Finite Group Databasebeta

← #15897237

Subgroups — #15897237

Order 3328.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719175C211yesmaximal, characteristic, complemented
2C2×#11718669C211yesmaximal, characteristic, complemented
3C2×#11718680C211yesmaximal, characteristic, complemented
4#13116032C212yesmaximal, complemented
5#13115839C212yesmaximal
6#13115784C211yesmaximal, characteristic, complemented
7#13115759C211yesmaximal, characteristic, complemented
8#13115194C212yesmaximal, complemented
9#13115103C212yesmaximal, complemented
10#13114816C211yesmaximal, characteristic, complemented
11C4×C13×#121C211yesmaximal, characteristic
12C2×C13×#121C2211yescharacteristic
13C2×C4×D4×C13—21no—
14C2×C4×D4×C13C2211yescharacteristic
15C22×C42×C13C2211yescharacteristic
16C22×#6105C2212yes—
17C2×#88180—21no—
18C2×#88178—21no—
19C2×#88178C2212yes—
20#11719175—24no—
21#11719175—24no—
22C2×#88124—21no—
23C2×#88123C2211yescharacteristic
24C2×#88123C2211yescharacteristic
25C2×C4×Dic26—21no—
26C2×C4×Dic26C2212yes—
27C13×#390C2211yescharacteristic
28C13×#389C2211yescharacteristic
29C13×#388—21no—
30C13×#386—21no—
31C13×#380C2211yescharacteristic
32C4×C13×C4⋊C4C2211yescharacteristic
33C4×C13×C22⋊C4—21no—
34C2×#88072C2212yes—
35C2×C22⋊#1055C2212yes—
36#11718680—24no—
37#11718669—24no—
38C2×#88050C2212yes—
39C2×#88050C2212yes—
40C2×#88050C2212yes—
41C2×#88048C2212yes—
42C2×C4⋊#1055C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44#11718121C2212yes—
45#11718121C2212yes—
46#11718116C2212yes—
47C22×D4×C13—21no—
48C22×D4×C13C2311yescharacteristic
49C23×C4×C13—21no—
50C23×C4×C13D411yescharacteristic
51C23×C4×C13D411yescharacteristic
52C23×C4×C13C2311yescharacteristic
53C22×Dic26—21no—
54C22×Dic26C2312yes—
55C13×#121—22no—
56C13×#121—22no—
57C4×D4×C13—22no—
58C4×D4×C13—22no—
59C4×D4×C13—44no—
60C2×C13×C4⋊C4—21no—
61C2×C13×C4⋊C4C2311yescharacteristic
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C22⋊C4—21no—
65C2×C13×C22⋊C4—21no—
66C2×C13×C22⋊C4—21no—
67C2×C13×C22⋊C4—21no—
68C2×C13×C22⋊C4—21no—
69C2×C42×C13—22no—
70C2×C42×C13—22no—
71C2×C42×C13—21no—
72C2×C42×C13D411yescharacteristic
73C2×C42×C13D411yescharacteristic
74C2×C42×C13C2311yescharacteristic
75C2×C42×C13C2311yescharacteristic
76C2×C22⋊Dic13—22no—
77C2×#6105—24no—
78C2×#6105—24no—
79C2×#6105—42no—
80#88180—28no—
81C2×C4⋊Dic13—21no—
82C2×C4⋊Dic13—21no—
83C2×C4⋊Dic13—21no—
84C2×C4⋊Dic13C2312yes—
85#88178—28no—
86#88178—24no—
87#88178—24no—
88C2×#6101—22no—
89C2×#6101—22no—
90C2×C4×Dic13—22no—
91C22×#1055C2312yes—
92C22×#1055C2312yes—
93#88124—28no—
94#88123—22no—
95#88123—22no—
96#88123D414yescomplemented
97#88123D414yescomplemented
98C4×Dic26—28no—
99C4×Dic26—24no—
100C4×Dic26—24no—
101C13×#95—21no—
102C13×#95—21no—
103C13×#95—21no—
104#88072—24no—
105C22⋊#1055—42no—
106#88050—24no—
107#88050—24no—
108#88050—42no—
109#88048—24no—
110C4⋊#1055—22no—
111C4⋊#1055—22no—
112#24151—131nomaximal, Sylow
113C24×C13—21no—
114C24×C13C2×D411yescharacteristic
115C2×D4×C13—24no—
116C2×D4×C13—21no—
117C2×D4×C13—21no—
118C2×D4×C13—21no—
119C2×D4×C13—21no—
120C2×D4×C13—21no—
121C2×D4×C13—42no—
122C2×D4×C13—42no—
123C2×D4×C13#4112yes—
124C22×C4×C13—22no—
125C22×C4×C13—22no—
126C22×C4×C13—21no—
127C22×C4×C13—22no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—41no—
137C22×C4×C13—41no—
138C22×C4×C13—42no—
139C22×C4×C13—42no—
140C22×C4×C13—42no—
141C22×C4×C13SD1614yescomplemented
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D411yescharacteristic
146C22×C4×C13C2×D411yescharacteristic
147C22×C4×C13C2411yescharacteristic
148C22×Dic13—22no—
149C22×Dic13—22no—
150C2×Dic26—22no—
151C2×Dic26—22no—
152C2×Dic26—22no—
153C2×Dic26—22no—
154C2×Dic26—24no—
155C2×Dic26—22no—
156C2×Dic26—42no—
157C2×Dic26—42no—
158C2×Dic26C2×D414yes—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4—21no—
164C13×C4⋊C4—42no—
165C13×C4⋊C4#4112yes—
166C13×C22⋊C4—24no—
167C13×C22⋊C4—42no—
168C13×C22⋊C4—42no—
169C13×C22⋊C4—42no—
170C13×C22⋊C4—42no—
171C42×C13—22no—
172C42×C13—22no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—42no—
176C42×C13C2×D412yes—
177C42×C13C2×D412yes—
178C22⋊Dic13—28no—
179#6105—48no—
180C4⋊Dic13—24no—
181C4⋊Dic13—24no—
182C4⋊Dic13—24no—
183C4⋊Dic13—22no—
184C4⋊Dic13C2×D414yes—
185#6101—28no—
186#6101—28no—
187C4×Dic13—28no—
188C2×#1055—22no—
189C2×#1055—22no—
190C2×#1055—22no—
191C2×#1055—22no—
192C2×#1055—22no—
193C2×#1055—22no—
194C2×#533—131no—
195C2×#476—131no—
196C2×#461—131no—
197C4×#121D1311yescharacteristic
198#2035—132no—
199#2000—132no—
200#1987—131no—
201#1981—131no—
202#1880—132no—
203#1865—132no—
204#1797—131no—
205C23×C13—21no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13—21no—
209C23×C13—21no—
210C23×C13—21no—
211C23×C13—22no—
212C23×C13—41no—
213C23×C13—41no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13—41no—
217C23×C13—41no—
218C23×C13C2×SD1612yes—
219C23×C13C22×D411yescharacteristic
220D4×C13—22no—
221D4×C13—22no—
222D4×C13—48no—
223D4×C13—41no—
224D4×C13—41no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—28no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—44no—
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×C13—41no—
247C2×C4×C13—41no—
248C2×C4×C13—41no—
249C2×C4×C13—41no—
250C2×C4×C13—84no—
251C2×C4×C13#12012yes—
252C2×C4×C13#12012yes—
253C2×C4×C13C2×SD1612yes—
254C2×C4×C13C2×SD1612yes—
255C2×C4×C13#13712yes—
256C2×C4×C13C22×D411yescharacteristic
257C2×C4×C13C22×D411yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×C4×C13C2×#4111yescharacteristic
260C2×C4×C13H(5,𝔽2)11yescharacteristic
261C2×C4×C132−1+411yescharacteristic
262C2×Dic13—24no—
263C2×Dic13—24no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268Dic26—24no—
269Dic26—24no—
270Dic26—48no—
271Dic26—42no—
272Dic26—42no—
273#1055—42no—
274#1055—42no—
275C22×SD16—132no—
276#533—264no—
277#533—264no—
278C2×C4⋊Q8—131no—
279C2×C4⋊Q8—131no—
280C2×#125—261no—
281C2×C22⋊Q8—132no—
282C2×C22⋊Q8—261no—
283C2×#121C2×D1311yescharacteristic
284C2×C4×Q8—132no—
285C2×C4×Q8—261no—
286C2×C4×D4—21no—
287C2×C4×D4C2×D1311yescharacteristic
288C2×#117—261no—
289C22×C42C2×D1311yescharacteristic
290#476—264no—
291#461—264no—
292C2×#106—132no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×#103—132no—
296C2×#103—132no—
297C2×#103—132no—
298C2×#102—132no—
299C2×C22⋊C8—132no—
300#390C2×D1311yescharacteristic
301#389C2×D1311yescharacteristic
302#388—21no—
303#386—21no—
304#380C2×D1311yescharacteristic
305C4×C4⋊C4C2×D1311yescharacteristic
306C4×C22⋊C4—21no—
307#340—132no—
308#340—132no—
309#336—132no—
310C22×C13—22no—
311C22×C13—22no—
312C22×C13—22no—
313C22×C13—41no—
314C22×C13—41no—
315C22×C13—41no—
316C22×C13—41no—
317C22×C13—42no—
318C22×C13—42no—
319C22×C13—42no—
320C22×C13—81no—
321C22×C13—81no—
322C22×C13—81no—
323C22×C13—81no—
324C22×C13C2×#12011yescharacteristic
325C22×C13#52511yescharacteristic
326C22×C13#53711yescharacteristic
327C22×C13C22×SD1611yescharacteristic
328C22×C13C2×#13711yescharacteristic
329C22×C13#57511yescharacteristic
330C22×C13#57511yescharacteristic
331C4×C13—22no—
332C4×C13—22no—
333C4×C13—42no—
334C4×C13—41no—
335C4×C13—41no—
336C4×C13—81no—
337C4×C13—81no—
338C4×C13#44814yes—
339C4×C13#54611yescharacteristic
340C4×C13#54711yescharacteristic
341C4×C13#54811yescharacteristic
342C4×C13#55011yescharacteristic
343Dic13—44no—
344Dic13—44no—
345C22×Q8—132no—
346C22×Q8—261no—
347C22×D4—21no—
348C22×D4C22×D1311yescharacteristic
349C23×C4—21no—
350C23×C4C22⋊D1311yescharacteristic
351C23×C4C22⋊D1311yescharacteristic
352C23×C4C22×D1311yescharacteristic
353C2×SD16—264no—
354C2×SD16—264no—
355C2×SD16—522no—
356C22×C8—132no—
357C22×C8—132no—
358C4⋊Q8—134no—
359C4⋊Q8—134no—
360C4⋊Q8—262no—
361C4⋊Q8—262no—
362#125—268no—
363C22⋊Q8—264no—
364C22⋊Q8—264no—
365C22⋊Q8—268no—
366#121—22no—
367#121—22no—
368C4×Q8—264no—
369C4×Q8—264no—
370C4×Q8—268no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—44no—
374#117—268no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—132no—
377C2×C4⋊C4—261no—
378C2×C4⋊C4—261no—
379C2×C4⋊C4—262no—
380C2×C4⋊C4—262no—
381C2×C4⋊C4—261no—
382C2×C4⋊C4C22×D1311yescharacteristic
383C2×C4⋊C4C22×D1311yescharacteristic
384C2×C4⋊C4C22×D1311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—262no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—262no—
395C2×C42C22⋊D1311yescharacteristic
396C2×C42C22⋊D1311yescharacteristic
397C2×C42C22×D1311yescharacteristic
398C2×C42C22×D1311yescharacteristic
399#106—264no—
400C4⋊C8—262no—
401C4⋊C8—262no—
402#103—264no—
403#103—264no—
404#103—522no—
405#102—264no—
406C22⋊C8—522no—
407#95—21no—
408#95—21no—
409#95—21no—
410C2×C13—42no—
411C2×C13—82no—
412C2×C13#240711yescharacteristic
413C2×C13C2×#44811yescharacteristic
414C2×C13#300511yescharacteristic
415C2×C13#310111yescharacteristic
416C2×C13#311611yescharacteristic
417C2×C13#324611yescharacteristic
418C2×C13#325211yescharacteristic
419C24—21no—
420C24C2×C22⋊D1311yescharacteristic
421C2×Q8—134no—
422C2×Q8—264no—
423C2×Q8—262no—
424C2×Q8—262no—
425C2×Q8—262no—
426C2×Q8—262no—
427C2×Q8—262no—
428C2×Q8—522no—
429C2×Q8—522no—
430C2×D4—24no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4#612712yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4#610514yescomplemented
459C22×C4C2×C22⋊D1311yescharacteristic
460C22×C4C2×C22⋊D1311yescharacteristic
461C22×C4C2×C22⋊D1311yescharacteristic
462C22×C4C2×C22⋊D1311yescharacteristic
463C22×C4C2×C22⋊D1311yescharacteristic
464C22×C4C23×D1311yescharacteristic
465SD16—528no—
466C2×C8—262no—
467C2×C8—262no—
468C2×C8—262no—
469C2×C8—262no—
470C2×C8—262no—
471C2×C8—262no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—134no—
479C4⋊C4—268no—
480C4⋊C4—268no—
481C4⋊C4—264no—
482C4⋊C4—264no—
483C4⋊C4—264no—
484C4⋊C4—262no—
485C4⋊C4#612712yes—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—268no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42—21no—
496C42—42no—
497C42—268no—
498C42D4×D1312yes—
499C42D4×D1312yes—
500C13#2415111yescharacteristic, complemented, Sylow
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#610512yes—
515C23C22×C22⋊D1311yescharacteristic
516Q8—264no—
517Q8—264no—
518Q8—528no—
519Q8—522no—
520Q8—522no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—264no—
553C2×C4—264no—
554C2×C4—522no—
555C2×C4—522no—
556C2×C4—522no—
557C2×C4—522no—
558C2×C4C2×#610512yes—
559C2×C4C2×#610512yes—
560C2×C4#8819112yes—
561C2×C4#8819112yes—
562C2×C4#8820512yes—
563C2×C4C2×#612711yescharacteristic
564C2×C4C2×#612711yescharacteristic
565C2×C4C2×D4×D1311yescharacteristic
566C2×C4C2×D4×D1311yescharacteristic
567C2×C4#8825611yescharacteristic
568C2×C4#8825711yescharacteristic
569C8—522no—
570C8—522no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#1171949611yescharacteristic
586C22C22×#610511yescharacteristic
587C22C2×#8819111yescharacteristic
588C22#1171952711yescharacteristic
589C22#1171952711yescharacteristic
590C22C2×#8820511yescharacteristic
591C22#1171953811yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—524no—
600C4—524no—
601C4#1171870114yes—
602C4#1171925511yescharacteristic
603C4#1171925611yescharacteristic
604C4#1171925711yescharacteristic
605C4#1171925811yescharacteristic
606C2—42no—
607C2—82no—
608C2#1311818111yescharacteristic
609C2C2×#1171870111yescharacteristic
610C2#1312407611yescharacteristic
611C2#1312484411yescharacteristic
612C2#1312499811yescharacteristic
613C2#1312615211yescharacteristic
614C2#1312620911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC4×C13×#121C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
hypercenterC4×#121D13
hypocenterC13#24151
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#11719175C2semidirect
2. C2×#11718669C2semidirect
3. C2×#11718680C2semidirect
4. #13116032C2semidirect
6. #13115784C2semidirect
7. #13115759C2semidirect
8. #13115194C2semidirect
9. #13115103C2semidirect
10. #13114816C2semidirect
96. #88123D4semidirect
97. #88123D4semidirect
141. C22×C4×C13SD16semidirect
458. C22×C4#6105semidirect
500. C13#24151semidirect

Maximal subgroups

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

subgroupindex
C2×#117191752
C2×#117186692
C2×#117186802
#131160322
#131158392
#131157842
#131157592
#131151942
#131151032
#131148162
C4×C13×#1212
#2415113