Finite Group Databasebeta

← #15866616

Subgroups — #15866616

Order 3328.

1526 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719521C211yesmaximal, characteristic
2C2×#11718627C211yesmaximal, characteristic
3C2×#11718886C211yesmaximal, characteristic, complemented
4C2×#11718774C211yesmaximal, characteristic
5C2×#11718762C211yesmaximal, characteristic, complemented
6#13117460C211yesmaximal, characteristic, complemented
7#13115654C212yesmaximal, complemented
8#13115649C212yesmaximal, complemented
9#13115644C212yesmaximal, complemented
10#13115636C212yesmaximal, complemented
11C22×C13×#103C211yesmaximal, characteristic, complemented
12C23×Q8×C13C2211yescharacteristic
13C22×#6127C2211yescharacteristic
14C23×C8×C13C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16#11719521—24no—
17#11719521C418yescomplemented
18C2×#88201C2212yes—
19C2×#88200—22no—
20C2×#88198C2212yes—
21C2×#88183—22no—
22C2×#88180C2211yescharacteristic
23C2×#88180C2211yescharacteristic
24C22×#6099C2211yescharacteristic
25C2×C13×#103—24no—
26C2×C13×#103C2212yes—
27C2×C13×#103C2212yes—
28#11718886—24no—
29C2×#88075C2212yes—
30C2×#88075C2212yes—
31#11718871—22no—
32#11718774—24no—
33C2×#88061C2211yescharacteristic
34C2×#88061C2211yescharacteristic
35C2×#88056C2211yescharacteristic
36C2×#88056C2211yescharacteristic
37#11718762—24no—
38C2×#88055C2212yes—
39C2×C8×Dic13C2212yes—
40#11718638C2212yes—
41#11718637C2212yes—
42#11718634—22no—
43C2×#88050C2212yes—
44#11718627—24no—
45C2×#88049C2212yes—
46C22×Q8×C13—21no—
47C22×Q8×C13—22no—
48C22×Q8×C13—22no—
49C22×Q8×C13—22no—
50C22×Q8×C13—22no—
51C22×Q8×C13—22no—
52C22×Q8×C13C2×C414yes—
53C22×Q8×C13C2312yes—
54C23×C4×C13—21no—
55C23×C4×C13—21no—
56C23×C4×C13C2311yescharacteristic
57C22×C22⋊D13—21no—
58C2×#6127—22no—
59C2×#6127—22no—
60C2×#6127—22no—
61C2×#6127—22no—
62C2×#6127—22no—
63C2×#6127C2×C414yes—
64C22×D52C2311yescharacteristic
65C22×C4×D13—21no—
66C22×Dic26C2311yescharacteristic
67C22×C8×C13—22no—
68C22×C8×C13—22no—
69C22×C8×C13D414yes—
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C2×C13×C4⋊C4—22no—
73C2×C13×C4⋊C4—22no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4C2312yes—
76C2×C22⋊Dic13—21no—
77C2×C22⋊Dic13D412yes—
78#88201—24no—
79#88201C2×C418yescomplemented
80#88200—216no—
81#88198—24no—
82#88198C2×C418yescomplemented
83#88183—216no—
84C2×#6103—24no—
85#88180—22no—
86#88180—22no—
87#88180C2×C414yes—
88#88180D414yes—
89C2×C4⋊Dic13D412yes—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×#6101—24no—
93C2×#6101—24no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13C2312yes—
96C2×#6099—22no—
97C2×#6099—22no—
98C2×#6099D414yes—
99C22×#1055C2312yes—
100C13×#103—24no—
101C13×#103—24no—
102#88075—24no—
103#88075—24no—
104#88071—24no—
105#88061—22no—
106#88061—22no—
107#88056—22no—
108#88056—22no—
109#88055—24no—
110C8×Dic13—24no—
111#88050—24no—
112#88049—24no—
113#21205—131nomaximal, Sylow
114C24×C13C2×D411yescharacteristic
115C23×D13—21no—
116C2×Q8×C13—24no—
117C2×Q8×C13—22no—
118C2×Q8×C13—28no—
119C2×Q8×C13—28no—
120C2×Q8×C13—22no—
121C2×Q8×C13—22no—
122C2×Q8×C13—22no—
123C2×Q8×C13—22no—
124C2×Q8×C13—44no—
125C2×Q8×C13—44no—
126C2×Q8×C13—42no—
127C2×Q8×C13—42no—
128C2×Q8×C13C22×C414yes—
129C22×C4×C13—24no—
130C22×C4×C13—22no—
131C22×C4×C13—22no—
132C22×C4×C13—42no—
133C22×C4×C13—42no—
134C22×C4×C13—42no—
135C22×C4×C13—82no—
136C22×C4×C13C22×C412yes—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D412yes—
139C22×C4×C13C2×D412yes—
140C22×C4×C13C2×D412yes—
141C22×C4×C13#4112yes—
142C22×C4×C13#4112yes—
143C22×C4×C13C2411yescharacteristic
144C2×C22⋊D13—24no—
145C2×C22⋊D13—42no—
146C2×C22⋊D13—42no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13—21no—
150C22×Dic13C2×D414yes—
151#6127—28no—
152#6127—28no—
153#6127—44no—
154#6127—44no—
155C2×D52—21no—
156C2×D52—21no—
157C2×D52—21no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52C22×C412yes—
161C2×C4×D13—24no—
162C2×C4×D13—42no—
163C2×C4×D13—42no—
164C2×Dic26—21no—
165C2×Dic26—21no—
166C2×Dic26—21no—
167C2×Dic26—21no—
168C2×Dic26—21no—
169C2×Dic26C22×C412yes—
170C2×C8×C13—28no—
171C2×C8×C13—21no—
172C2×C8×C13—21no—
173C2×C8×C13—21no—
174C2×C8×C13—21no—
175C2×C8×C13C2×D412yes—
176C2×C8×C13C2×D412yes—
177C13×C4⋊C4—22no—
178C13×C4⋊C4—22no—
179C13×C4⋊C4—42no—
180C22⋊Dic13—42no—
181C22⋊Dic13C22⋊C418yes—
182#6103—216no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13C22⋊C418yes—
188#6101—216no—
189#6101—84no—
190C4×Dic13—42no—
191C4×Dic13C22×C414yes—
192C4×Dic13C2×D414yes—
193#6099—28no—
194C2×#1055—22no—
195C2×#1055—22no—
196C2×#1055C2×D414yes—
197C2×#536—131no—
198C2×#417—131no—
199C2×#419—131no—
200C2×#416—131no—
201C22×#103D1311yescharacteristic, complemented
202C2×#405—131no—
203#2317—131no—
204#1960—132no—
205#1959—132no—
206#1958—132no—
207#1956—132no—
208C23×C13—22no—
209C23×C13—22no—
210C23×C13C2×C22⋊C412yes—
211C23×C13#12412yes—
212C23×C13#12712yes—
213C23×C13C22×D411yescharacteristic
214C22×D13—22no—
215C22×D13—41no—
216C22×D13—41no—
217C22×D13—42no—
218C22×D13—42no—
219Q8×C13—24no—
220Q8×C13—24no—
221Q8×C13—42no—
222Q8×C13—42no—
223Q8×C13—48no—
224C2×C4×C13—24no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—48no—
228C2×C4×C13—42no—
229C2×C4×C13—44no—
230C2×C4×C13—84no—
231C2×C4×C13C2×C22⋊C412yes—
232C2×C4×C13C4×D414yes—
233C2×C4×C13C4×D414yes—
234C2×C4×C13#12114yes—
235C2×C4×C13#12114yes—
236C2×C4×C13#12412yes—
237C2×C4×C13#12712yes—
238C2×C4×C13C23×C411yescharacteristic
239C2×C4×C13C22×D411yescharacteristic
240C2×C4×C13C22×D411yescharacteristic
241C2×C4×C13C22×D411yescharacteristic
242C2×C4×C13C2×#4111yescharacteristic
243C2×C4×C13C2×#4111yescharacteristic
244C22⋊D13—48no—
245C2×Dic13—24no—
246C2×Dic13—24no—
247C2×Dic13—22no—
248C2×Dic13—41no—
249C2×Dic13—41no—
250C2×Dic13—41no—
251C2×Dic13—41no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13C2×C22⋊C418yes—
255D52—22no—
256D52—22no—
257D52—41no—
258D52—41no—
259C4×D13—48no—
260Dic26—22no—
261Dic26—22no—
262Dic26—41no—
263Dic26—41no—
264C8×C13—22no—
265C8×C13—22no—
266#1055—24no—
267C22×#41—131no—
268C23×Q8C2×D1311yescharacteristic
269C22×M16—131no—
270C23×C8C2×D1311yescharacteristic
271#536—138no—
272#536—264no—
273C2×C4⋊Q8—132no—
274C2×#124—132no—
275C2×#123—262no—
276C2×C22⋊Q8—262no—
277C2×#117—131no—
278C2×#117—131no—
279C22×C4⋊C4C2×D1311yescharacteristic
280#419—264no—
281#417—264no—
282#416—264no—
283C2×#103—24no—
284C2×#103—132no—
285C2×#103—132no—
286C2×#103—131no—
287C2×#103—131no—
288C2×#103—132no—
289C2×#103C2×D1312yes—
290C2×#103C2×D1312yes—
291C2×#102—131no—
292C2×#102—131no—
293C2×#102—132no—
294#405—264no—
295C2×#97—132no—
296C2×C4×C8—132no—
297#389—262no—
298#385—262no—
299#384—132no—
300#382—132no—
301C22×C13—24no—
302C22×C13—24no—
303C22×C13—22no—
304C22×C13—22no—
305C22×C13#39014yes—
306C22×C13C22×C22⋊C411yescharacteristic
307C22×C13C2×#12411yescharacteristic
308C22×C13C2×#12711yescharacteristic
309C22×C13#57712yes—
310C22×C13#57812yes—
311C2×D13—44no—
312C2×D13—42no—
313C2×D13—82no—
314C2×D13—81no—
315C2×D13—81no—
316C4×C13—44no—
317C4×C13—82no—
318C4×C13#39014yes—
319C4×C13C2×C4×D411yescharacteristic
320C4×C13C2×C4×D411yescharacteristic
321C4×C13C2×#12111yescharacteristic
322C4×C13C2×#12111yescharacteristic
323Dic13—28no—
324Dic13—42no—
325Dic13—81no—
326Dic13—81no—
327C2×#41—134no—
328C2×#41—262no—
329C2×#41—262no—
330C2×#41—262no—
331C2×#41—262no—
332C2×#41—262no—
333C22×Q8—21no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—131no—
340C22×Q8C4×D1314yes—
341C22×Q8C22×D1312yes—
342C22×D4—131no—
343C22×D4—261no—
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—261no—
347C23×C4C22×D1311yescharacteristic
348C2×M16—134no—
349C2×M16—262no—
350C2×M16—262no—
351C22×C8—22no—
352C22×C8—22no—
353C22×C8—132no—
354C22×C8C22⋊D1314yes—
355C22×C8C22×D1311yescharacteristic
356C22×C8C22×D1311yescharacteristic
357C4⋊Q8—138no—
358C4⋊Q8—264no—
359#124—138no—
360#124—264no—
361#123—2616no—
362C22⋊Q8—2616no—
363#117—134no—
364#117—134no—
365#117—262no—
366#117—262no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—132no—
371C2×C4⋊C4—131no—
372C2×C4⋊C4—131no—
373C2×C4⋊C4—264no—
374C2×C4⋊C4—264no—
375C2×C4⋊C4C22×D1312yes—
376C2×C22⋊C4—132no—
377C2×C22⋊C4—264no—
378C2×C22⋊C4—261no—
379C2×C42—132no—
380C2×C42—261no—
381#103—24no—
382#103—24no—
383#103—264no—
384#103—264no—
385#103—264no—
386#103—262no—
387#103—262no—
388#102—264no—
389#102—262no—
390#102—262no—
391#97—264no—
392C4×C8—264no—
393#95—264no—
394C2×C13—24no—
395C2×C13C2×#39011yescharacteristic
396C2×C13#289812yes—
397C2×C13#313712yes—
398C2×C13#315411yescharacteristic
399C2×C13#315611yescharacteristic
400D13—82no—
401C24—261no—
402C24D4×D1311yescharacteristic
403#41—268no—
404#41—268no—
405#41—524no—
406#41—524no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—132no—
420C2×Q8—261no—
421C2×Q8—261no—
422C2×Q8—261no—
423C2×Q8—261no—
424C2×Q8—261no—
425C2×Q8C2×C4×D1314yes—
426C2×D4—132no—
427C2×D4—264no—
428C2×D4—261no—
429C2×D4—261no—
430C2×D4—261no—
431C2×D4—261no—
432C2×D4—261no—
433C2×D4—522no—
434C2×D4—522no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—134no—
443C22×C4—261no—
444C22×C4—264no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—522no—
448C22×C4—522no—
449C22×C4C2×C4×D1312yes—
450C22×C4#612712yes—
451C22×C4D4×D1311yescharacteristic
452C22×C4D4×D1312yes—
453C22×C4D4×D1312yes—
454C22×C4#612912yes—
455C22×C4C2×C22⋊D1312yes—
456C22×C4C23×D1311yescharacteristic
457M16—268no—
458C2×C8—28no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—134no—
464C2×C8—262no—
465C2×C8—262no—
466C2×C8C2×C22⋊D1312yes—
467C2×C8C2×C22⋊D1312yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—138no—
472C4⋊C4—2616no—
473C4⋊C4—261no—
474C4⋊C4—261no—
475C4⋊C4—261no—
476C4⋊C4—261no—
477C4⋊C4—1044no—
478C22⋊C4—138no—
479C22⋊C4—2616no—
480C22⋊C4—522no—
481C42—134no—
482C42—134no—
483C42—522no—
484C13#2120511yescharacteristic, complemented, Sylow
485C23—22no—
486C23—22no—
487C23—262no—
488C23—521no—
489C23—522no—
490C23—522no—
491C23—521no—
492C23C22⋊C4×D1312yes—
493C23#8813912yes—
494C23#8819412yes—
495C23C2×D4×D1311yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—42no—
499Q8—42no—
500Q8—48no—
501Q8—262no—
502Q8—262no—
503Q8—521no—
504Q8—521no—
505D4—262no—
506D4—262no—
507D4—528no—
508D4—521no—
509D4—521no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—138no—
518C2×C4—264no—
519C2×C4—264no—
520C2×C4—262no—
521C2×C4—521no—
522C2×C4—521no—
523C2×C4—528no—
524C2×C4—522no—
525C2×C4—522no—
526C2×C4—521no—
527C2×C4—521no—
528C2×C4C22⋊C4×D1312yes—
529C2×C4#8813514yes—
530C2×C4#8813814yes—
531C2×C4#8813912yes—
532C2×C4C4×C22⋊D1314yes—
533C2×C4#8819314yes—
534C2×C4#8819412yes—
535C2×C4C22×C4×D1311yescharacteristic
536C2×C4C2×#612711yescharacteristic
537C2×C4C2×D4×D1311yescharacteristic
538C2×C4C2×D4×D1311yescharacteristic
539C2×C4C2×#612911yescharacteristic
540C2×C4C22×C22⋊D1311yescharacteristic
541C8—22no—
542C8—22no—
543C8—264no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—524no—
549C22—522no—
550C22—1042no—
551C22—1041no—
552C22—1041no—
553C22#1171861814yes—
554C22C2×C22⋊C4×D1311yescharacteristic
555C22C2×#8813911yescharacteristic
556C22#1171946712yes—
557C22#1171947112yes—
558C22C2×#8819411yescharacteristic
559C4—44no—
560C4—82no—
561C4—268no—
562C4—522no—
563C4—1041no—
564C4—1041no—
565C4#1171861814yes—
566C4C2×#8813511yescharacteristic
567C4C2×#8813811yescharacteristic
568C4C2×C4×C22⋊D1311yescharacteristic
569C4C2×#8819311yescharacteristic
570C2—24no—
571C2—1042no—
572C2C2×#1171861811yescharacteristic
573C2#1312336412yes—
574C2#1312519912yes—
575C2#1312530611yescharacteristic
576C2#1312532511yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C13×#103C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC22×#103D13
hypocenterC13#21205
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C2×#11718886C2semidirect
5. C2×#11718762C2semidirect
6. #13117460C2semidirect
7. #13115654C2semidirect
8. #13115649C2semidirect
9. #13115644C2semidirect
10. #13115636C2semidirect
11. C22×C13×#103C2semidirect
17. #11719521C4semidirect
79. #88201C2×C4semidirect
82. #88198C2×C4semidirect
201. C22×#103D13semidirect
484. C13#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#117195212
C2×#117186272
C2×#117188862
C2×#117187742
C2×#117187622
#131174602
#131156542
#131156492
#131156442
#131156362
C22×C13×#1032
#2120513