Finite Group Databasebeta

← #15890495

Subgroups — #15890495

Order 3328.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719538C211yesmaximal, characteristic
2C22×#88048C211yesmaximal, characteristic, complemented
3C2×#11718627C211yesmaximal, characteristic, complemented
4C2×#11718663C211yesmaximal, characteristic, complemented
5C2×#11718895C211yesmaximal, characteristic, complemented
6#13116032C214yesmaximal, complemented
7#13116025C214yesmaximal, complemented
8#13114557C211yesmaximal, characteristic, complemented
9C13×#2311C211yesmaximal, characteristic, complemented
10C22×C13×#41C2211yescharacteristic
11C22×#6127C2211yescharacteristic
12C2×C13×#117C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14#11719538—24no—
15#11719538—24no—
16C2×#88200—21no—
17C2×#88193—22no—
18C2×#88192—21no—
19C2×#88184C2212yes—
20C22×C4⋊Dic13C2211yescharacteristic
21C2×#88178C2212yes—
22C22×#6099C2211yescharacteristic
23C23×#1055C2211yescharacteristic
24C13×#391—21no—
25C13×#390—21no—
26C13×#386—22no—
27C13×#380C2214yes—
28#11718895—24no—
29C2×C22⋊#1055C2214yes—
30#11718663—24no—
31C2×#88050C2212yes—
32C2×#88050C2212yes—
33#11718627—24no—
34C2×#88049C2212yes—
35C2×#88049C2212yes—
36C2×#88048—24no—
37C2×#88048C2214yes—
38#11718121C2214yes—
39C2×C13×#41—22no—
40C2×C13×#41—22no—
41C2×C13×#41—22no—
42C2×C13×#41—22no—
43C2×C13×#41—22no—
44C2×C13×#41—22no—
45C22×Q8×C13D411yescharacteristic
46C22×D4×C13D412yes—
47C22×D4×C13D411yescharacteristic
48C23×C4×C13—21no—
49C23×C4×C13C2312yes—
50C23×C4×C13C2311yescharacteristic
51C22×C22⋊D13—21no—
52C22×C22⋊D13—22no—
53C23×Dic13—21no—
54C2×#6127—22no—
55C2×#6127—22no—
56C2×#6127—22no—
57C2×#6127—22no—
58C2×#6127—22no—
59C2×#6127D414yes—
60C22×D52C2311yescharacteristic
61C22×C4×D13—21no—
62C22×Dic26C2311yescharacteristic
63C13×#117—22no—
64C13×#117—22no—
65C2×C13×C4⋊C4—22no—
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4C2312yes—
69C2×C13×C4⋊C4C2312yes—
70C2×C13×C22⋊C4—21no—
71C2×C13×C22⋊C4—24no—
72C2×C42×C13—21no—
73C2×C22⋊Dic13—22no—
74#88200—28no—
75#88193—216no—
76#88192—28no—
77#88184—24no—
78#88184—24no—
79C2×#6103—22no—
80C2×C4⋊Dic13—21no—
81C2×C4⋊Dic13—22no—
82C2×C4⋊Dic13D414yes—
83C2×C4⋊Dic13C2312yes—
84#88178—24no—
85#88178—24no—
86C2×#6101—22no—
87C2×#6099—22no—
88C2×#6099—22no—
89C2×#6099—22no—
90C22×#1055—22no—
91C22×#1055—22no—
92C22×#1055D414yescomplemented
93C22×#1055C2312yes—
94C22×#1055C2312yes—
95C13×#95—24no—
96C22⋊#1055—28no—
97#88050—24no—
98#88050—24no—
99#88049—24no—
100#88049—24no—
101#88048—28no—
102#23489—131nomaximal, Sylow
103C24×C13C2×D412yes—
104C24×C13C2×D411yescharacteristic
105C23×D13—21no—
106C13×#41—44no—
107C13×#41—44no—
108C13×#41—44no—
109C13×#41—44no—
110C2×Q8×C13—22no—
111C2×Q8×C13—22no—
112C2×Q8×C13—22no—
113C2×D4×C13—24no—
114C2×D4×C13—24no—
115C2×D4×C13—24no—
116C2×D4×C13—22no—
117C2×D4×C13—22no—
118C2×D4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—22no—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—22no—
127C22×C4×C13—42no—
128C22×C4×C13—82no—
129C22×C4×C13C2×D412yes—
130C22×C4×C13C2×D412yes—
131C22×C4×C13C2×D412yes—
132C22×C4×C13C2×D412yes—
133C22×C4×C13C2×D411yescharacteristic
134C22×C4×C13#4112yes—
135C22×C4×C13#4112yes—
136C22×C4×C13#4112yes—
137C22×C4×C13C2411yescharacteristic
138C2×C22⋊D13—24no—
139C2×C22⋊D13—28no—
140C2×C22⋊D13—42no—
141C2×C22⋊D13—42no—
142C2×C22⋊D13—84no—
143C22×Dic13—21no—
144C22×Dic13—22no—
145C22×Dic13—24no—
146C22×Dic13—82no—
147#6127—28no—
148#6127—28no—
149#6127—44no—
150#6127—44no—
151C2×D52—21no—
152C2×D52—21no—
153C2×D52—21no—
154C2×D52—21no—
155C2×D52—21no—
156C2×D52C2×D412yes—
157C2×C4×D13—24no—
158C2×C4×D13—42no—
159C2×C4×D13—42no—
160C2×Dic26—21no—
161C2×Dic26—21no—
162C2×Dic26—21no—
163C2×Dic26—21no—
164C2×Dic26—21no—
165C2×Dic26C2×D412yes—
166C13×C4⋊C4—22no—
167C13×C4⋊C4—22no—
168C13×C4⋊C4—22no—
169C13×C4⋊C4—22no—
170C13×C4⋊C4—42no—
171C13×C22⋊C4—42no—
172C13×C22⋊C4—84no—
173C42×C13—42no—
174C22⋊Dic13—28no—
175#6103—28no—
176C4⋊Dic13—24no—
177C4⋊Dic13—22no—
178C4⋊Dic13C2×D414yes—
179#6101—28no—
180#6099—44no—
181C2×#1055—28no—
182C2×#1055—22no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055—22no—
186C2×#1055—22no—
187C2×#1055C2×D414yes—
188C2×#537—131no—
189C22×#106—131no—
190C2×#417—131no—
191C2×#416—131no—
192C2×#408—131no—
193#2311D1311yescharacteristic, complemented
194#2035—134no—
195#2034—134no—
196#1736—131no—
197C23×C13—22no—
198C23×C13—22no—
199C23×C13—22no—
200C23×C13—22no—
201C23×C13—22no—
202C23×C13—24no—
203C23×C13—24no—
204C23×C13#12012yes—
205C23×C13#12112yes—
206C23×C13#12312yes—
207C23×C13C22×D411yescharacteristic
208C22×D13—22no—
209C22×D13—41no—
210C22×D13—41no—
211C22×D13—42no—
212C22×D13—42no—
213Q8×C13—44no—
214D4×C13—48no—
215D4×C13—44no—
216C2×C4×C13—22no—
217C2×C4×C13—22no—
218C2×C4×C13—22no—
219C2×C4×C13—42no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—44no—
227C2×C4×C13—42no—
228C2×C4×C13—84no—
229C2×C4×C13#12012yes—
230C2×C4×C13#12114yes—
231C2×C4×C13#12114yes—
232C2×C4×C13#12112yes—
233C2×C4×C13#12314yes—
234C2×C4×C13#12312yes—
235C2×C4×C13#12414yes—
236C2×C4×C13C22×D411yescharacteristic
237C2×C4×C13C22×D411yescharacteristic
238C2×C4×C13C22×D411yescharacteristic
239C2×C4×C13C2×#4111yescharacteristic
240C2×C4×C13C2×#4111yescharacteristic
241C2×C4×C13C2×#4111yescharacteristic
242C22⋊D13—48no—
243C22⋊D13—88no—
244C2×Dic13—22no—
245C2×Dic13—24no—
246C2×Dic13—41no—
247C2×Dic13—41no—
248C2×Dic13—44no—
249C2×Dic13—84no—
250D52—22no—
251D52—22no—
252D52—41no—
253D52—41no—
254C4×D13—48no—
255Dic26—22no—
256Dic26—22no—
257Dic26—41no—
258Dic26—41no—
259#1055—24no—
260#1055—42no—
261C22×#41—131no—
262C22×#41C2×D1311yescharacteristic
263C22×M16—131no—
264C23×C8—131no—
265#537—264no—
266#537—264no—
267C2×C22⋊Q8—132no—
268C2×C22⋊Q8—261no—
269C2×#121—132no—
270C2×#121—261no—
271C2×#121—262no—
272C2×#117C2×D1311yescharacteristic
273C22×C4⋊C4—131no—
274C22×C4⋊C4C2×D1311yescharacteristic
275C2×#106—134no—
276C2×#106—264no—
277#417—264no—
278#416—264no—
279C2×#103—132no—
280C2×#103—132no—
281C2×#102—132no—
282C2×#102—132no—
283#408—264no—
284C2×C22⋊C8—134no—
285#391—21no—
286#390—21no—
287#386—22no—
288#380C2×D1314yes—
289#340—134no—
290C22×C13—24no—
291C22×C13—22no—
292C22×C13—22no—
293C22×C13—24no—
294C22×C13—44no—
295C22×C13—42no—
296C22×C13—42no—
297C22×C13—42no—
298C22×C13—44no—
299C22×C13#39414yes—
300C22×C13C2×#12011yescharacteristic
301C22×C13C2×#12111yescharacteristic
302C22×C13C2×#12311yescharacteristic
303C22×C13#57712yes—
304C22×C13#57712yes—
305C2×D13—42no—
306C2×D13—42no—
307C2×D13—42no—
308C2×D13—82no—
309C2×D13—82no—
310C4×C13—42no—
311C4×C13—82no—
312C4×C13—82no—
313C4×C13#39414yes—
314C4×C13C2×#12111yescharacteristic
315C4×C13C2×#12111yescharacteristic
316C4×C13C2×#12311yescharacteristic
317C4×C13C2×#12411yescharacteristic
318Dic13—42no—
319Dic13—44no—
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—134no—
327C2×#41—262no—
328C2×#41—262no—
329C2×#41—262no—
330C2×#41—262no—
331C2×#41—262no—
332C22×Q8—131no—
333C22×Q8C22⋊D1311yescharacteristic
334C22×D4—131no—
335C22×D4—261no—
336C22×D4—262no—
337C22×D4C22⋊D1312yes—
338C22×D4C22⋊D1311yescharacteristic
339C23×C4—21no—
340C23×C4—261no—
341C23×C4—261no—
342C23×C4C22×D1312yes—
343C23×C4C22×D1311yescharacteristic
344C2×M16—262no—
345C2×M16—262no—
346C2×M16—262no—
347C22×C8—134no—
348C22×C8—132no—
349C22×C8—132no—
350C22×C8—262no—
351C22×C8—262no—
352C22⋊Q8—264no—
353C22⋊Q8—264no—
354C22⋊Q8—268no—
355#121—264no—
356#121—264no—
357#121—268no—
358#121—2616no—
359#117—22no—
360#117—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—134no—
365C2×C4⋊C4—132no—
366C2×C4⋊C4—262no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4—262no—
369C2×C4⋊C4C22×D1312yes—
370C2×C4⋊C4C22×D1312yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—262no—
374C2×C22⋊C4—262no—
375C2×C42—21no—
376#106—268no—
377#103—264no—
378#103—264no—
379#102—264no—
380#102—264no—
381C22⋊C8—268no—
382#95—24no—
383C2×C13—24no—
384C2×C13—82no—
385C2×C13C2×#39411yescharacteristic
386C2×C13#300911yescharacteristic
387C2×C13#301011yescharacteristic
388C2×C13#306012yes—
389C2×C13#309912yes—
390D13—82no—
391C24—261no—
392C24C2×C22⋊D1312yes—
393C24C2×C22⋊D1311yescharacteristic
394#41—44no—
395#41—44no—
396#41—44no—
397#41—44no—
398#41—268no—
399#41—268no—
400#41—524no—
401#41—524no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—132no—
406C2×Q8—261no—
407C2×Q8—261no—
408C2×Q8—261no—
409C2×Q8—261no—
410C2×Q8—261no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—132no—
418C2×D4—264no—
419C2×D4—268no—
420C2×D4—261no—
421C2×D4—261no—
422C2×D4—261no—
423C2×D4—261no—
424C2×D4—261no—
425C2×D4—522no—
426C2×D4—522no—
427C2×D4—1044no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—262no—
439C22×C4—261no—
440C22×C4—264no—
441C22×C4—264no—
442C22×C4—522no—
443C22×C4—522no—
444C22×C4—1042no—
445C22×C4#612712yes—
446C22×C4#612712yes—
447C22×C4D4×D1312yes—
448C22×C4D4×D1312yes—
449C22×C4D4×D1312yes—
450C22×C4#612912yes—
451C22×C4C2×C22⋊D1312yes—
452C22×C4C2×C22⋊D1311yescharacteristic
453C22×C4C23×D1311yescharacteristic
454M16—524no—
455C2×C8—134no—
456C2×C8—268no—
457C2×C8—262no—
458C2×C8—262no—
459C2×C8—262no—
460C2×C8—262no—
461C2×C8—262no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—134no—
468C4⋊C4—268no—
469C4⋊C4—264no—
470C4⋊C4—262no—
471C22⋊C4—42no—
472C22⋊C4—84no—
473C22⋊C4—268no—
474C22⋊C4—268no—
475C42—42no—
476C13#2348911yescharacteristic, complemented, Sylow
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—262no—
485C23—521no—
486C23—521no—
487C23—522no—
488C23—522no—
489C23#8818312yes—
490C23#8819112yes—
491C23#8819312yes—
492C23C22×C22⋊D1311yescharacteristic
493Q8—44no—
494Q8—262no—
495Q8—262no—
496Q8—521no—
497Q8—521no—
498D4—48no—
499D4—44no—
500D4—262no—
501D4—262no—
502D4—528no—
503D4—521no—
504D4—521no—
505D4—1048no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—264no—
520C2×C4—262no—
521C2×C4—521no—
522C2×C4—521no—
523C2×C4—524no—
524C2×C4—528no—
525C2×C4—1044no—
526C2×C4#8813714yes—
527C2×C4#8813814yes—
528C2×C4#8813814yes—
529C2×C4#8813914yes—
530C2×C4#8818312yes—
531C2×C4#8819112yes—
532C2×C4#8819312yes—
533C2×C4C2×#612711yescharacteristic
534C2×C4C2×#612711yescharacteristic
535C2×C4C2×D4×D1311yescharacteristic
536C2×C4C2×D4×D1311yescharacteristic
537C2×C4C2×D4×D1311yescharacteristic
538C2×C4C2×#612911yescharacteristic
539C8—264no—
540C8—522no—
541C22—22no—
542C22—22no—
543C22—24no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—522no—
551C22—522no—
552C22—522no—
553C22—1042no—
554C22—1042no—
555C22#1171862014yes—
556C22C2×#8818311yescharacteristic
557C22C2×#8819111yescharacteristic
558C22C2×#8819311yescharacteristic
559C22#1171954112yes—
560C22#1171954212yes—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—524no—
565C4—522no—
566C4#1171862014yes—
567C4C2×#8813711yescharacteristic
568C4C2×#8813811yescharacteristic
569C4C2×#8813811yescharacteristic
570C4C2×#8813911yescharacteristic
571C2—24no—
572C2—82no—
573C2—1042no—
574C2C2×#1171862011yescharacteristic
575C2#1312411111yescharacteristic
576C2#1312412011yescharacteristic
577C2#1312449512yes—
578C2#1312482512yes—
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2311C2
socleC23×C13C22×D4
prime radicalC2×#11719538C2
hypercenter#2311D13
hypocenterC13#23489
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C22×#88048C2semidirect
3. C2×#11718627C2semidirect
4. C2×#11718663C2semidirect
5. C2×#11718895C2semidirect
6. #13116032C2semidirect
7. #13116025C2semidirect
8. #13114557C2semidirect
9. C13×#2311C2semidirect
92. C22×#1055D4semidirect
193. #2311D13semidirect
476. C13#23489semidirect

Maximal subgroups

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

subgroupindex
C2×#117195382
C22×#880482
C2×#117186272
C2×#117186632
C2×#117188952
#131160322
#131160252
#131145572
C13×#23112
#2348913