Finite Group Databasebeta

← #29655136

Subgroups — #29655136

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21451940C211yesmaximal, characteristic, complemented
4#21444788C214yesmaximal, complemented
5#21444782C214yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840157—21no—
12#16840134—21no—
13#16840132—21no—
14#16840114—21no—
15#16839972—21no—
16#16839967—22no—
17#16839967—21no—
18#16839917C2212yes—
19#16839900C2214yes—
20#16839900C2212yes—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839692—22no—
24#16839682C2214yes—
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×Dic122C2312yes—
48C2×#11758202—22no—
49C2×#11758200—24no—
50C2×#11758200—22no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×#11758196—24no—
58C2×C4×Dic61—21no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519700—24no—
62#14519700—22no—
63#14519700—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—24no—
69#14519684—21no—
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—22no—
156C2×Dic122—22no—
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
196#11758202—82no—
197#11758200—84no—
198#11758200—82no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203#11758196—84no—
204C4×Dic61—24no—
205C2×#92418—22no—
206C2×#92418—22no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61C22×D411yescharacteristic
210Q8×C61—41no—
211Q8×C61—81no—
212Q8×C61—82no—
213D4×C61—42no—
214D4×C61—41no—
215D4×C61—44no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—81no—
219D4×C61—82no—
220D4×C61—82no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61—82no—
236C2×C4×C61—82no—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×Dic61—24no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—44no—
247C2×Dic61—42no—
248Dic122—44no—
249Dic122—42no—
250Dic122—42no—
251#92418—42no—
252#92418—42no—
253#18431—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—42no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—41no—
262C22×C61C2×#12011yescharacteristic
263C22×C61#57712yes—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61#45612yes—
276Dic61—44no—
277Dic61—82no—
278#3559D6111yescharacteristic
279#3218—611no—
280#3206—611no—
281#2660—611no—
282#1604—614no—
283#1603—614no—
284#1505—611no—
285#1500—611no—
286#1470—611no—
287C2×C61—22no—
288C2×C61—81no—
289C2×C61—82no—
290C2×C61#300911yescharacteristic
291C2×C61C2×#45611yescharacteristic
292C2×C61#302711yescharacteristic
293#563—21no—
294#562—21no—
295#554—21no—
296#553—21no—
297#553—21no—
298#547—22no—
299#547—21no—
300#544—21no—
301#540—21no—
302#539—22no—
303#539C2×D6112yes—
304#537C2×D6112yes—
305#532C2×D6111yescharacteristic
306C2×C4⋊Q8—611no—
307#529C2×D6111yescharacteristic
308C2×#125C2×D6111yescharacteristic
309#497—1221no—
310#492—1221no—
311#491—1221no—
312#487—1221no—
313#462—1221no—
314#461—1222no—
315#461—1221no—
316#451—612no—
317#448—614no—
318#448—612no—
319#432—611no—
320#431—611no—
321#398—1222no—
322#395—614no—
323#384—1221no—
324#382—1221no—
325#376—611no—
326#328—612no—
327#327—614no—
328#327—612no—
329#324—614no—
330C61#1843111yescharacteristic, complemented, Sylow
331C2×#41—21no—
332C2×#41—22no—
333C22×Q8—612no—
334C2×Q16—1222no—
335C2×SD16—1224no—
336C2×SD16—1222no—
337C4⋊Q8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—1221no—
340C4⋊Q8—1221no—
341C4⋊Q8—1221no—
342C4⋊Q8—1221no—
343C4⋊Q8C22⋊D6111yescharacteristic
344#127C22⋊D6111yescharacteristic
345#126—22no—
346#126—22no—
347#126—21no—
348#125—21no—
349#125—22no—
350#125—21no—
351#125—21no—
352#125—21no—
353#125—21no—
354#125C22⋊D6111yescharacteristic
355#124—21no—
356#124C22⋊D6111yescharacteristic
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—22no—
362#123—21no—
363#123—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C22⋊Q8C22×D6111yescharacteristic
371#121—21no—
372#121—22no—
373#121—21no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121—22no—
378#121C22×D6111yescharacteristic
379#121C22×D6112yes—
380C4×Q8C22⋊D6111yescharacteristic
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—21no—
387C4×D4C22⋊D6111yescharacteristic
388C4×D4C22⋊D6112yes—
389#117—21no—
390#117—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22⋊D6111yescharacteristic
397C2×C4⋊C4C22⋊D6112yes—
398C2×C4⋊C4C22⋊D6111yescharacteristic
399C2×C4⋊C4C22×D6112yes—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404#103—1224no—
405#103—1222no—
406#103—1222no—
407C22⋊C8—612no—
408C22⋊C8—612no—
409#97—1221no—
410C4×C8—1221no—
411#95—612no—
412#95—1221no—
413#95—1224no—
414#41—82no—
415#41—84no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—4882no—
421C2×Q8C2×C22⋊D6111yescharacteristic
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×C22⋊D6111yescharacteristic
429C2×D4C2×C22⋊D6112yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6112yes—
441C22×C4C23×D6111yescharacteristic
442Q16—4882no—
443SD16—4884no—
444SD16—4882no—
445C2×C8—1222no—
446C2×C8—1222no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—4884no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4C2×C22⋊D6112yes—
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C42—22no—
485C42—21no—
486C42—1224no—
487C42C2×C22⋊D6111yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C2×C22⋊D6111yescharacteristic
490C42C2×C22⋊D6111yescharacteristic
491C23—21no—
492C23—22no—
493C23C22×C22⋊D6111yescharacteristic
494Q8—41no—
495Q8—81no—
496Q8—82no—
497Q8—2444no—
498Q8—2442no—
499Q8—2442no—
500D4—42no—
501D4—41no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—81no—
506D4—82no—
507D4—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—1224no—
525C2×C4—2444no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4#1451974611yescharacteristic
530C2×C4#1451974611yescharacteristic
531C2×C4#1451974611yescharacteristic
532C2×C4#1451974611yescharacteristic
533C2×C4C22×C22⋊D6111yescharacteristic
534C2×C4C22×C22⋊D6111yescharacteristic
535C8—2442no—
536C8—2442no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—42no—
544C22—41no—
545C22C2×#1451974611yescharacteristic
546C22#1684057312yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—4882no—
559C4—2444no—
560C4#1683994512yes—
561C2—22no—
562C2—81no—
563C2—82no—
564C2#2145509611yescharacteristic
565C2C2×#1683994511yescharacteristic
566C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#18431
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. #21456843C2semidirect
2. #21456741C2semidirect
3. #21451940C2semidirect
4. #21444788C2semidirect
5. #21444782C2semidirect
6. #21444327C2semidirect
7. #21444305C2semidirect
8. #21444175C2semidirect
330. C61#18431semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214567412
#214519402
#214447882
#214447822
#214443272
#214443052
#214441752
C61×#35592
#1843161