Finite Group Databasebeta

← #15863919

Subgroups — #15863919

Order 3328.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719209C211yesmaximal, characteristic
2C2×#11718794C211yesmaximal, characteristic
3C2×#11718663C211yesmaximal, characteristic
4#13117452C211yesmaximal, characteristic, complemented
5#13115101C212yesmaximal, complemented
6#13115094C212yesmaximal, complemented
7#13115087C212yesmaximal, complemented
8#13115080C212yesmaximal, complemented
9C2×#11718121C211yesmaximal, characteristic, complemented
10#13114534C211yesmaximal, characteristic, complemented
11C2×C13×#428C211yesmaximal, characteristic, complemented
12C22×#6127C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C2×C13×#117C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×C4×C22⋊D13—22no—
17C2×#88180C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C23×#1055C2211yescharacteristic
20#11719209—24no—
21#11719209—24no—
22C2×#88147C2212yes—
23C2×#88146—21no—
24C2×C4⋊C4×D13—21no—
25C2×#88141C2212yes—
26C13×#428—24no—
27C2×C13×#107C2212yes—
28C2×C13×#106C2212yes—
29#11718794C418yescomplemented
30C2×#88061C2212yes—
31C2×#88056C2212yes—
32#11718663—24no—
33#11718634C2212yes—
34#11718634C2212yes—
35C2×#88050C2211yescharacteristic
36C2×#88050C2211yescharacteristic
37C2×#88049C2211yescharacteristic
38C2×#88049C2211yescharacteristic
39#11718603—22no—
40#11718589—21no—
41#11718589—21no—
42#11718142C2212yes—
43#11718142C2212yes—
44#11718121C2212yes—
45#11718121C2212yes—
46C23×C4×C13—21no—
47C23×C4×C13C2311yescharacteristic
48C22×C22⋊D13—21no—
49C23×Dic13—21no—
50C2×#6127—22no—
51C2×#6127—22no—
52C2×#6127—22no—
53C2×#6127C2×C414yes—
54C2×#6127D414yes—
55C2×#6127Q814yes—
56C22×D52C2311yescharacteristic
57C22×C4×D13—21no—
58C22×C4×D13—22no—
59C22×Dic26C2311yescharacteristic
60C2×C13×M16—22no—
61C2×C13×M16—22no—
62C2×C13×M16C2×C414yescomplemented
63C22×C8×C13C2312yes—
64C13×#117—22no—
65C13×#117—22no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4—22no—
68C2×C13×C4⋊C4D414yes—
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2312yes—
72C2×C13×C22⋊C4—21no—
73C2×C42×C13—21no—
74C2×C22⋊Dic13D412yes—
75C4×C22⋊D13—216no—
76C2×#6103—22no—
77#88180—22no—
78#88180—22no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13C2×C414yescomplemented
82C2×C4⋊Dic13D411yescharacteristic
83C2×C4⋊Dic13D411yescharacteristic
84C2×C4⋊Dic13C2312yes—
85C2×#6101—22no—
86C2×#6101—22no—
87C2×#6101—22no—
88C2×C4×Dic13—22no—
89C2×C4×Dic13C2312yes—
90C22×#1055—22no—
91C22×#1055—22no—
92C22×#1055—22no—
93C22×#1055C2311yescharacteristic
94C22×#1055C2311yescharacteristic
95#88147—24no—
96#88147—24no—
97#88146—28no—
98C4⋊C4×D13—28no—
99#88141—24no—
100#88141—24no—
101C13×#107—24no—
102C13×#106—24no—
103#88071—22no—
104#88071—22no—
105#88061C2×C418yescomplemented
106#88056C2×C418yescomplemented
107#88050—22no—
108#88050—22no—
109#88049—22no—
110#88049—22no—
111#20878—131nomaximal, Sylow
112C24×C13C2×D411yescharacteristic
113C23×D13—21no—
114C22×C4×C13—22no—
115C22×C4×C13—24no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
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⋊D13—24no—
131C22×Dic13—21no—
132C22×Dic13—22no—
133C22×Dic13—24no—
134C22×Dic13—81no—
135C22×Dic13—81no—
136C22×Dic13C2×D412yes—
137C22×Dic13C2×D412yes—
138#6127—28no—
139#6127—28no—
140#6127—28no—
141#6127C4⋊C4116yescomplemented
142C2×D52—21no—
143C2×D52—21no—
144C2×D52—21no—
145C2×D52C22×C412yes—
146C2×D52C2×D412yes—
147C2×D52C2×Q812yes—
148C2×C4×D13—24no—
149C2×C4×D13—24no—
150C2×C4×D13—24no—
151C2×C4×D13—28no—
152C2×C4×D13—82no—
153C2×C4×D13—82no—
154C2×Dic26—21no—
155C2×Dic26—21no—
156C2×Dic26—21no—
157C2×Dic26C22×C412yes—
158C2×Dic26C2×D412yes—
159C2×Dic26C2×Q812yes—
160C13×M16—28no—
161C2×C8×C13—22no—
162C2×C8×C13—22no—
163C2×C8×C13C22×C414yes—
164C13×C4⋊C4—24no—
165C13×C4⋊C4—22no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4C2×D414yes—
171C13×C22⋊C4—42no—
172C42×C13—42no—
173C22⋊Dic13—24no—
174#6103—28no—
175C4⋊Dic13—24no—
176C4⋊Dic13—22no—
177C4⋊Dic13—22no—
178C4⋊Dic13—22no—
179C4⋊Dic13C22×C414yes—
180#6101—28no—
181#6101—82no—
182#6101—82no—
183C4×Dic13—28no—
184C4×Dic13—22no—
185C4×Dic13—22no—
186C2×#1055—21no—
187C2×#1055—21no—
188C2×#1055—21no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—44no—
193C2×#520—131no—
194C2×#428D1311yescharacteristic, complemented
195C2×#417—131no—
196C2×#416—131no—
197#2316—131no—
198#1865—132no—
199#1864—132no—
200#1863—132no—
201#1862—132no—
202C2×#340—131no—
203#1729—131no—
204C23×C13—21no—
205C23×C13—21no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13C4×D412yes—
209C23×C13#12012yes—
210C23×C13C22⋊Q812yes—
211C23×C13C22×D411yescharacteristic
212C22×D13—22no—
213C22×D13—22no—
214C22×D13—22no—
215C22×D13—41no—
216C22×D13—41no—
217C22×D13—41no—
218C22×D13—41no—
219C2×C4×C13—24no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—41no—
223C2×C4×C13—41no—
224C2×C4×C13—41no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—84no—
229C2×C4×C13C2×C4⋊C414yes—
230C2×C4×C13C4×D414yes—
231C2×C4×C13C4×D412yes—
232C2×C4×C13#12012yes—
233C2×C4×C13C22⋊Q814yes—
234C2×C4×C13C22⋊Q812yes—
235C2×C4×C13#12314yes—
236C2×C4×C13C23×C411yescharacteristic
237C2×C4×C13C22×D411yescharacteristic
238C2×C4×C13C22×D411yescharacteristic
239C2×C4×C13C22×Q811yescharacteristic
240C2×C4×C13C2×#4111yescharacteristic
241C2×C4×C13C2×#4111yescharacteristic
242C22⋊D13—216no—
243C2×Dic13—22no—
244C2×Dic13—22no—
245C2×Dic13—22no—
246C2×Dic13—24no—
247C2×Dic13—22no—
248C2×Dic13—22no—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—42no—
254C2×Dic13—42no—
255C2×Dic13—84no—
256D52—22no—
257D52—22no—
258D52—22no—
259D52C2×C4⋊C414yes—
260C4×D13—216no—
261C4×D13—88no—
262Dic26—22no—
263Dic26—22no—
264Dic26—22no—
265Dic26C2×C4⋊C414yes—
266C8×C13—24no—
267#1055—41no—
268#1055—41no—
269C22×#41—131no—
270C22×M16C2×D1311yescharacteristic
271C23×C8—131no—
272#520—264no—
273#520—264no—
274C2×C4×Q8—132no—
275C2×C4×D4—132no—
276C2×C4×D4—262no—
277C2×#117—131no—
278C2×#117—261no—
279C2×#117C2×D1311yescharacteristic
280C22×C4⋊C4—131no—
281C22×C4⋊C4—261no—
282C22×C4⋊C4C2×D1311yescharacteristic
283#428—24no—
284C2×#107C2×D1312yes—
285C2×#106C2×D1312yes—
286#417—138no—
287#416—264no—
288C2×#103—132no—
289C2×#103—131no—
290C2×#103—131no—
291C2×#102—132no—
292C2×#102—131no—
293C2×#102—131no—
294#389—132no—
295#389—132no—
296#389—261no—
297#389—261no—
298#380—262no—
299#340—132no—
300#340—132no—
301#336—132no—
302#336—132no—
303C22×C13—22no—
304C22×C13—22no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13—22no—
308C22×C13—22no—
309C22×C13#38514yes—
310C22×C13C2×C4×D411yescharacteristic
311C22×C13C2×#12011yescharacteristic
312C22×C13C2×C22⋊Q811yescharacteristic
313C22×C13#57611yescharacteristic
314C22×C13#57711yescharacteristic
315C22×C13#57711yescharacteristic
316C22×C13#57811yescharacteristic
317C2×D13—24no—
318C2×D13—42no—
319C2×D13—42no—
320C2×D13—42no—
321C2×D13—42no—
322C2×D13—42no—
323C2×D13—42no—
324C4×C13—44no—
325C4×C13—81no—
326C4×C13—81no—
327C4×C13#38514yes—
328C4×C13C22×C4⋊C411yescharacteristic
329C4×C13C2×C4×D411yescharacteristic
330C4×C13C2×C22⋊Q811yescharacteristic
331C4×C13C2×#12311yescharacteristic
332Dic13—24no—
333Dic13—44no—
334Dic13—42no—
335Dic13—42no—
336C2×#41—134no—
337C2×#41—134no—
338C2×#41—134no—
339C2×#41—262no—
340C2×#41—262no—
341C2×#41—262no—
342C22×Q8—131no—
343C22×D4—131no—
344C22×D4—261no—
345C23×C4—21no—
346C23×C4—262no—
347C23×C4—261no—
348C23×C4—261no—
349C23×C4C22×D1311yescharacteristic
350C2×M16—22no—
351C2×M16—22no—
352C2×M16C4×D1314yescomplemented
353C22×C8—131no—
354C22×C8—131no—
355C22×C8—262no—
356C22×C8—262no—
357C22×C8—262no—
358C22×C8C22×D1312yes—
359C4×Q8—264no—
360C4×Q8—264no—
361C4×D4—264no—
362C4×D4—2616no—
363C4×D4—264no—
364#117—22no—
365#117—22no—
366#117—268no—
367#117—262no—
368#117—262no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—134no—
372C2×C4⋊C4—132no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—268no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—262no—
379C2×C4⋊C4—262no—
380C2×C4⋊C4—262no—
381C2×C4⋊C4C22⋊D1314yes—
382C2×C4⋊C4C22×D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1312yes—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—132no—
387C2×C22⋊C4—262no—
388C2×C42—21no—
389C2×C42—132no—
390C2×C42—262no—
391#107—24no—
392#106—24no—
393#103—138no—
394#103—262no—
395#103—262no—
396#102—138no—
397#102—262no—
398#102—262no—
399#95—262no—
400#95—262no—
401C2×C13—24no—
402C2×C13C2×#38511yescharacteristic
403C2×C13#294611yescharacteristic
404C2×C13#294711yescharacteristic
405C2×C13#301211yescharacteristic
406C2×C13#301311yescharacteristic
407C2×C13#308311yescharacteristic
408C2×C13#308411yescharacteristic
409D13—44no—
410C24—261no—
411C24D4×D1311yescharacteristic
412#41—1316no—
413#41—268no—
414#41—268no—
415#41—268no—
416C2×Q8—132no—
417C2×Q8—132no—
418C2×Q8—132no—
419C2×Q8—261no—
420C2×Q8—261no—
421C2×Q8—261no—
422C2×D4—132no—
423C2×D4—132no—
424C2×D4—132no—
425C2×D4—264no—
426C2×D4—264no—
427C2×D4—264no—
428C2×D4—261no—
429C2×D4—261no—
430C2×D4—261no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—132no—
438C22×C4—132no—
439C22×C4—268no—
440C22×C4—261no—
441C22×C4—264no—
442C22×C4—264no—
443C22×C4—264no—
444C22×C4—262no—
445C22×C4—264no—
446C22×C4—1042no—
447C22×C4—1042no—
448C22×C4—1041no—
449C22×C4—1041no—
450C22×C4C2×C4×D1312yes—
451C22×C4#612712yes—
452C22×C4D4×D1312yes—
453C22×C4D4×D1311yescharacteristic
454C22×C4Q8×D1312yes—
455C22×C4#613112yes—
456C22×C4C2×C22⋊D1312yes—
457C22×C4C23×D1311yescharacteristic
458M16—28no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C2×C8—261no—
467C2×C8—524no—
468C2×C8C2×C4×D1314yes—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—134no—
476C4⋊C4—268no—
477C4⋊C4—264no—
478C4⋊C4—262no—
479C4⋊C4—262no—
480C4⋊C4—262no—
481C4⋊C4—1042no—
482C4⋊C4—1042no—
483C4⋊C4C2×C22⋊D1314yes—
484C22⋊C4—42no—
485C22⋊C4—268no—
486C22⋊C4—264no—
487C42—42no—
488C42—268no—
489C42—262no—
490C42—262no—
491C13#2087811yescharacteristic, complemented, Sylow
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—262no—
497C23—262no—
498C23—262no—
499C23—521no—
500C23—521no—
501C23—521no—
502C23—521no—
503C23#8814712yes—
504C23#8815012yes—
505C23#8819112yes—
506C23C2×D4×D1311yescharacteristic
507Q8—134no—
508Q8—262no—
509Q8—262no—
510Q8—262no—
511D4—134no—
512D4—2616no—
513D4—262no—
514D4—262no—
515D4—262no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—262no—
527C2×C4—262no—
528C2×C4—262no—
529C2×C4—2616no—
530C2×C4—264no—
531C2×C4—262no—
532C2×C4—262no—
533C2×C4—262no—
534C2×C4—262no—
535C2×C4—262no—
536C2×C4—262no—
537C2×C4—522no—
538C2×C4—522no—
539C2×C4—1048no—
540C2×C4—1044no—
541C2×C4C4⋊C4×D1314yes—
542C2×C4#8814712yes—
543C2×C4#8814814yes—
544C2×C4#8815012yes—
545C2×C4C4×C22⋊D1314yes—
546C2×C4#8819112yes—
547C2×C4#8820014yes—
548C2×C4C22×C4×D1311yescharacteristic
549C2×C4C2×#612711yescharacteristic
550C2×C4C2×D4×D1311yescharacteristic
551C2×C4C2×Q8×D1311yescharacteristic
552C2×C4C2×#613111yescharacteristic
553C2×C4C22×C22⋊D1311yescharacteristic
554C8—24no—
555C8—521no—
556C8—521no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—264no—
564C22—522no—
565C22—522no—
566C22—522no—
567C22—522no—
568C22—522no—
569C22—522no—
570C22#1171865014yes—
571C22C2×#8814711yescharacteristic
572C22C2×#8815011yescharacteristic
573C22#1171947911yescharacteristic
574C22#1171948011yescharacteristic
575C22#1171948311yescharacteristic
576C22#1171948411yescharacteristic
577C22C2×#8819111yescharacteristic
578C4—44no—
579C4—81no—
580C4—81no—
581C4—264no—
582C4—522no—
583C4—522no—
584C4—524no—
585C4#1171865014yes—
586C4C2×C4⋊C4×D1311yescharacteristic
587C4C2×#8814811yescharacteristic
588C4C2×C4×C22⋊D1311yescharacteristic
589C4C2×#8820011yescharacteristic
590C2—24no—
591C2—524no—
592C2C2×#1171865011yescharacteristic
593C2#1312366611yescharacteristic
594C2#1312367711yescharacteristic
595C2#1312413911yescharacteristic
596C2#1312415011yescharacteristic
597C2#1312468611yescharacteristic
598C2#1312469711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#428C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC2×#428D13
hypocenterC13#20878
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #13117452C2semidirect
5. #13115101C2semidirect
6. #13115094C2semidirect
7. #13115087C2semidirect
8. #13115080C2semidirect
9. C2×#11718121C2semidirect
10. #13114534C2semidirect
11. C2×C13×#428C2semidirect
29. #11718794C4semidirect
62. C2×C13×M16C2×C4semidirect
81. C2×C4⋊Dic13C2×C4semidirect
105. #88061C2×C4semidirect
106. #88056C2×C4semidirect
141. #6127C4⋊C4semidirect
194. C2×#428D13semidirect
352. C2×M16C4×D13semidirect
491. C13#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#117192092
C2×#117187942
C2×#117186632
#131174522
#131151012
#131150942
#131150872
#131150802
C2×#117181212
#131145342
C2×C13×#4282
#2087813