Finite Group Databasebeta

← #15863695

Subgroups — #15863695

Order 3328.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719228C211yesmaximal, characteristic
2C2×#11718782C211yesmaximal, characteristic, complemented
3C2×#11718659C211yesmaximal, characteristic, complemented
4C2×#11718790C211yesmaximal, characteristic
5#13117429C211yesmaximal, characteristic
6#13115038C212yesmaximal, complemented
7#13115036C212yesmaximal, complemented
8#13115030C212yesmaximal, complemented
9#13115025C212yesmaximal, complemented
10C2×#11718121C211yesmaximal, characteristic, complemented
11C13×#1736C211yesmaximal, characteristic, complemented
12C22×#6127C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C2×C13×#117C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×#88180C2211yescharacteristic
17C22×C4⋊Dic13C2211yescharacteristic
18C23×#1055C2211yescharacteristic
19#11719228—24no—
20#11719228—24no—
21C2×#88146C2212yes—
22C2×C4⋊C4×D13C2212yes—
23C2×#88135—22no—
24C2×C4×D52—21no—
25C2×C4×Dic26—21no—
26#11718874—22no—
27#11718790C418yescomplemented
28#11718782—24no—
29C2×#88060C2212yes—
30C2×#88060C2212yes—
31C2×#88058C2212yes—
32C2×#88057C2212yes—
33#11718659—24no—
34#11718649—21no—
35#11718648C2212yes—
36#11718648C2212yes—
37#11718635—21no—
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C13×#340C2212yes—
43C13×#340C2212yes—
44#11718121C2212yes—
45#11718121C2212yes—
46C23×C4×C13—21no—
47C23×C4×C13C2311yescharacteristic
48C22×C22⋊D13D412yes—
49C23×Dic13—21no—
50C2×#6127—22no—
51C2×#6127—22no—
52C2×#6127—22no—
53C2×#6127—22no—
54C2×#6127C2×C414yes—
55C2×#6127Q814yes—
56C22×D52D411yescharacteristic
57C22×C4×D13—22no—
58C22×C4×D13C2312yes—
59C22×Dic26D411yescharacteristic
60C2×C13×M16—22no—
61C2×C13×M16—22no—
62C2×C13×M16C2×C414yescomplemented
63C22×C8×C13C2312yes—
64C13×#117—22no—
65C13×#117D414yes—
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—22no—
68C2×C13×C4⋊C4—21no—
69C2×C13×C4⋊C4C2312yes—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C22⋊C4—21no—
73C2×C42×C13—21no—
74C2×C22⋊Dic13—21no—
75C2×#6103—22no—
76C2×#6103—22no—
77C2×#6103—22no—
78#88180—22no—
79#88180—22no—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13D414yes—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2312yes—
86C2×#6101—22no—
87C2×C4×Dic13—22no—
88C2×C4×Dic13—21no—
89C22×#1055—22no—
90C22×#1055—22no—
91C22×#1055C2×C414yescomplemented
92C22×#1055C2311yescharacteristic
93C22×#1055C2311yescharacteristic
94#88146—24no—
95#88146—24no—
96C4⋊C4×D13—24no—
97C4⋊C4×D13—24no—
98#88135—216no—
99C4×D52—28no—
100C4×Dic26—28no—
101#88071—22no—
102#88071—22no—
103#88060C2×C418yescomplemented
104#88060C2×C418yescomplemented
105#88058—24no—
106#88057—24no—
107#88048—22no—
108#88048—22no—
109#88047—22no—
110#88047—22no—
111#20849—131nomaximal, Sylow
112C24×C13C2×D411yescharacteristic
113C23×D13C2×D412yes—
114C22×C4×C13—21no—
115C22×C4×C13—21no—
116C22×C4×C13—22no—
117C22×C4×C13—42no—
118C22×C4×C13—81no—
119C22×C4×C13—81no—
120C22×C4×C13C22×C412yes—
121C22×C4×C13C2×D412yes—
122C22×C4×C13C2×D411yescharacteristic
123C22×C4×C13C2×D412yes—
124C22×C4×C13C2×Q812yes—
125C22×C4×C13#4112yes—
126C22×C4×C13#4112yes—
127C22×C4×C13C2411yescharacteristic
128C2×C22⋊D13—24no—
129C2×C22⋊D13—24no—
130C2×C22⋊D13C4⋊C418yes—
131C22×Dic13—22no—
132C22×Dic13—24no—
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—81no—
136C22×Dic13—81no—
137C22×Dic13C2×D412yes—
138#6127—28no—
139#6127—28no—
140#6127—28no—
141#6127—28no—
142C2×D52—22no—
143C2×D52—22no—
144C2×D52C4⋊C414yes—
145C2×C4×D13—22no—
146C2×C4×D13—22no—
147C2×C4×D13—22no—
148C2×C4×D13—22no—
149C2×C4×D13—28no—
150C2×C4×D13—82no—
151C2×C4×D13—82no—
152C2×C4×D13C22×C414yes—
153C2×C4×D13C2×Q814yes—
154C2×Dic26—22no—
155C2×Dic26—22no—
156C2×Dic26C4⋊C414yes—
157C13×M16—28no—
158C2×C8×C13—22no—
159C2×C8×C13—22no—
160C2×C8×C13C22×C414yes—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—22no—
164C13×C4⋊C4—22no—
165C13×C4⋊C4—42no—
166C13×C4⋊C4C2×D412yes—
167C13×C4⋊C4C2×D412yes—
168C13×C22⋊C4—24no—
169C42×C13—24no—
170C22⋊Dic13—42no—
171#6103—28no—
172#6103—82no—
173#6103—82no—
174C4⋊Dic13—24no—
175C4⋊Dic13—22no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13C2×D414yes—
181#6101—28no—
182C4×Dic13—28no—
183C4×Dic13—42no—
184C2×#1055—28no—
185C2×#1055—21no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055C22×C412yes—
190C2×#1055C22×C412yes—
191C2×#520—131no—
192C2×#428—131no—
193C2×#427—131no—
194C2×#408—131no—
195#2311—131no—
196#1854—132no—
197#1854—132no—
198#1853—132no—
199#1852—132no—
200#1736D1311yescharacteristic, complemented
201C2×#340—131no—
202C23×C13—21no—
203C23×C13—21no—
204C23×C13—21no—
205C23×C13—21no—
206C23×C13C2×C4⋊C412yes—
207C23×C13#12012yes—
208C23×C13#12312yes—
209C23×C13C22×D411yescharacteristic
210C22×D13—22no—
211C22×D13—22no—
212C22×D13—22no—
213C22×D13—22no—
214C22×D13—22no—
215C22×D13—22no—
216C22×D13C2×C4⋊C414yes—
217C2×C4×C13—22no—
218C2×C4×C13—22no—
219C2×C4×C13—41no—
220C2×C4×C13—41no—
221C2×C4×C13—41no—
222C2×C4×C13—41no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—84no—
227C2×C4×C13C2×C4⋊C412yes—
228C2×C4×C13C4×D414yes—
229C2×C4×C13C4×D414yes—
230C2×C4×C13#12012yes—
231C2×C4×C13C22⋊Q814yes—
232C2×C4×C13C22⋊Q814yes—
233C2×C4×C13#12312yes—
234C2×C4×C13C23×C411yescharacteristic
235C2×C4×C13C22×D411yescharacteristic
236C2×C4×C13C22×D411yescharacteristic
237C2×C4×C13C22×Q811yescharacteristic
238C2×C4×C13C2×#4111yescharacteristic
239C2×C4×C13C2×#4111yescharacteristic
240C22⋊D13—216no—
241C2×Dic13—22no—
242C2×Dic13—22no—
243C2×Dic13—24no—
244C2×Dic13—42no—
245C2×Dic13—42no—
246C2×Dic13—41no—
247C2×Dic13—41no—
248C2×Dic13—41no—
249C2×Dic13—41no—
250C2×Dic13—41no—
251C2×Dic13—41no—
252C2×Dic13—84no—
253C2×Dic13C2×C4⋊C414yes—
254D52—28no—
255C4×D13—24no—
256C4×D13—24no—
257C4×D13—24no—
258C4×D13—24no—
259C4×D13—88no—
260Dic26—28no—
261C8×C13—24no—
262#1055—22no—
263#1055—22no—
264C22×#41—131no—
265C22×M16C2×D1311yescharacteristic
266C23×C8—131no—
267#520—264no—
268#520—264no—
269C2×C4×Q8—261no—
270C2×C4×D4—261no—
271C2×C4×D4—262no—
272C2×#117—131no—
273C2×#117—132no—
274C2×#117C2×D1311yescharacteristic
275C22×C4⋊C4—131no—
276C22×C4⋊C4—132no—
277C22×C4⋊C4C2×D1311yescharacteristic
278#428—264no—
279#427—264no—
280C2×#107—132no—
281C2×#107—131no—
282C2×#107—131no—
283C2×#106—132no—
284C2×#106—131no—
285C2×#106—131no—
286#408—138no—
287C2×C22⋊C8—132no—
288C2×C22⋊C8—132no—
289#391—261no—
290#390—261no—
291#386—262no—
292#380—132no—
293#380—132no—
294#340—132no—
295#340—132no—
296#340C2×D1312yes—
297#340C2×D1312yes—
298C22×C13—22no—
299C22×C13—22no—
300C22×C13—22no—
301C22×C13—22no—
302C22×C13—22no—
303C22×C13—22no—
304C22×C13#38514yes—
305C22×C13C22×C4⋊C411yescharacteristic
306C22×C13C2×#12011yescharacteristic
307C22×C13C2×#12311yescharacteristic
308C22×C13#57611yescharacteristic
309C22×C13#57711yescharacteristic
310C22×C13#57711yescharacteristic
311C22×C13#57811yescharacteristic
312C2×D13—24no—
313C2×D13—24no—
314C2×D13—24no—
315C2×D13—42no—
316C2×D13—42no—
317C2×D13—42no—
318C2×D13—42no—
319C4×C13—42no—
320C4×C13—42no—
321C4×C13—82no—
322C4×C13#38514yes—
323C4×C13C2×C4×D411yescharacteristic
324C4×C13C2×C4×D411yescharacteristic
325C4×C13C2×C22⋊Q811yescharacteristic
326C4×C13C2×C22⋊Q811yescharacteristic
327Dic13—24no—
328Dic13—44no—
329Dic13—81no—
330Dic13—81no—
331C2×#41—134no—
332C2×#41—134no—
333C2×#41—262no—
334C2×#41—262no—
335C2×#41—262no—
336C2×#41—262no—
337C22×Q8—131no—
338C22×D4—132no—
339C22×D4—131no—
340C23×C4—21no—
341C23×C4—132no—
342C23×C4—262no—
343C23×C4—261no—
344C23×C4C22×D1311yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16C4×D1314yescomplemented
348C22×C8—134no—
349C22×C8—131no—
350C22×C8—131no—
351C22×C8—262no—
352C22×C8—262no—
353C22×C8C22×D1312yes—
354C4×Q8—268no—
355C4×D4—268no—
356C4×D4—2616no—
357#117—22no—
358#117—264no—
359#117—264no—
360#117—262no—
361#117—262no—
362#117D5214yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—134no—
367C2×C4⋊C4—132no—
368C2×C4⋊C4—131no—
369C2×C4⋊C4—131no—
370C2×C4⋊C4—264no—
371C2×C4⋊C4—262no—
372C2×C4⋊C4—262no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4—264no—
375C2×C4⋊C4C22×D1312yes—
376C2×C4⋊C4C22×D1311yescharacteristic
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—262no—
380C2×C22⋊C4—261no—
381C2×C22⋊C4—262no—
382C2×C22⋊C4—262no—
383C2×C42—21no—
384C2×C42—262no—
385C2×C42—261no—
386#107—264no—
387#107—262no—
388#107—262no—
389#106—264no—
390#106—262no—
391#106—262no—
392C22⋊C8—138no—
393C22⋊C8—138no—
394#95—262no—
395#95—262no—
396C2×C13—24no—
397C2×C13C2×#38511yescharacteristic
398C2×C13#291211yescharacteristic
399C2×C13#291311yescharacteristic
400C2×C13#301211yescharacteristic
401C2×C13#301311yescharacteristic
402C2×C13#309611yescharacteristic
403C2×C13#309711yescharacteristic
404D13—44no—
405C24—132no—
406C24D4×D1311yescharacteristic
407#41—268no—
408#41—268no—
409#41—268no—
410#41—268no—
411C2×Q8—134no—
412C2×Q8—262no—
413C2×Q8—262no—
414C2×D4—138no—
415C2×D4—134no—
416C2×D4—264no—
417C2×D4—264no—
418C2×D4—262no—
419C2×D4—262no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—132no—
427C22×C4—134no—
428C22×C4—134no—
429C22×C4—268no—
430C22×C4—262no—
431C22×C4—262no—
432C22×C4—262no—
433C22×C4—262no—
434C22×C4—262no—
435C22×C4—264no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4—1042no—
439C22×C4—1042no—
440C22×C4—1041no—
441C22×C4—1041no—
442C22×C4C2×C4×D1312yes—
443C22×C4C2×D5212yes—
444C22×C4#612712yes—
445C22×C4D4×D1312yes—
446C22×C4D4×D1311yescharacteristic
447C22×C4#612912yes—
448C22×C4Q8×D1312yes—
449C22×C4C23×D1311yescharacteristic
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—132no—
454C2×C8—132no—
455C2×C8—268no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8C2×C4×D1314yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—134no—
467C4⋊C4—268no—
468C4⋊C4—264no—
469C4⋊C4—262no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4C2×D5212yes—
475C4⋊C4C2×D5212yes—
476C22⋊C4—24no—
477C22⋊C4—268no—
478C22⋊C4—522no—
479C22⋊C4—1042no—
480C22⋊C4—1042no—
481C42—24no—
482C42—268no—
483C42—522no—
484C13#2084911yescharacteristic, complemented, Sylow
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—134no—
490C23—262no—
491C23—262no—
492C23—262no—
493C23—262no—
494C23—262no—
495C23—262no—
496C23#8813612yes—
497C23#8813712yes—
498C23C4⋊C4×D1312yes—
499C23C2×D4×D1311yescharacteristic
500Q8—268no—
501D4—2616no—
502D4—268no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—134no—
514C2×C4—262no—
515C2×C4—262no—
516C2×C4—264no—
517C2×C4—264no—
518C2×C4—264no—
519C2×C4—264no—
520C2×C4—264no—
521C2×C4—522no—
522C2×C4—522no—
523C2×C4—521no—
524C2×C4—521no—
525C2×C4—521no—
526C2×C4—521no—
527C2×C4—521no—
528C2×C4—521no—
529C2×C4—1048no—
530C2×C4—1044no—
531C2×C4C4×D5214yes—
532C2×C4#8813514yes—
533C2×C4#8813612yes—
534C2×C4#8813712yes—
535C2×C4C4⋊C4×D1312yes—
536C2×C4#8815014yes—
537C2×C4#8815114yes—
538C2×C4C22×C4×D1311yescharacteristic
539C2×C4C22×D5211yescharacteristic
540C2×C4C2×#612711yescharacteristic
541C2×C4C2×D4×D1311yescharacteristic
542C2×C4C2×#612911yescharacteristic
543C2×C4C2×Q8×D1311yescharacteristic
544C8—24no—
545C8—262no—
546C8—262no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—264no—
554C22—264no—
555C22—264no—
556C22—522no—
557C22—522no—
558C22—522no—
559C22—522no—
560C22#1171832914yes—
561C22C2×#8813611yescharacteristic
562C22C2×#8813711yescharacteristic
563C22C2×C4⋊C4×D1311yescharacteristic
564C22#1171947911yescharacteristic
565C22#1171948011yescharacteristic
566C22#1171948311yescharacteristic
567C22#1171948411yescharacteristic
568C4—42no—
569C4—42no—
570C4—82no—
571C4—264no—
572C4—524no—
573C4—1041no—
574C4—1041no—
575C4#1171832914yes—
576C4C2×C4×D5211yescharacteristic
577C4C2×#8813511yescharacteristic
578C4C2×#8815011yescharacteristic
579C4C2×#8815111yescharacteristic
580C2—24no—
581C2—524no—
582C2C2×#1171832911yescharacteristic
583C2#1312343011yescharacteristic
584C2#1312343511yescharacteristic
585C2#1312414211yescharacteristic
586C2#1312415311yescharacteristic
587C2#1312479211yescharacteristic
588C2#1312480311yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1736C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenter#1736D13
hypocenterC13#20849
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#11718782C2semidirect
3. C2×#11718659C2semidirect
6. #13115038C2semidirect
7. #13115036C2semidirect
8. #13115030C2semidirect
9. #13115025C2semidirect
10. C2×#11718121C2semidirect
11. C13×#1736C2semidirect
27. #11718790C4semidirect
62. C2×C13×M16C2×C4semidirect
91. C22×#1055C2×C4semidirect
103. #88060C2×C4semidirect
104. #88060C2×C4semidirect
200. #1736D13semidirect
347. C2×M16C4×D13semidirect
484. C13#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#117192282
C2×#117187822
C2×#117186592
C2×#117187902
#131174292
#131150382
#131150362
#131150302
#131150252
C2×#117181212
C13×#17362
#2084913