Finite Group Databasebeta

← #15870552

Subgroups — #15870552

Order 3328.

988 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718420C211yesmaximal, characteristic
2#13117407C211yesmaximal, characteristic, complemented
3C4×D4×Dic13C211yesmaximal, characteristic
4#13115786C211yesmaximal, characteristic, complemented
5#13115596C211yesmaximal, characteristic, complemented
6#13115429C211yesmaximal, characteristic, complemented
7#13115425C211yesmaximal, characteristic, complemented
8#13115097C211yesmaximal, characteristic
9#13114834C211yesmaximal, characteristic
10#13114736C211yesmaximal, characteristic, complemented
11#13114732C211yesmaximal, characteristic, complemented
12#13114688C211yesmaximal, characteristic, complemented
13#13114663C211yesmaximal, characteristic
14#13114663C211yesmaximal, characteristic
15C13×#1864C211yesmaximal, characteristic
16C2×C4×D4×C13C2211yescharacteristic
17C2×D4×Dic13C2211yescharacteristic
18C2×D4×Dic13C2211yescharacteristic
19C22×C4×Dic13—21no—
20C2×#88142C2211yescharacteristic
21C2×#88141—21no—
22C2×C13×#106C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C13×#389C2211yescharacteristic
26C2×#88072C2211yescharacteristic
27C2×#88072C2211yescharacteristic
28C4×C22⋊Dic13—21no—
29C2×#88058C2211yescharacteristic
30C2×#88055C2211yescharacteristic
31C2×#88055C2211yescharacteristic
32C2×#88054C2211yescharacteristic
33C2×#88054C2211yescharacteristic
34#11718637C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39#11718633—21no—
40C2×#88048C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C22⋊C4×Dic13—21no—
43C22⋊C4×Dic13—21no—
44C4×C4⋊Dic13C2211yescharacteristic
45C42×Dic13C2211yescharacteristic
46#11718589C2211yescharacteristic
47#11718420C418yescomplemented
48#11718310—21no—
49#11718307—21no—
50C13×#340C2211yescharacteristic
51C13×#336C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C23×Dic13—21no—
63C23×Dic13—21no—
64C22×Dic26—21no—
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C4×D4×C13—22no—
68C4×D4×C13—22no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C22⋊C4—21no—
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—21no—
76C2×C22⋊Dic13—21no—
77D4×Dic13—22no—
78D4×Dic13—22no—
79D4×Dic13C2×C414yes—
80D4×Dic13Q814yescomplemented
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—44no—
95C2×C4×Dic13D411yescharacteristic
96C2×C4×Dic13D411yescharacteristic
97C2×C4×Dic13D411yescharacteristic
98C2×C4×Dic13D411yescharacteristic
99C2×C4×Dic13C2311yescharacteristic
100C2×C4×Dic13C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103#88142—22no—
104#88142C2×C414yescomplemented
105#88141—44no—
106C13×#106—22no—
107C13×#102—22no—
108C13×#102C2×C414yes—
109C13×#95—21no—
110#88072—22no—
111#88072C2×C414yes—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88058C2×C414yescomplemented
119#88055—22no—
120#88055—22no—
121#88054—22no—
122#88054C2×C414yescomplemented
123#88048—22no—
124#88047C2×C414yescomplemented
125#21613—131nomaximal, Sylow
126C24×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—21no—
129C2×D4×C13—21no—
130C2×D4×C13C22×C412yes—
131C2×D4×C13C2×Q812yes—
132C2×D4×C13#4112yes—
133C22×C4×C13—24no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—42no—
138C22×C4×C13—42no—
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2411yescharacteristic
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—24no—
146C22×Dic13—24no—
147C22×Dic13—21no—
148C22×Dic13—42no—
149C22×Dic13—42no—
150C22×Dic13—42no—
151C22×Dic13—42no—
152C22×Dic13C2×D411yescharacteristic
153C22×Dic13C2×D411yescharacteristic
154C22×Dic13C2×D411yescharacteristic
155C22×Dic13C2×D411yescharacteristic
156C2×Dic26—24no—
157C2×Dic26—42no—
158C2×Dic26—42no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13—21no—
164C2×C8×C13C22×C412yes—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—42no—
170C13×C4⋊C4C22×C412yes—
171C13×C4⋊C4#4112yes—
172C13×C22⋊C4—24no—
173C42×C13—21no—
174C42×C13—21no—
175C22⋊Dic13—24no—
176C22⋊Dic13—24no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13C22×C412yes—
183C4⋊Dic13C22×C412yes—
184C4⋊Dic13C2×Q812yes—
185#6101—24no—
186#6101—42no—
187#6101—42no—
188#6101—42no—
189C4×Dic13—22no—
190C4×Dic13—22no—
191C4×Dic13—22no—
192C4×Dic13—41no—
193C4×Dic13—42no—
194C4×Dic13—42no—
195C4×Dic13—42no—
196C4×Dic13—42no—
197C4×Dic13—42no—
198C4×Dic13C22×C412yes—
199C4×Dic13C2×Q812yes—
200C4×Dic13#4112yes—
201C4×Dic13#4112yes—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055C22×C412yes—
208C2×#474—131no—
209C4×C4⋊Q8—131no—
210C42×D4—131no—
211#1988—131no—
212#1948—131no—
213#1921—131no—
214#1921—131no—
215#1864—131no—
216#1864D1311yescharacteristic
217#1802—131no—
218#1776—131no—
219#1776—131no—
220#1767—131no—
221#1762—131no—
222#1762—131no—
223C23×C13—22no—
224C23×C13—22no—
225C23×C13—22no—
226C23×C13—41no—
227C23×C13—41no—
228C23×C13—41no—
229C23×C13—41no—
230C23×C13C22×D411yescharacteristic
231D4×C13—22no—
232D4×C13—22no—
233D4×C13—22no—
234D4×C13C4×Q814yescomplemented
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—22no—
238C2×C4×C13—21no—
239C2×C4×C13—22no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—48no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13#13712yes—
251C2×C4×C13C23×C411yescharacteristic
252C2×C4×C13C22×Q811yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C2×C4×C13C2×#4111yescharacteristic
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—22no—
266C2×Dic13—48no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—48no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13C4×D412yes—
278C2×Dic13C4×D412yes—
279C2×Dic13C22⋊Q812yes—
280C2×Dic13C22⋊Q812yes—
281C2×Dic13AGL(1,ℤ/8)12yes—
282C2×Dic13AGL(1,ℤ/8)12yes—
283Dic26—48no—
284C8×C13—22no—
285C8×C13—41no—
286#1055—22no—
287#1055—41no—
288C2×C4⋊Q8—131no—
289C2×C4×Q8—261no—
290C2×C4×D4—131no—
291C2×C4×D4—131no—
292C2×C4×D4C2×D1311yescharacteristic
293C22×C42—261no—
294#474—138no—
295C2×#107—131no—
296C2×#107—131no—
297C2×#106—131no—
298C2×#106C2×D1311yescharacteristic
299C2×C4⋊C8—131no—
300C2×C4⋊C8—131no—
301C2×#102—131no—
302C2×#102—131no—
303C2×#102C2×D1311yescharacteristic
304C2×#102C2×D1311yescharacteristic
305C2×#97—131no—
306C2×#97—131no—
307#391—261no—
308#389—131no—
309#389—261no—
310#389C2×D1311yescharacteristic
311#384—131no—
312C4×C4⋊C4—131no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—261no—
318C4×C22⋊C4—261no—
319C4×C22⋊C4—261no—
320C4×C22⋊C4—261no—
321C43—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340C2×D1311yescharacteristic
330#336—131no—
331#336C2×D1311yescharacteristic
332C22×C13—24no—
333C22×C13—42no—
334C22×C13—42no—
335C22×C13—42no—
336C22×C13—42no—
337C22×C13—42no—
338C22×C13—42no—
339C22×C13C2×C4×D411yescharacteristic
340C22×C13C2×C22⋊Q811yescharacteristic
341C22×C13#52511yescharacteristic
342C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
343C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C13C2×#13711yescharacteristic
345C22×C13#57511yescharacteristic
346C4×C13—22no—
347C4×C13—42no—
348C4×C13—41no—
349C4×C13—42no—
350C4×C13—81no—
351C4×C13C2×C4×Q811yescharacteristic
352C4×C13C4×#4111yescharacteristic
353C4×C13#52911yescharacteristic
354C4×C13#53311yescharacteristic
355Dic13—22no—
356Dic13—22no—
357Dic13—22no—
358Dic13—42no—
359Dic13—42no—
360Dic13—42no—
361Dic13—81no—
362Dic13#44214yes—
363C22×Q8—261no—
364C22×D4C22×D1311yescharacteristic
365C23×C4—21no—
366C23×C4—261no—
367C23×C4—261no—
368C22×C8—131no—
369C22×C8—131no—
370C22×C8C22×D1311yescharacteristic
371C22×C8C22×D1311yescharacteristic
372C4⋊Q8—134no—
373C4⋊Q8—262no—
374C4×Q8—524no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—134no—
378C4×D4—134no—
379C4×D4—262no—
380C4×D4—262no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—261no—
387C2×C4⋊C4—261no—
388C2×C4⋊C4—261no—
389C2×C4⋊C4—261no—
390C2×C4⋊C4C22×D1311yescharacteristic
391C2×C4⋊C4C22×D1311yescharacteristic
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—261no—
395C2×C22⋊C4—261no—
396C2×C42—131no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—131no—
400C2×C42—131no—
401C2×C42—131no—
402C2×C42—261no—
403C2×C42—261no—
404C2×C42—261no—
405C2×C42—261no—
406C2×C42—261no—
407C2×C42—524no—
408C2×C42C22×D1311yescharacteristic
409#107—134no—
410#107—134no—
411#106—22no—
412#106—262no—
413C4⋊C8—134no—
414C4⋊C8—262no—
415#102—22no—
416#102—134no—
417#102—262no—
418#102C4×D1314yes—
419#97—262no—
420#97—262no—
421#95—21no—
422#95—261no—
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427#95—261no—
428C2×C13—44no—
429C2×C13C4×C22⋊Q811yescharacteristic
430C2×C13C2×#44211yescharacteristic
431C2×C13C4×#13711yescharacteristic
432C2×C13#307711yescharacteristic
433C2×C13#307911yescharacteristic
434C2×C13#311011yescharacteristic
435C2×C13#311511yescharacteristic
436C24—21no—
437C2×Q8—264no—
438C2×Q8—522no—
439C2×Q8—522no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4C2×C4×D1312yes—
444C2×D4#612912yes—
445C2×D4Q8×D1312yes—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—131no—
453C22×C4—131no—
454C22×C4—131no—
455C22×C4—131no—
456C22×C4—264no—
457C22×C4—264no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4—522no—
463C22×C4—522no—
464C22×C4—522no—
465C22×C4—522no—
466C22×C4D4×D1311yescharacteristic
467C22×C4D4×D1311yescharacteristic
468C22×C4C23×D1311yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—132no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8C2×C4×D1312yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—132no—
487C4⋊C4—132no—
488C4⋊C4—132no—
489C4⋊C4—264no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—522no—
496C4⋊C4—522no—
497C4⋊C4—522no—
498C4⋊C4C2×C4×D1312yes—
499C4⋊C4#612912yes—
500C22⋊C4—24no—
501C22⋊C4—264no—
502C22⋊C4—264no—
503C42—21no—
504C42—21no—
505C42—132no—
506C42—132no—
507C42—132no—
508C42—132no—
509C42—262no—
510C42—262no—
511C42—262no—
512C42—521no—
513C42—522no—
514C42—522no—
515C42—522no—
516C42—522no—
517C42—522no—
518C13#2161311yescharacteristic, complemented, Sylow
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D1311yescharacteristic
527Q8—528no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#8814114yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—132no—
548C2×C4—132no—
549C2×C4—132no—
550C2×C4—132no—
551C2×C4—132no—
552C2×C4—132no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—528no—
563C2×C4—528no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4#8816912yes—
574C2×C4C22×C4×D1311yescharacteristic
575C2×C4C2×#612911yescharacteristic
576C2×C4C2×#612911yescharacteristic
577C2×C4C2×Q8×D1311yescharacteristic
578C2×C4#41×D1311yescharacteristic
579C2×C4#41×D1311yescharacteristic
580C8—22no—
581C8—41no—
582C8—262no—
583C8—521no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D1311yescharacteristic
592C22#1171929111yescharacteristic
593C22D13×C22⋊Q811yescharacteristic
594C22C2×#8816911yescharacteristic
595C22#1171947311yescharacteristic
596C22D13×AGL(1,ℤ/8)11yescharacteristic
597C22D13×AGL(1,ℤ/8)11yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—134no—
604C4—262no—
605C4—262no—
606C4—262no—
607C4—522no—
608C4—522no—
609C4—522no—
610C4—1041no—
611C4C2×#8814111yescharacteristic
612C4C4×#612911yescharacteristic
613C4#1171932711yescharacteristic
614C4#1171935411yescharacteristic
615C2—44no—
616C2#1311736811yescharacteristic
617C2D13×#44211yescharacteristic
618C2C4×#8816911yescharacteristic
619C2#1312463111yescharacteristic
620C2#1312464811yescharacteristic
621C2#1312493011yescharacteristic
622C2#1312498011yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1864C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenter#1864D13
hypocenterC13#21613
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #13117407C2semidirect
4. #13115786C2semidirect
5. #13115596C2semidirect
6. #13115429C2semidirect
7. #13115425C2semidirect
10. #13114736C2semidirect
11. #13114732C2semidirect
12. #13114688C2semidirect
47. #11718420C4semidirect
80. D4×Dic13Q8semidirect
104. #88142C2×C4semidirect
118. #88058C2×C4semidirect
122. #88054C2×C4semidirect
124. #88047C2×C4semidirect
234. D4×C13C4×Q8semidirect
518. C13#21613semidirect
531. D4#88141semidirect

Maximal subgroups

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

subgroupindex
C2×#117184202
#131174072
C4×D4×Dic132
#131157862
#131155962
#131154292
#131154252
#131150972
#131148342
#131147362
#131147322
#131146882
#131146632
C13×#18642
#2161313