Finite Group Databasebeta

← #14683069

Subgroups — #14683069

Order 2816.

1666 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622833C211yesmaximal, characteristic
2#13066877C212yesmaximal, complemented
3#13066863C212yesmaximal, complemented
4#13066857C211yesmaximal, characteristic
5#13066853C212yesmaximal, complemented
6#13066853C212yesmaximal, complemented
7#13066840C212yesmaximal, complemented
8#13066838C212yesmaximal, complemented
9C11×#1438C211yesmaximal, characteristic
10C23×D4×C11C2211yescharacteristic
11C24×Dic11C2211yescharacteristic
12C22×C11×C22⋊C4C2212yes—
13C22×C11×C22⋊C4C2211yescharacteristic
14C22×C11×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic11C2212yes—
16C22×C22⋊Dic11C2212yes—
17C22×C22⋊Dic11C2211yescharacteristic
18C22×C22⋊Dic11C2212yes—
19C22×C4×Dic11C2212yes—
20C2×C11×#95C2212yes—
21#10622833C418yescomplemented
22C2×C11⋊#95C2212yes—
23C2×C11⋊#95C2212yes—
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2212yes—
26C2×C11⋊#95C2212yes—
27C2×C11⋊#95C2211yescharacteristic
28C2×C11⋊#95C2212yes—
29C2×C11⋊#95C2212yes—
30C2×C11⋊#95C2211yescharacteristic
31C2×C11⋊#95C2212yes—
32C25×C11C2311yescharacteristic
33C25×C11C2311yescharacteristic
34C22×D4×C11—24no—
35C22×D4×C11—41no—
36C22×D4×C11—41no—
37C22×D4×C11—42no—
38C22×D4×C11C2×C414yes—
39C23×C4×C11C2312yes—
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2312yes—
42C23×Dic11—41no—
43C23×Dic11—41no—
44C23×Dic11—42no—
45C23×Dic11—42no—
46C23×Dic11C2×C414yes—
47C23×Dic11C2312yes—
48C23×Dic11C2312yes—
49C23×Dic11C2312yes—
50C23×Dic11C2312yes—
51C2×C11×C22⋊C4—24no—
52C2×C11×C22⋊C4—24no—
53C2×C11×C22⋊C4—24no—
54C2×C11×C22⋊C4—41no—
55C2×C11×C22⋊C4—42no—
56C2×C11×C22⋊C4—41no—
57C2×C11×C22⋊C4—42no—
58C2×C22⋊Dic11—24no—
59C2×C22⋊Dic11—24no—
60C2×C22⋊Dic11—24no—
61C2×C22⋊Dic11—42no—
62C2×C22⋊Dic11—42no—
63C2×C22⋊Dic11—42no—
64C2×C22⋊Dic11—42no—
65C2×C22⋊Dic11—42no—
66C2×C22⋊Dic11—42no—
67C2×C22⋊Dic11C2×C414yes—
68C2×C22⋊Dic11C2×C418yes—
69C2×C4×Dic11—24no—
70C2×C4×Dic11—42no—
71C2×C4×Dic11—42no—
72C11×#95—42no—
73C11⋊#95—24no—
74C11⋊#95—41no—
75C11⋊#95—42no—
76C11⋊#95—42no—
77C11⋊#95—42no—
78C11⋊#95—42no—
79C11⋊#95—42no—
80C11⋊#95—42no—
81C11⋊#95—41no—
82C11⋊#95C2×C418yes—
83#14516—111nomaximal, Sylow
84C24×C11—22no—
85C24×C11—22no—
86C24×C11—22no—
87C24×C11—22no—
88C24×C11—22no—
89C24×C11—21no—
90C24×C11—21no—
91C24×C11—22no—
92C24×C11—22no—
93C24×C11—22no—
94C24×C11—22no—
95C24×C11—22no—
96C24×C11—22no—
97C24×C11—21no—
98C24×C11—21no—
99C24×C11C22×C412yes—
100C24×C11C22×C412yes—
101C24×C11#4114yes—
102C24×C11C2411yescharacteristic
103C2×D4×C11—28no—
104C2×D4×C11—44no—
105C2×D4×C11—44no—
106C2×D4×C11—48no—
107C2×D4×C11—48no—
108C22×C4×C11—22no—
109C22×C4×C11—22no—
110C22×C4×C11—22no—
111C22×C4×C11—22no—
112C22×C4×C11—22no—
113C22×C4×C11—22no—
114C22×C4×C11—21no—
115C22×C4×C11—21no—
116C22×C4×C11—22no—
117C22×C4×C11—22no—
118C22×C4×C11—22no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—22no—
124C22×C4×C11—22no—
125C22×C4×C11C22×C412yes—
126C22×C4×C11#4114yes—
127C22×Dic11—22no—
128C22×Dic11—22no—
129C22×Dic11—22no—
130C22×Dic11—22no—
131C22×Dic11—22no—
132C22×Dic11—22no—
133C22×Dic11—22no—
134C22×Dic11—22no—
135C22×Dic11—22no—
136C22×Dic11—22no—
137C22×Dic11—22no—
138C22×Dic11—22no—
139C22×Dic11—22no—
140C22×Dic11—22no—
141C22×Dic11—22no—
142C22×Dic11—22no—
143C22×Dic11—22no—
144C22×Dic11—22no—
145C22×Dic11—22no—
146C22×Dic11—22no—
147C22×Dic11—22no—
148C22×Dic11—22no—
149C22×Dic11—22no—
150C22×Dic11—22no—
151C22×Dic11—22no—
152C22×Dic11—44no—
153C22×Dic11—48no—
154C22×Dic11—48no—
155C22×Dic11—44no—
156C22×Dic11—44no—
157C22×Dic11C22×C414yes—
158C22×Dic11C22×C414yes—
159C22×Dic11#4114yes—
160C11×C22⋊C4—44no—
161C11×C22⋊C4—48no—
162C11×C22⋊C4—44no—
163C22⋊Dic11—44no—
164C22⋊Dic11—48no—
165C22⋊Dic11—48no—
166C22⋊Dic11—48no—
167C4×Dic11—48no—
168C2×#386—111no—
169#1443—112no—
170#1440—112no—
171#1438—111no—
172#1438D1111yescharacteristic
173#1437—112no—
174#1437—112no—
175#1434—112no—
176C22⋊#95—112no—
177C23×C11—24no—
178C23×C11—22no—
179C23×C11—22no—
180C23×C11—24no—
181C23×C11—24no—
182C23×C11—24no—
183C23×C11—24no—
184C23×C11—24no—
185C23×C11—24no—
186C23×C11—24no—
187C23×C11—22no—
188C23×C11—24no—
189C23×C11—24no—
190C23×C11—22no—
191C23×C11—22no—
192C23×C11—42no—
193C23×C11—42no—
194C23×C11—42no—
195C23×C11—42no—
196C23×C11—41no—
197C23×C11—42no—
198C23×C11—41no—
199C23×C11—42no—
200C23×C11—41no—
201C23×C11—41no—
202C23×C11—42no—
203C23×C11—42no—
204C23×C11—42no—
205C23×C11—42no—
206C23×C11—42no—
207C23×C11—42no—
208C23×C11—41no—
209C23×C11—42no—
210C23×C11—42no—
211C23×C11—42no—
212C23×C11—42no—
213C23×C11—42no—
214C23×C11—41no—
215C23×C11—42no—
216C23×C11—41no—
217C23×C11—41no—
218C23×C11#11714yes—
219C23×C11C23×C411yescharacteristic
220C23×C11C2×#4112yes—
221C23×C11C2×#4112yes—
222C23×C11C2×#4112yes—
223C23×C11H(5,𝔽2)12yes—
224C23×C11H(5,𝔽2)11yescharacteristic
225C23×C11H(5,𝔽2)11yescharacteristic
226C23×C11H(5,𝔽2)12yes—
227C23×C11H(5,𝔽2)12yes—
228D4×C11—416no—
229C2×C4×C11—22no—
230C2×C4×C11—22no—
231C2×C4×C11—22no—
232C2×C4×C11—24no—
233C2×C4×C11—24no—
234C2×C4×C11—24no—
235C2×C4×C11—42no—
236C2×C4×C11—42no—
237C2×C4×C11—42no—
238C2×C4×C11—42no—
239C2×C4×C11—42no—
240C2×C4×C11—42no—
241C2×C4×C11—42no—
242C2×C4×C11—42no—
243C2×C4×C11—42no—
244C2×C4×C11—42no—
245C2×C4×C11—42no—
246C2×C4×C11—42no—
247C2×C4×C11—42no—
248C2×Dic11—24no—
249C2×Dic11—24no—
250C2×Dic11—24no—
251C2×Dic11—24no—
252C2×Dic11—24no—
253C2×Dic11—24no—
254C2×Dic11—24no—
255C2×Dic11—24no—
256C2×Dic11—24no—
257C2×Dic11—416no—
258C2×Dic11—42no—
259C2×Dic11—42no—
260C2×Dic11—42no—
261C2×Dic11—42no—
262C2×Dic11—42no—
263C2×Dic11—42no—
264C2×Dic11—42no—
265C2×Dic11—42no—
266C2×Dic11—42no—
267C2×Dic11—42no—
268C2×Dic11—42no—
269C2×Dic11—42no—
270C2×Dic11—42no—
271C2×Dic11—42no—
272C2×Dic11—42no—
273C2×Dic11—42no—
274C2×Dic11—42no—
275C2×Dic11—42no—
276C2×Dic11—42no—
277C23×D4C2×D1111yescharacteristic
278C24×C4—111no—
279C22×C22⋊C4—111no—
280C22×C22⋊C4—112no—
281C22×C22⋊C4—112no—
282C22×C22⋊C4—112no—
283C22×C22⋊C4C2×D1112yes—
284C22×C22⋊C4C2×D1111yescharacteristic
285C22×C22⋊C4C2×D1111yescharacteristic
286C22×C42—112no—
287#386—118no—
288C2×#95—112no—
289C2×#95—112no—
290C2×#95—112no—
291C2×#95—112no—
292C2×#95—112no—
293C2×#95—112no—
294C2×#95—111no—
295C2×#95—112no—
296C2×#95—112no—
297C2×#95—111no—
298C2×#95C2×D1112yes—
299C22×C11—24no—
300C22×C11—28no—
301C22×C11—44no—
302C22×C11—42no—
303C22×C11—42no—
304C22×C11—44no—
305C22×C11—44no—
306C22×C11—44no—
307C22×C11—44no—
308C22×C11—42no—
309C22×C11—44no—
310C22×C11—44no—
311C22×C11—42no—
312C22×C11—42no—
313C22×C11—82no—
314C22×C11—82no—
315C22×C11—82no—
316C22×C11—82no—
317C22×C11C2×#11711yescharacteristic
318C22×C11C4×#4112yes—
319C22×C11#51811yescharacteristic
320C22×C11#51811yescharacteristic
321C22×C11#51812yes—
322C22×C11#53812yes—
323C22×C11#53812yes—
324C22×C11#53812yes—
325C22×C11#53812yes—
326C22×C11#54012yes—
327C22×C11#54012yes—
328C22×C11#55112yes—
329C22×C11#55111yescharacteristic
330C22×C11#55112yes—
331C22×C11#55312yes—
332C22×C11#55312yes—
333C22×C11#55312yes—
334C22×C11#55911yescharacteristic
335C22×C11#56012yes—
336C22×C11#56011yescharacteristic
337C22×C11H(3,𝔽4)11yescharacteristic
338C4×C11—42no—
339C4×C11—44no—
340C4×C11—82no—
341Dic11—44no—
342Dic11—44no—
343Dic11—44no—
344Dic11—82no—
345C25C22×D1111yescharacteristic
346C25C22×D1111yescharacteristic
347C22×D4—24no—
348C22×D4—42no—
349C22×D4—41no—
350C22×D4—41no—
351C22×D4C4×D1114yes—
352C23×C4—112no—
353C23×C4—112no—
354C23×C4—112no—
355C23×C4—114no—
356C23×C4—112no—
357C23×C4—441no—
358C23×C4—441no—
359C23×C4—442no—
360C23×C4—442no—
361C23×C4C22×D1112yes—
362C23×C4C22×D1111yescharacteristic
363C23×C4C22×D1112yes—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—41no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—41no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4—114no—
372C2×C22⋊C4—118no—
373C2×C22⋊C4—224no—
374C2×C22⋊C4—224no—
375C2×C22⋊C4—224no—
376C2×C22⋊C4—442no—
377C2×C22⋊C4—442no—
378C2×C22⋊C4—442no—
379C2×C22⋊C4—442no—
380C2×C22⋊C4—442no—
381C2×C22⋊C4—442no—
382C2×C42—224no—
383C2×C42—442no—
384C2×C42—442no—
385#95—42no—
386#95—118no—
387#95—224no—
388#95—442no—
389#95—442no—
390#95—441no—
391#95—441no—
392#95—442no—
393#95—442no—
394#95—442no—
395#95—442no—
396C2×C11—44no—
397C2×C11—82no—
398C2×C11#231211yescharacteristic
399C2×C11#233312yes—
400C2×C11#235312yes—
401C2×C11#237411yescharacteristic
402C2×C11#237511yescharacteristic
403C2×C11#274912yes—
404C2×C11#275312yes—
405C2×C11#277812yes—
406C2×C11#279712yes—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—21no—
411C24—21no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—21no—
418C24—21no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24C2×C4×D1112yes—
423C24C2×C4×D1112yes—
424C24#434114yes—
425C24C23×D1111yescharacteristic
426C2×D4—28no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—48no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—114no—
449C22×C4—114no—
450C22×C4—114no—
451C22×C4—222no—
452C22×C4—222no—
453C22×C4—222no—
454C22×C4—222no—
455C22×C4—222no—
456C22×C4—222no—
457C22×C4—222no—
458C22×C4—222no—
459C22×C4—222no—
460C22×C4—222no—
461C22×C4—222no—
462C22×C4—222no—
463C22×C4—222no—
464C22×C4—222no—
465C22×C4—222no—
466C22×C4—222no—
467C22×C4—222no—
468C22×C4—222no—
469C22×C4—222no—
470C22×C4—222no—
471C22×C4—222no—
472C22×C4—222no—
473C22×C4—222no—
474C22×C4—222no—
475C22×C4—222no—
476C22×C4—444no—
477C22×C4—448no—
478C22×C4—444no—
479C22×C4—448no—
480C22×C4—444no—
481C22×C4C2×C4×D1112yes—
482C22×C4#434114yes—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—444no—
487C22⋊C4—448no—
488C22⋊C4—448no—
489C22⋊C4—448no—
490C42—448no—
491C11#1451611yescharacteristic, complemented, Sylow
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23#6693014yes—
534C23C22×C4×D1111yescharacteristic
535C23C2×#434112yes—
536C23C2×#434112yes—
537C23#6703712yes—
538C23#6703712yes—
539C23#6703711yescharacteristic
540C23#6703711yescharacteristic
541C23#41×D1112yes—
542C23#6704212yes—
543D4—416no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—224no—
564C2×C4—224no—
565C2×C4—224no—
566C2×C4—224no—
567C2×C4—224no—
568C2×C4—224no—
569C2×C4—224no—
570C2×C4—224no—
571C2×C4—224no—
572C2×C4—4416no—
573C2×C4—442no—
574C2×C4—442no—
575C2×C4—442no—
576C2×C4—442no—
577C2×C4—442no—
578C2×C4—442no—
579C2×C4—442no—
580C2×C4—442no—
581C2×C4—442no—
582C2×C4—442no—
583C2×C4—442no—
584C2×C4—442no—
585C2×C4—442no—
586C2×C4—442no—
587C2×C4—442no—
588C2×C4—442no—
589C2×C4—442no—
590C2×C4—442no—
591C2×C4—442no—
592C22—24no—
593C22—28no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#6693011yescharacteristic
611C22#1062301711yescharacteristic
612C22#1062301711yescharacteristic
613C22C4×#434112yes—
614C22#1062307612yes—
615C22#1062311811yescharacteristic
616C22#1062311812yes—
617C22#1062312412yes—
618C22#1062312412yes—
619C22#1062312511yescharacteristic
620C22#1062313212yes—
621C22#1062314512yes—
622C22#1062314612yes—
623C22#1062314812yes—
624C22#1062314912yes—
625C22#1062315112yes—
626C22#1062318912yes—
627C22#1062319511yescharacteristic
628C22#1062319711yescharacteristic
629C22#1062320912yes—
630C22#1062325012yes—
631C4—42no—
632C4—44no—
633C4—82no—
634C4—444no—
635C4—444no—
636C4—444no—
637C4—882no—
638C2—44no—
639C2—82no—
640C2#1307121211yescharacteristic
641C2#1307137512yes—
642C2#1307155412yes—
643C2#1307174911yescharacteristic
644C2#1307175411yescharacteristic
645C2#1307584112yes—
646C2#1307589512yes—
647C2#1307623612yes—
648C2#1307647912yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC11×#1438C2
socle, cosocleC24×C11C24
prime radicalC23×D4×C11C22
hypercenter#1438D11
hypocenterC11#14516
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #13066877C2semidirect
3. #13066863C2semidirect
5. #13066853C2semidirect
6. #13066853C2semidirect
7. #13066840C2semidirect
8. #13066838C2semidirect
21. #10622833C4semidirect
491. C11#14516semidirect

Maximal subgroups

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

subgroupindex
C2×#106228332
#130668772
#130668632
#130668572
#130668532
#130668402
#130668382
C11×#14382
#1451611