Finite Group Databasebeta

← #16805153

Subgroups — #16805153

Order 3328.

982 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
1C2×Q16×#261C211yesmaximal, characteristic, complemented
2C22×#88102C211yesmaximal, characteristic
3#13131360C212yesmaximal
4#13130982C211yesmaximal, characteristic, complemented
5#13130940C211yesmaximal, characteristic, complemented
6#13130916C212yesmaximal, complemented
7#13130821C212yesmaximal
8#13130800C212yesmaximal, complemented
9#107×#261C211yesmaximal, characteristic, complemented
10C8×C22⋊#261C211yesmaximal, characteristic
11D13×#467C211yesmaximal, characteristic
12C2×Q8×#261C2212yes—
13C22×C4⋊#261C2212yes—
14C2×Q16×D13C2211yescharacteristic
15C22×C8×D13C2211yescharacteristic
16D13×C22⋊Q8C2212yes—
17(C2×#261)⋊Q8C2212yes—
18C2×#88117C2212yes—
19C2×#88117C2212yes—
20C4×C22⋊#261C2211yescharacteristic
21D13⋊(C22⋊C8)C2211yescharacteristic
22Q16×#261—24no—
23C2×#88102—24no—
24C2×#88102C2212yes—
25C2×#88102C2211yescharacteristic
26C2×#88102C2211yescharacteristic
27C2×C8×#261C2211yescharacteristic
28C2×C8×#261C2211yescharacteristic
29#11719083—22no—
30#11719083C2212yes—
31#11719082—22no—
32C4⋊C4×#261C2212yes—
33#11719070—22no—
34C22⋊C4×#261—22no—
35C13×#467C411yescharacteristic, complemented
36#11718849C411yescharacteristic, complemented
37#11718777C411yescharacteristic, complemented
38#11718712C412yescomplemented
39#11718548C411yescharacteristic, complemented
40D13×#107C2211yescharacteristic
41#11718474C412yescomplemented
42D13×#103C2212yes—
43(C2×#261)⋊C8C2211yescharacteristic
44(C2×#261)⋊C8C2211yescharacteristic
45#11718246C2212yes—
46#11718246C2212yes—
47C23×#261—22no—
48C2×Q8×D13C2312yes—
49C22×C4×D13C2311yescharacteristic
50C2×C22⋊#261D411yescharacteristic
51C2×C22⋊#261D411yescharacteristic
52Q8×#261—24no—
53Q8×#261—42no—
54C2×C4⋊#261—24no—
55C2×C4⋊#261—22no—
56C2×C4⋊#261—22no—
57C2×C4⋊#261—22no—
58C2×C4⋊#261—42no—
59C2×C4⋊#261C2312yes—
60C2×C4⋊#261C2312yes—
61C2×C4×#261—22no—
62C2×C4×#261—22no—
63C2×C4×#261C2311yescharacteristic
64C2×C4×#261C2311yescharacteristic
65C2×D13⋊C8C2311yescharacteristic
66C2×D13⋊C8C2311yescharacteristic
67C2×C13×Q16C2×C411yescharacteristic, complemented
68C22×C8×C13C2×C411yescharacteristic
69C13×C22⋊Q8C2×C412yes—
70#88200C2×C412yes—
71C2×#6107C2×C412yescomplemented
72#88178C2×C412yes—
73C22×#1055C2×C411yescharacteristic
74Q16×D13—22no—
75Q16×D13—22no—
76C2×Dic52C2×C411yescharacteristic, complemented
77C2×C8×D13—22no—
78C2×C8×D13—22no—
79C2×C8×D13C2×C411yescharacteristic
80C2×C8×D13C2×C411yescharacteristic
81C2×C8×D13C2311yescharacteristic
82C2×C8×D13C2311yescharacteristic
83#88151C2×C412yes—
84#88150—22no—
85C4⋊C4×D13—22no—
86C4⋊C4×D13C2312yes—
87C22⋊C4×D13—22no—
88#88132—22no—
89C22⋊#1057—21no—
90#88117—24no—
91#88117—42no—
92(C2×#261)⋊C4—22no—
93(C2×#261)⋊C4—22no—
94(C2×#261)⋊C4—22no—
95(C2×#261)⋊C4—22no—
96(C2×#261)⋊C4D411yescharacteristic
97(C2×#261)⋊C4D411yescharacteristic
98#88111—21no—
99#88102—24no—
100#88102—22no—
101#88102D414yes—
102C8×#261—22no—
103C8×#261—22no—
104C13×#107C2×C411yescharacteristic, complemented
105C13×#103C2×C412yescomplemented
106#88075C2×C412yescomplemented
107#88058C2×C411yescharacteristic, complemented
108#88056C2×C412yescomplemented
109#88050C2×C412yescomplemented
110#88047C2×C412yescomplemented
111#23024—131nomaximal, Sylow
112C23×D13C2×D411yescharacteristic
113C22×#261—22no—
114C22×#261—22no—
115C22×#261—44no—
116C22×#261—44no—
117C22×#261—82no—
118C22×#261C2×D411yescharacteristic
119C22×#261C2×D411yescharacteristic
120C22×#261C2×D411yescharacteristic
121C22×#261C2×D411yescharacteristic
122C2×Q8×C13C22×C412yes—
123C22×C4×C13C22×C411yescharacteristic
124C22×Dic13#4111yescharacteristic
125Q8×D13—22no—
126Q8×D13—22no—
127Q8×D13—22no—
128Q8×D13—22no—
129C2×C4×D13—21no—
130C2×C4×D13—21no—
131C2×C4×D13—21no—
132C2×C4×D13—21no—
133C2×C4×D13—22no—
134C2×C4×D13—22no—
135C2×C4×D13C22×C411yescharacteristic
136C2×C4×D13C2×D411yescharacteristic
137C2×C4×D13#4111yescharacteristic
138C2×C4×D13C2411yescharacteristic
139C2×Dic26—22no—
140C2×Dic26C22×C412yes—
141C22⋊#261—22no—
142C22⋊#261Q1614yescomplemented
143C2×#1057—21no—
144C2×#1057—21no—
145C4⋊#261—22no—
146C4⋊#261—22no—
147C4⋊#261—22no—
148C4⋊#261—44no—
149C4⋊#261—44no—
150C4⋊#261—82no—
151C4⋊#261C2×D414yes—
152C4×#261—21no—
153C4×#261—21no—
154C4×#261—21no—
155C4×#261—21no—
156C4×#261—44no—
157C4×#261—82no—
158D13⋊C8—21no—
159D13⋊C8—21no—
160D13⋊C8—21no—
161D13⋊C8—21no—
162C13×Q16—22no—
163C2×C8×C13—22no—
164C2×C8×C13C22×C411yescharacteristic
165C2×C8×C13C22×C411yescharacteristic
166C13×C4⋊C4—22no—
167C13×C4⋊C4C22×C412yes—
168C13×C22⋊C4—22no—
169C22⋊Dic13—22no—
170#6107—24no—
171#6103—22no—
172#6103—22no—
173C4⋊Dic13—22no—
174C4⋊Dic13C22×C412yes—
175#6101—22no—
176#6101—22no—
177#6101—22no—
178C2×#1055—22no—
179C2×#1055C22×C411yescharacteristic
180C2×#1055C22×C411yescharacteristic
181Dic52—22no—
182C8×D13—21no—
183C8×D13—21no—
184C8×D13—22no—
185C8×D13—22no—
186C8×D13C2×D412yes—
187C8×D13#4112yes—
188C2×#467—131no—
189C2×C4×Q16—131no—
190C22×#107—131no—
191#2360—132no—
192#1969—131no—
193#1942—131no—
194#1928—132no—
195#1872—132no—
196#1863—132no—
197C4×#107—131no—
198C8×C22⋊C4—131no—
199C23×C13C4×D411yescharacteristic
200C22×D13—21no—
201C22×D13—21no—
202C22×D13—21no—
203C22×D13—21no—
204C22×D13C4×D411yescharacteristic
205C22×D13C2×Q1612yes—
206C22×D13#13512yes—
207C22×D13C22×D411yescharacteristic
208C2×#261—21no—
209C2×#261—21no—
210C2×#261—21no—
211C2×#261—21no—
212C2×#261—21no—
213C2×#261—24no—
214C2×#261—21no—
215C2×#261—21no—
216C2×#261—21no—
217C2×#261—21no—
218C2×#261—21no—
219C2×#261—42no—
220C2×#261—42no—
221C2×#261—42no—
222C2×#261—42no—
223C2×#261—42no—
224C2×#261—88no—
225C2×#261C2×Q1612yes—
226C2×#261C2×Q1612yes—
227Q8×C13—22no—
228Q8×C13—22no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13C4×D411yescharacteristic
234C2×C4×C13C23×C411yescharacteristic
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—22no—
238C2×Dic13—22no—
239C2×Dic13C4×D411yescharacteristic
240C2×Dic13C2×#4111yescharacteristic
241C4×D13—21no—
242C4×D13—21no—
243C4×D13—21no—
244C4×D13—21no—
245C4×D13—21no—
246C4×D13—21no—
247C4×D13—42no—
248C4×D13—42no—
249C4×D13—42no—
250C4×D13—42no—
251C4×D13C22×D411yescharacteristic
252C4×D13C2×#4111yescharacteristic
253C4×D13H(5,𝔽2)11yescharacteristic
254C4×D132−1+411yescharacteristic
255Dic26—22no—
256Dic26—22no—
257Dic26—42no—
258Dic26—42no—
259#1057—41no—
260#1057—41no—
261C8×C13—21no—
262C8×C13C4×D412yes—
263#1055—21no—
264#1055C4×D412yes—
265C22×Q16—131no—
266C23×C8—131no—
267C2×C22⋊Q8—132no—
268C2×C4×Q8—132no—
269C22×C4⋊C4—132no—
270#467—131no—
271#467—131no—
272#467—132no—
273#467—131no—
274#467—132no—
275#467#26111yescharacteristic, complemented
276C4×Q16—264no—
277C2×#107—131no—
278C2×#107—131no—
279C2×#107—132no—
280C2×#107—131no—
281C2×#107—264no—
282C2×#103—132no—
283C2×#103—132no—
284C2×#103—132no—
285C2×C22⋊C8—131no—
286C2×C4×C8—131no—
287C2×C4×C8—131no—
288#391—132no—
289#389—132no—
290#389—262no—
291#387—262no—
292#385—262no—
293C4×C4⋊C4—132no—
294C4×C22⋊C4—131no—
295C4×C22⋊C4—262no—
296#340—132no—
297#340—132no—
298#336—131no—
299#336—131no—
300C22×C13—21no—
301C22×C13—21no—
302C22×C13C4×Q1612yescomplemented
303C22×C13C2×C4×D411yescharacteristic
304C2×D13—21no—
305C2×D13—21no—
306C2×D13—22no—
307C2×D13—22no—
308C2×D13—22no—
309C2×D13—22no—
310C2×D13C4×Q1612yescomplemented
311C2×D13D4211yescharacteristic
312C2×D13#54711yescharacteristic
313C2×D13C22×Q1611yescharacteristic
314C2×D13C2×#13511yescharacteristic
315C2×D13#57611yescharacteristic
316C2×D13#57811yescharacteristic
317#261—22no—
318#261—22no—
319#261—41no—
320#261—41no—
321#261—44no—
322#261—82no—
323C4×C13—21no—
324C4×C13—42no—
325C4×C13—42no—
326C4×C13C2×C4×D411yescharacteristic
327C4×C13#52011yescharacteristic
328Dic13—21no—
329Dic13—42no—
330Dic13—42no—
331Dic13#54811yescharacteristic
332Dic13#55311yescharacteristic
333C22×Q8—132no—
334C23×C4—131no—
335C23×C4—262no—
336C2×Q16—132no—
337C2×Q16—131no—
338C2×Q16—262no—
339C2×Q16—262no—
340C2×Q16C2×#26111yescharacteristic, complemented
341C22×C8—131no—
342C22×C8—131no—
343C22×C8—131no—
344C22×C8—131no—
345C22×C8—131no—
346C22×C8—131no—
347C22×C8—131no—
348C22×C8—262no—
349C22×C8—262no—
350C22×C8C2×#26111yescharacteristic
351C22⋊Q8—132no—
352C22⋊Q8—132no—
353C22⋊Q8—132no—
354C22⋊Q8—262no—
355C22⋊Q8—262no—
356C22⋊Q8C2×#26112yes—
357C4×Q8—264no—
358C4×Q8—522no—
359C2×C4⋊C4—132no—
360C2×C4⋊C4—132no—
361C2×C4⋊C4—132no—
362C2×C4⋊C4—262no—
363C2×C4⋊C4—264no—
364C2×C4⋊C4—262no—
365C2×C4⋊C4—262no—
366C2×C4⋊C4—262no—
367C2×C4⋊C4—522no—
368C2×C22⋊C4—131no—
369C2×C22⋊C4—131no—
370C2×C22⋊C4—262no—
371C2×C42—131no—
372C2×C42—131no—
373C2×C42—262no—
374C2×C42—262no—
375#107—134no—
376#107—131no—
377#107—132no—
378#107—264no—
379#107—262no—
380#107C2×#26111yescharacteristic, complemented
381#103—132no—
382#103—132no—
383#103—132no—
384#103—264no—
385#103—522no—
386#103C2×#26112yescomplemented
387C22⋊C8—261no—
388C22⋊C8—261no—
389C4×C8—262no—
390C4×C8—262no—
391#95—131no—
392#95—131no—
393#95—262no—
394#95—262no—
395#95—262no—
396#95—262no—
397C2×C13—22no—
398C2×C13#236111yescharacteristic
399C2×C13C2×C4×Q1611yescharacteristic
400C2×C13#295111yescharacteristic
401D13—22no—
402D13#328011yescharacteristic
403D13D4×Q1611yescharacteristic
404D13#333611yescharacteristic
405D13#333811yescharacteristic
406C24—131no—
407C2×Q8—132no—
408C2×Q8—262no—
409C2×Q8—262no—
410C2×Q8—262no—
411C2×Q8—262no—
412C2×Q8—262no—
413C2×Q8C22×#26112yes—
414C22×C4—131no—
415C22×C4—131no—
416C22×C4—131no—
417C22×C4—131no—
418C22×C4—131no—
419C22×C4—131no—
420C22×C4—131no—
421C22×C4—131no—
422C22×C4—131no—
423C22×C4—261no—
424C22×C4—261no—
425C22×C4—261no—
426C22×C4—261no—
427C22×C4—262no—
428C22×C4—262no—
429C22×C4—262no—
430C22×C4—262no—
431C22×C4—524no—
432C22×C4—524no—
433C22×C4—1042no—
434C22×C4C22×#26111yescharacteristic
435Q16—22no—
436Q16—264no—
437Q16—262no—
438C2×C8—22no—
439C2×C8—131no—
440C2×C8—132no—
441C2×C8—132no—
442C2×C8—131no—
443C2×C8—262no—
444C2×C8—261no—
445C2×C8—261no—
446C2×C8—262no—
447C2×C8—262no—
448C2×C8—261no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8C22×#26111yescharacteristic
455C2×C8C22×#26111yescharacteristic
456C4⋊C4—22no—
457C4⋊C4—134no—
458C4⋊C4—132no—
459C4⋊C4—262no—
460C4⋊C4—262no—
461C4⋊C4—262no—
462C4⋊C4—262no—
463C4⋊C4—262no—
464C4⋊C4—262no—
465C4⋊C4—262no—
466C4⋊C4—524no—
467C4⋊C4—524no—
468C4⋊C4—1042no—
469C4⋊C4C22×#26112yes—
470C22⋊C4—22no—
471C22⋊C4—134no—
472C22⋊C4—262no—
473C22⋊C4—262no—
474C22⋊C4—262no—
475C22⋊C4—262no—
476C42—261no—
477C42—261no—
478C42—261no—
479C42—261no—
480C42—524no—
481C42—1042no—
482C13#2302411yescharacteristic, complemented, Sylow
483C23—131no—
484C23—132no—
485C23—132no—
486C23—131no—
487C23—261no—
488C23—261no—
489C23—261no—
490C23—261no—
491C23D4×#26111yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—262no—
495Q8—262no—
496Q8—522no—
497Q8—522no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—131no—
503C2×C4—131no—
504C2×C4—131no—
505C2×C4—131no—
506C2×C4—131no—
507C2×C4—132no—
508C2×C4—132no—
509C2×C4—131no—
510C2×C4—261no—
511C2×C4—261no—
512C2×C4—261no—
513C2×C4—261no—
514C2×C4—261no—
515C2×C4—261no—
516C2×C4—261no—
517C2×C4—261no—
518C2×C4—261no—
519C2×C4—261no—
520C2×C4—261no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—264no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—261no—
528C2×C4—261no—
529C2×C4—262no—
530C2×C4—262no—
531C2×C4—522no—
532C2×C4—522no—
533C2×C4—522no—
534C2×C4—522no—
535C2×C4—522no—
536C2×C4—522no—
537C2×C4—522no—
538C2×C4—522no—
539C2×C4—522no—
540C2×C4—1048no—
541C2×C4D4×#26111yescharacteristic
542C2×C4C23×#26111yescharacteristic
543C8—21no—
544C8—132no—
545C8—261no—
546C8—521no—
547C8—521no—
548C8D4×#26112yes—
549C22—21no—
550C22—21no—
551C22—131no—
552C22—131no—
553C22—131no—
554C22—131no—
555C22—131no—
556C22—131no—
557C22—132no—
558C22—262no—
559C22—262no—
560C22—262no—
561C22—262no—
562C22—261no—
563C22—261no—
564C22Q16×#26112yescomplemented
565C22C2×D4×#26111yescharacteristic
566C4—21no—
567C4—42no—
568C4—42no—
569C4—131no—
570C4—131no—
571C4—262no—
572C4—262no—
573C4—261no—
574C4—521no—
575C4—521no—
576C4—524no—
577C4—522no—
578C4—522no—
579C4—1042no—
580C4C2×D4×#26111yescharacteristic
581C4#1171963111yescharacteristic
582C2—22no—
583C2—131no—
584C2—131no—
585C2—131no—
586C2—131no—
587C2—262no—
588C2#1313136111yescharacteristic
589C2C2×Q16×#26111yescharacteristic
590C2#1313159811yescharacteristic
subgroupG/H
centerC22C2×D4×#261
commutatorC2×C4×C13C23×C4
FrattiniC2×C4C23×#261
FittingC13×#467C4
socleC22×C13C2×C4×D4
prime radicalC23×D13C2×D4
cosocleC2×C4×D13C24
hypercenter#467#261
hypocenterC13#23024
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×Q16×#261C2semidirect
4. #13130982C2semidirect
5. #13130940C2semidirect
6. #13130916C2semidirect
8. #13130800C2semidirect
9. #107×#261C2semidirect
35. C13×#467C4semidirect
36. #11718849C4semidirect
37. #11718777C4semidirect
38. #11718712C4semidirect
39. #11718548C4semidirect
41. #11718474C4semidirect
67. C2×C13×Q16C2×C4semidirect
71. C2×#6107C2×C4semidirect
76. C2×Dic52C2×C4semidirect
104. C13×#107C2×C4semidirect
105. C13×#103C2×C4semidirect
106. #88075C2×C4semidirect
107. #88058C2×C4semidirect
108. #88056C2×C4semidirect
109. #88050C2×C4semidirect
110. #88047C2×C4semidirect
142. C22⋊#261Q16semidirect
275. #467#261semidirect
302. C22×C13C4×Q16semidirect
310. C2×D13C4×Q16semidirect
340. C2×Q16C2×#261semidirect
380. #107C2×#261semidirect
386. #103C2×#261semidirect
482. C13#23024semidirect
564. C22Q16×#261semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#2612
C22×#881022
#131313602
#131309822
#131309402
#131309162
#131308212
#131308002
#107×#2612
C8×C22⋊#2612
D13×#4672
#2302413