Finite Group Databasebeta

← #15875006

Subgroups — #15875006

Order 3328.

796 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
1#125×Dic13C211yesmaximal, characteristic, complemented
2#13117301C211yesmaximal, characteristic
3#13115794C211yesmaximal, characteristic, complemented
4#13115606C211yesmaximal, characteristic, complemented
5#13115570C211yesmaximal, characteristic, complemented
6#13115565C211yesmaximal, characteristic, complemented
7#13115521C211yesmaximal, characteristic
8#13115521C211yesmaximal, characteristic
9#13115465C211yesmaximal, characteristic
10#13115458C211yesmaximal, characteristic
11#13115430C211yesmaximal, characteristic, complemented
12#13115423C211yesmaximal, characteristic, complemented
13#13114846C211yesmaximal, characteristic
14C4⋊C8×Dic13C211yesmaximal, characteristic, complemented
15C2×C13×#478C211yesmaximal, characteristic
16C2×C13×#125C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×#88180—21no—
19C13×#478C418yescomplemented
20C2×C13×#107C2211yescharacteristic
21C2×C13×#106C2211yescharacteristic
22C2×C13×C4⋊C8C2211yescharacteristic
23C2×C13×#102C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25#11718882C2211yescharacteristic
26#11718882C2211yescharacteristic
27C2×#88072C2211yescharacteristic
28C2×#88072C2211yescharacteristic
29C4×C22⋊Dic13—21no—
30C2×#88058C2211yescharacteristic
31C2×#88057C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33C2×C8×Dic13C2211yescharacteristic
34#11718646C2211yescharacteristic
35#11718646C2211yescharacteristic
36#11718646C2211yescharacteristic
37#11718646C2211yescharacteristic
38#11718642—21no—
39#11718642—21no—
40C4⋊C4×Dic13—21no—
41C4⋊C4×Dic13—21no—
42C4⋊C4×Dic13C2211yescharacteristic
43C4⋊C4×Dic13C2211yescharacteristic
44#11718612—21no—
45#11718612—21no—
46#11718597C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48C42×Dic13C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50#11718143C2211yescharacteristic
51#11718143C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C13×#125—22no—
65C13×#125C2×C414yes—
66C4×D4×C13—22no—
67C4×D4×C13C2×C414yes—
68C2×C13×C4⋊C4—21no—
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—
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13—21no—
79#88180—44no—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—21no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4⋊Dic13C2311yescharacteristic
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13D411yescharacteristic
92C2×C4×Dic13D411yescharacteristic
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13D411yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C13×#107C2×C414yes—
100C13×#106C2×C414yes—
101C13×C4⋊C8C2×C414yes—
102C13×#102C2×C414yes—
103C13×#102C2×C414yes—
104#88072—41no—
105#88072—41no—
106#88071—21no—
107#88071—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88071—21no—
115#88058—41no—
116#88057—41no—
117C8×Dic13—41no—
118C8×Dic13—41no—
119C4⋊#1055—22no—
120#22114—131nomaximal, Sylow
121C24×C13—21no—
122C2×D4×C13—21no—
123C2×D4×C13—21no—
124C2×D4×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13C22×C412yes—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—24no—
132C22×C4×C13—42no—
133C22×C4×C13—42no—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2411yescharacteristic
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13C2×D411yescharacteristic
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C22×Dic13C2×D411yescharacteristic
145C2×C8×C13—21no—
146C2×C8×C13—21no—
147C2×C8×C13—21no—
148C2×C8×C13—21no—
149C2×C8×C13C22×C412yes—
150C2×C8×C13C22×C412yes—
151C13×C4⋊C4—24no—
152C13×C4⋊C4—24no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4C22×C412yes—
157C13×C4⋊C4C22×C412yes—
158C13×C4⋊C4C22×C412yes—
159C13×C22⋊C4—24no—
160C42×C13C22×C412yes—
161C42×C13#4112yes—
162C22⋊Dic13—42no—
163C22⋊Dic13—42no—
164C22⋊Dic13—42no—
165C22⋊Dic13—42no—
166C4⋊Dic13—21no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—42no—
175C4⋊Dic13—42no—
176C4⋊Dic13—42no—
177C4⋊Dic13—42no—
178C4×Dic13—21no—
179C4×Dic13—21no—
180C4×Dic13—21no—
181C4×Dic13—21no—
182C4×Dic13—22no—
183C4×Dic13—22no—
184C4×Dic13—22no—
185C4×Dic13—41no—
186C4×Dic13—42no—
187C4×Dic13—42no—
188C4×Dic13—42no—
189C2×#1055—21no—
190C2×#1055—21no—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#478D1311yescharacteristic
196C4×#125—131no—
197#2293—131no—
198#1989—131no—
199#1949—131no—
200#1942—131no—
201#1941—131no—
202#1934—131no—
203#1934—131no—
204#1926—131no—
205#1925—131no—
206#1921—131no—
207#1920—131no—
208#1804—131no—
209C4×C4⋊C8—131no—
210C23×C13—22no—
211C23×C13—41no—
212C23×C13—41no—
213C23×C13—41no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13—41no—
217C23×C13C22×D411yescharacteristic
218D4×C13—22no—
219D4×C13—22no—
220D4×C13—41no—
221D4×C13—41no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—22no—
227C2×C4×C13—22no—
228C2×C4×C13—22no—
229C2×C4×C13—48no—
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×C13C4×D412yes—
237C2×C4×C13C4×D412yes—
238C2×C4×C13C23×C411yescharacteristic
239C2×C4×C13C2×#4111yescharacteristic
240C2×C4×C13H(5,𝔽2)11yescharacteristic
241C2×C4×C132−1+411yescharacteristic
242C2×C4×C132−1+411yescharacteristic
243C2×C4×C132−1+411yescharacteristic
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—41no—
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#13512yes—
266C2×Dic13#13512yes—
267C2×Dic13AGL(1,ℤ/8)12yes—
268C8×C13—22no—
269C8×C13—22no—
270#1055—41no—
271#1055—41no—
272C2×#125C2×D1311yescharacteristic
273C2×C4×D4C2×D1311yescharacteristic
274C2×#117—261no—
275#478Dic1318yescomplemented
276C2×#107—131no—
277C2×#107C2×D1311yescharacteristic
278C2×#106—131no—
279C2×#106C2×D1311yescharacteristic
280C2×C4⋊C8—131no—
281C2×C4⋊C8C2×D1311yescharacteristic
282C2×#102—131no—
283C2×#102—131no—
284C2×#102C2×D1311yescharacteristic
285C2×#102C2×D1311yescharacteristic
286C2×C4×C8—131no—
287C2×C4×C8—131no—
288#390—131no—
289#390—131no—
290#389—131no—
291#389—131no—
292#389—131no—
293#389—131no—
294#388—261no—
295#388—261no—
296#387—261no—
297#387—261no—
298#382—131no—
299C4×C4⋊C4—131no—
300C4×C4⋊C4—131no—
301C4×C4⋊C4—131no—
302C4×C4⋊C4—261no—
303C4×C4⋊C4—261no—
304C4×C22⋊C4—261no—
305C43—131no—
306#340—131no—
307#340—131no—
308#340—131no—
309#340—131no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#336—131no—
315#336—131no—
316C22×C13—42no—
317C22×C13—42no—
318C22×C13—42no—
319C22×C13—81no—
320C22×C13—81no—
321C22×C13—81no—
322C22×C13—81no—
323C22×C13C2×C4×D411yescharacteristic
324C22×C13#53611yescharacteristic
325C22×C13#53711yescharacteristic
326C22×C13C2×#13511yescharacteristic
327C22×C13C2×#13511yescharacteristic
328C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C13#57511yescharacteristic
330C4×C13—22no—
331C4×C13—22no—
332C4×C13—42no—
333C4×C13—42no—
334C4×C13—42no—
335C4×C13#51911yescharacteristic
336C4×C13#52011yescharacteristic
337C4×C13#54111yescharacteristic
338C4×C13#54211yescharacteristic
339Dic13—22no—
340Dic13—22no—
341Dic13—22no—
342Dic13—41no—
343Dic13—81no—
344Dic13—81no—
345Dic13—81no—
346Dic13—81no—
347C22×D4C22×D1311yescharacteristic
348C23×C4—21no—
349C22×C8—131no—
350C22×C8—131no—
351C22×C8C22×D1311yescharacteristic
352C22×C8C22×D1311yescharacteristic
353#125—22no—
354#125C2×Dic1314yes—
355C4×D4—22no—
356C4×D4C2×Dic1314yes—
357#117—524no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—261no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4C22×D1311yescharacteristic
369C2×C4⋊C4C22×D1311yescharacteristic
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—261no—
373C2×C22⋊C4—261no—
374C2×C22⋊C4—261no—
375C2×C22⋊C4—261no—
376C2×C42—131no—
377C2×C42—131no—
378C2×C42—131no—
379C2×C42—131no—
380C2×C42—131no—
381C2×C42—131no—
382C2×C42—261no—
383C2×C42—261no—
384C2×C42—261no—
385C2×C42C22×D1311yescharacteristic
386#107—521no—
387#107C2×Dic1314yes—
388#106—521no—
389#106C2×Dic1314yes—
390C4⋊C8—262no—
391C4⋊C8C2×Dic1314yes—
392#102—521no—
393#102—521no—
394#102C2×Dic1314yes—
395#102C2×Dic1314yes—
396C4×C8—521no—
397C4×C8—521no—
398#95—261no—
399#95—261no—
400#95—261no—
401#95—261no—
402#95—261no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407C2×C13—82no—
408C2×C13#232311yescharacteristic
409C2×C13C4×#13511yescharacteristic
410C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
411C2×C13#322811yescharacteristic
412C2×C13#323211yescharacteristic
413C2×C13#324711yescharacteristic
414C2×C13#325011yescharacteristic
415C24—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C22×Dic1312yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—131no—
429C22×C4—131no—
430C22×C4—131no—
431C22×C4—131no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4D4×D1311yescharacteristic
437C22×C4D4×D1311yescharacteristic
438C22×C4C23×D1311yescharacteristic
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—261no—
444C2×C8—261no—
445C2×C8—261no—
446C2×C8—261no—
447C2×C8—261no—
448C2×C8—261no—
449C2×C8C22×Dic1312yes—
450C2×C8C22×Dic1312yes—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—261no—
457C4⋊C4—261no—
458C4⋊C4—261no—
459C4⋊C4—261no—
460C4⋊C4—261no—
461C4⋊C4—261no—
462C4⋊C4—261no—
463C4⋊C4—261no—
464C4⋊C4—522no—
465C4⋊C4—522no—
466C4⋊C4—522no—
467C4⋊C4—522no—
468C4⋊C4C22×Dic1312yes—
469C4⋊C4C22×Dic1312yes—
470C4⋊C4C22×Dic1312yes—
471C22⋊C4—24no—
472C22⋊C4—522no—
473C22⋊C4—522no—
474C22⋊C4—522no—
475C22⋊C4—522no—
476C42—262no—
477C42—262no—
478C42—262no—
479C42—261no—
480C42—261no—
481C42—261no—
482C42—261no—
483C42—521no—
484C42—522no—
485C42—522no—
486C42—522no—
487C42#612912yes—
488C42C22×Dic1312yes—
489C13#2211411yescharacteristic, complemented, Sylow
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D1311yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—132no—
517C2×C4—132no—
518C2×C4—132no—
519C2×C4—261no—
520C2×C4—261no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—261no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—261no—
528C2×C4—521no—
529C2×C4—521no—
530C2×C4—521no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4D4×Dic1312yes—
541C2×C4D4×Dic1312yes—
542C2×C4C2×#612911yescharacteristic
543C2×C4#8825411yescharacteristic
544C2×C4#8825411yescharacteristic
545C2×C4#8825611yescharacteristic
546C2×C4#8825711yescharacteristic
547C2×C4C23×Dic1311yescharacteristic
548C8—22no—
549C8—22no—
550C8—521no—
551C8—521no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#1171933311yescharacteristic
560C22#1171933411yescharacteristic
561C22D13×#13511yescharacteristic
562C22D13×#13511yescharacteristic
563C22D13×AGL(1,ℤ/8)11yescharacteristic
564C22#1171947811yescharacteristic
565C22C2×D4×Dic1311yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—262no—
572C4—262no—
573C4—262no—
574C4—521no—
575C4—1041no—
576C4—1041no—
577C4—1041no—
578C4—1041no—
579C4#1171933011yescharacteristic
580C4#1171934811yescharacteristic
581C4#1171951711yescharacteristic
582C4#1171953211yescharacteristic
583C2—82no—
584C2#1311752011yescharacteristic
585C2#135×Dic1311yescharacteristic
586C2AGL(1,ℤ/8)×Dic1311yescharacteristic
587C2#1312597111yescharacteristic
588C2#1312600211yescharacteristic
589C2#1312616111yescharacteristic
590C2#1312618711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#478C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#478D13
hypocenterC13#22114
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic13C2semidirect
3. #13115794C2semidirect
4. #13115606C2semidirect
5. #13115570C2semidirect
6. #13115565C2semidirect
11. #13115430C2semidirect
12. #13115423C2semidirect
14. C4⋊C8×Dic13C2semidirect
19. C13×#478C4semidirect
275. #478Dic13semidirect
489. C13#22114semidirect

Maximal subgroups

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

subgroupindex
#125×Dic132
#131173012
#131157942
#131156062
#131155702
#131155652
#131155212
#131154652
#131154582
#131154302
#131154232
#131148462
C4⋊C8×Dic132
C2×C13×#4782
#2211413