Finite Group Databasebeta

← #29648277

Subgroups — #29648277

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459715C211yesmaximal, characteristic
2#21454839C211yesmaximal, characteristic, complemented
3#21444456C211yesmaximal, characteristic, complemented
4#21444448C212yesmaximal, complemented
5#21444437C212yesmaximal, complemented
6#21444427C212yesmaximal, complemented
7#21444418C211yesmaximal, characteristic, complemented
8#21444416C211yesmaximal, characteristic, complemented
9#21444413C211yesmaximal, characteristic, complemented
10#21444405C212yesmaximal, complemented
11C61×#2970C211yesmaximal, characteristic, complemented
12#16840495—21no—
13#16840495—21no—
14#16840484—21no—
15#16840484—21no—
16#16840441—22no—
17#16840441—22no—
18#16840383C412yes—
19#16840383C412yes—
20#16840333C411yescharacteristic
21#16840326C411yescharacteristic
22#16840319C2211yescharacteristic
23#16840296C2212yes—
24#16840296C2212yes—
25C2×#14519765C2211yescharacteristic
26#16840239C411yescharacteristic
27#16840236C411yescharacteristic
28Q8×#92418—21no—
29#16839840—21no—
30#16839840—22no—
31#16839819C2211yescharacteristic
32#16839819C2212yescomplemented
33C4×#11758194C2211yescharacteristic
34C4×#11758194C2212yescomplemented
35#16839503C2211yescharacteristic, complemented
36#16839503C2212yescomplemented
37#16839503C2211yescharacteristic, complemented
38#16839503C2211yescharacteristic, complemented
39#16839503C2211yescharacteristic, complemented
40#16839503C2212yescomplemented
41#16839501C2212yes—
42#16839501C2212yes—
43#16839501C2212yes—
44#16839501C2212yes—
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic
50C61×#432C2212yescomplemented
51C4×M16×C61C2211yescharacteristic
52C2×#97×C61C2212yescomplemented
53C22×Dic122C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C2×#11758194—21no—
56C2×#11758194—22no—
57#14519777C2×C411yescharacteristic
58#14519777C2×C411yescharacteristic
59#14519775—21no—
60#14519773—21no—
61#14519772—22no—
62#14519772—22no—
63#14519771—22no—
64#14519771—22no—
65#14519769C2311yescharacteristic
66#14519769C2311yescharacteristic
67#14519766—21no—
68#14519765—21no—
69#14519765—21no—
70#14519765C2×C412yes—
71#14519765C2312yes—
72#14519764—21no—
73#14519763—22no—
74#14519763—22no—
75#14519760—22no—
76#14519760—22no—
77#14519759—22no—
78#14519759—22no—
79#14519758—22no—
80#14519756—21no—
81#14519756—21no—
82#14519756C2×C412yes—
83#14519756C2×C412yes—
84#14519755—22no—
85#14519755—22no—
86#14519751C2×C412yes—
87#14519751C2×C412yes—
88#14519745—22no—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×Dic122C2×C412yes—
92C4×Dic122C2×C412yes—
93#14519742—21no—
94#14519742—21no—
95C4×D244C2×C412yes—
96C4×D244C2×C412yes—
97#14519739—21no—
98#14519739—21no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101#14519736C2×C411yescharacteristic
102#14519736C2×C411yescharacteristic
103#14519734—22no—
104#14519734—22no—
105#14519733—22no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C22⋊C4×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—22no—
111C22⋊#92418—21no—
112C22⋊#92418—22no—
113#14519688C2311yescharacteristic
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116C4×#92418—21no—
117C4×#92418—22no—
118C2×#41×C61C2×C411yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—22no—
121C4×Q8×C61D411yescharacteristic
122C4×Q8×C61D411yescharacteristic
123C4×D4×C61D411yescharacteristic
124C4×D4×C61D412yes—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127#117×C61C2×C411yescharacteristic
128#117×C61C2×C412yes—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2312yes—
131C4⋊C8×C61—21no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61—22no—
135#97×C61—22no—
136#97×C61C2311yescharacteristic
137#97×C61C2312yes—
138#97×C61C2311yescharacteristic
139C4×C8×C61—21no—
140C23×D61—21no—
141#41×C61—42no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C22×Dic61—21no—
154C2×D244—21no—
155C2×D244—42no—
156C2×D244C22×C412yes—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—42no—
163C2×Dic122C22×C412yes—
164M16×C61—42no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—22no—
168C2×C8×C61—44no—
169C4⋊C4×C61C22⋊C411yescharacteristic
170C4⋊C4×C61C22⋊C411yescharacteristic
171C4⋊C4×C61C22⋊C412yes—
172C4⋊C4×C61C22⋊C412yes—
173C22⋊C4×C61C22⋊C412yes—
174C22⋊C4×C61C22⋊C412yes—
175C22⋊C4×C61C22⋊C412yes—
176C42×C61C22×C412yes—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D412yes—
179C42×C61C2×D411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D412yes—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—21no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213#11758194—42no—
214#11758194—44no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—22no—
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C22×D61—22no—
221C22×D61—42no—
222C22×D61—81no—
223C22×D61—81no—
224Q8×C61—22no—
225D4×C61—22no—
226D4×C61—42no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C22⋊D61—82no—
243C22⋊D61—82no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—42no—
250D244—42no—
251D244—42no—
252C4×D61—82no—
253C4×D61—82no—
254Dic122—42no—
255Dic122—42no—
256Dic122—81no—
257Dic122—81no—
258C8×C61—22no—
259C8×C61—42no—
260C8×C61—41no—
261#92418—22no—
262#92418—41no—
263#92418—42no—
264#17691—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C22×C22⋊C411yescharacteristic
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—162no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61#41312yes—
285Dic61—42no—
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290#3510—611no—
291#2971—611no—
292#2970D6111yescharacteristic, complemented
293#1537—611no—
294#1535—612no—
295#1532—612no—
296#1529—612no—
297#1527—611no—
298#1527—611no—
299#1526—611no—
300#1524—612no—
301C2×C61—41no—
302C2×C61—42no—
303C2×C61C2×#41311yescharacteristic
304C2×C61#290111yescharacteristic
305C2×C61#290111yescharacteristic
306D61—82no—
307#559—1221no—
308#559—1221no—
309#556—1221no—
310#556—1221no—
311#551—1222no—
312#551—1222no—
313#542—612no—
314#542—612no—
315#536—611no—
316#535—611no—
317#533—611no—
318#529—612no—
319#529—612no—
320C2×#124—611no—
321#519—611no—
322#518—611no—
323C4×#41C2×D6111yescharacteristic
324#446—21no—
325C8×Q8—1221no—
326#436—1222no—
327#436—1221no—
328#435—22no—
329C8×D4—21no—
330#432—611no—
331#432—612no—
332#432C2×D6111yescharacteristic
333#432C2×D6112yescomplemented
334C4×M16—611no—
335C4×M16—612no—
336C4×M16C2×D6111yescharacteristic
337C2×#97C2×D6112yescomplemented
338#330—612no—
339#330—612no—
340#330—611no—
341#330—611no—
342#330—611no—
343#330—611no—
344#329—612no—
345#329—612no—
346#329—612no—
347#329—612no—
348C61#1769111yescharacteristic, complemented, Sylow
349C2×#41C4×D6111yescharacteristic
350C22×Q8—611no—
351C22×D4—611no—
352C2×M16—21no—
353C2×M16—1221no—
354C2×M16—1222no—
355C22×C8—22no—
356C4⋊Q8—611no—
357C4⋊Q8—611no—
358C4⋊Q8—1221no—
359#127—1221no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125—611no—
365#125—611no—
366#124—612no—
367#124—612no—
368#124—1221no—
369#124—1221no—
370#124—1222no—
371#124—1222no—
372#124—1221no—
373#124—1221no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—612no—
380C22⋊Q8—612no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385#121—612no—
386#121—612no—
387#120—1222no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8D24411yescharacteristic
395C4×Q8C22⋊D6111yescharacteristic
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4D24412yes—
403C4×D4D24411yescharacteristic
404C4×D4C22⋊D6112yes—
405C4×D4C22⋊D6111yescharacteristic
406#117—611no—
407#117—611no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117C4×D6111yescharacteristic
412#117C4×D6112yes—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1222no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6112yes—
418C4⋊C8—21no—
419C4⋊C8—22no—
420C4⋊C8—1222no—
421C4⋊C8—1221no—
422C22⋊C8—21no—
423C22⋊C8—22no—
424C22⋊C8—1221no—
425C22⋊C8—1222no—
426#97—22no—
427#97—611no—
428#97—612no—
429#97—611no—
430#97C22×D6111yescharacteristic
431#97C22×D6112yes—
432#97C22×D6111yescharacteristic
433C4×C8—21no—
434C4×C8—1222no—
435C4×C8—1221no—
436C24—1221no—
437#41—42no—
438C2×Q8—612no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—2442no—
443C2×Q8#1175819811yescharacteristic
444C2×D4—612no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—1221no—
448C2×D4—2442no—
449C2×D4#1175819811yescharacteristic
450C2×D4#1175819812yes—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C2×C4×D6112yes—
456C22×C4C2×D24411yescharacteristic
457C22×C4C2×D24412yes—
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6112yes—
460M16—42no—
461M16—2444no—
462M16—2442no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—44no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1221no—
470C4⋊C4—611no—
471C4⋊C4—611no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4#1175819811yescharacteristic
485C4⋊C4#1175819811yescharacteristic
486C4⋊C4#1175819812yes—
487C4⋊C4#1175819812yes—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4#1175819812yes—
501C22⋊C4#1175819812yes—
502C22⋊C4#1175819812yes—
503C42—1221no—
504C42—1221no—
505C42—1222no—
506C42—1222no—
507C42C2×C4×D6112yes—
508C42C2×C4×D6111yescharacteristic
509C42C2×D24412yes—
510C42C2×D24411yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6112yes—
513C42C23×D6111yescharacteristic
514C23—1222no—
515C23—4881no—
516C23—4881no—
517C23—2442no—
518C23C2×#1175819811yescharacteristic
519C23C2×#1175819812yes—
520Q8—22no—
521Q8—4881no—
522Q8—4881no—
523Q8—2442no—
524Q8—2442no—
525D4—22no—
526D4—42no—
527D4—4882no—
528D4—4882no—
529D4—2442no—
530D4—2442no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1222no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—4882no—
543C2×C4—4882no—
544C2×C4—2442no—
545C2×C4C2×#1175819812yes—
546C2×C4C2×#1175819812yes—
547C2×C4C2×#1175819811yescharacteristic
548C2×C4C2×#1175819811yescharacteristic
549C2×C4C2×#1175819812yes—
550C2×C4C2×#1175819811yescharacteristic
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4C22×D24411yescharacteristic
554C8—22no—
555C8—42no—
556C8—41no—
557C8—1222no—
558C8—2441no—
559C8—2442no—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—9762no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C22×#1175819811yescharacteristic
571C4—21no—
572C4—22no—
573C4—21no—
574C4—21no—
575C4—22no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4—2442no—
584C4#1683974412yes—
585C2—41no—
586C2—42no—
587C2—4882no—
588C2C2×#1683974411yescharacteristic
589C2#2145436411yescharacteristic
590C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2970C2
socleC22×C61C22×C22⋊C4
prime radical#16840326C4
cosocleC42×C61C24
hypercenter#2970D61
hypocenterC61#17691
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21454839C2semidirect
3. #21444456C2semidirect
4. #21444448C2semidirect
5. #21444437C2semidirect
6. #21444427C2semidirect
7. #21444418C2semidirect
8. #21444416C2semidirect
9. #21444413C2semidirect
10. #21444405C2semidirect
11. C61×#2970C2semidirect
32. #16839819C22semidirect
34. C4×#11758194C22semidirect
35. #16839503C22semidirect
36. #16839503C22semidirect
37. #16839503C22semidirect
38. #16839503C22semidirect
39. #16839503C22semidirect
40. #16839503C22semidirect
50. C61×#432C22semidirect
52. C2×#97×C61C22semidirect
292. #2970D61semidirect
333. #432C2×D61semidirect
337. C2×#97C2×D61semidirect
348. C61#17691semidirect

Maximal subgroups

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

subgroupindex
#214597152
#214548392
#214444562
#214444482
#214444372
#214444272
#214444182
#214444162
#214444132
#214444052
C61×#29702
#1769161