Finite Group Databasebeta

← #15867806

Subgroups — #15867806

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718366C211yesmaximal, characteristic
2#13117240C211yesmaximal, characteristic
3C4×C4⋊C4×D13C211yesmaximal, characteristic
4#13115598C211yesmaximal, characteristic, complemented
5#13115596C211yesmaximal, characteristic, complemented
6#13115593C211yesmaximal, characteristic, complemented
7#13115591C211yesmaximal, characteristic, complemented
8#13115030C211yesmaximal, characteristic
9#13115027C211yesmaximal, characteristic
10#13115025C211yesmaximal, characteristic
11#13115022C211yesmaximal, characteristic
12#13114734C211yesmaximal, characteristic, complemented
13#13114734C211yesmaximal, characteristic, complemented
14#13114733C211yesmaximal, characteristic, complemented
15C13×#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D13C2211yescharacteristic
17C2×C4⋊C4×D13C2211yescharacteristic
18C2×#88126—21no—
19C2×C42×D13—21no—
20C2×C42×D13C2211yescharacteristic
21C2×C13×#107C2211yescharacteristic
22C2×C13×#106C2211yescharacteristic
23C2×C13×#97C2211yescharacteristic
24C4×C13×C4⋊C4C2211yescharacteristic
25C4×C13×C4⋊C4C2211yescharacteristic
26C2×#88060C2211yescharacteristic
27C2×#88060C2211yescharacteristic
28C2×#88058C2211yescharacteristic
29C2×#88057C2211yescharacteristic
30C2×#88054C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32#11718648C2211yescharacteristic
33#11718648C2211yescharacteristic
34#11718637C2211yescharacteristic
35#11718637C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37C4⋊C4×Dic13C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88048C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C4×C4⋊Dic13C2211yescharacteristic
43C4×C4⋊Dic13C2211yescharacteristic
44C4×#6101—21no—
45C4×#6101—21no—
46C2×#88043C2211yescharacteristic
47#11718366C418yescomplemented
48#11718328—21no—
49D13×#95—21no—
50#11718311—21no—
51#11718310—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×C4×D13—21no—
61C22×C4×D13—21no—
62C22×C4×D13D411yescharacteristic
63C22×C4×D13D411yescharacteristic
64C22×C4×D13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C22×C8×C13C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C42×C13—21no—
72C2×C42×C13—21no—
73C2×C42×C13C2311yescharacteristic
74C2×#6103—21no—
75C2×#6103—21no—
76C2×C4⋊Dic13C2311yescharacteristic
77C2×C4⋊Dic13C2311yescharacteristic
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2311yescharacteristic
80C2×#6101—21no—
81C2×#6101—21no—
82C2×#6101—21no—
83C2×#6101—21no—
84C2×#6101—21no—
85C2×#6101—21no—
86C2×C4×Dic13—21no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13D411yescharacteristic
91C2×C4×Dic13D411yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C22×#1055C2311yescharacteristic
94C22×#1055C2311yescharacteristic
95C4⋊C4×D13—22no—
96C4⋊C4×D13—22no—
97C4⋊C4×D13—22no—
98C4⋊C4×D13—22no—
99#88126—44no—
100C42×D13—44no—
101C42×D13C2×C414yes—
102C42×D13Q814yes—
103C13×#107—22no—
104C13×#106—22no—
105C13×#97C2×C414yescomplemented
106C13×#95—21no—
107C13×#95—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88060C2×C414yes—
115#88060C2×C414yes—
116#88058—22no—
117#88057—22no—
118#88054C2×C414yescomplemented
119#88054C2×C414yescomplemented
120#88048—22no—
121#88048—22no—
122#88047—22no—
123#88047—22no—
124#88043C2×C414yescomplemented
125#21365—131nomaximal, Sylow
126C23×D13C2×D411yescharacteristic
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13C2×D411yescharacteristic
132C22×C4×C13C2×D411yescharacteristic
133C22×C4×C13C2411yescharacteristic
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13—21no—
137C22×Dic13—21no—
138C22×Dic13C2×D411yescharacteristic
139C22×Dic13C2×D411yescharacteristic
140C22×Dic13C2×D411yescharacteristic
141C2×C4×D13—22no—
142C2×C4×D13—22no—
143C2×C4×D13—21no—
144C2×C4×D13—21no—
145C2×C4×D13—21no—
146C2×C4×D13—21no—
147C2×C4×D13—22no—
148C2×C4×D13—22no—
149C2×C4×D13—24no—
150C2×C4×D13—24no—
151C2×C4×D13—42no—
152C2×C4×D13—42no—
153C2×C4×D13—42no—
154C2×C4×D13—42no—
155C2×C4×D13C4⋊C414yes—
156C2×C4×D13C4⋊C414yes—
157C2×C4×D13C22×C412yes—
158C2×C4×D13C2×Q812yes—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13—21no—
163C2×C8×C13C22×C412yes—
164C2×C8×C13C22×C412yes—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4#4112yes—
172C13×C4⋊C4#4112yes—
173C42×C13—42no—
174C42×C13—42no—
175C42×C13C22×C412yes—
176C42×C13C2×Q812yes—
177#6103—42no—
178#6103—42no—
179C4⋊Dic13—21no—
180C4⋊Dic13—21no—
181C4⋊Dic13—21no—
182C4⋊Dic13—21no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13#4112yes—
186C4⋊Dic13#4112yes—
187#6101—24no—
188#6101—24no—
189#6101—42no—
190#6101—42no—
191#6101—42no—
192#6101—42no—
193C4×Dic13—42no—
194C4×Dic13—42no—
195C4×Dic13—42no—
196C4×Dic13—42no—
197C4×Dic13C4⋊C414yes—
198C4×Dic13C4⋊C414yes—
199C4×Dic13C22×C412yes—
200C4×Dic13C2×Q812yes—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055C22×C412yes—
206C2×#1055C22×C412yes—
207C2×#432—131no—
208#2285—131no—
209C2×C4×C4⋊C4—131no—
210#1948—131no—
211#1948—131no—
212#1947—131no—
213#1947—131no—
214#1853—131no—
215#1853—131no—
216#1852—131no—
217#1852—131no—
218#1776—131no—
219#1776—131no—
220#1776—131no—
221#1776D1311yescharacteristic, complemented
222C23×C13C22×D411yescharacteristic
223C22×D13—21no—
224C22×D13—21no—
225C22×D13—21no—
226C22×D13—21no—
227C22×D13C2×C4⋊C412yes—
228C22×D13AGL(1,ℤ/8)12yes—
229C22×D13#13712yes—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—41no—
239C2×C4×C13—41no—
240C2×C4×C13—41no—
241C2×C4×C13—41no—
242C2×C4×C13—41no—
243C2×C4×C13—41no—
244C2×C4×C13—41no—
245C2×C4×C13—41no—
246C2×C4×C13C2×C4⋊C412yes—
247C2×C4×C13C2×C4⋊C412yes—
248C2×C4×C13C23×C411yescharacteristic
249C2×C4×C13C22×Q811yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13C2×#4111yescharacteristic
253C2×C4×C13C2×#4111yescharacteristic
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—22no—
259C2×Dic13—22no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13C2×C4⋊C412yes—
271C2×Dic13C2×C4⋊C412yes—
272C2×Dic13C2×C4⋊C412yes—
273C2×Dic13AGL(1,ℤ/8)12yes—
274C2×Dic13#13712yes—
275C4×D13—28no—
276C4×D13—22no—
277C4×D13—22no—
278C4×D13—22no—
279C4×D13—22no—
280C4×D13—28no—
281C4×D13—48no—
282C4×D13—48no—
283C8×C13—22no—
284C8×C13—22no—
285#1055—22no—
286#1055—22no—
287C2×#117—261no—
288C22×C4⋊C4—131no—
289C22×C4⋊C4—131no—
290C22×C42—131no—
291C22×C42—261no—
292#432—138no—
293C2×#107—131no—
294C2×#107—131no—
295C2×#107—131no—
296C2×#107C2×D1311yescharacteristic
297C2×#106—131no—
298C2×#106—131no—
299C2×#106—131no—
300C2×#106C2×D1311yescharacteristic
301C2×C4⋊C8—131no—
302C2×C4⋊C8—131no—
303C2×C22⋊C8—131no—
304C2×C22⋊C8—131no—
305C2×#97—131no—
306C2×#97C2×D1311yescharacteristic
307#384—131no—
308#384—131no—
309#382—261no—
310#380—131no—
311#380—131no—
312#380—261no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—261no—
318C4×C4⋊C4—261no—
319C4×C4⋊C4—261no—
320C4×C4⋊C4C2×D1311yescharacteristic
321C4×C4⋊C4C2×D1311yescharacteristic
322C2×#95—261no—
323#340—131no—
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340C2×D1311yescharacteristic
330#340C2×D1311yescharacteristic
331C22×C13C22×C4⋊C411yescharacteristic
332C22×C13#52511yescharacteristic
333C22×C13#52511yescharacteristic
334C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
335C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C13C2×#13711yescharacteristic
337C22×C13C2×#13711yescharacteristic
338C2×D13—22no—
339C2×D13—22no—
340C2×D13—22no—
341C2×D13—22no—
342C2×D13—22no—
343C2×D13—22no—
344C2×D13#42814yes—
345C4×C13—22no—
346C4×C13—22no—
347C4×C13—42no—
348C4×C13—42no—
349C4×C13—81no—
350C4×C13—81no—
351C4×C13C4×#4111yescharacteristic
352C4×C13C4×#4111yescharacteristic
353C4×C13#53311yescharacteristic
354C4×C13#53311yescharacteristic
355Dic13—22no—
356Dic13—22no—
357Dic13—42no—
358Dic13—42no—
359Dic13—81no—
360Dic13—81no—
361Dic13#42814yes—
362C23×C4—131no—
363C23×C4—131no—
364C23×C4—131no—
365C23×C4—261no—
366C23×C4—261no—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8C22×D1311yescharacteristic
370C22×C8C22×D1311yescharacteristic
371#117—524no—
372C2×C4⋊C4—131no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—131no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—262no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—262no—
381C2×C4⋊C4—262no—
382C2×C4⋊C4—261no—
383C2×C4⋊C4—261no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4C22×D1311yescharacteristic
387C2×C4⋊C4C22×D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C4⋊C4C22×D1311yescharacteristic
390C2×C22⋊C4—261no—
391C2×C22⋊C4—261no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—131no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—134no—
398C2×C42—134no—
399C2×C42—261no—
400C2×C42—261no—
401C2×C42—261no—
402C2×C42—261no—
403C2×C42—524no—
404C2×C42C22×D1311yescharacteristic
405#107—22no—
406#107—262no—
407#107—262no—
408#107—262no—
409#106—22no—
410#106—262no—
411#106—262no—
412#106—262no—
413C4⋊C8—134no—
414C4⋊C8—134no—
415C22⋊C8—134no—
416C22⋊C8—134no—
417#97—134no—
418#97C4×D1314yescomplemented
419#95—21no—
420#95—21no—
421#95—261no—
422#95—261no—
423#95—261no—
424#95—261no—
425#95—261no—
426#95—261no—
427C2×C13#229611yescharacteristic
428C2×C13C2×#42811yescharacteristic
429C2×C13C2×#42811yescharacteristic
430C2×C13#310911yescharacteristic
431C2×C13#311011yescharacteristic
432C2×C13#311211yescharacteristic
433C2×C13#311311yescharacteristic
434D13—24no—
435C24—131no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—131no—
441C22×C4—131no—
442C22×C4—131no—
443C22×C4—134no—
444C22×C4—134no—
445C22×C4—132no—
446C22×C4—132no—
447C22×C4—262no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—264no—
452C22×C4—264no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—261no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—261no—
461C22×C4—522no—
462C22×C4—522no—
463C22×C4—522no—
464C22×C4—522no—
465C22×C4D4×D1311yescharacteristic
466C22×C4D4×D1311yescharacteristic
467C22×C4C23×D1311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—132no—
473C2×C8—132no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8C2×C4×D1312yes—
479C2×C8C2×C4×D1312yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—132no—
487C4⋊C4—132no—
488C4⋊C4—264no—
489C4⋊C4—264no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—261no—
495C4⋊C4—261no—
496C4⋊C4—522no—
497C4⋊C4—522no—
498C4⋊C4—522no—
499C4⋊C4—522no—
500C4⋊C4#612712yes—
501C4⋊C4#612712yes—
502C22⋊C4—522no—
503C22⋊C4—522no—
504C42—42no—
505C42—42no—
506C42—134no—
507C42—134no—
508C42—132no—
509C42—132no—
510C42—522no—
511C42—522no—
512C42—522no—
513C42—522no—
514C42C2×C4×D1312yes—
515C42Q8×D1312yes—
516C13#2136511yescharacteristic, complemented, Sylow
517C23—132no—
518C23—132no—
519C23—132no—
520C23—261no—
521C23—261no—
522C23—261no—
523C23—261no—
524C23C2×D4×D1311yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—132no—
542C2×C4—132no—
543C2×C4—132no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—268no—
547C2×C4—268no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—262no—
553C2×C4—262no—
554C2×C4—262no—
555C2×C4—262no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—528no—
559C2×C4—528no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—521no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4C4⋊C4×D1312yes—
571C2×C4C4⋊C4×D1312yes—
572C2×C4C22×C4×D1311yescharacteristic
573C2×C4C2×#612711yescharacteristic
574C2×C4C2×#612711yescharacteristic
575C2×C4C2×Q8×D1311yescharacteristic
576C2×C4#41×D1311yescharacteristic
577C2×C4#41×D1311yescharacteristic
578C8—22no—
579C8—22no—
580C8—262no—
581C8—262no—
582C22—134no—
583C22—262no—
584C22—262no—
585C22—262no—
586C22—262no—
587C22—262no—
588C22—262no—
589C22C2×C4⋊C4×D1311yescharacteristic
590C22#1171925211yescharacteristic
591C22#1171925211yescharacteristic
592C22D13×AGL(1,ℤ/8)11yescharacteristic
593C22D13×AGL(1,ℤ/8)11yescharacteristic
594C22D13×#13711yescharacteristic
595C22D13×#13711yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—134no—
603C4—262no—
604C4—262no—
605C4—522no—
606C4—522no—
607C4—1041no—
608C4—1041no—
609C4C4×#612711yescharacteristic
610C4#1171922711yescharacteristic
611C4#1171928311yescharacteristic
612C4#1171928311yescharacteristic
613C2—264no—
614C2#1311731611yescharacteristic
615C2D13×#42811yescharacteristic
616C2D13×#42811yescharacteristic
617C2#1312492511yescharacteristic
618C2#1312493311yescharacteristic
619C2#1312495611yescharacteristic
620C2#1312496411yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1776C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1776D13
hypocenterC13#21365
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #13115598C2semidirect
5. #13115596C2semidirect
6. #13115593C2semidirect
7. #13115591C2semidirect
12. #13114734C2semidirect
13. #13114734C2semidirect
14. #13114733C2semidirect
15. C13×#1776C2semidirect
47. #11718366C4semidirect
105. C13×#97C2×C4semidirect
118. #88054C2×C4semidirect
119. #88054C2×C4semidirect
124. #88043C2×C4semidirect
221. #1776D13semidirect
418. #97C4×D13semidirect
516. C13#21365semidirect

Maximal subgroups

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

subgroupindex
C2×#117183662
#131172402
C4×C4⋊C4×D132
#131155982
#131155962
#131155932
#131155912
#131150302
#131150272
#131150252
#131150222
#131147342
#131147332
C13×#17762
#2136513