—4884no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C4⋊C4C2×C22⋊D6112yes—
464C4⋊C4C2×C22⋊D6111yescharacteristic
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C42—22no—
478C42—21no—
479C42—1224no—
480C42C2×C22⋊D6111yescharacteristic
481C42C2×C22⋊D6111yescharacteristic
482C42C2×C22⋊D6111yescharacteristic
483C42C2×C22⋊D6111yescharacteristic
484C23—21no—
485C23—22no—
486C23C22×C22⋊D6111yescharacteristic
487Q8—41no—
488Q8—81no—
489Q8—82no—
490Q8—4882no—
491Q8—4882no—
492D4—42no—
493D4—41no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—81no—
498D4—82no—
499D4—82no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—1224no—
517C2×C4—2442no—
518C2×C4—2444no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451974611yescharacteristic
522C2×C4#1451974611yescharacteristic
523C2×C4#1451974611yescharacteristic
524C2×C4#1451974611yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C8—2442no—
528C8—2442no—
529C22—21no—
530C22—21no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—42no—
536C22—41no—
537C22C2×#1451974611yescharacteristic
538C22#1684056512yes—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—42no—
546C4—41no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—4882no—
551C4—2444no—
552C4#1683994512yes—
553C2—22no—
554C2—81no—
555C2—82no—
556C2#2145508711yescharacteristic
557C2C2×#1683994511yescharacteristic
558C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#18440
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21457037C2semidirect
2. #21456937C2semidirect
3. #21451940C2semidirect
4. #21444822C2semidirect
5. #21444816C2semidirect
6. #21444327C2semidirect
7. #21444305C2semidirect
8. #21444175C2semidirect
326. C61#18440semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214569372
#214519402
#214448222
#214448162
#214443272
#214443052
#214441752
C61×#35592
#1844061