Finite Group Databasebeta

← #15873501

Subgroups — #15873501

Order 3328.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718381C211yesmaximal, characteristic
2#13117460C211yesmaximal, characteristic
3#13117443C211yesmaximal, characteristic
4#13115607C211yesmaximal, characteristic, complemented
5#13115605C211yesmaximal, characteristic, complemented
6#13115598C211yesmaximal, characteristic, complemented
7#13115596C211yesmaximal, characteristic, complemented
8#13115037C211yesmaximal, characteristic
9#13115033C211yesmaximal, characteristic
10#13115029C211yesmaximal, characteristic
11#13115001C211yesmaximal, characteristic, complemented
12#13114991C211yesmaximal, characteristic, complemented
13#13114968C211yesmaximal, characteristic, complemented
14#13114965C211yesmaximal, characteristic, complemented
15C13×#1853C211yesmaximal, characteristic
16C22×C13×C4⋊C4C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C2×#88180C2211yescharacteristic
19C2×C13×#106C2211yescharacteristic
20C2×C13×#106C2211yescharacteristic
21C2×C13×C22⋊C8C2211yescharacteristic
22C13×#380C2211yescharacteristic
23#11718873C2211yescharacteristic
24#11718871—21no—
25#11718871—21no—
26C2×C22⋊#1055C2211yescharacteristic
27C2×#88058C2211yescharacteristic
28C2×#88057C2211yescharacteristic
29C2×#88055C2211yescharacteristic
30C2×#88054C2211yescharacteristic
31C2×#88054C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718638C2211yescharacteristic
34#11718638C2211yescharacteristic
35#11718638C2211yescharacteristic
36#11718637C2211yescharacteristic
37#11718637C2211yescharacteristic
38#11718637C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41#11718634—21no—
42#11718634—21no—
43C2×#88048C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88048C2211yescharacteristic
46C2×#88047C2211yescharacteristic
47#11718606—21no—
48#11718605—21no—
49#11718381C418yescomplemented
50#11718315—21no—
51#11718314—21no—
52C13×#340C2211yescharacteristic
53C13×#340C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C23×C4×C13—21no—
61C23×C4×C13C2311yescharacteristic
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C2×C13×C4⋊C4—21no—
65C2×C13×C4⋊C4—41no—
66C2×C13×C4⋊C4—41no—
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2311yescharacteristic
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C22⋊C4—21no—
72C2×C22⋊Dic13—21no—
73C2×C22⋊Dic13—21no—
74C2×C22⋊Dic13D411yescharacteristic
75C2×C22⋊Dic13D411yescharacteristic
76#88180—41no—
77#88180—41no—
78#88180C2×C414yes—
79#88180Q814yes—
80C2×C4⋊Dic13D411yescharacteristic
81C2×C4⋊Dic13D411yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#106—41no—
100C13×#106—41no—
101C13×C22⋊C8C2×C414yes—
102C13×#95—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112C22⋊#1055C2×C414yes—
113#88058—41no—
114#88057—41no—
115#88055C2×C414yescomplemented
116#88054C2×C414yescomplemented
117#88054C2×C414yescomplemented
118C8×Dic13C2×C414yescomplemented
119#88048—41no—
120#88048—41no—
121#88048—41no—
122#88047—41no—
123#21990—131nomaximal, Sylow
124C24×C13C2×D411yescharacteristic
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—42no—
134C22×C4×C13—81no—
135C22×C4×C13—81no—
136C22×C4×C13C22×C412yes—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×Q812yes—
139C22×C4×C13C2411yescharacteristic
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13C2×D411yescharacteristic
145C22×Dic13C2×D411yescharacteristic
146C22×Dic13C2×D411yescharacteristic
147C22×Dic13C2×D411yescharacteristic
148C2×C8×C13—21no—
149C2×C8×C13—21no—
150C2×C8×C13—21no—
151C2×C8×C13—21no—
152C2×C8×C13C22×C412yes—
153C2×C8×C13C22×C412yes—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—42no—
163C13×C22⋊C4—81no—
164C22⋊Dic13—42no—
165C22⋊Dic13—81no—
166C22⋊Dic13C4⋊C414yes—
167C22⋊Dic13C4⋊C414yes—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13C4⋊C414yes—
177C4⋊Dic13C4⋊C414yes—
178#6101—81no—
179#6101—81no—
180#6101—81no—
181#6101—81no—
182#6101—81no—
183#6101—81no—
184C4×Dic13—42no—
185C4×Dic13—81no—
186C4×Dic13—81no—
187C4×Dic13C22×C412yes—
188C4×Dic13C22×C412yes—
189C4×Dic13C2×Q812yes—
190C4×Dic13C2×Q812yes—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055C22×C412yes—
196C2×#1055C22×C412yes—
197C2×#433—131no—
198#2317—131no—
199#2314—131no—
200#1949—131no—
201#1949—131no—
202#1948—131no—
203#1948—131no—
204#1854—131no—
205#1854—131no—
206#1853—131no—
207#1853D1311yescharacteristic
208#1847—131no—
209#1844—131no—
210#1836—131no—
211#1835—131no—
212C23×C13—21no—
213C23×C13—21no—
214C23×C13—21no—
215C23×C13—21no—
216C23×C13—21no—
217C23×C13—21no—
218C23×C13C2×C4⋊C412yes—
219C23×C13C22×D411yescharacteristic
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—22no—
223C2×C4×C13—22no—
224C2×C4×C13—22no—
225C2×C4×C13—22no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13—41no—
237C2×C4×C13—41no—
238C2×C4×C13—84no—
239C2×C4×C13C2×C4⋊C412yes—
240C2×C4×C13C23×C411yescharacteristic
241C2×C4×C13C22×Q811yescharacteristic
242C2×C4×C13H(5,𝔽2)11yescharacteristic
243C2×C4×C13H(5,𝔽2)11yescharacteristic
244C2×C4×C132−1+411yescharacteristic
245C2×C4×C132−1+411yescharacteristic
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—41no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—41no—
266C2×Dic13C2×C4⋊C412yes—
267C2×Dic13C2×C4⋊C412yes—
268C2×Dic13C2×C4⋊C412yes—
269C2×Dic13C2×C4⋊C412yes—
270C8×C13—22no—
271C8×C13—22no—
272#1055—22no—
273#1055—22no—
274C2×#117—131no—
275C2×#117—131no—
276C22×C4⋊C4C2×D1311yescharacteristic
277#433—138no—
278C2×#107—131no—
279C2×#107—131no—
280C2×#106—131no—
281C2×#106—131no—
282C2×#106—131no—
283C2×#106—131no—
284C2×#106C2×D1311yescharacteristic
285C2×#106C2×D1311yescharacteristic
286C2×C4⋊C8—131no—
287C2×C4⋊C8—131no—
288C2×C22⋊C8—131no—
289C2×C22⋊C8C2×D1311yescharacteristic
290C2×#97—131no—
291C2×C4×C8—131no—
292#389—261no—
293#389—261no—
294#388—261no—
295#388—261no—
296#387—261no—
297#387—261no—
298#385—261no—
299#385—261no—
300#384—131no—
301#384—131no—
302#384—131no—
303#382—131no—
304#382—131no—
305#382—131no—
306#380—131no—
307#380C2×D1311yescharacteristic
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340C2×D1311yescharacteristic
317#340C2×D1311yescharacteristic
318C22×C13—22no—
319C22×C13—22no—
320C22×C13—22no—
321C22×C13—41no—
322C22×C13—41no—
323C22×C13—41no—
324C22×C13—41no—
325C22×C13C22×C4⋊C411yescharacteristic
326C22×C13#53511yescharacteristic
327C22×C13#53611yescharacteristic
328C22×C13#57711yescharacteristic
329C22×C13#57711yescharacteristic
330C22×C13#57711yescharacteristic
331C22×C13#57711yescharacteristic
332C4×C13—22no—
333C4×C13—81no—
334C4×C13—81no—
335C4×C13—81no—
336C4×C13—81no—
337C4×C13#52011yescharacteristic
338C4×C13#52011yescharacteristic
339C4×C13#54411yescharacteristic
340C4×C13#54411yescharacteristic
341Dic13—22no—
342Dic13—22no—
343Dic13—22no—
344Dic13—22no—
345Dic13—81no—
346Dic13—81no—
347Dic13—81no—
348Dic13—81no—
349C23×C4—21no—
350C23×C4C22×D1311yescharacteristic
351C22×C8—131no—
352C22×C8—131no—
353C22×C8C22×D1311yescharacteristic
354C22×C8C22×D1311yescharacteristic
355#117—134no—
356#117—134no—
357#117—521no—
358#117—521no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—41no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—131no—
366C2×C4⋊C4—131no—
367C2×C4⋊C4—131no—
368C2×C4⋊C4—261no—
369C2×C4⋊C4—261no—
370C2×C4⋊C4—261no—
371C2×C4⋊C4—261no—
372C2×C4⋊C4—261no—
373C2×C4⋊C4—261no—
374C2×C4⋊C4C22×D1311yescharacteristic
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C4⋊C4C22×D1311yescharacteristic
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—131no—
380C2×C22⋊C4—131no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C42—131no—
384C2×C42—131no—
385C2×C42—261no—
386C2×C42—261no—
387C2×C42—261no—
388#107—521no—
389#107—521no—
390#106—41no—
391#106—41no—
392#106—521no—
393#106—521no—
394#106—521no—
395#106—521no—
396C4⋊C8—134no—
397C4⋊C8—134no—
398C22⋊C8—134no—
399C22⋊C8C4×D1314yes—
400#97—134no—
401C4×C8—134no—
402#95—21no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412C2×C13—42no—
413C2×C13#231711yescharacteristic
414C2×C13#291211yescharacteristic
415C2×C13#291211yescharacteristic
416C2×C13#322111yescharacteristic
417C2×C13#322211yescharacteristic
418C2×C13#324411yescharacteristic
419C2×C13#324411yescharacteristic
420C24D4×D1311yescharacteristic
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—131no—
433C22×C4—131no—
434C22×C4—131no—
435C22×C4—131no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4—261no—
439C22×C4—261no—
440C22×C4C2×C4×D1312yes—
441C22×C4D4×D1311yescharacteristic
442C22×C4Q8×D1312yes—
443C22×C4C23×D1311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—132no—
449C2×C8—132no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8C2×C4×D1312yes—
455C2×C8C2×C4×D1312yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—134no—
466C4⋊C4—134no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4—261no—
475C4⋊C4—1041no—
476C4⋊C4—1041no—
477C4⋊C4—1041no—
478C4⋊C4—1041no—
479C4⋊C4—1041no—
480C4⋊C4—1041no—
481C22⋊C4—81no—
482C22⋊C4—134no—
483C22⋊C4—134no—
484C22⋊C4—522no—
485C22⋊C4—1041no—
486C42—132no—
487C42—132no—
488C42—132no—
489C42—132no—
490C42—522no—
491C42—1041no—
492C42—1041no—
493C13#2199011yescharacteristic, complemented, Sylow
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D1312yes—
501C23C2×D4×D1311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—132no—
522C2×C4—132no—
523C2×C4—132no—
524C2×C4—132no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—261no—
528C2×C4—261no—
529C2×C4—261no—
530C2×C4—261no—
531C2×C4—261no—
532C2×C4—261no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—521no—
544C2×C4—521no—
545C2×C4C4⋊C4×D1312yes—
546C2×C4C22×C4×D1311yescharacteristic
547C2×C4#8825111yescharacteristic
548C2×C4#8825111yescharacteristic
549C2×C4C2×Q8×D1311yescharacteristic
550C2×C4#8825711yescharacteristic
551C2×C4#8825711yescharacteristic
552C8—22no—
553C8—22no—
554C8—262no—
555C8—262no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D1311yescharacteristic
564C22#1171929711yescharacteristic
565C22#1171935311yescharacteristic
566C22#1171946711yescharacteristic
567C22#1171946711yescharacteristic
568C22#1171948011yescharacteristic
569C22#1171948311yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—262no—
576C4—262no—
577C4—262no—
578C4—262no—
579C4—1041no—
580C4—1041no—
581C4—1041no—
582C4—1041no—
583C4#1171925111yescharacteristic
584C4#1171925111yescharacteristic
585C4#1171932811yescharacteristic
586C4#1171932811yescharacteristic
587C2—42no—
588C2#1311745511yescharacteristic
589C2#1312342711yescharacteristic
590C2#1312343011yescharacteristic
591C2#1312589311yescharacteristic
592C2#1312590411yescharacteristic
593C2#1312613011yescharacteristic
594C2#1312613811yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1853C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1853D13
hypocenterC13#21990
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #13115607C2semidirect
5. #13115605C2semidirect
6. #13115598C2semidirect
7. #13115596C2semidirect
11. #13115001C2semidirect
12. #13114991C2semidirect
13. #13114968C2semidirect
14. #13114965C2semidirect
49. #11718381C4semidirect
115. #88055C2×C4semidirect
116. #88054C2×C4semidirect
117. #88054C2×C4semidirect
118. C8×Dic13C2×C4semidirect
493. C13#21990semidirect

Maximal subgroups

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

subgroupindex
C2×#117183812
#131174602
#131174432
#131156072
#131156052
#131155982
#131155962
#131150372
#131150332
#131150292
#131150012
#131149912
#131149682
#131149652
C13×#18532
#2199013