Finite Group Databasebeta

← #14684834

Subgroups — #14684834

Order 2816.

1602 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
1#13066878C212yesmaximal, complemented
2#13066873C212yesmaximal, complemented
3#13066865C212yesmaximal, complemented
4#13066861C212yesmaximal, complemented
5#13066861C212yesmaximal, complemented
6#13066857C211yesmaximal, characteristic
7#13066857C211yesmaximal, characteristic
8#13066839C212yesmaximal, complemented
9C2×C11×#386C211yesmaximal, characteristic
10C23×D4×C11C2211yescharacteristic
11C24×C4×C11C2211yescharacteristic
12C22×C11×C22⋊C4C2211yescharacteristic
13C22×C11×C22⋊C4C2212yes—
14C22×C22⋊Dic11C2212yes—
15C22×C22⋊Dic11C2212yes—
16C22×C22⋊Dic11C2211yescharacteristic
17C22×C22⋊Dic11C2212yes—
18C22×C22⋊Dic11C2211yescharacteristic
19C22×C4⋊Dic11C2212yes—
20C11×#386C418yescomplemented
21C2×C11×#95C2212yes—
22C2×C11⋊#95C2212yes—
23C2×C11⋊#95C2212yes—
24C2×C11⋊#95C2212yes—
25C2×C11⋊#95C2212yes—
26C2×C11⋊#95C2211yescharacteristic
27C2×C11⋊#95C2212yes—
28C2×C11⋊#95C2211yescharacteristic
29C2×C11⋊#95C2212yes—
30C2×C11⋊#95C2212yes—
31C2×C11⋊#95C2212yes—
32C25×C11C2311yescharacteristic
33C25×C11C2311yescharacteristic
34C22×D4×C11—24no—
35C22×D4×C11—42no—
36C22×D4×C11—41no—
37C22×D4×C11—41no—
38C22×D4×C11C2×C414yes—
39C23×C4×C11—41no—
40C23×C4×C11—41no—
41C23×C4×C11—42no—
42C23×C4×C11—42no—
43C23×C4×C11C2×C414yes—
44C23×C4×C11C2312yes—
45C23×C4×C11C2312yes—
46C23×C4×C11C2311yescharacteristic
47C23×Dic11C2312yes—
48C23×Dic11C2312yes—
49C23×Dic11C2312yes—
50C23×Dic11C2312yes—
51C2×C11×C22⋊C4—42no—
52C2×C11×C22⋊C4—42no—
53C2×C11×C22⋊C4—42no—
54C2×C11×C22⋊C4C2×C414yes—
55C2×C11×C22⋊C4C2×C418yes—
56C2×C22⋊Dic11—24no—
57C2×C22⋊Dic11—24no—
58C2×C22⋊Dic11—42no—
59C2×C22⋊Dic11—42no—
60C2×C22⋊Dic11—42no—
61C2×C22⋊Dic11—42no—
62C2×C22⋊Dic11—41no—
63C2×C22⋊Dic11—41no—
64C2×C22⋊Dic11—42no—
65C2×C22⋊Dic11—42no—
66C2×C22⋊Dic11—42no—
67C2×C22⋊Dic11—42no—
68C2×C22⋊Dic11—42no—
69C2×C4⋊Dic11—42no—
70C2×C4⋊Dic11—42no—
71C2×C4⋊Dic11—42no—
72C11×#95C2×C418yes—
73C11⋊#95—41no—
74C11⋊#95—41no—
75C11⋊#95—42no—
76C11⋊#95—42no—
77C11⋊#95—42no—
78C11⋊#95—42no—
79C11⋊#95—42no—
80C11⋊#95—42no—
81C11⋊#95—42no—
82C11⋊#95—42no—
83#14666—111nomaximal, Sylow
84C24×C11—22no—
85C24×C11—22no—
86C24×C11—22no—
87C24×C11—22no—
88C24×C11—21no—
89C24×C11—21no—
90C24×C11—22no—
91C24×C11—22no—
92C24×C11—22no—
93C24×C11—22no—
94C24×C11—22no—
95C24×C11—21no—
96C24×C11—22no—
97C24×C11—21no—
98C24×C11—22no—
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—22no—
115C22×C4×C11—22no—
116C22×C4×C11—21no—
117C22×C4×C11—22no—
118C22×C4×C11—21no—
119C22×C4×C11—22no—
120C22×C4×C11—22no—
121C22×C4×C11—22no—
122C22×C4×C11—22no—
123C22×C4×C11—22no—
124C22×C4×C11—48no—
125C22×C4×C11—44no—
126C22×C4×C11—48no—
127C22×C4×C11—44no—
128C22×C4×C11—44no—
129C22×C4×C11C22×C412yes—
130C22×C4×C11C22×C414yes—
131C22×C4×C11C22×C414yes—
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—22no—
153C22×Dic11—22no—
154C22×Dic11—22no—
155C22×Dic11—22no—
156C22×Dic11—22no—
157C22×Dic11—22no—
158C22×Dic11—22no—
159C22×Dic11—22no—
160C11×C22⋊C4—48no—
161C11×C22⋊C4—44no—
162C22⋊Dic11—48no—
163C22⋊Dic11—48no—
164C22⋊Dic11—48no—
165C22⋊Dic11—44no—
166C22⋊Dic11—44no—
167C4⋊Dic11—48no—
168C2×#386D1111yescharacteristic
169#1443—112no—
170#1442—112no—
171#1440—112no—
172#1439—112no—
173#1439—112no—
174#1438—111no—
175#1438—111no—
176C22⋊#95—112no—
177C23×C11—22no—
178C23×C11—22no—
179C23×C11—24no—
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—22no—
190C23×C11—22no—
191C23×C11—24no—
192C23×C11—42no—
193C23×C11—41no—
194C23×C11—42no—
195C23×C11—42no—
196C23×C11—41no—
197C23×C11—41no—
198C23×C11—42no—
199C23×C11—42no—
200C23×C11—42no—
201C23×C11—41no—
202C23×C11—42no—
203C23×C11—42no—
204C23×C11—42no—
205C23×C11—42no—
206C23×C11—42no—
207C23×C11—42no—
208C23×C11—42no—
209C23×C11—41no—
210C23×C11—42no—
211C23×C11—41no—
212C23×C11—42no—
213C23×C11—41no—
214C23×C11—42no—
215C23×C11—42no—
216C23×C11—42no—
217C23×C11—41no—
218C23×C11#11714yes—
219C23×C11C23×C411yescharacteristic
220C23×C11C2×#4112yes—
221C23×C11H(5,𝔽2)11yescharacteristic
222C23×C11H(5,𝔽2)12yes—
223C23×C11H(5,𝔽2)11yescharacteristic
224C23×C11H(5,𝔽2)12yes—
225C23×C11H(5,𝔽2)12yes—
226C23×C11H(5,𝔽2)12yes—
227C23×C112−1+412yes—
228D4×C11—416no—
229C2×C4×C11—22no—
230C2×C4×C11—24no—
231C2×C4×C11—24no—
232C2×C4×C11—24no—
233C2×C4×C11—24no—
234C2×C4×C11—24no—
235C2×C4×C11—22no—
236C2×C4×C11—22no—
237C2×C4×C11—24no—
238C2×C4×C11—416no—
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×C4×C11—42no—
249C2×Dic11—42no—
250C2×Dic11—42no—
251C2×Dic11—42no—
252C2×Dic11—42no—
253C2×Dic11—42no—
254C2×Dic11—42no—
255C2×Dic11—42no—
256C2×Dic11—42no—
257C2×Dic11—42no—
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×C4C2×D1111yescharacteristic
279C22×C4⋊C4—112no—
280C22×C22⋊C4—112no—
281C22×C22⋊C4—112no—
282C22×C22⋊C4—111no—
283C22×C22⋊C4—112no—
284C22×C22⋊C4—111no—
285C22×C22⋊C4C2×D1111yescharacteristic
286C22×C22⋊C4C2×D1112yes—
287#386Dic1118yescomplemented
288C2×#95—112no—
289C2×#95—112no—
290C2×#95—112no—
291C2×#95—112no—
292C2×#95—111no—
293C2×#95—112no—
294C2×#95—111no—
295C2×#95—112no—
296C2×#95—112no—
297C2×#95—112no—
298C2×#95C2×D1112yes—
299C22×C11—24no—
300C22×C11—28no—
301C22×C11—44no—
302C22×C11—44no—
303C22×C11—44no—
304C22×C11—44no—
305C22×C11—42no—
306C22×C11—42no—
307C22×C11—44no—
308C22×C11—42no—
309C22×C11—44no—
310C22×C11—42no—
311C22×C11—42no—
312C22×C11—44no—
313C22×C11—82no—
314C22×C11—82no—
315C22×C11—82no—
316C22×C11—82no—
317C22×C11C2×#11711yescharacteristic
318C22×C11#51811yescharacteristic
319C22×C11#51811yescharacteristic
320C22×C11#51812yes—
321C22×C11#52012yes—
322C22×C11#53812yes—
323C22×C11#53812yes—
324C22×C11#53812yes—
325C22×C11#53812yes—
326C22×C11#53912yes—
327C22×C11#53912yes—
328C22×C11#56012yes—
329C22×C11#56011yescharacteristic
330C22×C11#56012yes—
331C22×C11#56012yes—
332C22×C11#56011yescharacteristic
333C22×C11H(3,𝔽4)11yescharacteristic
334C22×C11#56212yes—
335C22×C11#56212yes—
336C22×C11#56212yes—
337C22×C11#56311yescharacteristic
338C4×C11—42no—
339C4×C11—44no—
340C4×C11—44no—
341Dic11—82no—
342Dic11—82no—
343Dic11—82no—
344Dic11—82no—
345C25C22×D1111yescharacteristic
346C25C22×D1111yescharacteristic
347C22×D4—24no—
348C22×D4—42no—
349C22×D4—41no—
350C22×D4—41no—
351C22×D4C2×Dic1114yes—
352C23×C4—41no—
353C23×C4—41no—
354C23×C4—42no—
355C23×C4—42no—
356C23×C4—112no—
357C23×C4—112no—
358C23×C4—112no—
359C23×C4—112no—
360C23×C4C2×Dic1114yes—
361C23×C4C22×D1112yes—
362C23×C4C22×D1112yes—
363C23×C4C22×D1111yescharacteristic
364C2×C4⋊C4—442no—
365C2×C4⋊C4—442no—
366C2×C4⋊C4—442no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—224no—
371C2×C22⋊C4—224no—
372C2×C22⋊C4—442no—
373C2×C22⋊C4—442no—
374C2×C22⋊C4—442no—
375C2×C22⋊C4—442no—
376C2×C22⋊C4—441no—
377C2×C22⋊C4—441no—
378C2×C22⋊C4—442no—
379C2×C22⋊C4—442no—
380C2×C22⋊C4—442no—
381C2×C22⋊C4—442no—
382C2×C22⋊C4—442no—
383C2×C22⋊C4C2×Dic1114yes—
384C2×C22⋊C4C2×Dic1118yes—
385#95—442no—
386#95—442no—
387#95—442no—
388#95—442no—
389#95—442no—
390#95—442no—
391#95—441no—
392#95—441no—
393#95—442no—
394#95—442no—
395#95C2×Dic1118yes—
396C2×C11—44no—
397C2×C11—82no—
398C2×C11#231211yescharacteristic
399C2×C11#231812yes—
400C2×C11#237511yescharacteristic
401C2×C11#237611yescharacteristic
402C2×C11#237712yes—
403C2×C11#282412yes—
404C2×C11#282512yes—
405C2×C11#282612yes—
406C2×C11#282712yes—
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—
422C24#433914yes—
423C24C22×Dic1112yes—
424C24C22×Dic1112yes—
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—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
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—222no—
477C22×C4—222no—
478C22×C4—222no—
479C22×C4—222no—
480C22×C4C22×Dic1112yes—
481C22×C4C22×Dic1114yes—
482C22×C4C22×Dic1114yes—
483C4⋊C4—448no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—448no—
487C22⋊C4—448no—
488C22⋊C4—444no—
489C22⋊C4—448no—
490C22⋊C4—444no—
491C11#1466611yescharacteristic, complemented, Sylow
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23#6697914yes—
534C23C2×#433912yes—
535C23#6703711yescharacteristic
536C23#6703711yescharacteristic
537C23#6703712yes—
538C23#6703712yes—
539C23#6703712yes—
540C23#6704212yes—
541C23#6704312yes—
542C23C23×Dic1111yescharacteristic
543D4—416no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—416no—
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—42no—
564C2×C4—442no—
565C2×C4—442no—
566C2×C4—442no—
567C2×C4—442no—
568C2×C4—442no—
569C2×C4—442no—
570C2×C4—442no—
571C2×C4—442no—
572C2×C4—442no—
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—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22#1062302512yes—
611C22#1062302512yes—
612C22#1062309112yes—
613C22#1062309112yes—
614C22#1062309512yes—
615C22#1062309512yes—
616C22#1062312511yescharacteristic
617C22#1062312612yes—
618C22#1062312611yescharacteristic
619C22#1062315112yes—
620C22#1062315112yes—
621C22#1062315212yes—
622C22#1062315212yes—
623C22#1062318211yescharacteristic
624C22#1062319711yescharacteristic
625C22#1062319812yes—
626C22C2×#6697911yescharacteristic
627C22#1062332911yescharacteristic
628C22#1062332911yescharacteristic
629C22#1062332912yes—
630C22#1062335812yes—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—882no—
635C4—882no—
636C4—882no—
637C4—882no—
638C2—44no—
639C2—82no—
640C2#1307121411yescharacteristic
641C2#1307124912yes—
642C2#1307175711yescharacteristic
643C2#1307176411yescharacteristic
644C2#1307177312yes—
645C2#1307669312yes—
646C2#1307670212yes—
647C2#1307671112yes—
648C2#1307672012yes—
subgroupG/H
center, FrattiniC24C23×D11
commutatorC23×C11C23×C4
FittingC2×C11×#386C2
socle, cosocleC24×C11C24
prime radicalC23×D4×C11C22
hypercenterC2×#386D11
hypocenterC11#14666
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. #13066878C2semidirect
2. #13066873C2semidirect
3. #13066865C2semidirect
4. #13066861C2semidirect
5. #13066861C2semidirect
8. #13066839C2semidirect
20. C11×#386C4semidirect
287. #386Dic11semidirect
491. C11#14666semidirect

Maximal subgroups

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

subgroupindex
#130668782
#130668732
#130668652
#130668612
#130668572
#130668392
C2×C11×#3862
#1466611